Compartmental models in epidemiology

314: 25896:, where the population is effectively infinite. In stochastic models, the long-time endemic equilibrium derived above, does not hold, as there is a finite probability that the number of infected individuals drops below one in a system. In a true system then, the pathogen may not propagate, as no host will be infected. But, in deterministic mean-field models, the number of infected can take on real, namely, non-integer values of infected hosts, and the number of hosts in the model can be less than one, but more than zero, thereby allowing the pathogen in the model to propagate. The reliability of compartmental models is limited to compartmental applications. 302: 10918: 8062: 168: 15494: 14498: 10022: 8043: 815: 9283: 6411: 10870:(SIRVD) epidemic compartment model extends the SIR model to include the effects of vaccination campaigns and time-dependent fatality rates on epidemic outbreaks. It encompasses the SIR, SIRV, SIRD, and SI models as special cases, with individual time-dependent rates governing transitions between different fractions. This model uses the following system of differential equations for the population fractions 6275: 18202: 12403: 632: 8634: 12842: 14189: 6233: 15452:

links. Such Interacting Subpopulation SEIR models have been used for modeling the COVID-19 pandemic at continent scale to develop personalized, accelerated, subpopulation-targeted vaccination strategies that promise a shortening of the pandemic and a reduction of case and death counts in the setting of limited access to vaccines during a wave of virus Variants of Concern.

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This allowed to give a contribution to explain the poly-annual (typically biennial) epidemic outbreaks of some infectious diseases as interplay between the period of the contact rate oscillations and the pseudo-period of the damped oscillations near the endemic equilibrium. Remarkably, in some cases, the behavior may also be quasi-periodic or even chaotic.

9698: 25412: 16864: 5664: 1184:(also called basic reproduction ratio). This ratio is derived as the expected number of new infections (these new infections are sometimes called secondary infections) from a single infection in a population where all subjects are susceptible. This idea can probably be more readily seen if we say that the typical time between contacts is 22006:

operator that provides the next generation of infected people when applied to the present generation. Note that this operator (matrix) is responsible for the number of infected people, not all the compartments. Iteration of this operator describes the initial progression of infection within the heterogeneous population. So comparing the

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time, some people get tired/frustrated by the restrictions and stop following them (exhaustion), especially if the number of new cases drops down. After resting for some time, they can follow the restrictions again. But during this pause the second wave can come and become even stronger than the first one.

14493:{\displaystyle {\begin{aligned}{\frac {dM}{dt}}&=\Lambda -\delta M-\mu M\\{\frac {dS}{dt}}&=\delta M-{\frac {\beta SI}{N}}-\mu S\\{\frac {dE}{dt}}&={\frac {\beta SI}{N}}-(\varepsilon +\mu )E\\{\frac {dI}{dt}}&=\varepsilon E-(\gamma +\mu )I\\{\frac {dR}{dt}}&=\gamma I-\mu R\end{aligned}}} 12582: 647:, are often much faster than the dynamics of birth and death, therefore, birth and death are often omitted in simple compartmental models. The SIR system without so-called vital dynamics (birth and death, sometimes called demography) described above can be expressed by the following system of ordinary 5995: 17863:

This strategy repeatedly vaccinates a defined age-cohort (such as young children or the elderly) in a susceptible population over time. Using this strategy, the block of susceptible individuals is then immediately removed, making it possible to eliminate an infectious disease, (such as measles), from

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It is well known that the probability of getting a disease is not constant in time. As a pandemic progresses, reactions to the pandemic may change the contact rates which are assumed constant in the simpler models. Counter-measures such as masks, social distancing, and lockdown will alter the contact

11647: 7050:. The simulation first updates the infected from the susceptible and then the removed category is updated from the infected category for the next time point (t=1). This describes the flow persons between the three categories. During an epidemic the susceptible category is not shifted with this model, 25838:

In their model, it is assumed that the infected individuals can develop active TB by either direct progression (the disease develops immediately after infection) considered above as FAST tuberculosis or endogenous reactivation (the disease develops years after the infection) considered above as SLOW

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Dynamics of epidemics depend on how people's behavior changes in time. For example, at the beginning of the epidemic, people are ignorant and careless, then, after the outbreak of epidemics and alarm, they begin to comply with the various restrictions and the spreading of epidemics may decline. Over

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whereas if the integral is greater than one the disease will not die out and there may be such resonances. For example, considering the periodically varying contact rate as the 'input' of the system one has that the output is a periodic function whose period is a multiple of the period of the input.

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There are many modifications of the SIR model, including those that include births and deaths, where upon recovery there is no immunity (SIS model), where immunity lasts only for a short period of time (SIRS), where there is a latent period of the disease where the person is not infectious (SEIS and

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Using the differential equations of the SIR model and converting them to numerical discrete forms, one can set up the recursive equations and calculate the S, I, and R populations with any given initial conditions but accumulate errors over a long calculation time from the reference point. Sometimes

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corrects this dilution effect by multiplying the fraction of the susceptible population over the total population. It corrects the effective/transmissible interaction between an infectious person and the rest of the community when many of the interaction is immune in the middle to late stages of the

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but whose efficiencies are under debate. This debate is largely on the uncertainty of the number of days reduced from after infectious or detectable whichever comes first to before a symptom shows up for an infected susceptible person. If the person is infectious after symptoms show up, or detection

298:, the number of susceptible individuals falls rapidly as more of them are infected and thus enter the infectious and removed compartments. The disease cannot break out again until the number of susceptibles has built back up, e.g. as a result of offspring being born into the susceptible compartment. 25861:

Diseases transmitted from human to human indirectly, i.e. malaria spread by way of mosquitoes, are transmitted through a vector. In these cases, the infection transfers from human to insect and an epidemic model must include both species, generally requiring many more compartments than a model for

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Based on the classical SIR model, a Korteweg-de Vries (KdV)–SIR equation and its analytical solution have been proposed to illustrate the fundamental dynamics of an epidemic wave, the dependence of solutions on parameters, and the dependence of predictability horizons on various types of solutions.

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of discrete time steps e.g., use yesterday's principal and interest rate to calculate today's interest which assumes the interest rate is fixed during the day. The calculation contains projected errors if the analytical corrections on the numerical step size are not included, e.g. when the interest

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of COVID-19 can be practiced based on information from the different populations in a community. Numerical solution is a commonly used method to analyze complicated kinetic networks when the analytical solution is difficult to obtain or limited by requirements such as boundary conditions or special

7010:. These are the number of people in the susceptible, infected and removed categories at time equals zero. If the SIR model is assumed to hold at all times, these initial conditions are not independent. Subsequently, the flow model updates the three variables for every time point with set values for 139:

emoved (and immune) or deceased individuals. These are individuals who have been infected and have either recovered from the disease and entered the removed compartment, or died. It is assumed that the number of deaths is negligible with respect to the total population. This compartment may also be

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i.e. the baseline vaccination rate should be greater than the "mandatory vaccination" threshold, which, in case of exemption, cannot hold. Thus, "rational" exemption might be myopic since it is based only on the current low incidence due to high vaccine coverage, instead taking into account future

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may be read as the average number of infections caused by a single infectious subject in a wholly susceptible population, the above relationship biologically means that if this number is less than or equal to one the disease goes extinct, whereas if this number is greater than one the disease will

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characterizes the potential for initial spread of an infectious person in a naive population, assuming the change in the susceptible population is negligible during the initial spread. A linear system of ODEs can always be described by a matrix. So, the next step is to construct a linear positive

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Modern societies are facing the challenge of "rational" exemption, i.e. the family's decision to not vaccinate children as a consequence of a "rational" comparison between the perceived risk from infection and that from getting damages from the vaccine. In order to assess whether this behavior is

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As social contacts, disease severity and lethality, as well as the efficacy of prophylactic measures may differ substantially between interacting subpopulations, e.g., the elderly versus the young, separate SEIR models for each subgroup may be used that are mutually connected through interaction

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differs when considering the effects of vaccinations and when the real-time dependence of fatality and recovery rates diverge. These differences have been highlighted for stationary ratios and gradually decreasing fatality rates. The case of stationary ratios allows one to construct a diagnostics

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the mean infective period) of infectives are leaving this class per unit time to enter the removed class. These processes which occur simultaneously are referred to as the Law of Mass Action, a widely accepted idea that the rate of contact between two groups in a population is proportional to the

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is needed to estimate the errors. Given a set of initial conditions and the disease-spreading data, one can also fit the data with the SIR model and pull out the three reproduction numbers when the errors are usually negligible due to the short time step from the reference point. Any point of the

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In the case of some diseases such as AIDS and hepatitis B, it is possible for the offspring of infected parents to be born infected. This transmission of the disease down from the mother is referred to as vertical transmission. The influx of additional members into the infected category can be

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The SIR model has been studied on networks of various kinds in order to model a more realistic form of connection than the homogeneous mixing condition which is usually required. A simple model for epidemics on networks in which an individual has a probability p of being infected by each of his

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viruses, which are more prevalent during winter. With childhood diseases, such as measles, mumps, and rubella, there is a strong correlation with the school calendar, so that during the school holidays the probability of getting such a disease dramatically decreases. As a consequence, for many

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is needed to estimate the error. However, this error is usually acceptable for data fitting. When fitting a set of data with a close time step, the error is relatively small because the reference point is nearby compared to when predicting a long period of time after a reference point. Once the

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Spatiotemporal compartmental models describe not the total number, but the density of susceptible/infective/recovered persons. Consequently, they also allow to model the distribution of infected persons in space. In most cases, this is done by combining the SIR model with a diffusion equation

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the infection, whilst not suffering the disease themselves. They may then move back into the infectious compartment and suffer symptoms (as in tuberculosis) or they may continue to infect others in their carrier state, while not suffering symptoms. The most famous example of this is probably

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Whenever the integer nature of the number of agents is evident (populations with fewer than tens of thousands of individuals), inherent fluctuations in the disease spreading process caused by discrete agents result in uncertainties. In this scenario, the evolution of the disease predicted by

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Latent period is the transition time between contagion event and disease manifestation. In cases of diseases with varying latent periods, the basic reproduction number can be calculated as the sum of the reproduction numbers for each transition time into the disease. An example of this is

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In presence of a communicable diseases, one of the main tasks is that of eradicating it via prevention measures and, if possible, via the establishment of a mass vaccination program. Consider a disease for which the newborn are vaccinated (with a vaccine giving lifelong immunity) at a rate

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The only steady state solution to the classic SIR model as defined by the differential equations above is I=0, S and R can then take any values. The model can be changed while retaining three compartments to give a steady-state endemic solution by adding some input to the S compartment.

12456: 12128: 1989:, not all individuals of the population have been removed, so some must remain susceptible. A driving force leading to the end of an epidemic is a decline in the number of infectious individuals. The epidemic does not typically end because of a complete lack of susceptible individuals. 9489: 19285: 15098: 17864:

the entire population. Every T time units a constant fraction p of susceptible subjects is vaccinated in a relatively short (with respect to the dynamics of the disease) time. This leads to the following impulsive differential equations for the susceptible and vaccinated subjects:

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has to be changed compared to the simple SIR model.) Early models of this type have been used to model the spread of the black death in Europe. Extensions of this model have been used to incorporate, e.g., effects of nonpharmaceutical interventions such as social distancing.

5989: 21978:, the first step is to linearise around the disease-free equilibrium (DFE), but for the infected subsystem of non-linear ODEs which describe the production of new infections and changes in state among infected individuals. Epidemiologically, the linearisation reflects that 22177: 16651: 5471: 187:) represent the number of people in each compartment at a particular time. To represent that the number of susceptible, infectious, and removed individuals may vary over time (even if the total population size remains constant), we make the precise numbers a function of 15949:

The differences between countries are concentrated in two kinetic constants: the rate of mobilization and the rate of exhaustion calculated for COVID-19 epidemic in 13 countries. These constants for this epidemic in all countries can be extracted by the fitting of the

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changes, most probably the behaviors of people in the community have changed from their normal patterns before the outbreak, or the disease has mutated to a new form. Costive massive detection and isolation of susceptible close contacts have effects on reducing

3012: 14181: 25740:. The difference is due to a subtle difference in the underlying growth model; the matrix equation above assumes that newly infected patients are currently already contributing to infections, while in fact infections only occur due to the number infected at 3309: 14925:), i.e. a nonlinear set of differential equations with periodically varying parameters. It is well known that this class of dynamical systems may undergo very interesting and complex phenomena of nonlinear parametric resonance. It is easy to see that if: 21784: 21457: 21207:

alternate because, the infectious person has to ‘pass through’ a second gender before it can enter a new host of the first gender. In other words, it takes two generations to get back to the same type, and every two generations numbers are multiplied by

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i.e., independently from the initial size of the susceptible population the disease can never cause a proper epidemic outbreak. As a consequence, it is clear that both the basic reproduction number and the initial susceptibility are extremely important.

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Each member of the population typically progresses from susceptible to infectious to recovered. This can be shown as a flow diagram in which the boxes represent the different compartments and the arrows the transition between compartments (see diagram).

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is the compartment used for the individuals of the population who have been infected and then removed from the disease, either due to immunization or due to death. Those in this category are not able to be infected again or to transmit the infection to

16092: 15653: 18197:{\displaystyle {\begin{aligned}{\frac {dS}{dt}}&=\mu N-\mu S-\beta {\frac {I}{N}}S,\quad S(nT^{+})=(1-p)S(nT^{-}),&&n=0,1,2,\ldots \\{\frac {dV}{dt}}&=-\mu V,\quad V(nT^{+})=V(nT^{-})+pS(nT^{-}),&&n=0,1,2,\ldots \end{aligned}}} 17464:

the vaccination program is not successful in eradicating the disease, on the contrary, it will remain endemic, although at lower levels than the case of absence of vaccinations. This means that the mathematical model suggests that for a disease whose

13397: 12398:{\displaystyle {\begin{aligned}{\frac {dM}{dt}}&=\Lambda -\delta M-\mu M\\{\frac {dS}{dt}}&=\delta M-{\frac {\beta SI}{N}}-\mu S\\{\frac {dI}{dt}}&={\frac {\beta SI}{N}}-\gamma I-\mu I\\{\frac {dR}{dt}}&=\gamma I-\mu R\end{aligned}}} 4689:

and also the intensity/duration of the interaction like how close they interact for how long and whether or not they both wear masks, thus, it changes over time when the average behavior of the carriers and susceptible people changes. The model use

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Age has a deep influence on the disease spread rate in a population, especially the contact rate. This rate summarizes the effectiveness of contacts between susceptible and infectious subjects. Taking into account the ages of the epidemic classes

2370:. However, for large classes of communicable diseases it is more realistic to consider a force of infection that does not depend on the absolute number of infectious subjects, but on their fraction (with respect to the total constant population 13854: 6814: 24847: 23061: 22966: 17164: 23685: 8629:{\displaystyle {\begin{aligned}&{\frac {\beta }{\gamma }}\leq 1\Rightarrow \lim _{t\to +\infty }I(t)=0,\\&{\frac {\beta }{\gamma }}>1\Rightarrow \lim _{t\to +\infty }I(t)=\left(1-{\frac {\gamma }{\beta }}\right)N.\end{aligned}}} 14715: 12837:{\displaystyle {\begin{aligned}{\frac {dS}{dt}}&=\mu N-\mu S-{\frac {\beta IS}{N}}\\{\frac {dE}{dt}}&={\frac {\beta IS}{N}}-(\mu +a)E\\{\frac {dI}{dt}}&=aE-(\gamma +\mu )I\\{\frac {dR}{dt}}&=\gamma I-\mu R.\end{aligned}}} 657: 23570: 20370: 6228:{\displaystyle R_{0}>1,I(0)>0\Rightarrow \lim _{t\to \infty }(S(t),I(t),R(t))={\textrm {EE}}=\left({\frac {\gamma +\mu }{\beta }},{\frac {\mu }{\beta }}\left(R_{0}-1\right),{\frac {\gamma }{\beta }}\left(R_{0}-1\right)\right).} 11927:{\displaystyle {\begin{aligned}&{\frac {dS}{d\tau }}=-SI-b(\tau )S,\\&{\frac {dI}{d\tau }}=SI-I,\\&{\frac {dR}{d\tau }}=k(\tau )I,\\&{\frac {dV}{d\tau }}=b(\tau )S,\\&{\frac {dD}{d\tau }}=q(\tau )S\end{aligned}}} 1951: 19590: 19433: 15877: 14091:{\displaystyle {\begin{aligned}{\frac {dS}{dt}}&=\Lambda -{\frac {\beta SI}{N}}-\mu S+\gamma I\\{\frac {dE}{dt}}&={\frac {\beta SI}{N}}-(\epsilon +\mu )E\\{\frac {dI}{dt}}&=\varepsilon E-(\gamma +\mu )I\end{aligned}}} 5777: 11229:{\displaystyle {\begin{aligned}&{\frac {dS}{dt}}=-a(t)SI-v(t)S,\\&{\frac {dI}{dt}}=a(t)SI-\mu (t)I-\psi (t)I,\\&{\frac {dR}{dt}}=\mu (t)I,\\&{\frac {dV}{dt}}=v(t)S,\\&{\frac {dD}{dt}}=\psi (t)I\end{aligned}}} 13593: 10011:{\displaystyle {\begin{aligned}&{\frac {dS}{dt}}=-{\frac {\beta (t)IS}{N}}-v(t)S,\\&{\frac {dI}{dt}}={\frac {\beta (t)IS}{N}}-\gamma (t)I,\\&{\frac {dR}{dt}}=\gamma (t)I,\\&{\frac {dV}{dt}}=v(t)S,\end{aligned}}} 23825:

can be interpreted as the cumulative number of diagnoses (including individuals who have recovered) or the present number of infection cases; the logarithmic growth rate is the same for either definition. In order to estimate

22822: 17612: 15331:{\displaystyle {\begin{aligned}&\partial _{t}S=D_{S}\nabla ^{2}S-{\frac {\beta IS}{N}},\\&\partial _{t}I=D_{I}\nabla ^{2}I+{\frac {\beta IS}{N}}-\gamma I,\\&\partial _{t}R=D_{R}\nabla ^{2}R+\gamma I,\end{aligned}}} 7258:

Several assumptions were made in the formulation of these equations: First, an individual in the population must be considered as having an equal probability as every other individual of contracting the disease with a rate of

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is most useful in predicting future trends, for example, if we know the social interactions have reduced 50% frequently from that before the outbreak and the interaction intensities among people are the same, then we can set

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rate of annual collection is simplified to 12 times the monthly rate, a projected error is introduced. Thus the calculated results will carry accumulative errors when the time step is far away from the reference point and a

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ecovered). People may progress between compartments. The order of the labels usually shows the flow patterns between the compartments; for example SEIS means susceptible, exposed, infectious, then susceptible again.

12987: 9693:{\displaystyle {\begin{aligned}&{\frac {dS}{dt}}=-{\frac {\beta IS}{N}},\\&{\frac {dI}{dt}}={\frac {\beta IS}{N}}-\gamma I-\mu I,\\&{\frac {dR}{dt}}=\gamma I,\\&{\frac {dD}{dt}}=\mu I,\end{aligned}}} 6904:

should remain constant within the simulation, if a simulation is used to solve the SIR model. Alternatively, the analytic approximant can be used without performing a simulation. The model is started with values of

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meaning one infectious person on average infects 0.4 susceptible people per day and recovers in 1/0.2=5 days. Thus when this person recovered, there are two people still infectious directly got from this person and

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size of each of the groups concerned. Finally, it is assumed that the rate of infection and recovery is much faster than the time scale of births and deaths and therefore, these factors are ignored in this model.

14701: 3227: 1021: 24133: 19034: 14931: 27321: 25407:{\displaystyle {\frac {d}{dt}}{\begin{pmatrix}n_{E}\\n_{I}\end{pmatrix}}={\begin{pmatrix}-1/\tau _{E}&R_{0}/\tau _{I}\\1/\tau _{E}&-1/\tau _{I}\end{pmatrix}}{\begin{pmatrix}n_{E}\\n_{I}\end{pmatrix}}.} 20829: 16656: 16211: 15124: 13884: 5476: 4874:

does not tell us whether or not the spreading will speed up or slow down in the latter stages when the fraction of susceptible people in the community has dropped significantly after recovery or vaccination.

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In 2014, Harko and coauthors derived an exact so-called analytical solution (involving an integral that can only be calculated numerically) to the SIR model. In the case without vital dynamics setup, for

27135:"The basic reproduction number of the new coronavirus pandemic with mortality for India, the Syrian Arab Republic, the United States, Yemen, China, France, Nigeria and Russia with different rate of cases" 25836: 12447:

For many important infections, there is a significant latency period during which individuals have been infected but are not yet infectious themselves. During this period the individual is in compartment

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to represent these factors but it indeed is referenced to the initial stage when no action is taken to prevent the spread and all population is susceptible, thus all changes are absorbed by the change of

2703: 16859:{\displaystyle {\begin{aligned}{\frac {dS}{dt}}&=\nu N(1-P)-\mu S-\beta {\frac {I}{N}}S\\{\frac {dI}{dt}}&=\beta {\frac {I}{N}}S-(\mu +\gamma )I\\{\frac {dV}{dt}}&=\nu NP-\mu V\end{aligned}}} 5849: 17875: 17617: 14720: 13598: 8469: 8080: 5659:{\displaystyle {\begin{aligned}{\frac {dS}{dt}}&=\Lambda -\mu S-{\frac {\beta IS}{N}}\\{\frac {dI}{dt}}&={\frac {\beta IS}{N}}-\gamma I-\mu I\\{\frac {dR}{dt}}&=\gamma I-\mu R\end{aligned}}} 8749: 4425: 84:

These models are used to analyze the disease dynamics and to estimate the total number of infected people, the total number of recovered people, and to estimate epidemiological parameters such as the

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usceptible individuals. When a susceptible and an infectious individual come into "infectious contact", the susceptible individual contracts the disease and transitions to the infectious compartment.

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In the simplest case of equal fertilities in the three epidemic classes, we have that in order to have demographic equilibrium the following necessary and sufficient condition linking the fertility

4349: 4283: 13002: 9107: 8058:, do not confer any long-lasting immunity. Such infections may give temporary resistance but do not give long-term immunity upon recovery from infection, and individuals become susceptible again. 5838: 18806: 18423: 17001: 1103: 22558:

such that it includes only infections that are newly arising, but does not include terms which describe the transfer of infectious individuals from one infected compartment to another. Then if

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really rational, i.e. if it can equally lead to the eradication of the disease, one may simply assume that the vaccination rate is an increasing function of the number of infectious subjects:

12007: 7204: 2692: 19720: 7862: 1434: 18755: 18673: 18591: 1175: 21790: 17459: 8809: 7908: 10783: 5299:, the spreading slows down thus useful to set a prediction on the short-term trends. Also, it can be used to directly calculate the threshold population of vaccination/immunization for the 17541: 16950: 12436: 2839: 24556: 22388:{\displaystyle {\begin{cases}{\dot {S}}=\lambda -\mu S-\beta SI,\\\\{\dot {E}}=\beta SI-(\mu +\kappa )E,\\\\{\dot {I}}=\kappa E-(\mu +\gamma )I,\\\\{\dot {R}}=\gamma I-\mu R.\end{cases}}} 16152: 14118: 9467:. If there is no medicine or vaccination available, it is only possible to reduce the infection rate (often referred to as "flattening the curve") by measures such as "social distancing". 4629:

or real data fitted will yield a doubling of the number of infectious people faster than 5 days because the two infected people are infecting people. From the SIR model, we can tell that

3567:{\displaystyle {\begin{aligned}S(t)&=S(0)e^{-\xi (t)}\\I(t)&=N-S(t)-R(t)\\R(t)&=R(0)+\rho \xi (t)\\\xi (t)&={\frac {\beta }{N}}\int _{0}^{t}I(t^{*})\,dt^{*}\end{aligned}}} 21161: 20073: 23961: 11311: 7143: 25685: 24705: 24637: 23803: 12097: 8951:

compartmental equations deviates significantly from the observed results. These uncertainties may even cause the epidemic to end earlier than predicted by the compartmental equations.

7468:. For the second and third equations, consider the population leaving the susceptible class as equal to the number entering the infected class. However, a number equal to the fraction 7253: 2314: 2196: 410: 28034:

Diekmann O, Heesterbeek JA, Metz JA (1990). "On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations".

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can be written as a product of the infection rate near the disease-free equilibrium and average duration of infectiousness. It is used to find the peak and final size of an epidemic.

1645: 1595: 215:). For a specific disease in a specific population, these functions may be worked out in order to predict possible outbreaks and bring them under control. Note that in the SIR model, 19972: 14510:

An MSEIRS model is similar to the MSEIR, but the immunity in the R class would be temporary, so that individuals would regain their susceptibility when the temporary immunity ended.

4560: 3088: 2544: 2430: 25607: 22445: 21652: 21334: 20558: 10725: 10249: 9199: 4508: 2490: 12112:, babies are not born into the susceptible compartment but are immune to the disease for the first few months of life due to protection from maternal antibodies (passed across the 10505: 8895: 2253: 2135: 490:

is the fraction of those contacts between an infectious and susceptible individual which result in the susceptible person becoming infected. (This is mathematically similar to the

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The role of both the basic reproduction number and the initial susceptibility are extremely important. In fact, upon rewriting the equation for infectious individuals as follows:

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i.e., there will be a proper epidemic outbreak with an increase of the number of the infectious (which can reach a considerable fraction of the population). On the contrary, if

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Harko T, Lobo FS, Mak MK (2014). "Exact analytical solutions of the Susceptible-Infected-Recovered (SIR) epidemic model and of the SIR model with equal death and birth rates".

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SEIR), and where infants can be born with immunity (MSIR). Compartmental models can also be used to model multiple risk groups, and even the interaction of multiple pathogens.

1225: 24508: 23397: 23366: 22474: 22052:, there is a significant latency period during which individuals have been infected but are not yet infectious themselves. During this period the individual is in compartment 13403: 11452: 10301: 10059: 7651: 6882: 452:

is the average number of contacts per person per time, multiplied by the probability of disease transmission in a contact between a susceptible and an infectious subject, and

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The deterministic models presented here are valid only in case of sufficiently large populations, and as such should be used cautiously. These models are only valid in the

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Analogously, the steady-state number of infected persons is the number entering the infected state from the susceptible state (number susceptible, times rate of infection)

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in chemistry in which random collisions between molecules result in a chemical reaction and the fractional rate is proportional to the concentration of the two reactants.)

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can be found here ), to obtain its parameters from existing data, or to predict the future dynamics of an epidemics modeled by the SIR model. The approximant involves the

1545: 1317: 25499: 21069: 20220: 17389:{\displaystyle R_{0}(1-P)>1,\quad I(0)>0\Rightarrow \lim _{t\to +\infty }\left(S(t),I(t)\right)=EE=\left({\frac {N}{R_{0}(1-P)}},N\left(R_{0}(1-P)-1\right)\right).} 16395: 11556: 10555: 10396: 10197: 10147: 8945: 7440: 5341: 3997: 3970: 3943: 3265: 3115: 580: 24886: 21099: 16643: 12892: 11387: 10856: 9387: 9277: 9225: 6769: 5297: 5264: 5000: 4934: 3853: 3781: 2361: 25765: 25718: 24955: 24925: 24670: 23228: 22642: 22588: 21036: 14877:{\displaystyle {\begin{aligned}{\frac {dS}{dt}}&=\mu N-\mu S-\beta (t){\frac {I}{N}}S\\{\frac {dI}{dt}}&=\beta (t){\frac {I}{N}}S-(\gamma +\mu )I\end{aligned}}} 13585: 10912: 8978: 7514: 6515: 6405: 5415: 4839: 4810: 617: 22971: 22876: 22110: 12531: 12041: 9465: 8673: 7466: 6567: 5463: 3746: 804:{\displaystyle \left\{{\begin{aligned}&{\frac {dS}{dt}}=-\beta IS,\\&{\frac {dI}{dt}}=\beta IS-\gamma I,\\&{\frac {dR}{dt}}=\gamma I,\end{aligned}}\right.} 526: 488: 23575: 22170: 22150: 21583: 21550: 21517: 20249: 16455: 14923: 9439: 9413: 7486: 7088: 7048: 6541: 6379: 5374: 5056: 4967: 4754: 4627: 4594: 3604: 1987: 23336: 23306: 22130: 21585:. Compartmental models have a disease-free equilibrium (DFE) meaning that it is possible to find an equilibrium while setting the number of infected people to zero, 20989: 20859: 15936: 15900: 15440: 12511: 7317: 7068: 7028: 7008: 6973: 6938: 4732: 4647: 4203: 450: 25463: 25185: 25158: 25019: 24982: 24700: 24015: 23995: 23904: 23851: 22035: 22003: 21976: 21644: 21484: 20626: 20599: 16422: 16348: 15522: 15420: 15393: 15366: 12554: 10686:

gives the policy maker and general public a measure of the efficiency of social mitigation activities such as social distancing and face masking simply by dividing

10684: 10657: 10627: 10585: 10369: 9040: 7810: 7643: 7616: 6263: 5231: 5117: 5090: 5036: 4900: 4872: 4781: 4456: 3873: 3042: 269: 28846: 23725: 23501: 22090: 22060:

involves determining two compartments, infected and non-infected, since they are the populations that spread the infection. So we only need to model the exposed,

20706:

Consider a sexually transmitted disease. In a naive population where almost everyone is susceptible, but the infection seed, if the expected number of gender 1 is

20257: 11637: 10425: 6756: 6725: 6694: 6662: 6633: 6604: 5443: 5011:

time can be used as the initial condition to predict the future after it using this numerical model with assumption of time-evolved parameters such as population,

4177: 4148: 4119: 4090: 4061: 3913: 1495: 1466: 242: 25738: 24259: 21609: 9004: 7405: 6238:

The point EE is called the Endemic Equilibrium (the disease is not totally eradicated and remains in the population). With heuristic arguments, one may show that

23422: 15536:) model is organised as follows. The susceptible individuals (S) can be split in three subgroups by the types of behavior: ignorant or unaware of the epidemic (S 4711: 25436: 23874: 23823: 23745: 23272: 23248: 22868: 22846: 22663: 22609: 22555: 22495: 21329: 21293: 21246: 21226: 21205: 21185: 21009: 20959: 20939: 20919: 20899: 20879: 20767: 20744: 20724: 20657: 16993: 16973: 16887: 16596: 16475: 16368: 12912: 12574: 12481: 10823: 10803: 7377: 7357: 7337: 7297: 7277: 6902: 6727:

denotes the individuals of the population who have been infected with the disease and are capable of spreading the disease to those in the susceptible category.

4687: 4667: 3893: 3715: 3695: 3624: 2388: 1126: 900: 880: 860: 840: 546: 430: 24368: 4813:

only works for a person with symptoms, then these prevention methods are not necessary, and self-isolation and/or medical attention is the best way to cut the

1808: 29730: 27966: 25024: 19444: 19293: 15816: 13771:{\displaystyle {\begin{aligned}{\frac {dE_{1}}{dt}}&=\beta (t)I_{1}-(\gamma +a)E_{1}\\{\frac {dI_{1}}{dt}}&=aE_{1}-(\gamma +\mu )I_{1}\end{aligned}}} 152:

This model is reasonably predictive for infectious diseases that are transmitted from human to human, and where recovery confers lasting resistance, such as

17770:{\displaystyle {\begin{aligned}{\frac {dS}{dt}}&=\mu N(1-P)-\mu S-\rho S-\beta {\frac {I}{N}}S\\{\frac {dV}{dt}}&=\mu NP+\rho S-\mu V\end{aligned}}} 5675: 3677:

These solutions may be easily understood by noting that all of the terms on the right-hand sides of the original differential equations are proportional to

28226:

Blower SM, McLean AR, Porco TC, Small PM, Hopewell PC, Sanchez MA, Moss AR (August 1995). "The intrinsic transmission dynamics of tuberculosis epidemics".

4756:

is usually more stable over time assuming when the infectious person shows symptoms, she/he will seek medical attention or be self-isolated. So if we find

26967: 24020: 22731: 8205:{\displaystyle {\begin{aligned}{\frac {dS}{dt}}&=-{\frac {\beta SI}{N}}+\gamma I\\{\frac {dI}{dt}}&={\frac {\beta SI}{N}}-\gamma I\end{aligned}}} 20921:

is entirely susceptible. Diagonal elements are zero because people of the same gender cannot transmit the disease to each other but, for example, each

20772: 17606:

In case there also are vaccinations of non newborns at a rate ρ the equation for the susceptible and vaccinated subject has to be modified as follows:

23066: 9747:

is an extended SIR model that accounts for vaccination of the susceptible population. This model uses the following system of differential equations:

4000: 22112:

where a communicable disease is spreading. As in the previous example, we can use the transition rates between the compartments per capita such that

24138: 17786: 15524:

fitted using the SIR and the SIRV model are both shown. Note that although the SIR model can model an individual wave, a more complex model like SIR

282:, the model is dynamic in that the numbers in each compartment may fluctuate over time. The importance of this dynamic aspect is most obvious in an 81:(which are deterministic), but can also be used with a stochastic (random) framework, which is more realistic but much more complicated to analyze. 25778: 15736: 15664: 6577:

In 1927, W. O. Kermack and A. G. McKendrick created a model in which they considered a fixed population with only three compartments: susceptible,

22048:

As described in the example above, so many epidemic processes can be described with a SIR‌ model. However, for many important infections, such as

16221: 7953: 4845:

infectious period is in the order of one day from the symptoms showing up, making massive detection with typical frequency in a few days useless.

2366:

models the transition rate from the compartment of susceptible individuals to the compartment of infectious individuals, so that it is called the

1998: 1682: 20601:

is to average the expected number of new infections over all possible infected types. The next-generation method is a general method of deriving

8334: 23432: 18220: 8225: 582:. This is also equivalent to the assumption that the length of time spent by an individual in the infectious state is a random variable with an 30168: 23853:, assumptions are necessary about the time delay between infection and diagnosis and the time between infection and starting to be infectious. 9483:. The SIRD model has semi analytical solutions based on the four parts method. This model uses the following system of differential equations: 24930:

Latent infectious: an individual is infected, has no symptoms, but does infect others. The average duration of the latent infectious state is

13863:

In this model an infection does not leave any immunity thus individuals that have recovered return to being susceptible, moving back into the

12923: 5092:

is very useful to judge the control pressure, e.g., a large value meaning the disease will spread very fast and is very difficult to control.

16482: 2550: 23: 20381: 20081: 19280:{\displaystyle \partial _{t}i(t,a)+\partial _{a}i(t,a)=s(a,t)\int _{0}^{a_{M}}{k(a,a_{1};t)i(a_{1},t)da_{1}}-\mu (a)i(a,t)-\gamma (a)i(a,t)} 15093:{\displaystyle {\frac {1}{T}}\int _{0}^{T}{\frac {\beta (t)}{\mu +\gamma }}\,dt<1\Rightarrow \lim _{t\to +\infty }(S(t),I(t))=DFE=(N,0),} 8675:

and substituting this into the mean-field equations), such that the basic reproduction rate is greater than unity. The solution is given as

30603: 29722: 28839: 23698:. However, it is most frequently calculated using epidemiological models. During an epidemic, typically the number of diagnosed infections 21304: 18796:

is the maximum admissible age) and their dynamics is not described, as one might think, by "simple" partial differential equations, but by

14624: 3120: 934: 108:

is one of the simplest compartmental models, and many models are derivatives of this basic form. The model consists of three compartments:

30563: 27768:"On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations" 14590:{\displaystyle {\color {blue}{{\mathcal {M}}\to {\mathcal {S}}\to {\mathcal {E}}\to {\mathcal {I}}\to {\mathcal {R}}\to {\mathcal {S}}}}} 6696:

is used to represent the individuals not yet infected with the disease at time t, or those susceptible to the disease of the population.

26462:

Yang W, Zhang D, Peng L, Zhuge C, Hong L (2020). "Rational evaluation of various epidemic models based on the COVID-19 data of China".

16159: 4003:

and Schlickeiser, so that there is no need to perform a numerical integration to solve the SIR model (a simplified example practice on

2823:{\displaystyle t={\frac {N}{\beta }}\int _{u}^{1}{\frac {du^{*}}{u^{*}{\mathcal {I}}(u^{*})}},\quad \rho ={\frac {\gamma N}{\beta }},} 2435:

Capasso and, afterwards, other authors have proposed nonlinear forces of infection to model more realistically the contagion process.

24900:

Exposed: an individual is infected, but has no symptoms and does not yet infect others. The average duration of the exposed state is

20632:. (1990), can be used for models with underlying age structure or spatial structure, among other possibilities. In this picture, the 5984:{\displaystyle R_{0}\leq 1\Rightarrow \lim _{t\to \infty }(S(t),I(t),R(t))={\textrm {DFE}}=\left({\frac {\Lambda }{\mu }},0,0\right)} 586:. The "classical" SIR model may be modified by using more complex and realistic distributions for the I-R transition rate (e.g. the 22172:

be the rate at which a latent individual becomes infectious. Then, we can define the model dynamics using the following equations:

3303:

An equivalent so-called analytical solution (involving an integral that can only be calculated numerically) found by Miller yields

15560:(t). Symbolically, the social stress model can be presented by the "reaction scheme" (where I denotes the infected individuals): 4458:

represents the speed of reproduction rate at the beginning of the spreading when all populations are assumed susceptible, e.g. if

28832: 27322:"Personalized-dose Covid-19 vaccination in a wave of virus Variants of Concern: Trading individual efficacy for societal benefit" 23909: 21619:

known as an Endemic Equilibrium (EE) where the disease is not totally eradicated and remains in the population. Mathematically,

19646:

Complexity is added by the initial conditions for newborns (i.e. for a=0), that are straightforward for infectious and removed:

15422:

are diffusion constants. Thereby, one obtains a reaction-diffusion equation. (Note that, for dimensional reasons, the parameter

8681: 30731: 30503: 30490: 29104: 28860: 25936: 23750: 21615:. This solution, also usually ensures that the disease-free equilibrium is also an equilibrium of the system. There is another 4366: 13219:{\displaystyle \left(S(0),E(0),I(0),R(0)\right)\in \left\{(S,E,I,R)\in ^{4}:S\geq 0,E\geq 0,I\geq 0,R\geq 0,S+E+I+R=N\right\}} 326:

For the full specification of the model, the arrows should be labeled with the transition rates between compartments. Between

29775: 28777: 28741: 28722: 28013: 27976: 27011: 26976: 26166: 22669: 4300: 4234: 4121:

are coupled through the above relations, Kermack and McKendrick proposed to study the more general semi-time case, for which

29117: 26358:"Analytical solution of the SIR-model for the temporal evolution of epidemics. Part A: Time-independent reproduction factor" 21295:

is a non-negative matrix so it has single, unique, positive, real eigenvalue which is strictly greater than all the others.

19021:{\displaystyle \partial _{t}s(t,a)+\partial _{a}s(t,a)=-\mu (a)s(a,t)-s(a,t)\int _{0}^{a_{M}}k(a,a_{1};t)i(a_{1},t)\,da_{1}} 17151:{\displaystyle R_{0}(1-P)\leq 1\Rightarrow \lim _{t\to +\infty }\left(S(t),I(t)\right)=DFE=\left(N\left(1-P\right),0\right)} 12996:,0,0,0) and an Endemic Equilibrium EE, and one can show that, independently from biologically meaningful initial conditions 6569:. If there is no medicine or vaccination available, it is only possible to reduce the infection rate (often referred to as " 30767: 30292: 28756: 9048: 5792: 26935:

Padua RN, Tulang AB (2 December 2010). "A Density–Dependent Epidemiological Model for the Spread of Infectious Diseases".

19904:{\displaystyle s(t,0)=\int _{0}^{a_{M}}\left(\varphi _{s}(a)s(a,t)+\varphi _{i}(a)i(a,t)+\varphi _{r}(a)r(a,t)\right)\,da} 16578:

The SIR model can be modified to model vaccination. Typically these introduce an additional compartment to the SIR model,

12124:. This added detail can be shown by including an M class (for maternally derived immunity) at the beginning of the model. 4649:

is determined by the nature of the disease and also a function of the interactive frequency between the infectious person

29429: 18354: 1032: 129:

nfectious individuals. These are individuals who have been infected and are capable of infecting susceptible individuals.

28271:"Quantifying TB transmission: a systematic review of reproduction number and serial interval estimates for tuberculosis" 27473:

Hunziker P (2021-03-07). "Vaccination strategies for minimizing loss of life in Covid-19 in a Europe lacking vaccines".

21941:{\displaystyle R_{0}>1,I(0)>0\Rightarrow \lim _{t\to \infty }(C_{1}(t),C_{2}(t),\cdots ,C_{n}(t))={\textrm {EE}}.} 11947: 7151: 4205:. An analytic approximant and exact expressions for the final values are available for the semi-time SIR model as well. 2628: 30495: 28753:"Extracting transmission and recovery parameters for an adaptive global system dynamics model of the COVID-19 pandemic" 25931: 7818: 5843:

which has threshold properties. In fact, independently from biologically meaningful initial values, one can show that:

1331: 29499: 26200:"A box, a trough and marbles: How the Reed-Frost epidemic theory shaped epidemiological reasoning in the 20th century" 18681: 18599: 18517: 15659:(the autocatalytic form here means that the transition rate is proportional to the square of the infected fraction I); 3007:{\displaystyle ({\mathcal {S}}(1),{\mathcal {I}}(1),{\mathcal {R}}(1))=(S(0),N-R(0)-S(0),R(0)),\quad u_{T}<u<1,} 1137: 29925: 29920: 28703: 28361: 28332: 27644:

The COVID-19 Data Repository. The Center for Systems Science and Engineering (CSSE) at Johns Hopkins University (JHU)

27061:"Improved susceptible–infectious–susceptible epidemic equations based on uncertainties and autocorrelation functions" 27043: 26687: 26647: 26622: 25130: 17405: 14176:{\displaystyle \color {blue}{{\mathcal {M}}\to {\mathcal {S}}\to {\mathcal {E}}\to {\mathcal {I}}\to {\mathcal {R}}}} 8758: 7870: 29530: 12100:

method to extract analytically all SIRVD model parameters from measured COVID-19 data of a completed pandemic wave.

11610:

the model had been solved approximately, and exactly for some special cases, irrespective of the functional form of

10730: 30823: 30464: 30004: 29411: 29027: 16895: 13587:

the condition for the global attractiveness of DFE is that the following linear system with periodic coefficients:

3918:

A highly accurate analytic approximant of the SIR model as well as exact analytic expressions for the final values

27406:"Effects of social distancing and isolation on epidemic spreading modeled via dynamical density functional theory" 25870:

Other occurrences which may need to be considered when modeling an epidemic include things such as the following:

24513: 16102: 15497:

An example of using the numerical SIR model to fit the COVID-19 data (from U.S. CDC) in the state of Ohio, U.S.A.

14618:

classes of diseases, one should consider a force of infection with periodically ('seasonal') varying contact rate

7359:. This is the transmission probability times the contact rate. Besides, an infected individual makes contact with 4217:

spreading data. Three reproduction numbers can be pulled out from the data analyzed with numerical approximation,

30818: 30695: 30556: 29598: 28691: 26962: 21111: 19980: 12894:

but this is only constant because of the simplifying assumption that birth and death rates are equal; in general

11313:

are the infection, vaccination, recovery, and fatality rates, respectively. For the semi-time initial conditions

4179:

are both arbitrary. This latter version, denoted as semi-time SIR model, makes predictions only for future times

1128:. Note that the above relationship implies that one need only study the equation for two of the three variables. 27116: 26107:

Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character

26059:

Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character

26018:

Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character

25977:

Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character

24990:

after diagnosis: measures are taken to prevent further infections, for example by isolating the infected person.

14108:

For the case of a disease, with the factors of passive immunity, and a latency period there is the MSEIR model.

11242: 7096: 30339: 29937: 29656: 29634: 29065: 25899:

One of the possible extensions of mean-field models considers the spreading of epidemics on a network based on

25775:

tuberculosis (TB). Blower and coauthors calculated from a simple model of TB the following reproduction number:

25646: 24598: 21779:{\displaystyle R_{0}\leq 1\Rightarrow \lim _{t\to \infty }(C_{1}(t),C_{2}(t),\cdots ,C_{n}(t))={\textrm {DFE}}} 21452:{\displaystyle {\dot {C_{i}}}={\operatorname {d} \!C_{i} \over \operatorname {d} \!t}=f(C_{1},C_{2},...,C_{n})} 21308: 12046: 7212: 2264: 2146: 337: 78: 27992:

van den Driessche, P.; Watmough, James (2008), Brauer, Fred; van den Driessche, Pauline; Wu, Jianhong (eds.),

26870:

Smith DK, Lauro K, Kelly D, Fish J, Lintelman E, McEwen D, Smith C, Stecz M, Ambagaspitiya TD, Chen J (2022).

25903:

concepts. Stochastic epidemic models have been studied on different networks and more recently applied to the

17484: 4213:

Numerical solutions to the SIR model can be found in the literature. An example is using the model to analyze

1600: 1550: 290:

in 1968. Such diseases tend to occur in cycles of outbreaks due to the variation in number of susceptibles (S(

30307: 29005: 25941: 19917: 18797: 16087:{\displaystyle {\frac {d^{2}I}{dt}}-\sigma _{o}^{2}I+{\frac {3}{2}}{\frac {\sigma _{o}^{2}}{I_{max}}}I^{2}=0} 15648:{\displaystyle {\color {blue}{{\mathcal {S_{ign}}}+2{\mathcal {I}}\to {\mathcal {S_{res}}}+2{\mathcal {I}}}}} 7442:

susceptible persons, and therefore, the whole number of susceptibles infected by infectives per unit time is

4513: 3047: 2498: 2396: 914: 89: 63: 27896:"Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission" 25557: 22398: 20563:

A basic reproduction number can be calculated as the spectral radius of an appropriate functional operator.

20485: 16477:(for the number of infected persons). An analytical solution to the KdV-SIR equation is written as follows: 10689: 10202: 9127: 4596:, i.e. the number of infectious people doubled in one cycle of 5 days. The data simulated by the model with 4461: 2447: 313: 30664: 30618: 30613: 30580: 30334: 29487: 29250: 28965: 15548:) that do not react on the external stimuli (this is a sort of refractory period). In other words: S(t) = S 13392:{\displaystyle R_{0}\leq 1\Rightarrow \lim _{t\to +\infty }\left(S(t),E(t),I(t),R(t)\right)=DFE=(N,0,0,0),} 10430: 8846: 2207: 2089: 1956:

This shows that at the end of an epidemic that conforms to the simple assumptions of the SIR model, unless

275:= 0 whereas the latter describes the ratio between the frequency of contacts to the frequency of recovery. 29159: 26319: 25612: 24564: 18437: 10061:

are the rates of infection, recovery, and vaccination, respectively. For the semi-time initial conditions

8814: 7913: 30608: 30376: 30247: 30183: 30151: 30099: 29396: 29378: 25768: 23694:

The basic reproduction number can be estimated through examining detailed transmission chains or through

22501: 13548:{\displaystyle R_{0}>1,I(0)>0\Rightarrow \lim _{t\to +\infty }\left(S(t),E(t),I(t),R(t)\right)=EE.} 9706: 7779:{\displaystyle \operatorname {E} =\int _{0}^{\infty }e^{-(\mu +\delta )x}\,dx={\frac {1}{\mu +\delta }},} 28801: 25853:

considered within the model by including a fraction of the newborn members in the infected compartment.

25504: 18763: 17469:

may be as high as 18 one should vaccinate at least 94.4% of newborns in order to eradicate the disease.

10727:. Under massive vaccination, the goal of disease control is to reduce the effective reproduction number 9289: 8061: 7533: 6417: 6281: 1270:. From here it follows that, on average, the number of contacts by an infectious individual with others 1230: 30813: 30721: 30549: 29932: 29345: 29085: 16424:(R-naught) are the time-independent relative growth rate and basic reproduction number, respectively. 15462: 8423: 1187: 93: 27895: 26872:"Teaching Undergraduate Physical Chemistry Lab with Kinetic Analysis of COVID-19 in the United States" 26814:

Hart KD, Thompson C, Burger C, Hardwick D, Michaud AH, Bulushi A, Pridemore C, Ward C, Chen J (2021).

26518:"Effects of the infectious period distribution on predicted transitions in childhood disease dynamics" 26410:"Analytical solution of the SIR-model for the temporal evolution of epidemics. Part B: Semi-time case" 24470: 23373: 23342: 22450: 11392: 10254: 10032: 8639:

It is possible to find an analytical solution to this model (by making a transformation of variables:

6822: 74:

was also a significant and widely overlooked ancestor of modern epidemiological modelling approaches.

30593: 30436: 30426: 30121: 30057: 29780: 29745: 29548: 29525: 29311: 29124: 25921: 21616: 20662: 19652: 17552: 17466: 13849:{\displaystyle {\color {blue}{{\mathcal {S}}\to {\mathcal {E}}\to {\mathcal {I}}\to {\mathcal {S}}}}} 11561: 10630: 10306: 10064: 7090:. These variables determine the length of the epidemic and would have to be updated with each cycle. 5783: 5166: 4222: 1181: 505:, the transition rate is assumed to be proportional to the number of infectious individuals which is 85: 27189:

The first and second differential equations are transformed and brought to the same form as for the

23747:

is known. In the early stages of an epidemic, growth is exponential, with a logarithmic growth rate

22186: 11457: 5123: 3786: 3629: 30639: 30254: 30116: 29991: 29845: 28960: 28950: 27026:

Hethcote HW (1989). "Three Basic Epidemiological Models". In Levin SA, Hallam TG, Gross LJ (eds.).

26757:"Mathematical models of SIR disease spread with combined non-sexual and sexual transmission routes" 20628:

when more than one class of infectives is involved. This method, originally introduced by Diekmann

12463:

Assuming that the latency period is a random variable with exponential distribution with parameter

9230: 6809:{\displaystyle {\color {blue}{{\mathcal {S}}\rightarrow {\mathcal {I}}\rightarrow {\mathcal {R}}}}} 3270: 1650: 928:, however it is possible to derive its analytic solution in implicit form. Firstly note that from: 910: 583: 27210:"Analytical Modeling of the Temporal Evolution of Epidemics Outbreaks Accounting for Vaccinations" 24842:{\displaystyle \ln(n_{E}(t))=\ln(n_{E}(0))+{\frac {1}{\tau }}\int \limits _{0}^{t}\ln(R_{0}(t))dt} 21251: 19598: 11639:. This is achieved upon rewriting the above SIRVD model equations in equivalent, but reduced form 11316: 1500: 1277: 30441: 30359: 30225: 29755: 29553: 29260: 29198: 29032: 29000: 26485:"The SIR dynamic model of infectious disease transmission and its analogy with chemical kinetics" 25471: 24987: 23056:{\displaystyle V_{ij}={\partial \!\ V_{i}(\mathrm {x} ^{*}) \over \partial \!\ \mathrm {x} _{j}}} 22961:{\displaystyle F_{ij}={\partial \!\ F_{i}(\mathrm {x} ^{*}) \over \partial \!\ \mathrm {x} _{j}}} 21041: 20196: 16373: 15461:

infected neighbors in a given time step leads to results similar to giant component formation on

11509: 10510: 10374: 10152: 10114: 8900: 7645:

reflects the time in the susceptible state before becoming infected, which can be simplified to:

7410: 5306: 3975: 3948: 3921: 3232: 3093: 1791: 1322: 551: 286:

disease with a short infectious period, such as measles in the UK prior to the introduction of a

29112: 28000:, Lecture Notes in Mathematics, vol. 1945, Berlin, Heidelberg: Springer, pp. 159–178, 27661: 27647: 24852: 23680:{\displaystyle R_{0}=\rho (FV^{-1})={\frac {\kappa \beta S^{*}}{(\mu +\kappa )(\mu +\gamma )}}.} 21074: 16610: 12850: 11357: 10828: 9256: 9204: 5269: 5236: 4972: 4906: 3821: 3751: 2334: 30746: 30028: 29813: 29563: 29436: 29042: 28899: 28889: 25946: 25743: 25690: 24933: 24903: 24642: 23194: 22615: 22561: 21167:

of the expected number of each gender in the next-generation. Note that multiplication factors

21014: 20637: 13781:

is stable (i.e. it has its Floquet's eigenvalues inside the unit circle in the complex plane).

13561: 10873: 9366: 8957: 7789:

such that the number of susceptible persons is the number entering the susceptible compartment

7491: 5379: 4816: 4787: 3717:, and the time rescaled so that the differential operator on the left-hand side becomes simply 906: 648: 596: 283: 27949: 27479: 24017:(the serial interval) has passed, then the number of infectious individuals over time grows as 22095: 13789:

The SEIS model is like the SEIR model (above) except that no immunity is acquired at the end.

12516: 12017: 8642: 7445: 6494: 6384: 5448: 3720: 3606:

can be interpreted as the expected number of transmissions an individual has received by time

508: 455: 30808: 30772: 30716: 30349: 29981: 29850: 29358: 29265: 29184: 26055:"An application of the theory of probabilities to the study of a priori pathometry.—Part III" 23565:{\displaystyle V={\begin{pmatrix}\mu +\kappa &0\\-\kappa &\gamma +\mu \end{pmatrix}}} 22155: 22135: 22010:

of this operator to unity determines whether the generations of infected people grow or not.

21555: 21522: 21489: 20365:{\displaystyle 1=\int _{0}^{a_{M}}\varphi (a)\exp \left(-\int _{0}^{a}{\mu (q)dq}\right)\,da} 20225: 16427: 14890: 10589: 9444: 7471: 7073: 7033: 6546: 5465:, and where a communicable disease is spreading. The model with mass-action transmission is: 5346: 5041: 4939: 4739: 4599: 4566: 3580: 1959: 620: 29129: 26014:"An application of the theory of probabilities to the study of a priori pathometry.—Part II" 23311: 23281: 22115: 20964: 20834: 15921: 15885: 15425: 12486: 9418: 9392: 7302: 7053: 7013: 6978: 6943: 6908: 6520: 6358: 4717: 4632: 4182: 1131:

Secondly, we note that the dynamics of the infectious class depends on the following ratio:

435: 30726: 30572: 30431: 30324: 30270: 30198: 30188: 29880: 29735: 29608: 29215: 28824: 28633: 28578: 28509: 28490:

Pastor-Satorras R, Vespignani A (April 2001). "Epidemic spreading in scale-free networks".

28456: 28149: 28104: 27673: 27598: 27520: 27474: 27427: 27362: 27221: 27072: 26883: 26576: 26421: 26369: 26279: 26176: 26114: 26066: 26025: 25984: 25973:"An application of the theory of probabilities to the study of a priori pathometry.—Part I" 25441: 25163: 25136: 24997: 24960: 24678: 24000: 23973: 23882: 23829: 22013: 21981: 21954: 21622: 21462: 20604: 20577: 16400: 16326: 15500: 15398: 15371: 15344: 12536: 12420: 10662: 10635: 10605: 10563: 10347: 9013: 7792: 7621: 7594: 6570: 6241: 5209: 5095: 5068: 5014: 4878: 4850: 4759: 4434: 4031: 3858: 3020: 247: 71: 67: 26184: 23701: 22075: 11613: 10401: 6732: 6701: 6670: 6638: 6609: 6580: 5428: 4153: 4124: 4095: 4066: 4037: 3898: 1471: 1442: 218: 8: 30669: 30518: 30508: 30474: 30079: 29517: 29353: 29164: 25893: 25723: 23400:, times the average length of time an individual spends in a single visit to compartment 21588: 20991:

is a reproduction number, but one where who infects whom is accounted for. If generation

10917: 8983: 7382: 5163:. If social distancing and masks add another 50% cut in infection efficiency, we can set 4024: 3674:

Effectively the same result can be found in the original work by Kermack and McKendrick.

1946:{\displaystyle s_{\infty }=1-r_{\infty }=-R_{0}^{-1}\,W(-s_{0}R_{0}e^{-R_{0}(1-r_{0})}).} 636: 587: 301: 59: 28637: 28582: 28513: 28460: 28153: 27677: 27602: 27524: 27431: 27366: 27225: 27076: 26887: 26580: 26425: 26373: 26283: 26118: 26070: 26029: 25988: 23695: 23404: 12455: 12127: 10303:

the model had been solved approximately. The occurrence of a pandemic outburst requires

4969:, a steady state reached and no number of infectious people changes over time; and when 4693: 54:

The origin of such models is the early 20th century, with important works being that of

30762: 30659: 30649: 30242: 30237: 30146: 30126: 29760: 29603: 29568: 29422: 29388: 28938: 28783: 28752: 28669: 28656: 28621: 28602: 28568: 28541: 28499: 28429: 28421: 28295: 28270: 28251: 28170: 28137: 28113: 28069: 27931: 27868: 27835: 27813: 27743: 27716: 27697: 27621: 27588: 27576: 27554: 27541: 27510: 27498: 27450: 27417: 27405: 27386: 27249: 27167: 27134: 27093: 27060: 27002: 26917: 26904: 26871: 26840: 26816:"Remote Learning of COVID-19 Kinetic Analysis in a Physical Chemistry Laboratory Class" 26815: 26781: 26756: 26729: 26704: 26592: 26544: 26517: 26463: 26387: 26295: 26269: 26232: 26199: 25900: 25421: 23859: 23808: 23730: 23257: 23251: 23233: 22853: 22831: 22648: 22594: 22540: 22480: 21314: 21278: 21231: 21211: 21190: 21170: 20994: 20944: 20924: 20904: 20884: 20864: 20752: 20729: 20709: 20642: 19585:{\displaystyle F(a,t,i(\cdot ,\cdot ))=\int _{0}^{a_{M}}k(a,a_{1};t)i(a_{1},t)\,da_{1}} 19428:{\displaystyle \partial _{t}r(t,a)+\partial _{a}r(t,a)=-\mu (a)r(a,t)+\gamma (a)i(a,t)} 16978: 16958: 16872: 16581: 16460: 16353: 15872:{\displaystyle {\color {blue}{{\mathcal {S_{...}}}+{\mathcal {I}}\to {\mathcal {2I}}}}} 12897: 12559: 12466: 10808: 10788: 7362: 7342: 7322: 7282: 7262: 6887: 4672: 4652: 4012: 3878: 3700: 3680: 3609: 2373: 2367: 1795: 1111: 885: 865: 845: 825: 531: 491: 415: 29174: 28769: 28088: 27911: 5772:{\displaystyle \left(S(t),I(t),R(t)\right)=\left({\frac {\Lambda }{\mu }},0,0\right).} 30674: 30654: 30634: 30528: 30469: 30329: 30282: 30084: 29586: 29255: 29245: 29235: 29220: 29037: 29020: 28884: 28787: 28773: 28737: 28718: 28699: 28673: 28661: 28594: 28533: 28472: 28357: 28328: 28300: 28243: 28175: 28118: 28061: 28009: 27972: 27923: 27915: 27873: 27855: 27805: 27797: 27748: 27717:"Analysis of an SIR epidemic model with pulse vaccination and distributed time delay" 27701: 27689: 27626: 27558: 27546: 27455: 27378: 27253: 27172: 27154: 27098: 27039: 27007: 26972: 26921: 26909: 26845: 26786: 26734: 26683: 26643: 26618: 26549: 26391: 26237: 26219: 26162: 25926: 25904: 10588:

parameters. It uses recursive equations to calculate the next step by converting the

8417: 3818:. The differential equations are now all linear, and the third equation, of the form 1497:

are the initial numbers of, respectively, susceptible and removed subjects. Writing

28606: 28545: 28433: 28255: 28195:"Qualitative Analysis of Delayed SIR Epidemic Model with a Saturated Incidence Rate" 28073: 27935: 27817: 26596: 26299: 19595:

is the force of infection, which, of course, will depend, though the contact kernel

12992:

Similarly to the SIR model, also, in this case, we have a Disease-Free-Equilibrium (

10560:

The numerical solution of this model to calculate the real-time reproduction number

7530:

For example, one may postulate that the expected duration of susceptibility will be

1647:

for the proportion of susceptible and removed individuals respectively in the limit

30787: 30782: 30700: 30523: 30459: 30446: 30275: 30264: 30156: 30136: 29971: 29885: 29860: 29591: 29538: 29477: 29368: 29015: 28943: 28765: 28651: 28641: 28586: 28525: 28517: 28464: 28413: 28290: 28282: 28235: 28206: 28165: 28157: 28108: 28100: 28051: 28043: 28001: 27907: 27863: 27847: 27787: 27779: 27738: 27728: 27681: 27616: 27606: 27536: 27528: 27445: 27435: 27390: 27370: 27333: 27297: 27287: 27239: 27229: 27162: 27146: 27088: 27080: 27031: 26997: 26944: 26899: 26891: 26835: 26827: 26776: 26768: 26724: 26716: 26584: 26539: 26529: 26496: 26439: 26429: 26377: 26287: 26227: 26211: 26180: 26154: 26146: 26122: 26074: 26033: 25992: 22817:{\displaystyle V_{i}(\mathrm {x} )=V_{i}^{-}(\mathrm {x} )-V_{i}^{+}(\mathrm {x} )} 15468: 12121: 12014:

is a reduced, dimensionless time. The temporal dependence of the infected fraction

10598: 5059: 5007: 635:

A single realization of the SIR epidemic as produced with an implementation of the

593:

For the special case in which there is no removal from the infectious compartment (

167: 28136:

Lipsitch M, Cohen T, Cooper B, Robins JM, Ma S, James L, et al. (June 2003).

25133:. It follows from the differential equation for the number of exposed individuals 9479:(meaning specifically individuals having survived the disease and now immune) and 30690: 30598: 30391: 30386: 30230: 30178: 29986: 29963: 29808: 29785: 29169: 29134: 29010: 28955: 28349: 27499:"Spreading of infections on random graphs: A percolation-type model for COVID-19" 26172: 25951: 22007: 21105: 20633: 15479: 5058:. However, away from this reference point, errors will accumulate over time thus 4008: 317:

Flow diagram of states in an SIR epidemic model and transition rates between them

171:

Spatial SIR model simulation. Each cell can infect its eight immediate neighbors.

29139: 28521: 28005: 27532: 27276:"Mathematics of Epidemics: General solution of SIRVD, SIRV, SIRD and SIR Models" 27035: 26895: 26318:

Beckley R, Weatherspoon C, Alexander M, Chandler M, Johnson A, Bhatt GS (2013).

23181:{\displaystyle \mathrm {x} ^{*}=(S^{*},E^{*},I^{*},R^{*})=(\lambda /\mu ,0,0,0)} 4019:

which is part of all basic data visualization software such as Microsoft Excel,

639:

and the numerical solution of the ordinary differential equation system (dashed)

271:

are different quantities – the former describes the number of recovered at

30777: 30741: 30588: 30421: 30354: 30297: 30069: 29958: 29950: 29875: 29828: 29823: 29770: 29494: 29275: 29270: 29230: 29179: 29144: 28982: 28918: 28626:

Proceedings of the National Academy of Sciences of the United States of America

28590: 28468: 28447:

May RM, Lloyd AL (December 2001). "Infection dynamics on scale-free networks".

27611: 27440: 27302: 27244: 27150: 26992:

von Csefalvay, Chris (2023-01-01), von Csefalvay, Chris (ed.), "Host factors",

26444: 26434: 26409: 26382: 26357: 26215: 21164: 20746:, we can know how many would be infected in the next-generation. Such that the 15483: 28807: 28286: 27993: 27685: 26772: 26720: 26588: 26291: 26158: 17848:{\displaystyle P\geq 1-\left(1+{\frac {\rho }{\mu }}\right){\frac {1}{R_{0}}}} 30802: 30736: 30644: 30302: 30259: 30218: 30111: 30094: 30052: 29765: 29316: 29285: 28906: 27919: 27859: 27801: 27693: 27662:"A KdV–SIR Equation and Its Analytical Solutions for Solitary Epidemic Waves" 27643: 27158: 26831: 26223: 23877: 23308:

a matrix which represents the average duration of infectiousness. Therefore,

22043: 21307:, the dynamics of spreading is usually described through a set of non-linear 18511:(to limit ourselves to the susceptible-infectious-removed scheme) such that: 15792:{\displaystyle {\color {blue}{{\mathcal {S_{exh}}}\to {\mathcal {S_{ign}}}}}} 15720:{\displaystyle {\color {blue}{{\mathcal {S_{res}}}\to {\mathcal {S_{exh}}}}}} 15487: 12429: 5300: 29152: 28646: 28161: 26642:. Oxford Series in Ecology and Evolution. Oxford : Oxford University Press. 25842: 24358:{\displaystyle {\frac {dn_{E}(t)}{dt}}=n_{E}(t){\frac {\ln(R_{0})}{\tau }}.} 16314:{\displaystyle I_{max}={\frac {S_{o}}{2}}{\frac {(R_{o}-1)^{2}}{R_{o}^{2}}}} 8897:. As the system is assumed to be closed, the susceptible population is then 8013:{\displaystyle I={\frac {\mu N}{\mu +\lambda }}\lambda {\frac {1}{\mu +v}}.} 5420: 2073:{\displaystyle {\frac {dI}{dt}}=\left(R_{0}{\frac {S}{N}}-1\right)\gamma I,} 1780:{\displaystyle s_{\infty }=1-r_{\infty }=s_{0}e^{-R_{0}(r_{\infty }-r_{0})}} 30513: 30451: 30213: 30193: 30141: 30089: 30040: 29840: 29790: 29401: 28972: 28665: 28598: 28537: 28476: 28304: 28179: 28122: 27927: 27877: 27851: 27752: 27630: 27550: 27459: 27234: 27209: 27176: 27102: 26913: 26849: 26790: 26738: 26553: 26534: 26501: 26241: 26127: 26102: 26079: 26054: 26038: 26013: 25997: 25972: 21612: 12416: 8405:{\displaystyle {\frac {dI}{dt}}=(\beta -\gamma )I-{\frac {\beta }{N}}I^{2}} 4208: 28378: 28247: 28211: 28194: 28065: 27809: 27382: 27353:

Noble JV (August 1974). "Geographic and temporal development of plagues".

26948: 23491:{\displaystyle F={\begin{pmatrix}0&\beta S^{*}\\0&0\end{pmatrix}}} 22056:(for exposed). Here, the formation of the next-generation matrix from the 18338:{\displaystyle S^{*}(t)=1-{\frac {p}{1-(1-p)E^{-\mu T}}}E^{-\mu MOD(t,T)}} 13871:) compartment. The following differential equations describe this model: 8317:{\displaystyle {\frac {dS}{dt}}+{\frac {dI}{dt}}=0\Rightarrow S(t)+I(t)=N} 30406: 30208: 30131: 30106: 29855: 29835: 29740: 29700: 29624: 28995: 28990: 28911: 28894: 28812: 28573: 28504: 27733: 27292: 27275: 25916: 24460:{\displaystyle {\frac {d\ln(n_{E}(t))}{dt}}={\frac {\ln(R_{0})}{\tau }}.} 14614: 12425: 10593: 8051: 7299:

of people that an individual makes contact with per unit time. Then, let

4034:

considered the so-called all-time SIR model where the initial conditions

918: 287: 55: 28138:"Transmission dynamics and control of severe acute respiratory syndrome" 27338: 27084: 12982:{\displaystyle R_{0}={\frac {a}{\mu +a}}{\frac {\beta }{\mu +\gamma }}.} 9415:. The animation shows the effect of reducing the rate of infection from 9112:

In the long run, in the SI model, all individuals will become infected.

6517:. The animation shows the effect of reducing the rate of infection from 26:. The population is assigned to compartments with labels – for example, 30541: 30344: 30319: 30287: 30064: 30045: 30035: 30023: 29870: 29482: 29454: 29280: 29240: 28529: 28425: 28239: 28047: 27783: 26484: 25415: 25118:{\displaystyle R_{0}=1+K(\tau _{E}+\tau _{I})+K^{2}\tau _{E}\tau _{I}.} 18214:

one obtains that the dynamics of the susceptible subjects is given by:

16560:{\displaystyle I=I_{max}sech^{2}\left({\frac {\sigma _{o}}{2}}t\right)} 15493: 8954:

As a special case, one obtains the usual logistic function by assuming

2617:{\displaystyle {\mathcal {I}}(u)=N-{\mathcal {R}}(u)-{\mathcal {S}}(u)} 925: 26664: 24891: 21646:

is a threshold for stability of a disease-free equilibrium such that:

20469:{\displaystyle n^{*}(a)=C\exp \left(-\int _{0}^{a}\mu (q)\,dq\right),} 20183:{\displaystyle \partial _{t}n(t,a)+\partial _{a}n(t,a)=-\mu (a)n(a,t)} 5233:

will perfectly correlate with the waves of the spreading and whenever

882:

is the stock of removed population (either by death or recovery), and

22:

are a very general modelling technique. They are often applied to the

30416: 29895: 28928: 28923: 28856: 28056: 27792: 27767: 27374: 26468: 14696:{\displaystyle F=\beta (t){\frac {I}{N}},\quad \beta (t+T)=\beta (t)} 12135:

To indicate this mathematically, an additional compartment is added,

12117: 10021: 8055: 3222:{\displaystyle u_{T}=e^{-(R_{\infty }-R(0))/\rho }(=S_{\infty }/S(0)} 1016:{\displaystyle {\frac {dS}{dt}}+{\frac {dI}{dt}}+{\frac {dR}{dt}}=0,} 644: 28417: 27577:"Social stress drives the multi-wave dynamics of COVID-19 outbreaks" 27030:. Biomathematics. Vol. 18. Berlin: Springer. pp. 119–144. 26317: 22072:, compartments. Consider a population characterized by a death rate 12513:), and also assuming the presence of vital dynamics with birth rate 814: 30399: 30314: 29890: 29865: 29818: 29363: 28933: 28876: 28354:

Population Dynamics of Infectious Diseases: Theory and Applications

27593: 27574: 27515: 27422: 26615:

The mathematical theory of infectious diseases and its applications

25126: 24128:{\displaystyle n_{E}(t)=n_{E}(0)\,R_{0}^{t/\tau }=n_{E}(0)\,e^{Kt}} 23275: 22049: 20479:

automatically ensuring the existence of the disease-free solution:

16889:

is the class of vaccinated subjects. It is immediate to show that:

12113: 9735:

are the rates of infection, recovery, and mortality, respectively.

8042: 4842: 4214: 4004: 1790:(note that the infectious compartment empties in this limit). This 295: 27766:

Diekmann, O.; Heesterbeek, J. A. P.; Metz, J. A. J. (1990-06-01).

26274: 25887: 21298: 9282: 7407:

of them are susceptible. Thus, we have every infective can infect

5062:

is needed to find an optimal time step for more accurate results.

30203: 30074: 29327: 28817: 28559:

Newman ME (July 2002). "Spread of epidemic disease on networks".

26153:. Vol. 4. University of California Press. pp. 149–165. 24896:

In this model, an individual infection has the following stages:

20824:{\displaystyle G={\begin{pmatrix}0&f\\m&0\end{pmatrix}},} 16206:{\displaystyle R_{o}={\frac {\beta }{\gamma }}{\frac {S_{o}}{N}}} 12109: 6664:. The compartments used for this model consist of three classes: 6410: 631: 623:

solution, in which every individual eventually becomes infected.

161: 153: 27968:

Mathematical Tools for Understanding Infectious Disease Dynamics

27575:

Kastalskiy, IA; Pankratova, EV; Mirkes, EM; et al. (2021).

26640:

Infectious diseases in primates: behavior, ecology and evolution

21248:. The average per generation multiplication factor is therefore 21108:

of the next-generation matrix is the basic reproduction number,

619:), the SIR model reduces to a very simple SI model, which has a 30411: 29976: 29629: 29225: 23254:

which represents the infection rates near the equilibrium, and

19714:

but that are nonlocal for the density of susceptible newborns:

4020: 28404:

Bartlett MS (1957). "Measles periodicity and community size".

26147:"Deterministic and Stochastic Epidemics in Closed Populations" 24249:{\displaystyle \ln(n_{E}(t))=\ln(n_{E}(0))+\ln(R_{0})t/\tau .} 18428: 15861: 15836: 15833: 15779: 15776: 15756: 15753: 15707: 15704: 15684: 15681: 15622: 15619: 15586: 15583: 2438: 29640: 27965:

Diekmann, Odo; Heesterbeek, Hans; Britton, Tom (2012-11-18).

26567:

Hethcote H (2000). "The Mathematics of Infectious Diseases".

25831:{\displaystyle R_{0}=R_{0}^{\text{FAST}}+R_{0}^{\text{SLOW}}} 6274: 2492:, etc., it corresponds to the following time parametrization 1108:

expressing in mathematical terms the constancy of population

157: 28854: 528:. If an individual is infectious for an average time period 29093: 28750: 28269:

Ma Y, Horsburgh CR, White LF, Jenkins HE (September 2018).

26403: 26401: 22536:

be the rate of appearance of new infections in compartment

22381: 19977:

Moreover, defining now the density of the total population

10398:

of the susceptible compartment remains relatively close to

1547:

for the initial proportion of susceptible individuals, and

798: 28325:

Mathematical Models in Population Biology and Epidemiology

27964: 27403: 26813: 26637: 23970:

If an individual, after getting infected, infects exactly

22044:

The SEIR model with vital dynamics and constant population

15473:

model - combination of SIR with modelling of social stress

8744:{\displaystyle I(t)={\frac {I_{\infty }}{1+Ve^{-\chi t}}}} 1321:

By dividing the first differential equation by the third,

28734:

Mathematical Structures of Epidemic Systems. 2nd Printing

28489: 28135: 27991: 27893: 25843:

Other considerations within compartmental epidemic models

20861:

is the expected number of secondary infections of gender

10858:, the spreading decays and daily infected cases go down. 5421:

The SIR model with vital dynamics and constant population

4420:{\displaystyle R_{e}={\frac {\beta _{t}S}{\gamma _{t}N}}} 28622:"Evidence that coronavirus superspreading is fat-tailed" 28379:"COVID-19: Data-Driven Mean-Field-Type Game Perspective" 26398: 26103:"A Contribution to the Mathematical Theory of Epidemics" 22590:

is the rate of transfer of individuals into compartment

21611:. In other words, as a rule, there is an infection-free 10805:

is the number of susceptible population at the time and

4209:

Numerical solutions to the SIR model with approximations

626: 28268: 28033: 27765: 27058: 25767:

ago. A more correct treatment would require the use of

22721:{\displaystyle F_{i}(\mathrm {x} )-V_{i}(\mathrm {x} )} 22476:

denotes the number or proportion of individuals in the

16598:, for vaccinated individuals. Below are some examples. 15446: 12415:

Some people who have had an infectious disease such as

9120: 4344:{\displaystyle R_{t}={\frac {\beta _{t}}{\gamma _{t}}}} 4278:{\displaystyle R_{0}={\frac {\beta _{0}}{\gamma _{0}}}} 16:

Type of mathematical model used for infectious diseases

28225: 27889: 27887: 27404:

Te Vrugt M, Bickmann J, Wittkowski R (November 2020).

27059:

Nakamura, G.M.; Cardoso, G.C.; Martinez, A.S. (2020).

25366: 25257: 25214: 23516: 23447: 23191:

We now can form the next-generation matrix (operator)

20787: 12147:

This results in the following differential equations:

9447: 9421: 9395: 9369: 9292: 7918: 7881: 6549: 6523: 6497: 6420: 6387: 6361: 6284: 28322: 27894:

van den Driessche, P.; Watmough, James (2002-11-01).

26971:(Revised ed.). Oxford: Oxford University Press. 25781: 25746: 25726: 25693: 25649: 25615: 25560: 25507: 25474: 25444: 25424: 25193: 25166: 25139: 25027: 25000: 24963: 24936: 24906: 24888:

below 0, time-averaged, to avoid exponential growth.

24855: 24708: 24681: 24645: 24601: 24567: 24516: 24473: 24371: 24262: 24141: 24023: 24003: 23976: 23912: 23885: 23862: 23832: 23811: 23753: 23733: 23704: 23578: 23504: 23435: 23407: 23376: 23345: 23314: 23284: 23260: 23236: 23197: 23069: 22974: 22879: 22856: 22834: 22734: 22672: 22651: 22618: 22597: 22564: 22543: 22504: 22483: 22453: 22401: 22395:

Here we have 4 compartments and we can define vector

22180: 22158: 22138: 22118: 22098: 22078: 22016: 21984: 21957: 21793: 21655: 21625: 21591: 21558: 21525: 21492: 21465: 21337: 21317: 21281: 21254: 21234: 21214: 21193: 21173: 21114: 21077: 21044: 21017: 20997: 20967: 20947: 20927: 20907: 20887: 20867: 20837: 20775: 20755: 20732: 20712: 20665: 20645: 20607: 20580: 20488: 20384: 20260: 20228: 20199: 20084: 19983: 19920: 19723: 19655: 19601: 19447: 19296: 19037: 18809: 18766: 18684: 18602: 18520: 18440: 18357: 18223: 17873: 17789: 17615: 17555: 17487: 17408: 17167: 17004: 16981: 16961: 16898: 16875: 16654: 16613: 16584: 16485: 16463: 16430: 16403: 16376: 16356: 16329: 16224: 16162: 16105: 15971: 15924: 15888: 15819: 15739: 15667: 15569: 15503: 15428: 15401: 15374: 15347: 15122: 14934: 14893: 14718: 14627: 14613:

In addition, Some diseases are seasonal, such as the

14523: 14192: 14121: 13882: 13802: 13596: 13564: 13406: 13238: 13005: 12926: 12900: 12853: 12585: 12562: 12539: 12519: 12489: 12469: 12158: 12049: 12020: 11950: 11650: 11616: 11564: 11512: 11460: 11395: 11360: 11319: 11245: 10931: 10876: 10831: 10811: 10791: 10733: 10692: 10665: 10638: 10608: 10566: 10513: 10433: 10404: 10377: 10350: 10309: 10257: 10205: 10155: 10117: 10067: 10035: 9756: 9709: 9492: 9259: 9233: 9207: 9130: 9102:{\displaystyle {\frac {dI}{dt}}\propto I\cdot (N-I).} 9051: 9016: 8986: 8960: 8903: 8849: 8817: 8761: 8684: 8645: 8467: 8426: 8337: 8228: 8078: 7956: 7916: 7873: 7821: 7795: 7654: 7624: 7597: 7536: 7494: 7474: 7448: 7413: 7385: 7365: 7345: 7325: 7305: 7285: 7265: 7215: 7154: 7099: 7076: 7056: 7036: 7016: 6981: 6946: 6911: 6890: 6825: 6772: 6763:

The flow of this model may be considered as follows:

6735: 6704: 6673: 6641: 6612: 6583: 6573:") by appropriate measures such as social distancing. 6244: 5998: 5852: 5833:{\displaystyle R_{0}={\frac {\beta }{\mu +\gamma }},} 5795: 5678: 5474: 5451: 5431: 5382: 5349: 5309: 5272: 5239: 5212: 5169: 5126: 5098: 5071: 5044: 5017: 4975: 4942: 4909: 4881: 4853: 4819: 4790: 4762: 4742: 4720: 4696: 4675: 4655: 4635: 4602: 4569: 4516: 4464: 4437: 4369: 4303: 4237: 4185: 4156: 4127: 4098: 4069: 4040: 3978: 3951: 3924: 3901: 3881: 3861: 3824: 3789: 3754: 3723: 3703: 3683: 3632: 3612: 3583: 3312: 3273: 3235: 3123: 3096: 3050: 3023: 2842: 2706: 2631: 2553: 2501: 2450: 2399: 2376: 2337: 2267: 2210: 2149: 2092: 2001: 1962: 1811: 1685: 1653: 1603: 1553: 1503: 1474: 1445: 1334: 1280: 1233: 1190: 1140: 1114: 1035: 937: 888: 868: 848: 828: 660: 599: 554: 534: 511: 458: 438: 418: 340: 305:

Yellow=Susceptible, Maroon=Infectious, Teal=Recovered

250: 221: 28751:

Carlson CS, Rubin DM, Heikkilä V, Postema M (2021).

27834:

Heffernan, J.M; Smith, R.J; Wahl, L.M (2005-09-22).

26612: 26351: 26349: 26347: 26345: 26343: 26096: 26094: 26092: 26090: 21459:

which shows how the number of people in compartment

8980:. This can be also considered in the SIR model with 8032: 7070:

changes over the course of the epidemic and so does

5425:

Consider a population characterized by a death rate

28696:

Infectious diseases of humans: dynamics and control

28086: 27884: 26968:

Infectious Diseases of Humans: Dynamics and Control

26869: 24892:

Latent infectious period, isolation after diagnosis

22827:We can now make matrices of partial derivatives of 18418:{\displaystyle R_{0}\int _{0}^{T}S^{*}(t)\,dt<1} 14610:rate in a way to reduce the speed of the pandemic. 7488:(which represents the mean recovery/death rate, or 1098:{\displaystyle S(t)+I(t)+R(t)={\text{constant}}=N,} 28406:Journal of the Royal Statistical Society, Series A 28087:Wohl S, Schaffner SF, Sabeti PC (September 2016). 27833: 27714: 27273: 27207: 26705:"A note on the derivation of epidemic final sizes" 26662: 26461: 26407: 26355: 25830: 25759: 25732: 25712: 25679: 25635: 25601: 25542: 25493: 25457: 25430: 25406: 25179: 25152: 25117: 25013: 24976: 24949: 24919: 24880: 24841: 24694: 24664: 24631: 24587: 24550: 24502: 24459: 24357: 24248: 24127: 24009: 23989: 23955: 23898: 23868: 23845: 23817: 23797: 23739: 23719: 23679: 23564: 23490: 23416: 23391: 23360: 23330: 23300: 23266: 23242: 23222: 23180: 23055: 22960: 22862: 22840: 22816: 22720: 22657: 22644:is the rate of transfer of individuals out of the 22636: 22603: 22582: 22549: 22528: 22489: 22468: 22439: 22387: 22164: 22144: 22124: 22104: 22084: 22029: 21997: 21970: 21940: 21778: 21638: 21603: 21577: 21544: 21511: 21478: 21451: 21323: 21287: 21267: 21240: 21220: 21199: 21179: 21155: 21093: 21063: 21030: 21003: 20983: 20953: 20933: 20913: 20893: 20873: 20853: 20823: 20761: 20738: 20718: 20696: 20651: 20620: 20593: 20552: 20468: 20364: 20243: 20214: 20182: 20067: 19966: 19903: 19703: 19635: 19584: 19427: 19279: 19020: 18788: 18749: 18667: 18585: 18503: 18417: 18337: 18196: 17847: 17769: 17586: 17546:In such a case the eradication condition becomes: 17535: 17453: 17388: 17150: 16987: 16967: 16955:thus we shall deal with the long term behavior of 16944: 16881: 16858: 16637: 16590: 16559: 16469: 16449: 16416: 16389: 16362: 16350:indicates the initial value of the state variable 16342: 16313: 16205: 16146: 16086: 15930: 15894: 15871: 15791: 15719: 15647: 15516: 15434: 15414: 15387: 15360: 15330: 15092: 14917: 14876: 14695: 14589: 14492: 14175: 14090: 13848: 13770: 13579: 13547: 13391: 13218: 12981: 12917:For this model, the basic reproduction number is: 12906: 12886: 12836: 12568: 12548: 12525: 12505: 12475: 12397: 12091: 12035: 12002:{\displaystyle \tau (t)=\int _{0}^{t}a(\xi )d\xi } 12001: 11926: 11631: 11602: 11550: 11498: 11446: 11381: 11346: 11305: 11228: 10906: 10868:susceptible-infected-recovered-vaccinated-deceased 10850: 10817: 10797: 10777: 10719: 10678: 10651: 10621: 10579: 10549: 10499: 10419: 10390: 10363: 10336: 10295: 10243: 10191: 10141: 10103: 10053: 10010: 9727: 9692: 9459: 9433: 9407: 9381: 9355: 9271: 9245: 9219: 9193: 9101: 9034: 8998: 8972: 8939: 8889: 8835: 8803: 8743: 8667: 8628: 8450: 8404: 8316: 8204: 8012: 7939: 7902: 7856: 7804: 7778: 7637: 7610: 7583: 7508: 7480: 7460: 7434: 7399: 7371: 7351: 7331: 7311: 7291: 7271: 7247: 7199:{\displaystyle {\frac {dI}{dt}}=\beta SI-\gamma I} 7198: 7137: 7082: 7062: 7042: 7022: 7002: 6967: 6932: 6896: 6876: 6808: 6750: 6719: 6688: 6656: 6627: 6598: 6561: 6535: 6509: 6483: 6399: 6373: 6347: 6257: 6227: 5983: 5832: 5771: 5658: 5457: 5437: 5409: 5368: 5335: 5291: 5258: 5225: 5198: 5155: 5111: 5084: 5050: 5030: 4994: 4961: 4928: 4894: 4866: 4833: 4804: 4775: 4748: 4726: 4705: 4681: 4661: 4641: 4621: 4588: 4554: 4502: 4450: 4419: 4343: 4277: 4197: 4171: 4142: 4113: 4084: 4055: 3991: 3964: 3937: 3907: 3887: 3867: 3847: 3810: 3775: 3740: 3709: 3689: 3663: 3618: 3598: 3566: 3292: 3259: 3221: 3109: 3082: 3036: 3006: 2822: 2687:{\displaystyle {\mathcal {R}}(u)=R(0)-\rho \ln(u)} 2686: 2616: 2538: 2484: 2424: 2382: 2355: 2308: 2247: 2190: 2129: 2072: 1981: 1945: 1779: 1668: 1639: 1589: 1539: 1489: 1460: 1428: 1311: 1262: 1219: 1169: 1120: 1097: 1015: 894: 874: 854: 834: 803: 611: 574: 540: 520: 482: 444: 424: 404: 263: 236: 26340: 26100: 26087: 23034: 22997: 22939: 22902: 21382: 21366: 20881:caused by a single infected individual of gender 17598:resurgence of infection due to coverage decline. 10344:and there is a critical reduced vaccination rate 9745:Susceptible-Infectious-Recovered-Vaccinated model 7857:{\displaystyle S={\frac {\mu N}{\mu +\lambda }}.} 5669:for which the disease-free equilibrium (DFE) is: 4936:, we will see an exponential-like outbreak; when 1429:{\displaystyle S(t)=S(0)e^{-R_{0}(R(t)-R(0))/N},} 30800: 27994:"Further Notes on the Basic Reproduction Number" 26320:"Modeling epidemics with differential equations" 25160:and the number of latent infectious individuals 24256:The underlying matching differential equation is 23338:gives the rate at which infected individuals in 21835: 21676: 21486:changes over time. For example, in a SIR model, 20726:and the expected number of infected gender 2 is 18750:{\displaystyle R(t)=\int _{0}^{a_{M}}r(t,a)\,da} 18668:{\displaystyle I(t)=\int _{0}^{a_{M}}i(t,a)\,da} 18586:{\displaystyle S(t)=\int _{0}^{a_{M}}s(t,a)\,da} 17780:leading to the following eradication condition: 17225: 17040: 16900: 15006: 13448: 13259: 9286:Animation of the SIRD model with initial values 8557: 8493: 7664: 7546: 6040: 5873: 4357:and the real-time effective reproduction number: 1170:{\displaystyle R_{0}={\frac {\beta }{\gamma }},} 28715:An Introduction to Infectious Disease Modelling 28347:For more information on this type of model see 28262: 28199:International Journal of Differential Equations 26865: 26863: 26861: 26859: 26151:Contributions to Biology and Problems of Health 25888:Deterministic versus stochastic epidemic models 21299:Next-generation matrix for compartmental models 17454:{\displaystyle P<P^{*}=1-{\frac {1}{R_{0}}}} 15731:due to fatigue from anti-epidemic restrictions; 14887:(the dynamics of recovered easily follows from 10921:This file provides a cartoon of the SIRVD model 9473:Susceptible-Infectious-Recovered-Deceased model 9006:, i.e. no removal will take place. That is the 8804:{\displaystyle I_{\infty }=(1-\gamma /\beta )N} 7903:{\displaystyle \lambda ={\tfrac {\beta I}{N}},} 7379:persons per unit time whereas only a fraction, 6884:in the three functions resolves that the value 6414:Animation of the SIR model with initial values 5002:, the disease decays and fades away over time. 3697:. The equations may thus be divided through by 27836:"Perspectives on the basic reproductive ratio" 27666:International Journal of Bifurcation and Chaos 26809: 26807: 26805: 26803: 17601: 17472: 10778:{\displaystyle R_{e}={\frac {R_{t}S}{N}}<1} 10507:can be mapped to the solved special case with 9124:Diagram of the SIRD model with initial values 8416:i.e. the dynamics of infectious is ruled by a 7618:reflects the time alive (life expectancy) and 6266:remain permanently endemic in the population. 905:This model was for the first time proposed by 643:The dynamics of an epidemic, for example, the 30557: 29206: 28840: 28712: 27496: 26991: 26608: 26606: 25418:of the matrix is the logarithmic growth rate 17858: 16945:{\displaystyle \lim _{t\to +\infty }V(t)=NP,} 16457:presents the maximum of the state variables 13558:In case of periodically varying contact rate 8050:Some infections, for example, those from the 8023: 6278:Diagram of the SIR model with initial values 917:, and followed work McKendrick had done with 24:mathematical modelling of infectious diseases 27019: 26994:Computational Modeling of Infectious Disease 26856: 26560: 26259: 25880:Distributions that are spatially non-uniform 24551:{\displaystyle K={\frac {\ln R_{0}}{\tau }}} 23427:Finally, for this SEIR process we can have: 21305:mathematical modelling of infectious disease 16147:{\displaystyle \sigma _{o}=\gamma (R_{o}-1)} 15963:The KdV-SIR equation is written as follows: 28802:SIR model: Online experiments with JSXGraph 28690: 28619: 28403: 28219: 28193:Rihan, Fathalla A.; Anwar, M. Naim (2012). 27570: 27568: 26961: 26934: 26800: 26515: 26509: 26457: 26455: 26204:History and Philosophy of the Life Sciences 25125:This estimation method has been applied to 21156:{\displaystyle R_{0}=\rho (G)={\sqrt {mf}}} 20068:{\displaystyle n(t,a)=s(t,a)+i(t,a)+r(t,a)} 18429:The influence of age: age-structured models 16570:which represents a solitary wave solution. 15490:complement the classical epidemics models. 15107: 2439:Exact analytical solutions to the SIR model 2328:Note that in the above model the function: 30564: 30550: 28847: 28833: 28192: 27721:Journal of Biomedicine & Biotechnology 27715:Gao S, Teng Z, Nieto JJ, Torres A (2007). 27659: 26937:Liceo Journal of Higher Education Research 26680:Mathematical Structure of Epidemic Systems 26603: 26566: 26327:Tennessee State University Internal Report 26052: 26011: 23997:new individuals only after exactly a time 23956:{\displaystyle K={\frac {\ln(2)}{T_{d}}}.} 11306:{\displaystyle a(t),v(t),\mu (t),\psi (t)} 10427:. Arbitrary initial conditions satisfying 7138:{\displaystyle {\frac {dS}{dt}}=-\beta SI} 28655: 28645: 28572: 28503: 28294: 28210: 28169: 28112: 28055: 27867: 27791: 27742: 27732: 27620: 27610: 27592: 27540: 27514: 27478: 27449: 27439: 27421: 27337: 27301: 27291: 27243: 27233: 27166: 27092: 27001: 26903: 26839: 26780: 26728: 26543: 26533: 26500: 26467: 26443: 26433: 26381: 26313: 26311: 26309: 26273: 26231: 26197: 26140: 26138: 26126: 26078: 26037: 25996: 25680:{\displaystyle K=0.183~\mathrm {d} ^{-1}} 24849:showing that it may be important to keep 24632:{\displaystyle K=0.183~\mathrm {d} ^{-1}} 24111: 24065: 23798:{\displaystyle K:={\frac {d\ln(N)}{dt}}.} 20901:, assuming that the population of gender 20569: 20451: 20355: 19894: 19568: 19004: 18740: 18658: 18576: 18402: 14989: 14604: 12432:. The carrier compartment is labelled C. 12419:never completely recover and continue to 12092:{\displaystyle j(\tau )=S(\tau )I(\tau )} 10629:is pulled out, one can compare it to the 9010:. The differential equation system using 7745: 7520: 7248:{\displaystyle {\frac {dR}{dt}}=\gamma I} 3546: 2323: 2309:{\displaystyle {\frac {dI}{dt}}(0)<0,} 2191:{\displaystyle {\frac {dI}{dt}}(0)>0,} 1865: 405:{\displaystyle d(S/N)/dt=-\beta SI/N^{2}} 30571: 28446: 28348: 28027: 27660:Paxson, Wei; Shen, Bo-Wen (2022-10-01). 27565: 27472: 27319: 27269: 27267: 27265: 27263: 27132: 27025: 26631: 26452: 25847: 20961:, on average. Meaning that each element 17536:{\displaystyle P=P(I),\quad P'(I)>0.} 16573: 15492: 15455: 10916: 10020: 9281: 9119: 8041: 6409: 6273: 5065:Among these three reproduction numbers, 1640:{\displaystyle r_{\infty }=R(\infty )/N} 1590:{\displaystyle s_{\infty }=S(\infty )/N} 1227:, and the typical time until removal is 842:is the stock of susceptible population, 813: 630: 312: 300: 166: 96:may affect the outcome of the epidemic. 29921:Ear-Nose-Throat/Upper respiratory tract 29094:Access to water, sanitation and hygiene 28731: 28376: 27397: 27203: 27201: 27199: 27139:Clinical Epidemiology and Global Health 26677: 26671: 26522:Journal of the Royal Society, Interface 26255: 26253: 26251: 26144: 19967:{\displaystyle \varphi _{j}(a),j=s,i,r} 18348:and that the eradication condition is: 16601: 15957: 4555:{\displaystyle \gamma _{0}=0.2day^{-1}} 3083:{\displaystyle {\mathcal {I}}(u_{T})=0} 2539:{\displaystyle {\mathcal {S}}(u)=S(0)u} 2425:{\displaystyle F=\beta {\frac {I}{N}}.} 334:, the transition rate is assumed to be 278:As implied by the variable function of 30801: 30732:Construction and management simulation 30504:Infectious disease (medical specialty) 30396:Antimicrobial resistance surveillance 28558: 28105:10.1146/annurev-virology-110615-035747 27840:Journal of the Royal Society Interface 26754: 26702: 26306: 26135: 25937:Mathematical modelling in epidemiology 25856: 25602:{\displaystyle R_{0}=\exp(K\tau _{E})} 22728:gives the rate of change of such that 22440:{\displaystyle \mathrm {x} =(S,E,I,R)} 20553:{\displaystyle DFS(a)=(n^{*}(a),0,0).} 19643:on the interactions between the ages. 15882:has different reaction rate constants 10720:{\displaystyle {\frac {R_{t}}{R_{0}}}} 10244:{\displaystyle k=\gamma (t)/\beta (t)} 9194:{\displaystyle S(0)=997,I(0)=3,R(0)=0} 8811:is the endemic infectious population, 7812:times the duration of susceptibility: 5266:, the spreading accelerates, and when 4503:{\displaystyle \beta _{0}=0.4day^{-1}} 2485:{\displaystyle {\mathcal {S}}(u)=S(t)} 913:as a special case of what we now call 30545: 28828: 28356:. London-New York: Chapman and Hall. 28318: 28316: 28314: 28089:"Genomic Analysis of Viral Outbreaks" 27829: 27827: 27497:Croccolo F, Roman HE (October 2020). 27492: 27490: 27352: 27346: 27315: 27313: 27260: 26638:Sonia Altizer; Nunn, Charles (2006). 26482: 25609:). For example, with the same values 25021:may be written in the following form 24984:other individuals during this period. 23689: 22666:-th compartment, then the difference 20659:gives the basic reproduction number, 10500:{\displaystyle S(0)+I(0)+R(0)+V(0)=N} 8890:{\displaystyle V=I_{\infty }/I_{0}-1} 7910:times the duration of infectiousness 3090:. By the transcendental equation for 2248:{\displaystyle R_{0}\cdot S(0)<N,} 2130:{\displaystyle R_{0}\cdot S(0)>N,} 627:The SIR model without birth and death 92:. Such models can show how different 30768:List of computer simulation software 29751:Compartmental models in epidemiology 28620:Wong F, Collins JJ (November 2020). 28323:Brauer F, Castillo-Chávez C (2001). 27196: 26248: 25970: 25636:{\displaystyle \tau =5~\mathrm {d} } 24588:{\displaystyle \tau =5~\mathrm {d} } 18504:{\displaystyle s(t,a),i(t,a),r(t,a)} 15801:slow relaxation to the initial state 15447:Interacting Subpopulation SEIR Model 12483:(i.e. the average latency period is 8836:{\displaystyle \chi =\beta -\gamma } 8219:the total population it holds that: 7940:{\displaystyle {\tfrac {1}{\mu +v}}} 3915:above) are simply linearly related. 3626:. The two solutions are related by 1274:the infectious has been removed is: 28717:. Oxford: Oxford University Press. 28713:Vynnycky E, White RG, eds. (2010). 28698:. Oxford: Oxford University Press. 26996:, Academic Press, pp. 93–119, 26262:Applied Mathematics and Computation 23278:for linear transition terms making 22529:{\displaystyle F_{i}(\mathrm {x} )} 22057: 20375:and the demographic equilibrium is 19974:are the fertilities of the adults. 15532:The simplest SIR-social stress (SIR 14505: 10371:beyond which the steady-state size 9728:{\displaystyle \beta ,\gamma ,\mu } 9356:{\textstyle S(0)=997,I(0)=3,R(0)=0} 8046:Yellow=Susceptible, Maroon=Infected 6484:{\textstyle S(0)=997,I(0)=3,R(0)=0} 6348:{\textstyle S(0)=997,I(0)=3,R(0)=0} 321: 77:The models are most often run with 13: 28684: 28311: 27985: 27942: 27824: 27646:; the preprocessed data are here: 27487: 27310: 27003:10.1016/b978-0-32-395389-4.00012-8 26101:Kermack WO, McKendrick AG (1927). 25932:List of COVID-19 simulation models 25664: 25629: 25543:{\displaystyle R_{0}=1+K\tau _{E}} 24616: 24581: 23379: 23348: 23072: 23040: 23031: 23016: 22994: 22945: 22936: 22921: 22899: 22807: 22778: 22749: 22711: 22687: 22519: 22456: 22403: 21845: 21686: 21379: 21363: 20117: 20086: 19329: 19298: 19070: 19039: 18842: 18811: 18789:{\displaystyle a_{M}\leq +\infty } 18783: 18207:It is easy to see that by setting 17238: 17053: 16913: 15954:model to publicly available data 15858: 15848: 15830: 15826: 15821: 15773: 15769: 15750: 15746: 15741: 15701: 15697: 15678: 15674: 15669: 15637: 15616: 15612: 15601: 15580: 15576: 15571: 15528:would better model multiple waves. 15300: 15274: 15223: 15197: 15155: 15129: 15019: 14579: 14569: 14559: 14549: 14539: 14529: 14525: 14224: 14166: 14156: 14146: 14136: 14126: 14122: 14103: 13914: 13838: 13828: 13818: 13808: 13804: 13461: 13272: 12520: 12454: 12190: 12126: 10861: 10383: 8861: 8767: 8707: 8570: 8506: 8427: 8060: 7712: 7655: 7584:{\displaystyle \operatorname {E} } 7537: 6798: 6788: 6778: 6774: 6269: 6050: 5956: 5883: 5741: 5506: 5452: 4291:the real-time reproduction number: 3984: 3957: 3930: 3279: 3197: 3153: 3102: 3053: 2886: 2867: 2848: 2768: 2634: 2600: 2581: 2556: 2504: 2453: 1836: 1817: 1754: 1710: 1691: 1660: 1623: 1609: 1573: 1559: 1263:{\displaystyle T_{r}=\gamma ^{-1}} 14: 30835: 28795: 28770:10.1109/AFRICON51333.2021.9570946 27274:Schlickeiser R, Kröger M (2024). 27208:Schlickeiser R, Kröger M (2021). 26617:(2nd ed.). London: Griffin. 26408:Schlickeiser R, Kröger M (2021). 26356:Kröger M, Schlickeiser R (2020). 25883:Diseases caused by macroparasites 25551: 23188:is the disease-free equilibrium. 14706:with period T equal to one year. 13784: 12576:is constant), we have the model: 12442: 12103: 9738: 9115: 8451:{\displaystyle \forall I(0)>0} 8033:Variations on the basic SIR model 4841:values. The typical onset of the 1220:{\displaystyle T_{c}=\beta ^{-1}} 30500:Economics of Infectious Diseases 26709:Bulletin of Mathematical Biology 26516:Krylova O, Earn DJ (July 2013). 25720:, rather than the true value of 24503:{\displaystyle R_{0}=e^{K\tau }} 23392:{\displaystyle \mathrm {x} _{i}} 23361:{\displaystyle \mathrm {x} _{j}} 22469:{\displaystyle \mathrm {x} _{i}} 12435: 12410: 11447:{\displaystyle R(0)=V(0)=D(0)=0} 10296:{\displaystyle b=v(t)/\beta (t)} 10054:{\displaystyle \beta ,\gamma ,v} 8037: 6877:{\displaystyle N=S(t)+I(t)+R(t)} 99: 30696:Integrated assessment modelling 28613: 28552: 28483: 28440: 28397: 28370: 28341: 28186: 28129: 28080: 28036:Journal of Mathematical Biology 27958: 27772:Journal of Mathematical Biology 27759: 27708: 27653: 27637: 27466: 27183: 27126: 27109: 27052: 26985: 26955: 26928: 26748: 26696: 26656: 26476: 26198:Engelmann, Lukas (2021-08-30). 26012:Ross R, Hudson H (3 May 1917). 23965: 21309:ordinary differential equations 20697:{\displaystyle R_{0}=\rho (G).} 19704:{\displaystyle i(t,0)=r(t,0)=0} 18081: 17942: 17587:{\displaystyle P(0)\geq P^{*},} 17509: 17202: 15803:(end of the refractory period). 14659: 12108:For many infections, including 12043:and the rate of new infections 11603:{\displaystyle q=\psi (t)/a(t)} 10337:{\displaystyle k+b<1-2\eta } 10104:{\displaystyle S(0)=(1-\eta )N} 5199:{\displaystyle R_{t}=0.25R_{0}} 2978: 2795: 1794:has a solution in terms of the 79:ordinary differential equations 30509:Infectious disease informatics 30340:Transmission-based precautions 27971:. Princeton University Press. 26191: 26046: 26005: 25964: 25596: 25580: 25550:, which is different from the 25076: 25050: 24875: 24862: 24830: 24827: 24821: 24808: 24771: 24768: 24762: 24749: 24737: 24734: 24728: 24715: 24445: 24432: 24406: 24403: 24397: 24384: 24343: 24330: 24318: 24312: 24285: 24279: 24229: 24216: 24204: 24201: 24195: 24182: 24170: 24167: 24161: 24148: 24108: 24102: 24062: 24056: 24040: 24034: 23934: 23928: 23778: 23772: 23714: 23708: 23668: 23656: 23653: 23641: 23614: 23595: 23175: 23143: 23137: 23085: 23026: 23011: 22931: 22916: 22811: 22803: 22782: 22774: 22753: 22745: 22715: 22707: 22691: 22683: 22523: 22515: 22434: 22410: 22329: 22317: 22277: 22265: 21922: 21919: 21913: 21891: 21885: 21869: 21863: 21850: 21842: 21831: 21822: 21816: 21763: 21760: 21754: 21732: 21726: 21710: 21704: 21691: 21683: 21672: 21446: 21395: 21137: 21131: 20688: 20682: 20544: 20529: 20523: 20510: 20504: 20498: 20448: 20442: 20401: 20395: 20340: 20334: 20298: 20292: 20238: 20232: 20209: 20203: 20177: 20165: 20159: 20153: 20141: 20129: 20110: 20098: 20062: 20050: 20041: 20029: 20020: 20008: 19999: 19987: 19937: 19931: 19886: 19874: 19868: 19862: 19846: 19834: 19828: 19822: 19806: 19794: 19788: 19782: 19739: 19727: 19692: 19680: 19671: 19659: 19630: 19605: 19565: 19546: 19540: 19515: 19484: 19481: 19469: 19451: 19422: 19410: 19404: 19398: 19389: 19377: 19371: 19365: 19353: 19341: 19322: 19310: 19274: 19262: 19256: 19250: 19241: 19229: 19223: 19217: 19194: 19175: 19169: 19144: 19115: 19103: 19094: 19082: 19063: 19051: 19001: 18982: 18976: 18951: 18923: 18911: 18902: 18890: 18884: 18878: 18866: 18854: 18835: 18823: 18798:integro-differential equations 18737: 18725: 18694: 18688: 18655: 18643: 18612: 18606: 18573: 18561: 18530: 18524: 18498: 18486: 18477: 18465: 18456: 18444: 18399: 18393: 18330: 18318: 18276: 18264: 18240: 18234: 18154: 18138: 18126: 18110: 18101: 18085: 18002: 17986: 17980: 17968: 17962: 17946: 17665: 17653: 17565: 17559: 17524: 17518: 17503: 17497: 17364: 17352: 17325: 17313: 17272: 17266: 17257: 17251: 17232: 17221: 17212: 17206: 17190: 17178: 17087: 17081: 17072: 17066: 17047: 17036: 17027: 17015: 16927: 16921: 16907: 16797: 16785: 16704: 16692: 16632: 16620: 16284: 16264: 16141: 16122: 15853: 15763: 15691: 15606: 15084: 15072: 15054: 15051: 15045: 15036: 15030: 15024: 15013: 15002: 14972: 14966: 14864: 14852: 14833: 14827: 14777: 14771: 14690: 14684: 14675: 14663: 14643: 14637: 14574: 14564: 14554: 14544: 14534: 14434: 14422: 14376: 14364: 14161: 14151: 14141: 14131: 14078: 14066: 14020: 14008: 13833: 13823: 13813: 13751: 13739: 13672: 13660: 13644: 13638: 13574: 13568: 13525: 13519: 13510: 13504: 13495: 13489: 13480: 13474: 13455: 13444: 13435: 13429: 13383: 13359: 13336: 13330: 13321: 13315: 13306: 13300: 13291: 13285: 13266: 13255: 13124: 13111: 13105: 13081: 13065: 13059: 13050: 13044: 13035: 13029: 13020: 13014: 12775: 12763: 12717: 12705: 12428:, who infected 22 people with 12086: 12080: 12074: 12068: 12059: 12053: 12030: 12024: 11990: 11984: 11960: 11954: 11914: 11908: 11868: 11862: 11822: 11816: 11776: 11773: 11767: 11758: 11752: 11746: 11700: 11694: 11626: 11620: 11597: 11591: 11580: 11574: 11545: 11539: 11528: 11522: 11499:{\displaystyle k=\mu (t)/a(t)} 11493: 11487: 11476: 11470: 11435: 11429: 11420: 11414: 11405: 11399: 11370: 11364: 11329: 11323: 11300: 11294: 11285: 11279: 11270: 11264: 11255: 11249: 11216: 11210: 11170: 11164: 11124: 11118: 11078: 11072: 11060: 11054: 11039: 11033: 10993: 10987: 10972: 10966: 10825:is the total population. When 10538: 10532: 10523: 10517: 10488: 10482: 10473: 10467: 10458: 10452: 10443: 10437: 10414: 10408: 10290: 10284: 10273: 10267: 10238: 10232: 10221: 10215: 10180: 10174: 10165: 10159: 10127: 10121: 10095: 10083: 10077: 10071: 9995: 9989: 9949: 9943: 9903: 9897: 9876: 9870: 9827: 9821: 9800: 9794: 9344: 9338: 9323: 9317: 9302: 9296: 9182: 9176: 9161: 9155: 9140: 9134: 9093: 9081: 8934: 8928: 8913: 8907: 8795: 8775: 8694: 8688: 8584: 8578: 8564: 8553: 8520: 8514: 8500: 8489: 8439: 8433: 8373: 8361: 8305: 8299: 8290: 8284: 8278: 7737: 7725: 7696: 7693: 7667: 7661: 7578: 7575: 7549: 7543: 6997: 6985: 6962: 6950: 6927: 6915: 6871: 6865: 6856: 6850: 6841: 6835: 6793: 6783: 6745: 6739: 6714: 6708: 6683: 6677: 6651: 6645: 6622: 6616: 6593: 6587: 6472: 6466: 6451: 6445: 6430: 6424: 6336: 6330: 6315: 6309: 6294: 6288: 6100: 6097: 6091: 6082: 6076: 6067: 6061: 6055: 6047: 6036: 6027: 6021: 5933: 5930: 5924: 5915: 5909: 5900: 5894: 5888: 5880: 5869: 5782:In this case, we can derive a 5723: 5717: 5708: 5702: 5693: 5687: 5156:{\displaystyle R_{t}=0.5R_{0}} 4903:disease spreading. Thus, when 4166: 4160: 4137: 4131: 4108: 4102: 4079: 4073: 4050: 4044: 3811:{\displaystyle \tau =\int Idt} 3664:{\displaystyle e^{-\xi (t)}=u} 3650: 3644: 3593: 3587: 3543: 3530: 3492: 3486: 3476: 3470: 3458: 3452: 3439: 3433: 3423: 3417: 3408: 3402: 3383: 3377: 3365: 3359: 3345: 3339: 3326: 3320: 3254: 3245: 3239: 3216: 3210: 3186: 3173: 3170: 3164: 3145: 3071: 3058: 2972: 2969: 2963: 2954: 2948: 2939: 2933: 2918: 2912: 2906: 2900: 2897: 2891: 2878: 2872: 2859: 2853: 2843: 2786: 2773: 2681: 2675: 2660: 2654: 2645: 2639: 2611: 2605: 2592: 2586: 2567: 2561: 2530: 2524: 2515: 2509: 2479: 2473: 2464: 2458: 2294: 2288: 2233: 2227: 2176: 2170: 2115: 2109: 1937: 1932: 1913: 1869: 1772: 1746: 1657: 1626: 1620: 1576: 1570: 1526: 1520: 1484: 1478: 1455: 1449: 1410: 1407: 1401: 1392: 1386: 1380: 1359: 1353: 1344: 1338: 1075: 1069: 1060: 1054: 1045: 1039: 358: 344: 231: 225: 1: 29926:Chest/Lower respiratory tract 29701:Ocular (Eye) mucosal membrane 29062:Behavioral/lifestyle factors 28816:. March 27, 2020 – via 27955:. Stanford University (2007). 27912:10.1016/S0025-5564(02)00108-6 26876:Journal of Chemical Education 26613:Bailey, Norman T. J. (1975). 25957: 25942:Modifiable areal unit problem 9246:{\displaystyle \gamma =0.035} 3293:{\displaystyle I_{\infty }=0} 1669:{\displaystyle t\to \infty ,} 90:effective reproduction number 30665:Hydrological transport model 30619:Protein structure prediction 30614:Modelling biological systems 29488:Aerosol-generating procedure 29266:Silent/Subclinical infection 28966:Multidrug-resistant bacteria 27503:Chaos, Solitons and Fractals 27028:Applied Mathematical Ecology 26761:Infectious Disease Modelling 26703:Miller JC (September 2012). 25769:delay differential equations 21268:{\displaystyle {\sqrt {mf}}} 20941:can transmit the disease to 19636:{\displaystyle k(a,a_{1};t)} 11347:{\displaystyle S(0)=1-\eta } 10025:A cartoon for the SIRV model 4669:with the susceptible people 1540:{\displaystyle s_{0}=S(0)/N} 1312:{\displaystyle T_{r}/T_{c}.} 7: 30609:Metabolic network modelling 30152:Vaccine-preventable disease 30100:Monoclonal antibody therapy 28522:10.1103/PhysRevLett.86.3200 28006:10.1007/978-3-540-78911-6_6 27533:10.1016/j.chaos.2020.110077 27480:10.1101/2021.01.29.21250747 27036:10.1007/978-3-642-61317-3_5 26965:, Anderson B (1992-09-24). 26896:10.1021/acs.jchemed.2c00416 26755:Miller JC (February 2017). 26665:"Mathematica, Version 12.1" 25910: 25494:{\displaystyle \tau _{I}=0} 21064:{\displaystyle \phi _{a+1}} 20215:{\displaystyle \varphi (.)} 17602:Vaccination of non-newborns 17473:Vaccination and information 16995:, for which it holds that: 16390:{\displaystyle \sigma _{o}} 11551:{\displaystyle b=v(t)/a(t)} 10550:{\displaystyle R(0)=V(0)=0} 10391:{\displaystyle S_{\infty }} 10192:{\displaystyle R(0)=V(0)=0} 10142:{\displaystyle I(0)=\eta N} 9201:and the rates of infection 8940:{\displaystyle S(t)=N-I(t)} 7435:{\displaystyle abS=\beta S} 5336:{\displaystyle R_{t}=R_{0}} 4007:numerical simulation using 3992:{\displaystyle R_{\infty }} 3965:{\displaystyle I_{\infty }} 3938:{\displaystyle S_{\infty }} 3260:{\displaystyle S(0)\neq 0)} 3110:{\displaystyle R_{\infty }} 902:is the sum of these three. 575:{\displaystyle \gamma =1/D} 94:public health interventions 10: 30840: 30722:Business process modelling 30293:Respiratory source control 28591:10.1103/PhysRevE.66.016128 28469:10.1103/PhysRevE.64.066112 28275:Epidemiology and Infection 27612:10.1038/s41598-021-01317-z 27441:10.1038/s41467-020-19024-0 27151:10.1016/j.cegh.2020.08.005 27065:Royal Society Open Science 26216:10.1007/s40656-021-00445-z 25971:Ross R (1 February 1916). 25438:, which can be solved for 24881:{\displaystyle \ln(R_{0})} 23369:produce new infections in 22152:be the recovery rate, and 21331:coupled equations of form 21311:(ODE). So there is always 21094:{\displaystyle G\phi _{a}} 17859:Pulse vaccination strategy 16638:{\displaystyle P\in (0,1)} 15540:), rationally resistant (S 15482:should be considered. The 12887:{\displaystyle S+E+I+R=N,} 12556:(so that the total number 11382:{\displaystyle I(0)=\eta } 10851:{\displaystyle R_{e}<1} 10659:. Before the vaccination, 9382:{\textstyle \gamma =0.035} 9272:{\displaystyle \mu =0.005} 9220:{\displaystyle \beta =0.4} 8024:Other compartmental models 6819:Using a fixed population, 6355:, and rates for infection 5292:{\displaystyle R_{e}<1} 5259:{\displaystyle R_{e}>1} 4995:{\displaystyle R_{e}<1} 4929:{\displaystyle R_{e}>1} 3848:{\displaystyle dR/d\tau =} 3776:{\displaystyle d\tau =Idt} 2356:{\displaystyle F=\beta I,} 862:is the stock of infected, 30755: 30709: 30683: 30627: 30594:Chemical process modeling 30579: 30483: 30437:Global Health Initiatives 30427:Evolutionary epidemiology 30375: 30167: 30122:Post-exposure prophylaxis 30058:Antimicrobial stewardship 30016: 30003: 29908: 29799: 29781:Multiplicity of infection 29746:Basic reproduction number 29721: 29693: 29679: 29665: 29649: 29617: 29579: 29522:Percutaneous inoculation 29508: 29470: 29463: 29445: 29377: 29344: 29337: 29301: 29294: 29197: 29084: 29006:Host–pathogen interaction 28981: 28875: 28868: 28287:10.1017/S0950268818001760 28093:Annual Review of Virology 27998:Mathematical Epidemiology 27686:10.1142/S0218127422501991 27320:Hunziker P (2021-07-24). 27133:Al-Raeei, Marwan (2021). 26773:10.1016/j.idm.2016.12.003 26721:10.1007/s11538-012-9749-6 26589:10.1137/s0036144500371907 26292:10.1016/j.amc.2014.03.030 26159:10.1525/9780520350717-011 26053:Ross R, Hudson H (1917). 25922:Basic reproduction number 25865: 25760:{\displaystyle \tau _{E}} 25713:{\displaystyle R_{0}=1.9} 24994:This is a SEIR model and 24957:. The individual infects 24950:{\displaystyle \tau _{I}} 24920:{\displaystyle \tau _{E}} 24665:{\displaystyle R_{0}=2.5} 23223:{\displaystyle G=FV^{-1}} 22637:{\displaystyle V_{i}^{-}} 22583:{\displaystyle V_{i}^{+}} 21038:then the next generation 21031:{\displaystyle \phi _{a}} 17467:basic reproduction number 13580:{\displaystyle \beta (t)} 12116:and additionally through 10907:{\displaystyle S,I,R,V,D} 10631:basic reproduction number 8973:{\displaystyle \gamma =0} 7509:{\displaystyle 1/\gamma } 7319:be the multiplication of 6510:{\textstyle \gamma =0.04} 6400:{\textstyle \gamma =0.04} 5784:basic reproduction number 5410:{\displaystyle S=N/R_{0}} 4834:{\displaystyle 1/\gamma } 4805:{\displaystyle 1/\gamma } 4223:basic reproduction number 1182:basic reproduction number 915:Kermack–McKendrick theory 612:{\displaystyle \gamma =0} 432:is the total population, 86:basic reproduction number 30640:Chemical transport model 30604:Infectious disease model 30255:Protective sequestration 30117:Pre-exposure prophylaxis 29408:Generational difference 28961:Horizontal gene transfer 28951:Antimicrobial resistance 28808:"Simulating an epidemic" 28736:. Heidelberg: Springer. 27900:Mathematical Biosciences 26832:10.1021/acsomega.1c04842 26489:PeerJ Physical Chemistry 26435:10.1088/1751-8121/abed66 26383:10.1088/1751-8121/abc65d 25501:, this model results in 23805:For exponential growth, 22105:{\displaystyle \lambda } 15108:SIR model with diffusion 14709:Thus, our model becomes 12526:{\displaystyle \Lambda } 12459:SEIR compartmental model 12131:MSIR compartmental model 12036:{\displaystyle I(\tau )} 9460:{\textstyle \beta =0.12} 9363:, and rates of recovery 8668:{\displaystyle I=y^{-1}} 8215:Note that denoting with 7461:{\displaystyle \beta SI} 6562:{\textstyle \beta =0.12} 5458:{\displaystyle \Lambda } 3741:{\displaystyle d/d\tau } 2833:with initial conditions 1323:separating the variables 911:Anderson Gray McKendrick 584:exponential distribution 521:{\displaystyle \gamma I} 483:{\displaystyle SI/N^{2}} 294:)) over time. During an 30824:Mathematics in medicine 30442:Microbial phylogenetics 30360:Wastewater surveillance 29881:Geographic distribution 29756:Critical community size 29554:Surgical site infection 29402:Iatrogenic/Medical care 29321:Microbial translocation 29317:Endogenous reactivation 29312:Normal flora overgrowth 29261:Opportunistic infection 29033:Opportunistic infection 28647:10.1073/pnas.2018490117 28492:Physical Review Letters 28162:10.1126/science.1086616 27672:(13): 2250199–2250780. 26663:Wolfram Research, Inc. 23856:In exponential growth, 22612:by all other means and 22165:{\displaystyle \kappa } 22145:{\displaystyle \gamma } 22132:be the infection rate, 21578:{\displaystyle C_{3}=R} 21545:{\displaystyle C_{2}=I} 21512:{\displaystyle C_{1}=S} 20244:{\displaystyle \mu (a)} 16450:{\displaystyle I_{max}} 14918:{\displaystyle R=N-S-I} 9475:differentiates between 9434:{\textstyle \beta =0.5} 9408:{\textstyle \mu =0.005} 8065:SIS compartmental model 7481:{\displaystyle \gamma } 7083:{\displaystyle \gamma } 7043:{\displaystyle \gamma } 6536:{\textstyle \beta =0.5} 6491:, and rate of recovery 6374:{\textstyle \beta =0.4} 5369:{\displaystyle R_{E}=1} 5051:{\displaystyle \gamma } 4962:{\displaystyle R_{e}=1} 4749:{\displaystyle \gamma } 4622:{\displaystyle R_{0}=2} 4589:{\displaystyle R_{0}=2} 3599:{\displaystyle \xi (t)} 3117:above, it follows that 1982:{\displaystyle s_{0}=0} 1792:transcendental equation 1325:and integrating we get 30819:Differential equations 30514:Microbial bioterrorism 30137:efficacy/effectiveness 29545:Surgical intervention 29437:Breakthrough infection 29059:Genetic predisposition 29043:Susceptible individual 28890:Germ theory of disease 28694:, Anderson RM (1991). 27852:10.1098/rsif.2005.0042 27326:Precision Nanomedicine 27235:10.3390/physics3020028 27117:"(p. 19) The SI Model" 26535:10.1098/rsif.2013.0098 26502:10.7717/peerj-pchem.14 26128:10.1098/rspa.1927.0118 26080:10.1098/rspa.1917.0015 26039:10.1098/rspa.1917.0014 25998:10.1098/rspa.1916.0007 25947:Next-generation matrix 25874:Non-homogeneous mixing 25832: 25761: 25734: 25714: 25681: 25637: 25603: 25544: 25495: 25459: 25432: 25408: 25181: 25154: 25119: 25015: 24978: 24951: 24921: 24882: 24843: 24801: 24696: 24666: 24633: 24589: 24552: 24504: 24461: 24359: 24250: 24129: 24011: 23991: 23957: 23900: 23870: 23847: 23819: 23799: 23741: 23721: 23681: 23566: 23492: 23418: 23393: 23362: 23332: 23331:{\displaystyle G_{ij}} 23302: 23301:{\displaystyle V^{-1}} 23268: 23244: 23224: 23182: 23057: 22962: 22864: 22842: 22818: 22722: 22659: 22638: 22605: 22584: 22551: 22530: 22491: 22470: 22441: 22389: 22166: 22146: 22126: 22125:{\displaystyle \beta } 22106: 22086: 22031: 21999: 21972: 21942: 21780: 21640: 21605: 21579: 21546: 21513: 21480: 21453: 21325: 21289: 21269: 21242: 21222: 21201: 21181: 21157: 21095: 21065: 21032: 21005: 20985: 20984:{\displaystyle g_{ij}} 20955: 20935: 20915: 20895: 20875: 20855: 20854:{\displaystyle g_{ij}} 20825: 20763: 20748:next-generation matrix 20740: 20720: 20698: 20653: 20638:next-generation matrix 20622: 20595: 20570:Next-generation method 20554: 20470: 20366: 20245: 20216: 20184: 20069: 19968: 19905: 19705: 19637: 19586: 19429: 19281: 19022: 18790: 18751: 18669: 18587: 18505: 18419: 18339: 18198: 17849: 17771: 17588: 17537: 17455: 17390: 17152: 16989: 16969: 16946: 16883: 16860: 16639: 16592: 16561: 16471: 16451: 16418: 16391: 16364: 16344: 16315: 16207: 16148: 16088: 15932: 15931:{\displaystyle \beta } 15896: 15895:{\displaystyle \beta } 15873: 15793: 15721: 15649: 15529: 15518: 15436: 15435:{\displaystyle \beta } 15416: 15389: 15362: 15332: 15094: 14919: 14878: 14697: 14605:Variable contact rates 14591: 14494: 14177: 14092: 13850: 13772: 13581: 13549: 13393: 13220: 12983: 12908: 12888: 12838: 12570: 12550: 12527: 12507: 12506:{\displaystyle a^{-1}} 12477: 12460: 12399: 12132: 12093: 12037: 12003: 11928: 11633: 11604: 11552: 11500: 11448: 11383: 11348: 11307: 11230: 10922: 10908: 10852: 10819: 10799: 10779: 10721: 10680: 10653: 10623: 10581: 10551: 10501: 10421: 10392: 10365: 10338: 10297: 10245: 10193: 10143: 10105: 10055: 10026: 10012: 9729: 9694: 9468: 9461: 9435: 9409: 9383: 9357: 9279: 9273: 9247: 9221: 9195: 9103: 9036: 9000: 8974: 8941: 8891: 8837: 8805: 8745: 8669: 8630: 8452: 8406: 8318: 8206: 8066: 8047: 8014: 7941: 7904: 7858: 7806: 7780: 7639: 7612: 7585: 7521:Steady-state solutions 7510: 7482: 7462: 7436: 7401: 7373: 7353: 7333: 7313: 7312:{\displaystyle \beta } 7293: 7279:and an equal fraction 7273: 7249: 7200: 7139: 7084: 7064: 7063:{\displaystyle \beta } 7044: 7024: 7023:{\displaystyle \beta } 7004: 7003:{\displaystyle R(t=0)} 6969: 6968:{\displaystyle I(t=0)} 6934: 6933:{\displaystyle S(t=0)} 6898: 6878: 6810: 6752: 6721: 6690: 6658: 6629: 6600: 6574: 6563: 6537: 6511: 6485: 6407: 6401: 6375: 6349: 6259: 6229: 5985: 5834: 5773: 5660: 5459: 5439: 5411: 5370: 5337: 5293: 5260: 5227: 5200: 5157: 5113: 5086: 5052: 5032: 4996: 4963: 4930: 4896: 4868: 4835: 4806: 4777: 4750: 4728: 4727:{\displaystyle \beta } 4707: 4683: 4663: 4643: 4642:{\displaystyle \beta } 4623: 4590: 4556: 4504: 4452: 4421: 4345: 4279: 4199: 4198:{\displaystyle t>0} 4173: 4144: 4115: 4086: 4057: 3993: 3966: 3939: 3909: 3889: 3869: 3849: 3812: 3777: 3742: 3711: 3691: 3665: 3620: 3600: 3568: 3294: 3261: 3223: 3111: 3084: 3038: 3008: 2824: 2688: 2618: 2540: 2486: 2426: 2384: 2357: 2324:The force of infection 2310: 2249: 2192: 2131: 2074: 1983: 1947: 1781: 1670: 1641: 1591: 1541: 1491: 1462: 1430: 1313: 1264: 1221: 1171: 1122: 1099: 1017: 907:William Ogilvy Kermack 896: 876: 856: 836: 819: 805: 649:differential equations 640: 613: 576: 542: 522: 484: 446: 445:{\displaystyle \beta } 426: 406: 318: 306: 265: 238: 172: 64:Kermack and McKendrick 30773:Mathematical modeling 30717:Biopsychosocial model 30350:Universal precautions 29308:Endogenous overgrowth 29160:Poultry and livestock 27648:World in Data project 27410:Nature Communications 26949:10.7828/ljher.v6i2.62 26667:. Champaign IL, 2020. 25862:direct transmission. 25848:Vertical transmission 25833: 25762: 25735: 25715: 25682: 25638: 25604: 25545: 25496: 25460: 25458:{\displaystyle R_{0}} 25433: 25409: 25182: 25180:{\displaystyle n_{I}} 25155: 25153:{\displaystyle n_{E}} 25120: 25016: 25014:{\displaystyle R_{0}} 24979: 24977:{\displaystyle R_{0}} 24952: 24922: 24883: 24844: 24787: 24697: 24695:{\displaystyle R_{0}} 24667: 24634: 24590: 24553: 24505: 24462: 24360: 24251: 24130: 24012: 24010:{\displaystyle \tau } 23992: 23990:{\displaystyle R_{0}} 23958: 23901: 23899:{\displaystyle T_{d}} 23871: 23848: 23846:{\displaystyle R_{0}} 23820: 23800: 23742: 23722: 23682: 23567: 23493: 23419: 23394: 23363: 23333: 23303: 23269: 23245: 23225: 23183: 23058: 22963: 22865: 22843: 22819: 22723: 22660: 22639: 22606: 22585: 22552: 22531: 22498:-th compartment. Let 22492: 22471: 22442: 22390: 22167: 22147: 22127: 22107: 22087: 22032: 22030:{\displaystyle R_{0}} 22000: 21998:{\displaystyle R_{0}} 21973: 21971:{\displaystyle R_{0}} 21943: 21781: 21641: 21639:{\displaystyle R_{0}} 21606: 21580: 21547: 21514: 21481: 21479:{\displaystyle C_{i}} 21454: 21326: 21290: 21270: 21243: 21223: 21202: 21182: 21158: 21096: 21066: 21033: 21006: 20986: 20956: 20936: 20916: 20896: 20876: 20856: 20826: 20764: 20741: 20721: 20699: 20654: 20623: 20621:{\displaystyle R_{0}} 20596: 20594:{\displaystyle R_{0}} 20574:One way to calculate 20555: 20471: 20367: 20246: 20217: 20185: 20070: 19969: 19906: 19706: 19638: 19587: 19430: 19282: 19023: 18791: 18752: 18670: 18588: 18506: 18420: 18340: 18199: 17850: 17772: 17589: 17538: 17456: 17391: 17153: 16990: 16970: 16947: 16884: 16861: 16640: 16593: 16574:Modelling vaccination 16562: 16472: 16452: 16419: 16417:{\displaystyle R_{o}} 16392: 16365: 16345: 16343:{\displaystyle S_{o}} 16316: 16208: 16149: 16089: 15933: 15897: 15874: 15809:SIR epidemic reaction 15794: 15722: 15657:mobilization reaction 15650: 15519: 15517:{\displaystyle R_{t}} 15496: 15456:SIR Model on Networks 15437: 15417: 15415:{\displaystyle D_{R}} 15390: 15388:{\displaystyle D_{I}} 15363: 15361:{\displaystyle D_{S}} 15333: 15095: 14920: 14879: 14698: 14592: 14495: 14178: 14093: 13851: 13773: 13582: 13550: 13394: 13221: 12984: 12909: 12889: 12839: 12571: 12551: 12549:{\displaystyle N\mu } 12528: 12508: 12478: 12458: 12400: 12130: 12094: 12038: 12004: 11929: 11634: 11605: 11553: 11501: 11449: 11384: 11349: 11308: 11231: 10920: 10909: 10853: 10820: 10800: 10780: 10722: 10681: 10679:{\displaystyle R_{t}} 10654: 10652:{\displaystyle R_{0}} 10624: 10622:{\displaystyle R_{t}} 10590:numerical integration 10582: 10580:{\displaystyle R_{t}} 10552: 10502: 10422: 10393: 10366: 10364:{\displaystyle b_{c}} 10339: 10298: 10246: 10194: 10144: 10106: 10056: 10024: 10013: 9730: 9695: 9462: 9436: 9410: 9384: 9358: 9285: 9274: 9248: 9222: 9196: 9123: 9104: 9037: 9035:{\displaystyle S=N-I} 9001: 8975: 8942: 8892: 8838: 8806: 8746: 8670: 8631: 8453: 8407: 8319: 8207: 8064: 8045: 8015: 7942: 7905: 7859: 7807: 7805:{\displaystyle \mu N} 7781: 7640: 7638:{\displaystyle T_{S}} 7613: 7611:{\displaystyle T_{L}} 7586: 7511: 7483: 7463: 7437: 7402: 7374: 7354: 7334: 7314: 7294: 7274: 7250: 7201: 7140: 7085: 7065: 7045: 7025: 7005: 6970: 6935: 6899: 6879: 6811: 6753: 6722: 6691: 6659: 6630: 6601: 6564: 6538: 6512: 6486: 6413: 6402: 6376: 6350: 6277: 6260: 6258:{\displaystyle R_{0}} 6230: 5986: 5835: 5774: 5661: 5460: 5440: 5412: 5371: 5338: 5294: 5261: 5228: 5226:{\displaystyle R_{e}} 5201: 5158: 5114: 5112:{\displaystyle R_{t}} 5087: 5085:{\displaystyle R_{0}} 5053: 5033: 5031:{\displaystyle R_{t}} 4997: 4964: 4931: 4897: 4895:{\displaystyle R_{e}} 4869: 4867:{\displaystyle R_{t}} 4836: 4807: 4778: 4776:{\displaystyle R_{t}} 4751: 4729: 4708: 4684: 4664: 4644: 4624: 4591: 4557: 4505: 4453: 4451:{\displaystyle R_{0}} 4422: 4346: 4280: 4200: 4174: 4145: 4116: 4087: 4058: 3994: 3967: 3940: 3910: 3890: 3870: 3868:{\displaystyle \tau } 3850: 3813: 3778: 3743: 3712: 3692: 3666: 3621: 3601: 3569: 3295: 3262: 3224: 3112: 3085: 3039: 3037:{\displaystyle u_{T}} 3009: 2825: 2689: 2619: 2541: 2487: 2427: 2385: 2358: 2311: 2250: 2193: 2132: 2075: 1984: 1948: 1782: 1671: 1642: 1592: 1542: 1492: 1463: 1431: 1314: 1265: 1222: 1172: 1123: 1100: 1018: 897: 877: 857: 837: 817: 806: 634: 614: 577: 543: 523: 485: 447: 427: 407: 316: 304: 266: 264:{\displaystyle R_{0}} 239: 170: 30727:Catastrophe modeling 30573:Scientific modelling 30452:Genomic reassortment 30432:Genetic epidemiology 30271:Health communication 30199:Flattening the curve 30189:Disease surveillance 29972:Systemic/Generalized 29736:Animal disease model 29216:Asymptomatic carrier 28764:. pp. 456–459. 27293:10.3390/math12070941 26682:. Berlin: Springer. 26483:Simon, Cory (2020). 26414:Journal of Physics A 26362:Journal of Physics A 25877:Variable infectivity 25779: 25744: 25724: 25691: 25647: 25613: 25558: 25505: 25472: 25468:In the special case 25442: 25422: 25191: 25164: 25137: 25025: 24998: 24961: 24934: 24904: 24853: 24706: 24679: 24643: 24599: 24565: 24514: 24471: 24369: 24260: 24139: 24021: 24001: 23974: 23910: 23883: 23860: 23830: 23809: 23751: 23731: 23720:{\displaystyle N(t)} 23702: 23576: 23502: 23433: 23405: 23374: 23343: 23312: 23282: 23258: 23234: 23195: 23067: 22972: 22877: 22854: 22832: 22732: 22670: 22649: 22616: 22595: 22562: 22541: 22502: 22481: 22451: 22399: 22178: 22156: 22136: 22116: 22096: 22085:{\displaystyle \mu } 22076: 22014: 21982: 21955: 21791: 21653: 21623: 21589: 21556: 21523: 21490: 21463: 21335: 21315: 21279: 21252: 21232: 21212: 21191: 21171: 21163:, that is here, the 21112: 21075: 21042: 21015: 21011:is represented with 20995: 20965: 20945: 20925: 20905: 20885: 20865: 20835: 20773: 20753: 20730: 20710: 20663: 20643: 20605: 20578: 20486: 20382: 20258: 20226: 20197: 20082: 19981: 19918: 19721: 19653: 19599: 19445: 19294: 19035: 18807: 18764: 18682: 18600: 18518: 18438: 18355: 18221: 17871: 17787: 17613: 17553: 17485: 17406: 17165: 17002: 16979: 16959: 16896: 16873: 16652: 16611: 16602:Vaccinating newborns 16582: 16483: 16461: 16428: 16401: 16374: 16354: 16327: 16222: 16160: 16103: 15969: 15958:The KdV-SIR equation 15922: 15886: 15817: 15737: 15665: 15567: 15501: 15426: 15399: 15372: 15345: 15120: 14932: 14891: 14716: 14625: 14521: 14190: 14119: 13880: 13800: 13594: 13562: 13404: 13236: 13003: 12924: 12898: 12851: 12583: 12560: 12537: 12533:equal to death rate 12517: 12487: 12467: 12156: 12047: 12018: 11948: 11648: 11632:{\displaystyle a(t)} 11614: 11562: 11510: 11458: 11454:and constant ratios 11393: 11358: 11317: 11243: 10929: 10874: 10829: 10809: 10789: 10731: 10690: 10663: 10636: 10606: 10564: 10511: 10431: 10420:{\displaystyle S(0)} 10402: 10375: 10348: 10307: 10255: 10203: 10199:and constant ratios 10153: 10115: 10065: 10033: 9754: 9707: 9490: 9445: 9419: 9393: 9367: 9290: 9257: 9231: 9205: 9128: 9049: 9014: 8984: 8958: 8901: 8847: 8815: 8759: 8682: 8643: 8465: 8424: 8335: 8226: 8076: 7954: 7914: 7871: 7819: 7793: 7652: 7622: 7595: 7534: 7492: 7472: 7446: 7411: 7383: 7363: 7343: 7323: 7303: 7283: 7263: 7213: 7152: 7097: 7074: 7054: 7034: 7014: 6979: 6944: 6909: 6888: 6823: 6770: 6751:{\displaystyle R(t)} 6733: 6720:{\displaystyle I(t)} 6702: 6689:{\displaystyle S(t)} 6671: 6657:{\displaystyle R(t)} 6639: 6628:{\displaystyle I(t)} 6610: 6599:{\displaystyle S(t)} 6581: 6571:flattening the curve 6547: 6521: 6495: 6418: 6385: 6359: 6282: 6242: 5996: 5850: 5793: 5676: 5472: 5449: 5438:{\displaystyle \mu } 5429: 5380: 5347: 5307: 5270: 5237: 5210: 5167: 5124: 5096: 5069: 5042: 5015: 4973: 4940: 4907: 4879: 4851: 4817: 4788: 4760: 4740: 4718: 4694: 4673: 4653: 4633: 4600: 4567: 4514: 4462: 4435: 4367: 4301: 4235: 4183: 4172:{\displaystyle I(0)} 4154: 4143:{\displaystyle S(0)} 4125: 4114:{\displaystyle R(0)} 4096: 4085:{\displaystyle I(0)} 4067: 4056:{\displaystyle S(0)} 4038: 3976: 3949: 3922: 3908:{\displaystyle \xi } 3899: 3879: 3859: 3822: 3787: 3752: 3721: 3701: 3681: 3630: 3610: 3581: 3310: 3271: 3233: 3121: 3094: 3048: 3021: 2840: 2704: 2629: 2551: 2499: 2448: 2397: 2374: 2335: 2265: 2208: 2147: 2090: 1999: 1960: 1809: 1683: 1651: 1601: 1551: 1501: 1490:{\displaystyle R(0)} 1472: 1461:{\displaystyle S(0)} 1443: 1332: 1278: 1231: 1188: 1138: 1112: 1033: 935: 886: 866: 846: 826: 658: 597: 552: 532: 509: 456: 436: 416: 338: 248: 237:{\displaystyle R(0)} 219: 20:Compartmental models 30670:Modular Ocean Model 30519:Pandemic prevention 30475:Viral phylodynamics 30456:Re-emerging disease 30345:Travel restrictions 29549:Postoperative wound 29518:Blood-borne disease 29500:Respiratory droplet 29412:Vertical/Congenital 29397:Nosocomial/Hospital 29354:Spillover infection 28638:2020PNAS..11729416W 28632:(47): 29416–29418. 28583:2002PhRvE..66a6128N 28514:2001PhRvL..86.3200P 28461:2001PhRvE..64f6112M 28385:. 11, 51 (4): 1–111 28377:Tembine, H (2020). 28212:10.1155/2012/408637 28154:2003Sci...300.1966L 27678:2022IJBC...3250199P 27603:2021NatSR..1122497K 27525:2020CSF...13910077C 27432:2020NatCo..11.5576T 27367:1974Natur.250..726N 27339:10.33218/001c.26101 27303:20.500.11850/665745 27245:20.500.11850/487253 27226:2021Physi...3..386S 27085:10.1098/rsos.191504 27077:2020RSOS....791504N 26888:2022JChEd..99.3471S 26826:(43): 29223–29232. 26581:2000SIAMR..42..599H 26445:20.500.11850/479548 26426:2021JPhA...54q5601S 26374:2020JPhA...53X5601K 26284:2014arXiv1403.2160H 26145:Kendall DG (1956). 26119:1927RSPSA.115..700K 26071:1917RSPSA..93..225R 26030:1917RSPSA..93..212R 25989:1916RSPSA..92..204R 25894:thermodynamic limit 25857:Vector transmission 25827: 25809: 25733:{\displaystyle 2.5} 24088: 23252:non-negative matrix 22802: 22773: 22633: 22579: 21604:{\displaystyle I=0} 20831:where each element 20438: 20329: 20288: 20222:with the mortality 19766: 19511: 19139: 18947: 18721: 18639: 18557: 18382: 17399:In other words, if 16308: 16049: 16016: 15938:is lower than for S 15914:. Presumably, for S 15544:), and exhausted (S 14959: 11980: 8999:{\displaystyle R=0} 8069:We have the model: 7716: 7400:{\displaystyle S/N} 3855:const., shows that 3526: 2737: 2083:it yields that if: 1864: 637:Gillespie algorithm 588:Erlang distribution 30763:Data visualization 30747:Input–output model 30660:Hydrological model 30650:Geologic modelling 30465:Selection pressure 29761:Force of infection 29389:Contagious disease 29324:Endogenous seeding 29221:Chain of infection 29145:Injection drug use 29056:Vaccination status 28939:Case fatality rate 28857:infectious disease 28732:Capasso V (2008). 28567:(1 Pt 2): 016128. 28455:(6 Pt 2): 066112. 28240:10.1038/nm0895-815 28048:10.1007/BF00178324 27784:10.1007/BF00178324 27734:10.1155/2007/64870 27581:Scientific Reports 26678:Capasso V (1993). 25901:percolation theory 25828: 25813: 25795: 25757: 25730: 25710: 25677: 25633: 25599: 25540: 25491: 25455: 25428: 25404: 25395: 25355: 25243: 25177: 25150: 25115: 25011: 24974: 24947: 24917: 24878: 24839: 24702:is time dependent 24692: 24662: 24629: 24585: 24561:For example, with 24548: 24500: 24457: 24355: 24246: 24125: 24066: 24007: 23987: 23953: 23896: 23876:is related to the 23866: 23843: 23815: 23795: 23737: 23717: 23696:genomic sequencing 23690:Estimation methods 23677: 23562: 23556: 23488: 23482: 23417:{\displaystyle j.} 23414: 23389: 23358: 23328: 23298: 23264: 23240: 23220: 23178: 23053: 22958: 22860: 22838: 22814: 22788: 22759: 22718: 22655: 22634: 22619: 22601: 22580: 22565: 22547: 22526: 22487: 22466: 22437: 22385: 22380: 22162: 22142: 22122: 22102: 22082: 22027: 21995: 21968: 21938: 21849: 21776: 21690: 21636: 21601: 21575: 21542: 21509: 21476: 21449: 21321: 21285: 21265: 21238: 21218: 21197: 21177: 21153: 21091: 21061: 21028: 21001: 20981: 20951: 20931: 20911: 20891: 20871: 20851: 20821: 20812: 20769:can be written as: 20759: 20736: 20716: 20694: 20649: 20618: 20591: 20550: 20466: 20424: 20362: 20315: 20267: 20241: 20212: 20180: 20065: 19964: 19901: 19745: 19701: 19633: 19582: 19490: 19425: 19277: 19118: 19018: 18926: 18786: 18747: 18700: 18665: 18618: 18583: 18536: 18501: 18415: 18368: 18335: 18194: 18192: 17845: 17767: 17765: 17584: 17533: 17451: 17386: 17242: 17148: 17057: 16985: 16965: 16942: 16917: 16879: 16856: 16854: 16635: 16588: 16557: 16467: 16447: 16414: 16387: 16360: 16340: 16311: 16294: 16203: 16144: 16084: 16035: 16002: 15928: 15892: 15869: 15867: 15789: 15787: 15729:exhaustion process 15717: 15715: 15645: 15643: 15530: 15514: 15432: 15412: 15385: 15358: 15328: 15326: 15090: 15023: 14945: 14915: 14874: 14872: 14693: 14587: 14585: 14490: 14488: 14173: 14172: 14088: 14086: 13846: 13844: 13768: 13766: 13577: 13545: 13465: 13389: 13276: 13216: 12979: 12904: 12884: 12834: 12832: 12566: 12546: 12523: 12503: 12473: 12461: 12395: 12393: 12133: 12120:). This is called 12089: 12033: 11999: 11966: 11924: 11922: 11629: 11600: 11548: 11496: 11444: 11379: 11344: 11303: 11226: 11224: 10923: 10904: 10848: 10815: 10795: 10775: 10717: 10676: 10649: 10619: 10577: 10547: 10497: 10417: 10388: 10361: 10334: 10293: 10241: 10189: 10139: 10101: 10051: 10027: 10008: 10006: 9725: 9690: 9688: 9469: 9457: 9431: 9405: 9379: 9353: 9280: 9269: 9243: 9217: 9191: 9099: 9032: 8996: 8970: 8937: 8887: 8833: 8801: 8741: 8665: 8626: 8624: 8574: 8510: 8448: 8402: 8314: 8202: 8200: 8067: 8048: 8010: 7937: 7935: 7900: 7895: 7854: 7802: 7776: 7702: 7635: 7608: 7581: 7506: 7478: 7458: 7432: 7397: 7369: 7349: 7329: 7309: 7289: 7269: 7245: 7196: 7135: 7080: 7060: 7040: 7020: 7000: 6965: 6930: 6894: 6874: 6806: 6804: 6748: 6717: 6686: 6654: 6625: 6596: 6575: 6559: 6533: 6507: 6481: 6408: 6397: 6371: 6345: 6255: 6225: 6054: 5981: 5887: 5830: 5769: 5656: 5654: 5455: 5435: 5407: 5366: 5333: 5289: 5256: 5223: 5196: 5153: 5109: 5082: 5048: 5028: 4992: 4959: 4926: 4892: 4864: 4831: 4802: 4773: 4746: 4724: 4706:{\displaystyle SI} 4703: 4679: 4659: 4639: 4619: 4586: 4552: 4500: 4448: 4417: 4341: 4275: 4195: 4169: 4140: 4111: 4082: 4053: 3989: 3962: 3935: 3905: 3885: 3865: 3845: 3808: 3773: 3738: 3707: 3687: 3661: 3616: 3596: 3564: 3562: 3512: 3290: 3257: 3219: 3107: 3080: 3034: 3004: 2820: 2723: 2684: 2614: 2536: 2482: 2422: 2380: 2368:force of infection 2353: 2306: 2245: 2188: 2127: 2070: 1979: 1943: 1847: 1777: 1666: 1637: 1587: 1537: 1487: 1458: 1426: 1309: 1260: 1217: 1167: 1118: 1095: 1013: 892: 872: 852: 832: 820: 801: 796: 641: 609: 572: 538: 518: 492:law of mass action 480: 442: 422: 402: 319: 307: 261: 234: 173: 135:for the number of 58:in 1916, Ross and 30814:Scientific models 30796: 30795: 30675:Wildfire modeling 30655:Groundwater model 30635:Atmospheric model 30539: 30538: 30529:Tropical medicine 30470:Synthetic biology 30371: 30370: 30330:Social distancing 30283:Outbreak response 29904: 29903: 29731:Agent-based model 29717: 29716: 29713: 29712: 29709: 29708: 29256:Natural reservoir 29246:Infectious period 29236:Incubation period 29193: 29192: 29149:Natural disaster 29118:Tropical diseases 29101:Biodiversity loss 29038:Risk of infection 29028:Microbiome health 29021:Immunosuppression 28779:978-1-6654-1984-0 28758:2021 IEEE Africon 28743:978-3-540-56526-0 28724:978-0-19-856576-5 28561:Physical Review E 28498:(14): 3200–3203. 28449:Physical Review E 28281:(12): 1478–1494. 28148:(5627): 1966–70. 28015:978-3-540-78911-6 27978:978-0-691-15539-5 27361:(5469): 726–729. 27013:978-0-323-95389-4 26978:978-0-19-854040-3 26882:(10): 3471–3477. 26168:978-0-520-35071-7 25927:Flatten the curve 25905:COVID-19 pandemic 25825: 25807: 25661: 25627: 25431:{\displaystyle K} 25207: 24785: 24613: 24579: 24546: 24452: 24418: 24350: 24297: 23948: 23869:{\displaystyle K} 23818:{\displaystyle N} 23790: 23740:{\displaystyle t} 23672: 23267:{\displaystyle V} 23243:{\displaystyle F} 23051: 23037: 23000: 22956: 22942: 22905: 22863:{\displaystyle V} 22841:{\displaystyle F} 22658:{\displaystyle i} 22604:{\displaystyle i} 22550:{\displaystyle i} 22490:{\displaystyle i} 22354: 22302: 22247: 22198: 21932: 21834: 21773: 21675: 21387: 21354: 21324:{\displaystyle n} 21288:{\displaystyle G} 21263: 21241:{\displaystyle f} 21221:{\displaystyle m} 21200:{\displaystyle m} 21180:{\displaystyle f} 21151: 21004:{\displaystyle a} 20954:{\displaystyle m} 20934:{\displaystyle f} 20914:{\displaystyle i} 20894:{\displaystyle j} 20874:{\displaystyle i} 20762:{\displaystyle G} 20739:{\displaystyle m} 20719:{\displaystyle f} 20652:{\displaystyle G} 18296: 18060: 17934: 17896: 17843: 17821: 17727: 17700: 17638: 17449: 17329: 17224: 17039: 16988:{\displaystyle I} 16968:{\displaystyle S} 16899: 16882:{\displaystyle V} 16825: 16777: 16757: 16730: 16677: 16591:{\displaystyle V} 16547: 16470:{\displaystyle I} 16363:{\displaystyle S} 16309: 16259: 16201: 16184: 16066: 16031: 15997: 15254: 15186: 15005: 14987: 14943: 14844: 14815: 14788: 14741: 14654: 14462: 14404: 14359: 14334: 14301: 14267: 14215: 14048: 14003: 13978: 13936: 13905: 13714: 13626: 13447: 13258: 12974: 12956: 12907:{\displaystyle N} 12803: 12745: 12700: 12675: 12651: 12608: 12569:{\displaystyle N} 12476:{\displaystyle a} 12367: 12325: 12300: 12267: 12233: 12181: 11900: 11854: 11808: 11732: 11674: 11202: 11156: 11110: 11025: 10955: 10818:{\displaystyle N} 10798:{\displaystyle S} 10767: 10715: 9981: 9935: 9889: 9859: 9813: 9780: 9672: 9635: 9589: 9568: 9540: 9516: 9070: 9042:thus reduces to: 8739: 8609: 8556: 8545: 8492: 8481: 8418:logistic function 8390: 8356: 8328:It follows that: 8270: 8247: 8187: 8162: 8129: 8101: 8005: 7984: 7934: 7894: 7849: 7771: 7372:{\displaystyle b} 7352:{\displaystyle b} 7332:{\displaystyle a} 7292:{\displaystyle b} 7272:{\displaystyle a} 7234: 7173: 7118: 6897:{\displaystyle N} 6635:; and recovered, 6381:and for recovery 6189: 6150: 6137: 6110: 6039: 5962: 5943: 5872: 5825: 5747: 5628: 5586: 5561: 5537: 5497: 5303:stage by setting 4682:{\displaystyle S} 4662:{\displaystyle I} 4415: 4339: 4273: 3999:were provided by 3888:{\displaystyle R} 3710:{\displaystyle I} 3690:{\displaystyle I} 3619:{\displaystyle t} 3510: 2815: 2790: 2721: 2417: 2383:{\displaystyle N} 2286: 2168: 2048: 2020: 1162: 1121:{\displaystyle N} 1084: 1026:it follows that: 1002: 979: 956: 895:{\displaystyle N} 875:{\displaystyle R} 855:{\displaystyle I} 835:{\displaystyle S} 780: 731: 688: 541:{\displaystyle D} 425:{\displaystyle N} 175:These variables ( 30831: 30788:Visual analytics 30783:Systems thinking 30701:Population model 30566: 30559: 30552: 30543: 30542: 30524:Tropical disease 30460:Reverse zoonosis 30447:One Health Model 30276:Health education 30265:Community health 30184:Cordon sanitaire 30157:Ring vaccination 30127:Repurposed drugs 30014: 30013: 29933:Gastrointestinal 29776:Machine learning 29580:Gastrointestinal 29531:Intravenous line 29468: 29467: 29382:/Cross-infection 29369:Reverse zoonosis 29342: 29341: 29299: 29298: 29204: 29203: 29185:War and conflict 29053:Nutrition status 29016:Immunodeficiency 28873: 28872: 28849: 28842: 28835: 28826: 28825: 28821: 28791: 28763: 28747: 28728: 28709: 28678: 28677: 28659: 28649: 28617: 28611: 28610: 28576: 28574:cond-mat/0205009 28556: 28550: 28549: 28507: 28505:cond-mat/0010317 28487: 28481: 28480: 28444: 28438: 28437: 28401: 28395: 28394: 28392: 28390: 28374: 28368: 28367: 28345: 28339: 28338: 28327:. NY: Springer. 28320: 28309: 28308: 28298: 28266: 28260: 28259: 28223: 28217: 28216: 28214: 28190: 28184: 28183: 28173: 28133: 28127: 28126: 28116: 28084: 28078: 28077: 28059: 28031: 28025: 28024: 28023: 28022: 27989: 27983: 27982: 27962: 27956: 27946: 27940: 27939: 27891: 27882: 27881: 27871: 27831: 27822: 27821: 27795: 27763: 27757: 27756: 27746: 27736: 27712: 27706: 27705: 27657: 27651: 27641: 27635: 27634: 27624: 27614: 27596: 27572: 27563: 27562: 27544: 27518: 27494: 27485: 27484: 27482: 27470: 27464: 27463: 27453: 27443: 27425: 27401: 27395: 27394: 27375:10.1038/250726a0 27350: 27344: 27343: 27341: 27317: 27308: 27307: 27305: 27295: 27271: 27258: 27257: 27247: 27237: 27205: 27194: 27187: 27181: 27180: 27170: 27130: 27124: 27123: 27121: 27113: 27107: 27106: 27096: 27056: 27050: 27049: 27023: 27017: 27016: 27005: 26989: 26983: 26982: 26959: 26953: 26952: 26932: 26926: 26925: 26907: 26867: 26854: 26853: 26843: 26811: 26798: 26797: 26784: 26752: 26746: 26745: 26732: 26715:(9): 2125–2141. 26700: 26694: 26693: 26675: 26669: 26668: 26660: 26654: 26653: 26635: 26629: 26628: 26610: 26601: 26600: 26564: 26558: 26557: 26547: 26537: 26528:(84): 20130098. 26513: 26507: 26506: 26504: 26480: 26474: 26473: 26471: 26459: 26450: 26449: 26447: 26437: 26405: 26396: 26395: 26385: 26353: 26338: 26337: 26335: 26333: 26324: 26315: 26304: 26303: 26277: 26257: 26246: 26245: 26235: 26195: 26189: 26188: 26142: 26133: 26132: 26130: 26113:(772): 700–721. 26098: 26085: 26084: 26082: 26065:(621): 225–240. 26050: 26044: 26043: 26041: 26024:(650): 212–225. 26009: 26003: 26002: 26000: 25983:(638): 204–230. 25968: 25837: 25835: 25834: 25829: 25826: 25823: 25821: 25808: 25805: 25803: 25791: 25790: 25766: 25764: 25763: 25758: 25756: 25755: 25739: 25737: 25736: 25731: 25719: 25717: 25716: 25711: 25703: 25702: 25687:, we would find 25686: 25684: 25683: 25678: 25676: 25675: 25667: 25659: 25642: 25640: 25639: 25634: 25632: 25625: 25608: 25606: 25605: 25600: 25595: 25594: 25570: 25569: 25549: 25547: 25546: 25541: 25539: 25538: 25517: 25516: 25500: 25498: 25497: 25492: 25484: 25483: 25464: 25462: 25461: 25456: 25454: 25453: 25437: 25435: 25434: 25429: 25413: 25411: 25410: 25405: 25400: 25399: 25392: 25391: 25378: 25377: 25360: 25359: 25352: 25351: 25342: 25329: 25328: 25319: 25307: 25306: 25297: 25292: 25291: 25280: 25279: 25270: 25248: 25247: 25240: 25239: 25226: 25225: 25208: 25206: 25195: 25186: 25184: 25183: 25178: 25176: 25175: 25159: 25157: 25156: 25151: 25149: 25148: 25124: 25122: 25121: 25116: 25111: 25110: 25101: 25100: 25091: 25090: 25075: 25074: 25062: 25061: 25037: 25036: 25020: 25018: 25017: 25012: 25010: 25009: 24983: 24981: 24980: 24975: 24973: 24972: 24956: 24954: 24953: 24948: 24946: 24945: 24926: 24924: 24923: 24918: 24916: 24915: 24887: 24885: 24884: 24879: 24874: 24873: 24848: 24846: 24845: 24840: 24820: 24819: 24800: 24795: 24786: 24778: 24761: 24760: 24727: 24726: 24701: 24699: 24698: 24693: 24691: 24690: 24671: 24669: 24668: 24663: 24655: 24654: 24639:, we would find 24638: 24636: 24635: 24630: 24628: 24627: 24619: 24611: 24594: 24592: 24591: 24586: 24584: 24577: 24557: 24555: 24554: 24549: 24547: 24542: 24541: 24540: 24524: 24509: 24507: 24506: 24501: 24499: 24498: 24483: 24482: 24466: 24464: 24463: 24458: 24453: 24448: 24444: 24443: 24424: 24419: 24417: 24409: 24396: 24395: 24373: 24364: 24362: 24361: 24356: 24351: 24346: 24342: 24341: 24322: 24311: 24310: 24298: 24296: 24288: 24278: 24277: 24264: 24255: 24253: 24252: 24247: 24239: 24228: 24227: 24194: 24193: 24160: 24159: 24134: 24132: 24131: 24126: 24124: 24123: 24101: 24100: 24087: 24083: 24074: 24055: 24054: 24033: 24032: 24016: 24014: 24013: 24008: 23996: 23994: 23993: 23988: 23986: 23985: 23962: 23960: 23959: 23954: 23949: 23947: 23946: 23937: 23920: 23905: 23903: 23902: 23897: 23895: 23894: 23875: 23873: 23872: 23867: 23852: 23850: 23849: 23844: 23842: 23841: 23824: 23822: 23821: 23816: 23804: 23802: 23801: 23796: 23791: 23789: 23781: 23761: 23746: 23744: 23743: 23738: 23726: 23724: 23723: 23718: 23686: 23684: 23683: 23678: 23673: 23671: 23639: 23638: 23637: 23621: 23613: 23612: 23588: 23587: 23571: 23569: 23568: 23563: 23561: 23560: 23497: 23495: 23494: 23489: 23487: 23486: 23467: 23466: 23423: 23421: 23420: 23415: 23398: 23396: 23395: 23390: 23388: 23387: 23382: 23367: 23365: 23364: 23359: 23357: 23356: 23351: 23337: 23335: 23334: 23329: 23327: 23326: 23307: 23305: 23304: 23299: 23297: 23296: 23273: 23271: 23270: 23265: 23249: 23247: 23246: 23241: 23229: 23227: 23226: 23221: 23219: 23218: 23187: 23185: 23184: 23179: 23153: 23136: 23135: 23123: 23122: 23110: 23109: 23097: 23096: 23081: 23080: 23075: 23062: 23060: 23059: 23054: 23052: 23050: 23049: 23048: 23043: 23035: 23029: 23025: 23024: 23019: 23010: 23009: 22998: 22992: 22987: 22986: 22967: 22965: 22964: 22959: 22957: 22955: 22954: 22953: 22948: 22940: 22934: 22930: 22929: 22924: 22915: 22914: 22903: 22897: 22892: 22891: 22869: 22867: 22866: 22861: 22847: 22845: 22844: 22839: 22823: 22821: 22820: 22815: 22810: 22801: 22796: 22781: 22772: 22767: 22752: 22744: 22743: 22727: 22725: 22724: 22719: 22714: 22706: 22705: 22690: 22682: 22681: 22664: 22662: 22661: 22656: 22643: 22641: 22640: 22635: 22632: 22627: 22610: 22608: 22607: 22602: 22589: 22587: 22586: 22581: 22578: 22573: 22556: 22554: 22553: 22548: 22535: 22533: 22532: 22527: 22522: 22514: 22513: 22496: 22494: 22493: 22488: 22475: 22473: 22472: 22467: 22465: 22464: 22459: 22446: 22444: 22443: 22438: 22406: 22394: 22392: 22391: 22386: 22384: 22383: 22356: 22355: 22347: 22341: 22304: 22303: 22295: 22289: 22249: 22248: 22240: 22234: 22200: 22199: 22191: 22171: 22169: 22168: 22163: 22151: 22149: 22148: 22143: 22131: 22129: 22128: 22123: 22111: 22109: 22108: 22103: 22091: 22089: 22088: 22083: 22066:, and infected, 22036: 22034: 22033: 22028: 22026: 22025: 22004: 22002: 22001: 21996: 21994: 21993: 21977: 21975: 21974: 21969: 21967: 21966: 21947: 21945: 21944: 21939: 21934: 21933: 21930: 21912: 21911: 21884: 21883: 21862: 21861: 21848: 21803: 21802: 21785: 21783: 21782: 21777: 21775: 21774: 21771: 21753: 21752: 21725: 21724: 21703: 21702: 21689: 21665: 21664: 21645: 21643: 21642: 21637: 21635: 21634: 21610: 21608: 21607: 21602: 21584: 21582: 21581: 21576: 21568: 21567: 21551: 21549: 21548: 21543: 21535: 21534: 21518: 21516: 21515: 21510: 21502: 21501: 21485: 21483: 21482: 21477: 21475: 21474: 21458: 21456: 21455: 21450: 21445: 21444: 21420: 21419: 21407: 21406: 21388: 21386: 21377: 21376: 21375: 21361: 21356: 21355: 21350: 21349: 21340: 21330: 21328: 21327: 21322: 21294: 21292: 21291: 21286: 21274: 21272: 21271: 21266: 21264: 21256: 21247: 21245: 21244: 21239: 21227: 21225: 21224: 21219: 21206: 21204: 21203: 21198: 21186: 21184: 21183: 21178: 21162: 21160: 21159: 21154: 21152: 21144: 21124: 21123: 21100: 21098: 21097: 21092: 21090: 21089: 21070: 21068: 21067: 21062: 21060: 21059: 21037: 21035: 21034: 21029: 21027: 21026: 21010: 21008: 21007: 21002: 20990: 20988: 20987: 20982: 20980: 20979: 20960: 20958: 20957: 20952: 20940: 20938: 20937: 20932: 20920: 20918: 20917: 20912: 20900: 20898: 20897: 20892: 20880: 20878: 20877: 20872: 20860: 20858: 20857: 20852: 20850: 20849: 20830: 20828: 20827: 20822: 20817: 20816: 20768: 20766: 20765: 20760: 20745: 20743: 20742: 20737: 20725: 20723: 20722: 20717: 20703: 20701: 20700: 20695: 20675: 20674: 20658: 20656: 20655: 20650: 20627: 20625: 20624: 20619: 20617: 20616: 20600: 20598: 20597: 20592: 20590: 20589: 20559: 20557: 20556: 20551: 20522: 20521: 20475: 20473: 20472: 20467: 20462: 20458: 20437: 20432: 20394: 20393: 20371: 20369: 20368: 20363: 20354: 20350: 20349: 20328: 20323: 20287: 20286: 20285: 20275: 20250: 20248: 20247: 20242: 20221: 20219: 20218: 20213: 20189: 20187: 20186: 20181: 20125: 20124: 20094: 20093: 20074: 20072: 20071: 20066: 19973: 19971: 19970: 19965: 19930: 19929: 19910: 19908: 19907: 19902: 19893: 19889: 19861: 19860: 19821: 19820: 19781: 19780: 19765: 19764: 19763: 19753: 19710: 19708: 19707: 19702: 19642: 19640: 19639: 19634: 19623: 19622: 19591: 19589: 19588: 19583: 19581: 19580: 19558: 19557: 19533: 19532: 19510: 19509: 19508: 19498: 19434: 19432: 19431: 19426: 19337: 19336: 19306: 19305: 19286: 19284: 19283: 19278: 19210: 19209: 19208: 19187: 19186: 19162: 19161: 19138: 19137: 19136: 19126: 19078: 19077: 19047: 19046: 19027: 19025: 19024: 19019: 19017: 19016: 18994: 18993: 18969: 18968: 18946: 18945: 18944: 18934: 18850: 18849: 18819: 18818: 18795: 18793: 18792: 18787: 18776: 18775: 18756: 18754: 18753: 18748: 18720: 18719: 18718: 18708: 18674: 18672: 18671: 18666: 18638: 18637: 18636: 18626: 18592: 18590: 18589: 18584: 18556: 18555: 18554: 18544: 18510: 18508: 18507: 18502: 18424: 18422: 18421: 18416: 18392: 18391: 18381: 18376: 18367: 18366: 18344: 18342: 18341: 18336: 18334: 18333: 18297: 18295: 18294: 18293: 18253: 18233: 18232: 18213: 18203: 18201: 18200: 18195: 18193: 18161: 18153: 18152: 18125: 18124: 18100: 18099: 18061: 18059: 18051: 18043: 18009: 18001: 18000: 17961: 17960: 17935: 17927: 17897: 17895: 17887: 17879: 17854: 17852: 17851: 17846: 17844: 17842: 17841: 17829: 17827: 17823: 17822: 17814: 17776: 17774: 17773: 17768: 17766: 17728: 17726: 17718: 17710: 17701: 17693: 17639: 17637: 17629: 17621: 17593: 17591: 17590: 17585: 17580: 17579: 17542: 17540: 17539: 17534: 17517: 17460: 17458: 17457: 17452: 17450: 17448: 17447: 17435: 17424: 17423: 17395: 17393: 17392: 17387: 17382: 17378: 17377: 17373: 17351: 17350: 17330: 17328: 17312: 17311: 17298: 17279: 17275: 17241: 17177: 17176: 17157: 17155: 17154: 17149: 17147: 17143: 17136: 17132: 17094: 17090: 17056: 17014: 17013: 16994: 16992: 16991: 16986: 16974: 16972: 16971: 16966: 16951: 16949: 16948: 16943: 16916: 16888: 16886: 16885: 16880: 16865: 16863: 16862: 16857: 16855: 16826: 16824: 16816: 16808: 16778: 16770: 16758: 16756: 16748: 16740: 16731: 16723: 16678: 16676: 16668: 16660: 16644: 16642: 16641: 16636: 16597: 16595: 16594: 16589: 16566: 16564: 16563: 16558: 16556: 16552: 16548: 16543: 16542: 16533: 16526: 16525: 16507: 16506: 16476: 16474: 16473: 16468: 16456: 16454: 16453: 16448: 16446: 16445: 16423: 16421: 16420: 16415: 16413: 16412: 16396: 16394: 16393: 16388: 16386: 16385: 16369: 16367: 16366: 16361: 16349: 16347: 16346: 16341: 16339: 16338: 16320: 16318: 16317: 16312: 16310: 16307: 16302: 16293: 16292: 16291: 16276: 16275: 16262: 16260: 16255: 16254: 16245: 16240: 16239: 16212: 16210: 16209: 16204: 16202: 16197: 16196: 16187: 16185: 16177: 16172: 16171: 16153: 16151: 16150: 16145: 16134: 16133: 16115: 16114: 16093: 16091: 16090: 16085: 16077: 16076: 16067: 16065: 16064: 16048: 16043: 16034: 16032: 16024: 16015: 16010: 15998: 15996: 15988: 15984: 15983: 15973: 15937: 15935: 15934: 15929: 15901: 15899: 15898: 15893: 15878: 15876: 15875: 15870: 15868: 15866: 15865: 15864: 15852: 15851: 15842: 15841: 15840: 15839: 15798: 15796: 15795: 15790: 15788: 15786: 15785: 15784: 15783: 15782: 15762: 15761: 15760: 15759: 15726: 15724: 15723: 15718: 15716: 15714: 15713: 15712: 15711: 15710: 15690: 15689: 15688: 15687: 15654: 15652: 15651: 15646: 15644: 15642: 15641: 15640: 15628: 15627: 15626: 15625: 15605: 15604: 15592: 15591: 15590: 15589: 15523: 15521: 15520: 15515: 15513: 15512: 15441: 15439: 15438: 15433: 15421: 15419: 15418: 15413: 15411: 15410: 15394: 15392: 15391: 15386: 15384: 15383: 15367: 15365: 15364: 15359: 15357: 15356: 15337: 15335: 15334: 15329: 15327: 15308: 15307: 15298: 15297: 15282: 15281: 15271: 15255: 15250: 15239: 15231: 15230: 15221: 15220: 15205: 15204: 15194: 15187: 15182: 15171: 15163: 15162: 15153: 15152: 15137: 15136: 15126: 15099: 15097: 15096: 15091: 15022: 14988: 14986: 14975: 14961: 14958: 14953: 14944: 14936: 14924: 14922: 14921: 14916: 14883: 14881: 14880: 14875: 14873: 14845: 14837: 14816: 14814: 14806: 14798: 14789: 14781: 14742: 14740: 14732: 14724: 14702: 14700: 14699: 14694: 14655: 14647: 14596: 14594: 14593: 14588: 14586: 14584: 14583: 14582: 14573: 14572: 14563: 14562: 14553: 14552: 14543: 14542: 14533: 14532: 14506:The MSEIRS model 14499: 14497: 14496: 14491: 14489: 14463: 14461: 14453: 14445: 14405: 14403: 14395: 14387: 14360: 14355: 14344: 14335: 14333: 14325: 14317: 14302: 14297: 14286: 14268: 14266: 14258: 14250: 14216: 14214: 14206: 14198: 14182: 14180: 14179: 14174: 14171: 14170: 14169: 14160: 14159: 14150: 14149: 14140: 14139: 14130: 14129: 14097: 14095: 14094: 14089: 14087: 14049: 14047: 14039: 14031: 14004: 13999: 13988: 13979: 13977: 13969: 13961: 13937: 13932: 13921: 13906: 13904: 13896: 13888: 13855: 13853: 13852: 13847: 13845: 13843: 13842: 13841: 13832: 13831: 13822: 13821: 13812: 13811: 13777: 13775: 13774: 13769: 13767: 13763: 13762: 13735: 13734: 13715: 13713: 13705: 13704: 13703: 13690: 13684: 13683: 13656: 13655: 13627: 13625: 13617: 13616: 13615: 13602: 13586: 13584: 13583: 13578: 13554: 13552: 13551: 13546: 13532: 13528: 13464: 13416: 13415: 13398: 13396: 13395: 13390: 13343: 13339: 13275: 13248: 13247: 13225: 13223: 13222: 13217: 13215: 13211: 13132: 13131: 13072: 13068: 12988: 12986: 12985: 12980: 12975: 12973: 12959: 12957: 12955: 12941: 12936: 12935: 12913: 12911: 12910: 12905: 12893: 12891: 12890: 12885: 12843: 12841: 12840: 12835: 12833: 12804: 12802: 12794: 12786: 12746: 12744: 12736: 12728: 12701: 12696: 12685: 12676: 12674: 12666: 12658: 12652: 12647: 12636: 12609: 12607: 12599: 12591: 12575: 12573: 12572: 12567: 12555: 12553: 12552: 12547: 12532: 12530: 12529: 12524: 12512: 12510: 12509: 12504: 12502: 12501: 12482: 12480: 12479: 12474: 12439: 12404: 12402: 12401: 12396: 12394: 12368: 12366: 12358: 12350: 12326: 12321: 12310: 12301: 12299: 12291: 12283: 12268: 12263: 12252: 12234: 12232: 12224: 12216: 12182: 12180: 12172: 12164: 12122:passive immunity 12098: 12096: 12095: 12090: 12042: 12040: 12039: 12034: 12008: 12006: 12005: 12000: 11979: 11974: 11933: 11931: 11930: 11925: 11923: 11901: 11899: 11891: 11883: 11880: 11855: 11853: 11845: 11837: 11834: 11809: 11807: 11799: 11791: 11788: 11733: 11731: 11723: 11715: 11712: 11675: 11673: 11665: 11657: 11654: 11638: 11636: 11635: 11630: 11609: 11607: 11606: 11601: 11587: 11557: 11555: 11554: 11549: 11535: 11505: 11503: 11502: 11497: 11483: 11453: 11451: 11450: 11445: 11388: 11386: 11385: 11380: 11353: 11351: 11350: 11345: 11312: 11310: 11309: 11304: 11235: 11233: 11232: 11227: 11225: 11203: 11201: 11193: 11185: 11182: 11157: 11155: 11147: 11139: 11136: 11111: 11109: 11101: 11093: 11090: 11026: 11024: 11016: 11008: 11005: 10956: 10954: 10946: 10938: 10935: 10913: 10911: 10910: 10905: 10857: 10855: 10854: 10849: 10841: 10840: 10824: 10822: 10821: 10816: 10804: 10802: 10801: 10796: 10784: 10782: 10781: 10776: 10768: 10763: 10759: 10758: 10748: 10743: 10742: 10726: 10724: 10723: 10718: 10716: 10714: 10713: 10704: 10703: 10694: 10685: 10683: 10682: 10677: 10675: 10674: 10658: 10656: 10655: 10650: 10648: 10647: 10628: 10626: 10625: 10620: 10618: 10617: 10599:convergence test 10586: 10584: 10583: 10578: 10576: 10575: 10556: 10554: 10553: 10548: 10506: 10504: 10503: 10498: 10426: 10424: 10423: 10418: 10397: 10395: 10394: 10389: 10387: 10386: 10370: 10368: 10367: 10362: 10360: 10359: 10343: 10341: 10340: 10335: 10302: 10300: 10299: 10294: 10280: 10250: 10248: 10247: 10242: 10228: 10198: 10196: 10195: 10190: 10148: 10146: 10145: 10140: 10110: 10108: 10107: 10102: 10060: 10058: 10057: 10052: 10017: 10015: 10014: 10009: 10007: 9982: 9980: 9972: 9964: 9961: 9936: 9934: 9926: 9918: 9915: 9890: 9885: 9865: 9860: 9858: 9850: 9842: 9839: 9814: 9809: 9789: 9781: 9779: 9771: 9763: 9760: 9734: 9732: 9731: 9726: 9699: 9697: 9696: 9691: 9689: 9673: 9671: 9663: 9655: 9652: 9636: 9634: 9626: 9618: 9615: 9590: 9585: 9574: 9569: 9567: 9559: 9551: 9548: 9541: 9536: 9525: 9517: 9515: 9507: 9499: 9496: 9466: 9464: 9463: 9458: 9440: 9438: 9437: 9432: 9414: 9412: 9411: 9406: 9388: 9386: 9385: 9380: 9362: 9360: 9359: 9354: 9278: 9276: 9275: 9270: 9252: 9250: 9249: 9244: 9226: 9224: 9223: 9218: 9200: 9198: 9197: 9192: 9108: 9106: 9105: 9100: 9071: 9069: 9061: 9053: 9041: 9039: 9038: 9033: 9005: 9003: 9002: 8997: 8979: 8977: 8976: 8971: 8946: 8944: 8943: 8938: 8896: 8894: 8893: 8888: 8880: 8879: 8870: 8865: 8864: 8842: 8840: 8839: 8834: 8810: 8808: 8807: 8802: 8791: 8771: 8770: 8750: 8748: 8747: 8742: 8740: 8738: 8737: 8736: 8711: 8710: 8701: 8674: 8672: 8671: 8666: 8664: 8663: 8635: 8633: 8632: 8627: 8625: 8615: 8611: 8610: 8602: 8573: 8546: 8538: 8535: 8509: 8482: 8474: 8471: 8457: 8455: 8454: 8449: 8411: 8409: 8408: 8403: 8401: 8400: 8391: 8383: 8357: 8355: 8347: 8339: 8323: 8321: 8320: 8315: 8271: 8269: 8261: 8253: 8248: 8246: 8238: 8230: 8211: 8209: 8208: 8203: 8201: 8188: 8183: 8172: 8163: 8161: 8153: 8145: 8130: 8125: 8114: 8102: 8100: 8092: 8084: 8019: 8017: 8016: 8011: 8006: 8004: 7990: 7985: 7983: 7972: 7964: 7946: 7944: 7943: 7938: 7936: 7933: 7919: 7909: 7907: 7906: 7901: 7896: 7890: 7882: 7863: 7861: 7860: 7855: 7850: 7848: 7837: 7829: 7811: 7809: 7808: 7803: 7785: 7783: 7782: 7777: 7772: 7770: 7756: 7744: 7743: 7715: 7710: 7692: 7691: 7679: 7678: 7644: 7642: 7641: 7636: 7634: 7633: 7617: 7615: 7614: 7609: 7607: 7606: 7590: 7588: 7587: 7582: 7574: 7573: 7561: 7560: 7515: 7513: 7512: 7507: 7502: 7487: 7485: 7484: 7479: 7467: 7465: 7464: 7459: 7441: 7439: 7438: 7433: 7406: 7404: 7403: 7398: 7393: 7378: 7376: 7375: 7370: 7358: 7356: 7355: 7350: 7338: 7336: 7335: 7330: 7318: 7316: 7315: 7310: 7298: 7296: 7295: 7290: 7278: 7276: 7275: 7270: 7254: 7252: 7251: 7246: 7235: 7233: 7225: 7217: 7205: 7203: 7202: 7197: 7174: 7172: 7164: 7156: 7144: 7142: 7141: 7136: 7119: 7117: 7109: 7101: 7089: 7087: 7086: 7081: 7069: 7067: 7066: 7061: 7049: 7047: 7046: 7041: 7029: 7027: 7026: 7021: 7009: 7007: 7006: 7001: 6974: 6972: 6971: 6966: 6939: 6937: 6936: 6931: 6903: 6901: 6900: 6895: 6883: 6881: 6880: 6875: 6815: 6813: 6812: 6807: 6805: 6803: 6802: 6801: 6792: 6791: 6782: 6781: 6757: 6755: 6754: 6749: 6726: 6724: 6723: 6718: 6695: 6693: 6692: 6687: 6663: 6661: 6660: 6655: 6634: 6632: 6631: 6626: 6605: 6603: 6602: 6597: 6568: 6566: 6565: 6560: 6542: 6540: 6539: 6534: 6516: 6514: 6513: 6508: 6490: 6488: 6487: 6482: 6406: 6404: 6403: 6398: 6380: 6378: 6377: 6372: 6354: 6352: 6351: 6346: 6264: 6262: 6261: 6256: 6254: 6253: 6234: 6232: 6231: 6226: 6221: 6217: 6216: 6212: 6205: 6204: 6190: 6182: 6177: 6173: 6166: 6165: 6151: 6143: 6138: 6133: 6122: 6112: 6111: 6108: 6053: 6008: 6007: 5990: 5988: 5987: 5982: 5980: 5976: 5963: 5955: 5945: 5944: 5941: 5886: 5862: 5861: 5839: 5837: 5836: 5831: 5826: 5824: 5810: 5805: 5804: 5778: 5776: 5775: 5770: 5765: 5761: 5748: 5740: 5730: 5726: 5665: 5663: 5662: 5657: 5655: 5629: 5627: 5619: 5611: 5587: 5582: 5571: 5562: 5560: 5552: 5544: 5538: 5533: 5522: 5498: 5496: 5488: 5480: 5464: 5462: 5461: 5456: 5444: 5442: 5441: 5436: 5416: 5414: 5413: 5408: 5406: 5405: 5396: 5375: 5373: 5372: 5367: 5359: 5358: 5342: 5340: 5339: 5334: 5332: 5331: 5319: 5318: 5298: 5296: 5295: 5290: 5282: 5281: 5265: 5263: 5262: 5257: 5249: 5248: 5232: 5230: 5229: 5224: 5222: 5221: 5205: 5203: 5202: 5197: 5195: 5194: 5179: 5178: 5162: 5160: 5159: 5154: 5152: 5151: 5136: 5135: 5118: 5116: 5115: 5110: 5108: 5107: 5091: 5089: 5088: 5083: 5081: 5080: 5060:convergence test 5057: 5055: 5054: 5049: 5037: 5035: 5034: 5029: 5027: 5026: 5008:convergence test 5001: 4999: 4998: 4993: 4985: 4984: 4968: 4966: 4965: 4960: 4952: 4951: 4935: 4933: 4932: 4927: 4919: 4918: 4901: 4899: 4898: 4893: 4891: 4890: 4873: 4871: 4870: 4865: 4863: 4862: 4840: 4838: 4837: 4832: 4827: 4811: 4809: 4808: 4803: 4798: 4782: 4780: 4779: 4774: 4772: 4771: 4755: 4753: 4752: 4747: 4733: 4731: 4730: 4725: 4712: 4710: 4709: 4704: 4688: 4686: 4685: 4680: 4668: 4666: 4665: 4660: 4648: 4646: 4645: 4640: 4628: 4626: 4625: 4620: 4612: 4611: 4595: 4593: 4592: 4587: 4579: 4578: 4561: 4559: 4558: 4553: 4551: 4550: 4526: 4525: 4509: 4507: 4506: 4501: 4499: 4498: 4474: 4473: 4457: 4455: 4454: 4449: 4447: 4446: 4426: 4424: 4423: 4418: 4416: 4414: 4410: 4409: 4399: 4395: 4394: 4384: 4379: 4378: 4350: 4348: 4347: 4342: 4340: 4338: 4337: 4328: 4327: 4318: 4313: 4312: 4284: 4282: 4281: 4276: 4274: 4272: 4271: 4262: 4261: 4252: 4247: 4246: 4204: 4202: 4201: 4196: 4178: 4176: 4175: 4170: 4149: 4147: 4146: 4141: 4120: 4118: 4117: 4112: 4091: 4089: 4088: 4083: 4062: 4060: 4059: 4054: 4016: 3998: 3996: 3995: 3990: 3988: 3987: 3971: 3969: 3968: 3963: 3961: 3960: 3944: 3942: 3941: 3936: 3934: 3933: 3914: 3912: 3911: 3906: 3894: 3892: 3891: 3886: 3874: 3872: 3871: 3866: 3854: 3852: 3851: 3846: 3835: 3817: 3815: 3814: 3809: 3782: 3780: 3779: 3774: 3747: 3745: 3744: 3739: 3731: 3716: 3714: 3713: 3708: 3696: 3694: 3693: 3688: 3670: 3668: 3667: 3662: 3654: 3653: 3625: 3623: 3622: 3617: 3605: 3603: 3602: 3597: 3573: 3571: 3570: 3565: 3563: 3559: 3558: 3542: 3541: 3525: 3520: 3511: 3503: 3369: 3368: 3299: 3297: 3296: 3291: 3283: 3282: 3266: 3264: 3263: 3258: 3228: 3226: 3225: 3220: 3206: 3201: 3200: 3185: 3184: 3180: 3157: 3156: 3133: 3132: 3116: 3114: 3113: 3108: 3106: 3105: 3089: 3087: 3086: 3081: 3070: 3069: 3057: 3056: 3043: 3041: 3040: 3035: 3033: 3032: 3013: 3011: 3010: 3005: 2988: 2987: 2890: 2889: 2871: 2870: 2852: 2851: 2829: 2827: 2826: 2821: 2816: 2811: 2803: 2791: 2789: 2785: 2784: 2772: 2771: 2765: 2764: 2754: 2753: 2752: 2739: 2736: 2731: 2722: 2714: 2693: 2691: 2690: 2685: 2638: 2637: 2623: 2621: 2620: 2615: 2604: 2603: 2585: 2584: 2560: 2559: 2545: 2543: 2542: 2537: 2508: 2507: 2491: 2489: 2488: 2483: 2457: 2456: 2431: 2429: 2428: 2423: 2418: 2410: 2389: 2387: 2386: 2381: 2362: 2360: 2359: 2354: 2315: 2313: 2312: 2307: 2287: 2285: 2277: 2269: 2254: 2252: 2251: 2246: 2220: 2219: 2197: 2195: 2194: 2189: 2169: 2167: 2159: 2151: 2136: 2134: 2133: 2128: 2102: 2101: 2079: 2077: 2076: 2071: 2060: 2056: 2049: 2041: 2039: 2038: 2021: 2019: 2011: 2003: 1988: 1986: 1985: 1980: 1972: 1971: 1952: 1950: 1949: 1944: 1936: 1935: 1931: 1930: 1912: 1911: 1894: 1893: 1884: 1883: 1863: 1855: 1840: 1839: 1821: 1820: 1799: 1786: 1784: 1783: 1778: 1776: 1775: 1771: 1770: 1758: 1757: 1745: 1744: 1727: 1726: 1714: 1713: 1695: 1694: 1675: 1673: 1672: 1667: 1646: 1644: 1643: 1638: 1633: 1613: 1612: 1596: 1594: 1593: 1588: 1583: 1563: 1562: 1546: 1544: 1543: 1538: 1533: 1513: 1512: 1496: 1494: 1493: 1488: 1467: 1465: 1464: 1459: 1435: 1433: 1432: 1427: 1422: 1421: 1417: 1379: 1378: 1318: 1316: 1315: 1310: 1305: 1304: 1295: 1290: 1289: 1269: 1267: 1266: 1261: 1259: 1258: 1243: 1242: 1226: 1224: 1223: 1218: 1216: 1215: 1200: 1199: 1176: 1174: 1173: 1168: 1163: 1155: 1150: 1149: 1127: 1125: 1124: 1119: 1104: 1102: 1101: 1096: 1085: 1082: 1022: 1020: 1019: 1014: 1003: 1001: 993: 985: 980: 978: 970: 962: 957: 955: 947: 939: 901: 899: 898: 893: 881: 879: 878: 873: 861: 859: 858: 853: 841: 839: 838: 833: 810: 808: 807: 802: 800: 797: 781: 779: 771: 763: 760: 732: 730: 722: 714: 711: 689: 687: 679: 671: 668: 618: 616: 615: 610: 581: 579: 578: 573: 568: 547: 545: 544: 539: 527: 525: 524: 519: 489: 487: 486: 481: 479: 478: 469: 451: 449: 448: 443: 431: 429: 428: 423: 411: 409: 408: 403: 401: 400: 391: 365: 354: 322:Transition rates 270: 268: 267: 262: 260: 259: 243: 241: 240: 235: 125:: The number of 115:: The number of 72:Reed–Frost model 30839: 30838: 30834: 30833: 30832: 30830: 30829: 30828: 30799: 30798: 30797: 30792: 30751: 30705: 30691:Energy modeling 30679: 30623: 30599:Ecosystem model 30575: 30570: 30540: 30535: 30479: 30392:Antigenic shift 30387:Antigenic drift 30378: 30367: 30231:Barrier nursing 30179:Contact tracing 30170: 30163: 30008: 30006: 29999: 29910: 29900: 29801: 29795: 29786:Serial interval 29705: 29689: 29680:Cervico-vaginal 29675: 29666:Trans-placental 29661: 29645: 29613: 29575: 29511:Vascular system 29510: 29504: 29459: 29447: 29441: 29381: 29373: 29333: 29290: 29208: 29189: 29080: 29011:Immune response 28977: 28956:Drug resistance 28900:Infectious dose 28864: 28853: 28806: 28798: 28780: 28761: 28744: 28725: 28706: 28687: 28685:Further reading 28682: 28681: 28618: 28614: 28557: 28553: 28488: 28484: 28445: 28441: 28418:10.2307/2342553 28402: 28398: 28388: 28386: 28375: 28371: 28364: 28346: 28342: 28335: 28321: 28312: 28267: 28263: 28228:Nature Medicine 28224: 28220: 28191: 28187: 28134: 28130: 28085: 28081: 28032: 28028: 28020: 28018: 28016: 27990: 27986: 27979: 27963: 27959: 27953: 27947: 27943: 27892: 27885: 27832: 27825: 27764: 27760: 27713: 27709: 27658: 27654: 27642: 27638: 27573: 27566: 27495: 27488: 27471: 27467: 27402: 27398: 27351: 27347: 27318: 27311: 27272: 27261: 27206: 27197: 27188: 27184: 27131: 27127: 27119: 27115: 27114: 27110: 27057: 27053: 27046: 27024: 27020: 27014: 26990: 26986: 26979: 26960: 26956: 26933: 26929: 26868: 26857: 26812: 26801: 26753: 26749: 26701: 26697: 26690: 26676: 26672: 26661: 26657: 26650: 26636: 26632: 26625: 26611: 26604: 26565: 26561: 26514: 26510: 26481: 26477: 26460: 26453: 26406: 26399: 26354: 26341: 26331: 26329: 26322: 26316: 26307: 26258: 26249: 26196: 26192: 26169: 26143: 26136: 26099: 26088: 26051: 26047: 26010: 26006: 25969: 25965: 25960: 25952:Risk assessment 25913: 25890: 25868: 25859: 25850: 25845: 25822: 25817: 25804: 25799: 25786: 25782: 25780: 25777: 25776: 25751: 25747: 25745: 25742: 25741: 25725: 25722: 25721: 25698: 25694: 25692: 25689: 25688: 25668: 25663: 25662: 25648: 25645: 25644: 25628: 25614: 25611: 25610: 25590: 25586: 25565: 25561: 25559: 25556: 25555: 25534: 25530: 25512: 25508: 25506: 25503: 25502: 25479: 25475: 25473: 25470: 25469: 25449: 25445: 25443: 25440: 25439: 25423: 25420: 25419: 25394: 25393: 25387: 25383: 25380: 25379: 25373: 25369: 25362: 25361: 25354: 25353: 25347: 25343: 25338: 25330: 25324: 25320: 25315: 25309: 25308: 25302: 25298: 25293: 25287: 25283: 25281: 25275: 25271: 25266: 25253: 25252: 25242: 25241: 25235: 25231: 25228: 25227: 25221: 25217: 25210: 25209: 25199: 25194: 25192: 25189: 25188: 25171: 25167: 25165: 25162: 25161: 25144: 25140: 25138: 25135: 25134: 25106: 25102: 25096: 25092: 25086: 25082: 25070: 25066: 25057: 25053: 25032: 25028: 25026: 25023: 25022: 25005: 25001: 24999: 24996: 24995: 24968: 24964: 24962: 24959: 24958: 24941: 24937: 24935: 24932: 24931: 24911: 24907: 24905: 24902: 24901: 24894: 24869: 24865: 24854: 24851: 24850: 24815: 24811: 24796: 24791: 24777: 24756: 24752: 24722: 24718: 24707: 24704: 24703: 24686: 24682: 24680: 24677: 24676: 24650: 24646: 24644: 24641: 24640: 24620: 24615: 24614: 24600: 24597: 24596: 24580: 24566: 24563: 24562: 24536: 24532: 24525: 24523: 24515: 24512: 24511: 24491: 24487: 24478: 24474: 24472: 24469: 24468: 24439: 24435: 24425: 24423: 24410: 24391: 24387: 24374: 24372: 24370: 24367: 24366: 24337: 24333: 24323: 24321: 24306: 24302: 24289: 24273: 24269: 24265: 24263: 24261: 24258: 24257: 24235: 24223: 24219: 24189: 24185: 24155: 24151: 24140: 24137: 24136: 24116: 24112: 24096: 24092: 24079: 24075: 24070: 24050: 24046: 24028: 24024: 24022: 24019: 24018: 24002: 23999: 23998: 23981: 23977: 23975: 23972: 23971: 23968: 23942: 23938: 23921: 23919: 23911: 23908: 23907: 23890: 23886: 23884: 23881: 23880: 23861: 23858: 23857: 23837: 23833: 23831: 23828: 23827: 23810: 23807: 23806: 23782: 23762: 23760: 23752: 23749: 23748: 23732: 23729: 23728: 23703: 23700: 23699: 23692: 23640: 23633: 23629: 23622: 23620: 23605: 23601: 23583: 23579: 23577: 23574: 23573: 23555: 23554: 23543: 23534: 23533: 23528: 23512: 23511: 23503: 23500: 23499: 23481: 23480: 23475: 23469: 23468: 23462: 23458: 23453: 23443: 23442: 23434: 23431: 23430: 23406: 23403: 23402: 23383: 23378: 23377: 23375: 23372: 23371: 23352: 23347: 23346: 23344: 23341: 23340: 23319: 23315: 23313: 23310: 23309: 23289: 23285: 23283: 23280: 23279: 23259: 23256: 23255: 23235: 23232: 23231: 23211: 23207: 23196: 23193: 23192: 23149: 23131: 23127: 23118: 23114: 23105: 23101: 23092: 23088: 23076: 23071: 23070: 23068: 23065: 23064: 23044: 23039: 23038: 23030: 23020: 23015: 23014: 23005: 23001: 22993: 22991: 22979: 22975: 22973: 22970: 22969: 22949: 22944: 22943: 22935: 22925: 22920: 22919: 22910: 22906: 22898: 22896: 22884: 22880: 22878: 22875: 22874: 22855: 22852: 22851: 22833: 22830: 22829: 22806: 22797: 22792: 22777: 22768: 22763: 22748: 22739: 22735: 22733: 22730: 22729: 22710: 22701: 22697: 22686: 22677: 22673: 22671: 22668: 22667: 22650: 22647: 22646: 22628: 22623: 22617: 22614: 22613: 22596: 22593: 22592: 22574: 22569: 22563: 22560: 22559: 22542: 22539: 22538: 22518: 22509: 22505: 22503: 22500: 22499: 22482: 22479: 22478: 22460: 22455: 22454: 22452: 22449: 22448: 22402: 22400: 22397: 22396: 22379: 22378: 22346: 22345: 22342: 22339: 22338: 22294: 22293: 22290: 22287: 22286: 22239: 22238: 22235: 22232: 22231: 22190: 22189: 22182: 22181: 22179: 22176: 22175: 22157: 22154: 22153: 22137: 22134: 22133: 22117: 22114: 22113: 22097: 22094: 22093: 22092:and birth rate 22077: 22074: 22073: 22046: 22040: 22021: 22017: 22015: 22012: 22011: 22008:spectral radius 21989: 21985: 21983: 21980: 21979: 21962: 21958: 21956: 21953: 21952: 21929: 21928: 21907: 21903: 21879: 21875: 21857: 21853: 21838: 21798: 21794: 21792: 21789: 21788: 21770: 21769: 21748: 21744: 21720: 21716: 21698: 21694: 21679: 21660: 21656: 21654: 21651: 21650: 21630: 21626: 21624: 21621: 21620: 21590: 21587: 21586: 21563: 21559: 21557: 21554: 21553: 21530: 21526: 21524: 21521: 21520: 21497: 21493: 21491: 21488: 21487: 21470: 21466: 21464: 21461: 21460: 21440: 21436: 21415: 21411: 21402: 21398: 21378: 21371: 21367: 21362: 21360: 21345: 21341: 21339: 21338: 21336: 21333: 21332: 21316: 21313: 21312: 21301: 21280: 21277: 21276: 21255: 21253: 21250: 21249: 21233: 21230: 21229: 21213: 21210: 21209: 21192: 21189: 21188: 21172: 21169: 21168: 21143: 21119: 21115: 21113: 21110: 21109: 21106:spectral radius 21085: 21081: 21076: 21073: 21072: 21049: 21045: 21043: 21040: 21039: 21022: 21018: 21016: 21013: 21012: 20996: 20993: 20992: 20972: 20968: 20966: 20963: 20962: 20946: 20943: 20942: 20926: 20923: 20922: 20906: 20903: 20902: 20886: 20883: 20882: 20866: 20863: 20862: 20842: 20838: 20836: 20833: 20832: 20811: 20810: 20805: 20799: 20798: 20793: 20783: 20782: 20774: 20771: 20770: 20754: 20751: 20750: 20731: 20728: 20727: 20711: 20708: 20707: 20670: 20666: 20664: 20661: 20660: 20644: 20641: 20640: 20634:spectral radius 20612: 20608: 20606: 20603: 20602: 20585: 20581: 20579: 20576: 20575: 20572: 20566: 20517: 20513: 20487: 20484: 20483: 20433: 20428: 20420: 20416: 20389: 20385: 20383: 20380: 20379: 20330: 20324: 20319: 20311: 20307: 20281: 20277: 20276: 20271: 20259: 20256: 20255: 20227: 20224: 20223: 20198: 20195: 20194: 20120: 20116: 20089: 20085: 20083: 20080: 20079: 19982: 19979: 19978: 19925: 19921: 19919: 19916: 19915: 19856: 19852: 19816: 19812: 19776: 19772: 19771: 19767: 19759: 19755: 19754: 19749: 19722: 19719: 19718: 19654: 19651: 19650: 19618: 19614: 19600: 19597: 19596: 19576: 19572: 19553: 19549: 19528: 19524: 19504: 19500: 19499: 19494: 19446: 19443: 19442: 19332: 19328: 19301: 19297: 19295: 19292: 19291: 19204: 19200: 19182: 19178: 19157: 19153: 19140: 19132: 19128: 19127: 19122: 19073: 19069: 19042: 19038: 19036: 19033: 19032: 19012: 19008: 18989: 18985: 18964: 18960: 18940: 18936: 18935: 18930: 18845: 18841: 18814: 18810: 18808: 18805: 18804: 18771: 18767: 18765: 18762: 18761: 18714: 18710: 18709: 18704: 18683: 18680: 18679: 18632: 18628: 18627: 18622: 18601: 18598: 18597: 18550: 18546: 18545: 18540: 18519: 18516: 18515: 18439: 18436: 18435: 18431: 18387: 18383: 18377: 18372: 18362: 18358: 18356: 18353: 18352: 18302: 18298: 18283: 18279: 18257: 18252: 18228: 18224: 18222: 18219: 18218: 18208: 18191: 18190: 18160: 18148: 18144: 18120: 18116: 18095: 18091: 18062: 18052: 18044: 18042: 18039: 18038: 18008: 17996: 17992: 17956: 17952: 17926: 17898: 17888: 17880: 17878: 17874: 17872: 17869: 17868: 17861: 17837: 17833: 17828: 17813: 17806: 17802: 17788: 17785: 17784: 17764: 17763: 17729: 17719: 17711: 17709: 17706: 17705: 17692: 17640: 17630: 17622: 17620: 17616: 17614: 17611: 17610: 17604: 17575: 17571: 17554: 17551: 17550: 17510: 17486: 17483: 17482: 17475: 17443: 17439: 17434: 17419: 17415: 17407: 17404: 17403: 17346: 17342: 17341: 17337: 17307: 17303: 17302: 17297: 17296: 17292: 17247: 17243: 17228: 17172: 17168: 17166: 17163: 17162: 17122: 17118: 17114: 17110: 17062: 17058: 17043: 17009: 17005: 17003: 17000: 16999: 16980: 16977: 16976: 16960: 16957: 16956: 16903: 16897: 16894: 16893: 16874: 16871: 16870: 16853: 16852: 16827: 16817: 16809: 16807: 16804: 16803: 16769: 16759: 16749: 16741: 16739: 16736: 16735: 16722: 16679: 16669: 16661: 16659: 16655: 16653: 16650: 16649: 16612: 16609: 16608: 16604: 16583: 16580: 16579: 16576: 16538: 16534: 16532: 16531: 16527: 16521: 16517: 16496: 16492: 16484: 16481: 16480: 16462: 16459: 16458: 16435: 16431: 16429: 16426: 16425: 16408: 16404: 16402: 16399: 16398: 16397:(σ-naught) and 16381: 16377: 16375: 16372: 16371: 16355: 16352: 16351: 16334: 16330: 16328: 16325: 16324: 16303: 16298: 16287: 16283: 16271: 16267: 16263: 16261: 16250: 16246: 16244: 16229: 16225: 16223: 16220: 16219: 16192: 16188: 16186: 16176: 16167: 16163: 16161: 16158: 16157: 16129: 16125: 16110: 16106: 16104: 16101: 16100: 16072: 16068: 16054: 16050: 16044: 16039: 16033: 16023: 16011: 16006: 15989: 15979: 15975: 15974: 15972: 15970: 15967: 15966: 15960: 15953: 15945: 15941: 15923: 15920: 15919: 15917: 15913: 15909: 15905: 15887: 15884: 15883: 15857: 15856: 15847: 15846: 15829: 15825: 15824: 15823: 15822: 15820: 15818: 15815: 15814: 15772: 15768: 15767: 15766: 15749: 15745: 15744: 15743: 15742: 15740: 15738: 15735: 15734: 15700: 15696: 15695: 15694: 15677: 15673: 15672: 15671: 15670: 15668: 15666: 15663: 15662: 15636: 15635: 15615: 15611: 15610: 15609: 15600: 15599: 15579: 15575: 15574: 15573: 15572: 15570: 15568: 15565: 15564: 15559: 15555: 15551: 15547: 15543: 15539: 15535: 15527: 15508: 15504: 15502: 15499: 15498: 15480:Social dynamics 15475: 15472: 15465:random graphs. 15458: 15449: 15427: 15424: 15423: 15406: 15402: 15400: 15397: 15396: 15379: 15375: 15373: 15370: 15369: 15352: 15348: 15346: 15343: 15342: 15325: 15324: 15303: 15299: 15293: 15289: 15277: 15273: 15269: 15268: 15240: 15238: 15226: 15222: 15216: 15212: 15200: 15196: 15192: 15191: 15172: 15170: 15158: 15154: 15148: 15144: 15132: 15128: 15123: 15121: 15118: 15117: 15110: 15009: 14976: 14962: 14960: 14954: 14949: 14935: 14933: 14930: 14929: 14892: 14889: 14888: 14871: 14870: 14836: 14817: 14807: 14799: 14797: 14794: 14793: 14780: 14743: 14733: 14725: 14723: 14719: 14717: 14714: 14713: 14646: 14626: 14623: 14622: 14607: 14578: 14577: 14568: 14567: 14558: 14557: 14548: 14547: 14538: 14537: 14528: 14527: 14526: 14524: 14522: 14519: 14518: 14508: 14487: 14486: 14464: 14454: 14446: 14444: 14441: 14440: 14406: 14396: 14388: 14386: 14383: 14382: 14345: 14343: 14336: 14326: 14318: 14316: 14313: 14312: 14287: 14285: 14269: 14259: 14251: 14249: 14246: 14245: 14217: 14207: 14199: 14197: 14193: 14191: 14188: 14187: 14165: 14164: 14155: 14154: 14145: 14144: 14135: 14134: 14125: 14124: 14123: 14120: 14117: 14116: 14106: 14104:The MSEIR model 14085: 14084: 14050: 14040: 14032: 14030: 14027: 14026: 13989: 13987: 13980: 13970: 13962: 13960: 13957: 13956: 13922: 13920: 13907: 13897: 13889: 13887: 13883: 13881: 13878: 13877: 13837: 13836: 13827: 13826: 13817: 13816: 13807: 13806: 13805: 13803: 13801: 13798: 13797: 13787: 13765: 13764: 13758: 13754: 13730: 13726: 13716: 13706: 13699: 13695: 13691: 13689: 13686: 13685: 13679: 13675: 13651: 13647: 13628: 13618: 13611: 13607: 13603: 13601: 13597: 13595: 13592: 13591: 13563: 13560: 13559: 13470: 13466: 13451: 13411: 13407: 13405: 13402: 13401: 13281: 13277: 13262: 13243: 13239: 13237: 13234: 13233: 13229:it holds that: 13127: 13123: 13080: 13076: 13010: 13006: 13004: 13001: 13000: 12963: 12958: 12945: 12940: 12931: 12927: 12925: 12922: 12921: 12914:is a variable. 12899: 12896: 12895: 12852: 12849: 12848: 12831: 12830: 12805: 12795: 12787: 12785: 12782: 12781: 12747: 12737: 12729: 12727: 12724: 12723: 12686: 12684: 12677: 12667: 12659: 12657: 12654: 12653: 12637: 12635: 12610: 12600: 12592: 12590: 12586: 12584: 12581: 12580: 12561: 12558: 12557: 12538: 12535: 12534: 12518: 12515: 12514: 12494: 12490: 12488: 12485: 12484: 12468: 12465: 12464: 12452:(for exposed). 12445: 12413: 12392: 12391: 12369: 12359: 12351: 12349: 12346: 12345: 12311: 12309: 12302: 12292: 12284: 12282: 12279: 12278: 12253: 12251: 12235: 12225: 12217: 12215: 12212: 12211: 12183: 12173: 12165: 12163: 12159: 12157: 12154: 12153: 12106: 12048: 12045: 12044: 12019: 12016: 12015: 11975: 11970: 11949: 11946: 11945: 11921: 11920: 11892: 11884: 11882: 11878: 11877: 11846: 11838: 11836: 11832: 11831: 11800: 11792: 11790: 11786: 11785: 11724: 11716: 11714: 11710: 11709: 11666: 11658: 11656: 11651: 11649: 11646: 11645: 11615: 11612: 11611: 11583: 11563: 11560: 11559: 11531: 11511: 11508: 11507: 11479: 11459: 11456: 11455: 11394: 11391: 11390: 11359: 11356: 11355: 11318: 11315: 11314: 11244: 11241: 11240: 11223: 11222: 11194: 11186: 11184: 11180: 11179: 11148: 11140: 11138: 11134: 11133: 11102: 11094: 11092: 11088: 11087: 11017: 11009: 11007: 11003: 11002: 10947: 10939: 10937: 10932: 10930: 10927: 10926: 10875: 10872: 10871: 10864: 10862:The SIRVD model 10836: 10832: 10830: 10827: 10826: 10810: 10807: 10806: 10790: 10787: 10786: 10754: 10750: 10749: 10747: 10738: 10734: 10732: 10729: 10728: 10709: 10705: 10699: 10695: 10693: 10691: 10688: 10687: 10670: 10666: 10664: 10661: 10660: 10643: 10639: 10637: 10634: 10633: 10613: 10609: 10607: 10604: 10603: 10571: 10567: 10565: 10562: 10561: 10512: 10509: 10508: 10432: 10429: 10428: 10403: 10400: 10399: 10382: 10378: 10376: 10373: 10372: 10355: 10351: 10349: 10346: 10345: 10308: 10305: 10304: 10276: 10256: 10253: 10252: 10224: 10204: 10201: 10200: 10154: 10151: 10150: 10116: 10113: 10112: 10066: 10063: 10062: 10034: 10031: 10030: 10005: 10004: 9973: 9965: 9963: 9959: 9958: 9927: 9919: 9917: 9913: 9912: 9866: 9864: 9851: 9843: 9841: 9837: 9836: 9790: 9788: 9772: 9764: 9762: 9757: 9755: 9752: 9751: 9741: 9708: 9705: 9704: 9687: 9686: 9664: 9656: 9654: 9650: 9649: 9627: 9619: 9617: 9613: 9612: 9575: 9573: 9560: 9552: 9550: 9546: 9545: 9526: 9524: 9508: 9500: 9498: 9493: 9491: 9488: 9487: 9446: 9443: 9442: 9420: 9417: 9416: 9394: 9391: 9390: 9368: 9365: 9364: 9291: 9288: 9287: 9258: 9255: 9254: 9232: 9229: 9228: 9206: 9203: 9202: 9129: 9126: 9125: 9118: 9062: 9054: 9052: 9050: 9047: 9046: 9015: 9012: 9011: 8985: 8982: 8981: 8959: 8956: 8955: 8902: 8899: 8898: 8875: 8871: 8866: 8860: 8856: 8848: 8845: 8844: 8816: 8813: 8812: 8787: 8766: 8762: 8760: 8757: 8756: 8726: 8722: 8712: 8706: 8702: 8700: 8683: 8680: 8679: 8656: 8652: 8644: 8641: 8640: 8623: 8622: 8601: 8594: 8590: 8560: 8537: 8533: 8532: 8496: 8473: 8468: 8466: 8463: 8462: 8425: 8422: 8421: 8396: 8392: 8382: 8348: 8340: 8338: 8336: 8333: 8332: 8262: 8254: 8252: 8239: 8231: 8229: 8227: 8224: 8223: 8199: 8198: 8173: 8171: 8164: 8154: 8146: 8144: 8141: 8140: 8115: 8113: 8103: 8093: 8085: 8083: 8079: 8077: 8074: 8073: 8040: 8035: 8026: 7994: 7989: 7973: 7965: 7963: 7955: 7952: 7951: 7923: 7917: 7915: 7912: 7911: 7883: 7880: 7872: 7869: 7868: 7838: 7830: 7828: 7820: 7817: 7816: 7794: 7791: 7790: 7760: 7755: 7721: 7717: 7711: 7706: 7687: 7683: 7674: 7670: 7653: 7650: 7649: 7629: 7625: 7623: 7620: 7619: 7602: 7598: 7596: 7593: 7592: 7569: 7565: 7556: 7552: 7535: 7532: 7531: 7529: 7523: 7498: 7493: 7490: 7489: 7473: 7470: 7469: 7447: 7444: 7443: 7412: 7409: 7408: 7389: 7384: 7381: 7380: 7364: 7361: 7360: 7344: 7341: 7340: 7324: 7321: 7320: 7304: 7301: 7300: 7284: 7281: 7280: 7264: 7261: 7260: 7226: 7218: 7216: 7214: 7211: 7210: 7165: 7157: 7155: 7153: 7150: 7149: 7110: 7102: 7100: 7098: 7095: 7094: 7075: 7072: 7071: 7055: 7052: 7051: 7035: 7032: 7031: 7015: 7012: 7011: 6980: 6977: 6976: 6945: 6942: 6941: 6910: 6907: 6906: 6889: 6886: 6885: 6824: 6821: 6820: 6797: 6796: 6787: 6786: 6777: 6776: 6775: 6773: 6771: 6768: 6767: 6734: 6731: 6730: 6703: 6700: 6699: 6672: 6669: 6668: 6640: 6637: 6636: 6611: 6608: 6607: 6582: 6579: 6578: 6548: 6545: 6544: 6522: 6519: 6518: 6496: 6493: 6492: 6419: 6416: 6415: 6386: 6383: 6382: 6360: 6357: 6356: 6283: 6280: 6279: 6272: 6249: 6245: 6243: 6240: 6239: 6200: 6196: 6195: 6191: 6181: 6161: 6157: 6156: 6152: 6142: 6123: 6121: 6120: 6116: 6107: 6106: 6043: 6003: 5999: 5997: 5994: 5993: 5954: 5953: 5949: 5940: 5939: 5876: 5857: 5853: 5851: 5848: 5847: 5814: 5809: 5800: 5796: 5794: 5791: 5790: 5739: 5738: 5734: 5683: 5679: 5677: 5674: 5673: 5653: 5652: 5630: 5620: 5612: 5610: 5607: 5606: 5572: 5570: 5563: 5553: 5545: 5543: 5540: 5539: 5523: 5521: 5499: 5489: 5481: 5479: 5475: 5473: 5470: 5469: 5450: 5447: 5446: 5445:and birth rate 5430: 5427: 5426: 5423: 5401: 5397: 5392: 5381: 5378: 5377: 5354: 5350: 5348: 5345: 5344: 5327: 5323: 5314: 5310: 5308: 5305: 5304: 5277: 5273: 5271: 5268: 5267: 5244: 5240: 5238: 5235: 5234: 5217: 5213: 5211: 5208: 5207: 5190: 5186: 5174: 5170: 5168: 5165: 5164: 5147: 5143: 5131: 5127: 5125: 5122: 5121: 5103: 5099: 5097: 5094: 5093: 5076: 5072: 5070: 5067: 5066: 5043: 5040: 5039: 5022: 5018: 5016: 5013: 5012: 4980: 4976: 4974: 4971: 4970: 4947: 4943: 4941: 4938: 4937: 4914: 4910: 4908: 4905: 4904: 4886: 4882: 4880: 4877: 4876: 4858: 4854: 4852: 4849: 4848: 4823: 4818: 4815: 4814: 4794: 4789: 4786: 4785: 4767: 4763: 4761: 4758: 4757: 4741: 4738: 4737: 4719: 4716: 4715: 4695: 4692: 4691: 4674: 4671: 4670: 4654: 4651: 4650: 4634: 4631: 4630: 4607: 4603: 4601: 4598: 4597: 4574: 4570: 4568: 4565: 4564: 4543: 4539: 4521: 4517: 4515: 4512: 4511: 4491: 4487: 4469: 4465: 4463: 4460: 4459: 4442: 4438: 4436: 4433: 4432: 4405: 4401: 4400: 4390: 4386: 4385: 4383: 4374: 4370: 4368: 4365: 4364: 4333: 4329: 4323: 4319: 4317: 4308: 4304: 4302: 4299: 4298: 4267: 4263: 4257: 4253: 4251: 4242: 4238: 4236: 4233: 4232: 4211: 4184: 4181: 4180: 4155: 4152: 4151: 4126: 4123: 4122: 4097: 4094: 4093: 4068: 4065: 4064: 4039: 4036: 4035: 4014: 4009:Microsoft Excel 3983: 3979: 3977: 3974: 3973: 3956: 3952: 3950: 3947: 3946: 3929: 3925: 3923: 3920: 3919: 3900: 3897: 3896: 3880: 3877: 3876: 3860: 3857: 3856: 3831: 3823: 3820: 3819: 3788: 3785: 3784: 3753: 3750: 3749: 3727: 3722: 3719: 3718: 3702: 3699: 3698: 3682: 3679: 3678: 3637: 3633: 3631: 3628: 3627: 3611: 3608: 3607: 3582: 3579: 3578: 3561: 3560: 3554: 3550: 3537: 3533: 3521: 3516: 3502: 3495: 3480: 3479: 3442: 3427: 3426: 3386: 3371: 3370: 3352: 3348: 3329: 3313: 3311: 3308: 3307: 3278: 3274: 3272: 3269: 3268: 3234: 3231: 3230: 3202: 3196: 3192: 3176: 3152: 3148: 3141: 3137: 3128: 3124: 3122: 3119: 3118: 3101: 3097: 3095: 3092: 3091: 3065: 3061: 3052: 3051: 3049: 3046: 3045: 3028: 3024: 3022: 3019: 3018: 2983: 2979: 2885: 2884: 2866: 2865: 2847: 2846: 2841: 2838: 2837: 2804: 2802: 2780: 2776: 2767: 2766: 2760: 2756: 2755: 2748: 2744: 2740: 2738: 2732: 2727: 2713: 2705: 2702: 2701: 2633: 2632: 2630: 2627: 2626: 2599: 2598: 2580: 2579: 2555: 2554: 2552: 2549: 2548: 2503: 2502: 2500: 2497: 2496: 2452: 2451: 2449: 2446: 2445: 2441: 2409: 2398: 2395: 2394: 2375: 2372: 2371: 2336: 2333: 2332: 2326: 2278: 2270: 2268: 2266: 2263: 2262: 2215: 2211: 2209: 2206: 2205: 2160: 2152: 2150: 2148: 2145: 2144: 2097: 2093: 2091: 2088: 2087: 2040: 2034: 2030: 2029: 2025: 2012: 2004: 2002: 2000: 1997: 1996: 1967: 1963: 1961: 1958: 1957: 1926: 1922: 1907: 1903: 1899: 1895: 1889: 1885: 1879: 1875: 1856: 1851: 1835: 1831: 1816: 1812: 1810: 1807: 1806: 1797: 1766: 1762: 1753: 1749: 1740: 1736: 1732: 1728: 1722: 1718: 1709: 1705: 1690: 1686: 1684: 1681: 1680: 1652: 1649: 1648: 1629: 1608: 1604: 1602: 1599: 1598: 1579: 1558: 1554: 1552: 1549: 1548: 1529: 1508: 1504: 1502: 1499: 1498: 1473: 1470: 1469: 1444: 1441: 1440: 1413: 1374: 1370: 1366: 1362: 1333: 1330: 1329: 1300: 1296: 1291: 1285: 1281: 1279: 1276: 1275: 1251: 1247: 1238: 1234: 1232: 1229: 1228: 1208: 1204: 1195: 1191: 1189: 1186: 1185: 1154: 1145: 1141: 1139: 1136: 1135: 1113: 1110: 1109: 1081: 1034: 1031: 1030: 994: 986: 984: 971: 963: 961: 948: 940: 938: 936: 933: 932: 924:This system is 887: 884: 883: 867: 864: 863: 847: 844: 843: 827: 824: 823: 795: 794: 772: 764: 762: 758: 757: 723: 715: 713: 709: 708: 680: 672: 670: 665: 661: 659: 656: 655: 629: 598: 595: 594: 564: 553: 550: 549: 533: 530: 529: 510: 507: 506: 474: 470: 465: 457: 454: 453: 437: 434: 433: 417: 414: 413: 396: 392: 387: 361: 350: 339: 336: 335: 324: 255: 251: 249: 246: 245: 220: 217: 216: 102: 17: 12: 11: 5: 30837: 30827: 30826: 30821: 30816: 30811: 30794: 30793: 30791: 30790: 30785: 30780: 30778:Systems theory 30775: 30770: 30765: 30759: 30757: 30756:Related topics 30753: 30752: 30750: 30749: 30744: 30742:Economic model 30739: 30734: 30729: 30724: 30719: 30713: 30711: 30707: 30706: 30704: 30703: 30698: 30693: 30687: 30685: 30684:Sustainability 30681: 30680: 30678: 30677: 30672: 30667: 30662: 30657: 30652: 30647: 30642: 30637: 30631: 30629: 30625: 30624: 30622: 30621: 30616: 30611: 30606: 30601: 30596: 30591: 30589:Cellular model 30585: 30583: 30577: 30576: 30569: 30568: 30561: 30554: 30546: 30537: 30536: 30534: 30533: 30532: 30531: 30521: 30516: 30511: 30506: 30501: 30498: 30493: 30487: 30485: 30481: 30480: 30478: 30477: 30472: 30467: 30462: 30457: 30454: 30449: 30444: 30439: 30434: 30429: 30424: 30422:Emergent virus 30419: 30414: 30409: 30404: 30403: 30402: 30394: 30389: 30383: 30381: 30373: 30372: 30369: 30368: 30366: 30365: 30362: 30357: 30355:Vector control 30352: 30347: 30342: 30337: 30332: 30327: 30322: 30317: 30312: 30311: 30310: 30305: 30300: 30298:N95 respirator 30290: 30285: 30280: 30279: 30278: 30273: 30268: 30257: 30252: 30251: 30250: 30240: 30235: 30234: 30233: 30223: 30222: 30221: 30216: 30211: 30201: 30196: 30191: 30186: 30181: 30175: 30173: 30171:pharmaceutical 30165: 30164: 30162: 30161: 30160: 30159: 30154: 30149: 30144: 30139: 30129: 30124: 30119: 30114: 30109: 30104: 30103: 30102: 30092: 30087: 30082: 30077: 30072: 30067: 30062: 30061: 30060: 30050: 30049: 30048: 30038: 30033: 30032: 30031: 30020: 30018: 30017:Pharmaceutical 30011: 30001: 30000: 29998: 29997: 29994: 29989: 29984: 29979: 29974: 29969: 29968:Cardiovascular 29966: 29961: 29956: 29953: 29948: 29947:Nervous system 29945: 29942: 29941: 29940: 29930: 29929: 29928: 29923: 29914: 29912: 29906: 29905: 29902: 29901: 29899: 29898: 29893: 29888: 29883: 29878: 29873: 29868: 29863: 29858: 29853: 29848: 29843: 29838: 29833: 29832: 29831: 29826: 29816: 29811: 29805: 29803: 29797: 29796: 29794: 29793: 29788: 29783: 29778: 29773: 29771:Infection rate 29768: 29763: 29758: 29753: 29748: 29743: 29738: 29733: 29727: 29725: 29719: 29718: 29715: 29714: 29711: 29710: 29707: 29706: 29704: 29703: 29697: 29695: 29691: 29690: 29688: 29687: 29683: 29681: 29677: 29676: 29674: 29673: 29669: 29667: 29663: 29662: 29660: 29659: 29653: 29651: 29647: 29646: 29644: 29643: 29637: 29632: 29627: 29621: 29619: 29615: 29614: 29612: 29611: 29606: 29601: 29596: 29595: 29594: 29583: 29581: 29577: 29576: 29574: 29573: 29572: 29571: 29566: 29558: 29557: 29556: 29551: 29543: 29542: 29541: 29536: 29533: 29528: 29526:Injection site 29520: 29514: 29512: 29506: 29505: 29503: 29502: 29497: 29495:Dental aerosol 29492: 29491: 29490: 29480: 29474: 29472: 29465: 29461: 29460: 29458: 29457: 29451: 29449: 29443: 29442: 29440: 29439: 29434: 29433: 29432: 29427: 29426: 29425: 29420: 29417: 29406: 29405: 29404: 29399: 29391: 29385: 29383: 29379:Human-to-human 29375: 29374: 29372: 29371: 29366: 29361: 29356: 29350: 29348: 29339: 29335: 29334: 29332: 29331: 29325: 29322: 29319: 29314: 29309: 29305: 29303: 29296: 29292: 29291: 29289: 29288: 29283: 29278: 29276:Super-spreader 29273: 29271:Superinfection 29268: 29263: 29258: 29253: 29248: 29243: 29238: 29233: 29228: 29223: 29218: 29212: 29210: 29201: 29195: 29194: 29191: 29190: 29188: 29187: 29182: 29180:Vector control 29177: 29172: 29167: 29162: 29157: 29156: 29155: 29147: 29142: 29137: 29132: 29127: 29122: 29121: 29120: 29115: 29109:Climate zones 29107: 29105:Climate change 29102: 29099: 29096: 29090: 29088: 29082: 29081: 29079: 29078: 29077: 29076: 29073: 29070: 29069: 29068: 29060: 29057: 29054: 29051: 29048: 29040: 29035: 29030: 29025: 29024: 29023: 29018: 29008: 29003: 28998: 28993: 28987: 28985: 28979: 28978: 28976: 28975: 28970: 28969: 28968: 28963: 28958: 28948: 28947: 28946: 28941: 28936: 28931: 28921: 28919:Quorum sensing 28916: 28915: 28914: 28904: 28903: 28902: 28892: 28887: 28881: 28879: 28870: 28866: 28865: 28852: 28851: 28844: 28837: 28829: 28823: 28822: 28804: 28797: 28796:External links 28794: 28793: 28792: 28778: 28748: 28742: 28729: 28723: 28710: 28704: 28686: 28683: 28680: 28679: 28612: 28551: 28482: 28439: 28396: 28369: 28362: 28352:, ed. (1982). 28340: 28333: 28310: 28261: 28218: 28185: 28128: 28099:(1): 173–195. 28079: 28026: 28014: 27984: 27977: 27957: 27951: 27941: 27883: 27846:(4): 281–293. 27823: 27778:(4): 365–382. 27758: 27707: 27652: 27636: 27564: 27486: 27465: 27396: 27345: 27332:(3): 805–820. 27309: 27259: 27195: 27182: 27125: 27108: 27051: 27044: 27018: 27012: 26984: 26977: 26954: 26927: 26855: 26799: 26747: 26695: 26688: 26670: 26655: 26648: 26630: 26623: 26602: 26575:(4): 599–653. 26559: 26508: 26475: 26451: 26420:(17): 175601. 26397: 26368:(50): 505601. 26339: 26305: 26247: 26190: 26167: 26134: 26086: 26045: 26004: 25962: 25961: 25959: 25956: 25955: 25954: 25949: 25944: 25939: 25934: 25929: 25924: 25919: 25912: 25909: 25889: 25886: 25885: 25884: 25881: 25878: 25875: 25867: 25864: 25858: 25855: 25849: 25846: 25844: 25841: 25839:tuberculosis. 25820: 25816: 25812: 25802: 25798: 25794: 25789: 25785: 25754: 25750: 25729: 25709: 25706: 25701: 25697: 25674: 25671: 25666: 25658: 25655: 25652: 25631: 25624: 25621: 25618: 25598: 25593: 25589: 25585: 25582: 25579: 25576: 25573: 25568: 25564: 25537: 25533: 25529: 25526: 25523: 25520: 25515: 25511: 25490: 25487: 25482: 25478: 25452: 25448: 25427: 25403: 25398: 25390: 25386: 25382: 25381: 25376: 25372: 25368: 25367: 25365: 25358: 25350: 25346: 25341: 25337: 25334: 25331: 25327: 25323: 25318: 25314: 25311: 25310: 25305: 25301: 25296: 25290: 25286: 25282: 25278: 25274: 25269: 25265: 25262: 25259: 25258: 25256: 25251: 25246: 25238: 25234: 25230: 25229: 25224: 25220: 25216: 25215: 25213: 25205: 25202: 25198: 25174: 25170: 25147: 25143: 25114: 25109: 25105: 25099: 25095: 25089: 25085: 25081: 25078: 25073: 25069: 25065: 25060: 25056: 25052: 25049: 25046: 25043: 25040: 25035: 25031: 25008: 25004: 24992: 24991: 24985: 24971: 24967: 24944: 24940: 24928: 24914: 24910: 24893: 24890: 24877: 24872: 24868: 24864: 24861: 24858: 24838: 24835: 24832: 24829: 24826: 24823: 24818: 24814: 24810: 24807: 24804: 24799: 24794: 24790: 24784: 24781: 24776: 24773: 24770: 24767: 24764: 24759: 24755: 24751: 24748: 24745: 24742: 24739: 24736: 24733: 24730: 24725: 24721: 24717: 24714: 24711: 24689: 24685: 24661: 24658: 24653: 24649: 24626: 24623: 24618: 24610: 24607: 24604: 24583: 24576: 24573: 24570: 24545: 24539: 24535: 24531: 24528: 24522: 24519: 24497: 24494: 24490: 24486: 24481: 24477: 24467:In this case, 24456: 24451: 24447: 24442: 24438: 24434: 24431: 24428: 24422: 24416: 24413: 24408: 24405: 24402: 24399: 24394: 24390: 24386: 24383: 24380: 24377: 24354: 24349: 24345: 24340: 24336: 24332: 24329: 24326: 24320: 24317: 24314: 24309: 24305: 24301: 24295: 24292: 24287: 24284: 24281: 24276: 24272: 24268: 24245: 24242: 24238: 24234: 24231: 24226: 24222: 24218: 24215: 24212: 24209: 24206: 24203: 24200: 24197: 24192: 24188: 24184: 24181: 24178: 24175: 24172: 24169: 24166: 24163: 24158: 24154: 24150: 24147: 24144: 24122: 24119: 24115: 24110: 24107: 24104: 24099: 24095: 24091: 24086: 24082: 24078: 24073: 24069: 24064: 24061: 24058: 24053: 24049: 24045: 24042: 24039: 24036: 24031: 24027: 24006: 23984: 23980: 23967: 23964: 23952: 23945: 23941: 23936: 23933: 23930: 23927: 23924: 23918: 23915: 23893: 23889: 23865: 23840: 23836: 23814: 23794: 23788: 23785: 23780: 23777: 23774: 23771: 23768: 23765: 23759: 23756: 23736: 23716: 23713: 23710: 23707: 23691: 23688: 23676: 23670: 23667: 23664: 23661: 23658: 23655: 23652: 23649: 23646: 23643: 23636: 23632: 23628: 23625: 23619: 23616: 23611: 23608: 23604: 23600: 23597: 23594: 23591: 23586: 23582: 23559: 23553: 23550: 23547: 23544: 23542: 23539: 23536: 23535: 23532: 23529: 23527: 23524: 23521: 23518: 23517: 23515: 23510: 23507: 23485: 23479: 23476: 23474: 23471: 23470: 23465: 23461: 23457: 23454: 23452: 23449: 23448: 23446: 23441: 23438: 23413: 23410: 23386: 23381: 23355: 23350: 23325: 23322: 23318: 23295: 23292: 23288: 23263: 23239: 23217: 23214: 23210: 23206: 23203: 23200: 23177: 23174: 23171: 23168: 23165: 23162: 23159: 23156: 23152: 23148: 23145: 23142: 23139: 23134: 23130: 23126: 23121: 23117: 23113: 23108: 23104: 23100: 23095: 23091: 23087: 23084: 23079: 23074: 23047: 23042: 23033: 23028: 23023: 23018: 23013: 23008: 23004: 22996: 22990: 22985: 22982: 22978: 22952: 22947: 22938: 22933: 22928: 22923: 22918: 22913: 22909: 22901: 22895: 22890: 22887: 22883: 22859: 22837: 22813: 22809: 22805: 22800: 22795: 22791: 22787: 22784: 22780: 22776: 22771: 22766: 22762: 22758: 22755: 22751: 22747: 22742: 22738: 22717: 22713: 22709: 22704: 22700: 22696: 22693: 22689: 22685: 22680: 22676: 22654: 22631: 22626: 22622: 22600: 22577: 22572: 22568: 22546: 22525: 22521: 22517: 22512: 22508: 22486: 22463: 22458: 22436: 22433: 22430: 22427: 22424: 22421: 22418: 22415: 22412: 22409: 22405: 22382: 22377: 22374: 22371: 22368: 22365: 22362: 22359: 22353: 22350: 22344: 22343: 22340: 22337: 22334: 22331: 22328: 22325: 22322: 22319: 22316: 22313: 22310: 22307: 22301: 22298: 22292: 22291: 22288: 22285: 22282: 22279: 22276: 22273: 22270: 22267: 22264: 22261: 22258: 22255: 22252: 22246: 22243: 22237: 22236: 22233: 22230: 22227: 22224: 22221: 22218: 22215: 22212: 22209: 22206: 22203: 22197: 22194: 22188: 22187: 22185: 22161: 22141: 22121: 22101: 22081: 22045: 22042: 22024: 22020: 21992: 21988: 21965: 21961: 21949: 21948: 21937: 21927: 21924: 21921: 21918: 21915: 21910: 21906: 21902: 21899: 21896: 21893: 21890: 21887: 21882: 21878: 21874: 21871: 21868: 21865: 21860: 21856: 21852: 21847: 21844: 21841: 21837: 21833: 21830: 21827: 21824: 21821: 21818: 21815: 21812: 21809: 21806: 21801: 21797: 21786: 21768: 21765: 21762: 21759: 21756: 21751: 21747: 21743: 21740: 21737: 21734: 21731: 21728: 21723: 21719: 21715: 21712: 21709: 21706: 21701: 21697: 21693: 21688: 21685: 21682: 21678: 21674: 21671: 21668: 21663: 21659: 21633: 21629: 21600: 21597: 21594: 21574: 21571: 21566: 21562: 21541: 21538: 21533: 21529: 21508: 21505: 21500: 21496: 21473: 21469: 21448: 21443: 21439: 21435: 21432: 21429: 21426: 21423: 21418: 21414: 21410: 21405: 21401: 21397: 21394: 21391: 21385: 21381: 21374: 21370: 21365: 21359: 21353: 21348: 21344: 21320: 21300: 21297: 21284: 21262: 21259: 21237: 21217: 21196: 21176: 21165:geometric mean 21150: 21147: 21142: 21139: 21136: 21133: 21130: 21127: 21122: 21118: 21088: 21084: 21080: 21058: 21055: 21052: 21048: 21025: 21021: 21000: 20978: 20975: 20971: 20950: 20930: 20910: 20890: 20870: 20848: 20845: 20841: 20820: 20815: 20809: 20806: 20804: 20801: 20800: 20797: 20794: 20792: 20789: 20788: 20786: 20781: 20778: 20758: 20735: 20715: 20693: 20690: 20687: 20684: 20681: 20678: 20673: 20669: 20648: 20615: 20611: 20588: 20584: 20571: 20568: 20561: 20560: 20549: 20546: 20543: 20540: 20537: 20534: 20531: 20528: 20525: 20520: 20516: 20512: 20509: 20506: 20503: 20500: 20497: 20494: 20491: 20477: 20476: 20465: 20461: 20457: 20454: 20450: 20447: 20444: 20441: 20436: 20431: 20427: 20423: 20419: 20415: 20412: 20409: 20406: 20403: 20400: 20397: 20392: 20388: 20373: 20372: 20361: 20358: 20353: 20348: 20345: 20342: 20339: 20336: 20333: 20327: 20322: 20318: 20314: 20310: 20306: 20303: 20300: 20297: 20294: 20291: 20284: 20280: 20274: 20270: 20266: 20263: 20240: 20237: 20234: 20231: 20211: 20208: 20205: 20202: 20191: 20190: 20179: 20176: 20173: 20170: 20167: 20164: 20161: 20158: 20155: 20152: 20149: 20146: 20143: 20140: 20137: 20134: 20131: 20128: 20123: 20119: 20115: 20112: 20109: 20106: 20103: 20100: 20097: 20092: 20088: 20064: 20061: 20058: 20055: 20052: 20049: 20046: 20043: 20040: 20037: 20034: 20031: 20028: 20025: 20022: 20019: 20016: 20013: 20010: 20007: 20004: 20001: 19998: 19995: 19992: 19989: 19986: 19963: 19960: 19957: 19954: 19951: 19948: 19945: 19942: 19939: 19936: 19933: 19928: 19924: 19912: 19911: 19900: 19897: 19892: 19888: 19885: 19882: 19879: 19876: 19873: 19870: 19867: 19864: 19859: 19855: 19851: 19848: 19845: 19842: 19839: 19836: 19833: 19830: 19827: 19824: 19819: 19815: 19811: 19808: 19805: 19802: 19799: 19796: 19793: 19790: 19787: 19784: 19779: 19775: 19770: 19762: 19758: 19752: 19748: 19744: 19741: 19738: 19735: 19732: 19729: 19726: 19712: 19711: 19700: 19697: 19694: 19691: 19688: 19685: 19682: 19679: 19676: 19673: 19670: 19667: 19664: 19661: 19658: 19632: 19629: 19626: 19621: 19617: 19613: 19610: 19607: 19604: 19593: 19592: 19579: 19575: 19571: 19567: 19564: 19561: 19556: 19552: 19548: 19545: 19542: 19539: 19536: 19531: 19527: 19523: 19520: 19517: 19514: 19507: 19503: 19497: 19493: 19489: 19486: 19483: 19480: 19477: 19474: 19471: 19468: 19465: 19462: 19459: 19456: 19453: 19450: 19436: 19435: 19424: 19421: 19418: 19415: 19412: 19409: 19406: 19403: 19400: 19397: 19394: 19391: 19388: 19385: 19382: 19379: 19376: 19373: 19370: 19367: 19364: 19361: 19358: 19355: 19352: 19349: 19346: 19343: 19340: 19335: 19331: 19327: 19324: 19321: 19318: 19315: 19312: 19309: 19304: 19300: 19288: 19287: 19276: 19273: 19270: 19267: 19264: 19261: 19258: 19255: 19252: 19249: 19246: 19243: 19240: 19237: 19234: 19231: 19228: 19225: 19222: 19219: 19216: 19213: 19207: 19203: 19199: 19196: 19193: 19190: 19185: 19181: 19177: 19174: 19171: 19168: 19165: 19160: 19156: 19152: 19149: 19146: 19143: 19135: 19131: 19125: 19121: 19117: 19114: 19111: 19108: 19105: 19102: 19099: 19096: 19093: 19090: 19087: 19084: 19081: 19076: 19072: 19068: 19065: 19062: 19059: 19056: 19053: 19050: 19045: 19041: 19029: 19028: 19015: 19011: 19007: 19003: 19000: 18997: 18992: 18988: 18984: 18981: 18978: 18975: 18972: 18967: 18963: 18959: 18956: 18953: 18950: 18943: 18939: 18933: 18929: 18925: 18922: 18919: 18916: 18913: 18910: 18907: 18904: 18901: 18898: 18895: 18892: 18889: 18886: 18883: 18880: 18877: 18874: 18871: 18868: 18865: 18862: 18859: 18856: 18853: 18848: 18844: 18840: 18837: 18834: 18831: 18828: 18825: 18822: 18817: 18813: 18785: 18782: 18779: 18774: 18770: 18758: 18757: 18746: 18743: 18739: 18736: 18733: 18730: 18727: 18724: 18717: 18713: 18707: 18703: 18699: 18696: 18693: 18690: 18687: 18676: 18675: 18664: 18661: 18657: 18654: 18651: 18648: 18645: 18642: 18635: 18631: 18625: 18621: 18617: 18614: 18611: 18608: 18605: 18594: 18593: 18582: 18579: 18575: 18572: 18569: 18566: 18563: 18560: 18553: 18549: 18543: 18539: 18535: 18532: 18529: 18526: 18523: 18500: 18497: 18494: 18491: 18488: 18485: 18482: 18479: 18476: 18473: 18470: 18467: 18464: 18461: 18458: 18455: 18452: 18449: 18446: 18443: 18430: 18427: 18426: 18425: 18414: 18411: 18408: 18405: 18401: 18398: 18395: 18390: 18386: 18380: 18375: 18371: 18365: 18361: 18346: 18345: 18332: 18329: 18326: 18323: 18320: 18317: 18314: 18311: 18308: 18305: 18301: 18292: 18289: 18286: 18282: 18278: 18275: 18272: 18269: 18266: 18263: 18260: 18256: 18251: 18248: 18245: 18242: 18239: 18236: 18231: 18227: 18205: 18204: 18189: 18186: 18183: 18180: 18177: 18174: 18171: 18168: 18165: 18162: 18159: 18156: 18151: 18147: 18143: 18140: 18137: 18134: 18131: 18128: 18123: 18119: 18115: 18112: 18109: 18106: 18103: 18098: 18094: 18090: 18087: 18084: 18080: 18077: 18074: 18071: 18068: 18065: 18063: 18058: 18055: 18050: 18047: 18041: 18040: 18037: 18034: 18031: 18028: 18025: 18022: 18019: 18016: 18013: 18010: 18007: 18004: 17999: 17995: 17991: 17988: 17985: 17982: 17979: 17976: 17973: 17970: 17967: 17964: 17959: 17955: 17951: 17948: 17945: 17941: 17938: 17933: 17930: 17925: 17922: 17919: 17916: 17913: 17910: 17907: 17904: 17901: 17899: 17894: 17891: 17886: 17883: 17877: 17876: 17860: 17857: 17856: 17855: 17840: 17836: 17832: 17826: 17820: 17817: 17812: 17809: 17805: 17801: 17798: 17795: 17792: 17778: 17777: 17762: 17759: 17756: 17753: 17750: 17747: 17744: 17741: 17738: 17735: 17732: 17730: 17725: 17722: 17717: 17714: 17708: 17707: 17704: 17699: 17696: 17691: 17688: 17685: 17682: 17679: 17676: 17673: 17670: 17667: 17664: 17661: 17658: 17655: 17652: 17649: 17646: 17643: 17641: 17636: 17633: 17628: 17625: 17619: 17618: 17603: 17600: 17595: 17594: 17583: 17578: 17574: 17570: 17567: 17564: 17561: 17558: 17544: 17543: 17532: 17529: 17526: 17523: 17520: 17516: 17513: 17508: 17505: 17502: 17499: 17496: 17493: 17490: 17474: 17471: 17462: 17461: 17446: 17442: 17438: 17433: 17430: 17427: 17422: 17418: 17414: 17411: 17397: 17396: 17385: 17381: 17376: 17372: 17369: 17366: 17363: 17360: 17357: 17354: 17349: 17345: 17340: 17336: 17333: 17327: 17324: 17321: 17318: 17315: 17310: 17306: 17301: 17295: 17291: 17288: 17285: 17282: 17278: 17274: 17271: 17268: 17265: 17262: 17259: 17256: 17253: 17250: 17246: 17240: 17237: 17234: 17231: 17227: 17223: 17220: 17217: 17214: 17211: 17208: 17205: 17201: 17198: 17195: 17192: 17189: 17186: 17183: 17180: 17175: 17171: 17159: 17158: 17146: 17142: 17139: 17135: 17131: 17128: 17125: 17121: 17117: 17113: 17109: 17106: 17103: 17100: 17097: 17093: 17089: 17086: 17083: 17080: 17077: 17074: 17071: 17068: 17065: 17061: 17055: 17052: 17049: 17046: 17042: 17038: 17035: 17032: 17029: 17026: 17023: 17020: 17017: 17012: 17008: 16984: 16964: 16953: 16952: 16941: 16938: 16935: 16932: 16929: 16926: 16923: 16920: 16915: 16912: 16909: 16906: 16902: 16878: 16867: 16866: 16851: 16848: 16845: 16842: 16839: 16836: 16833: 16830: 16828: 16823: 16820: 16815: 16812: 16806: 16805: 16802: 16799: 16796: 16793: 16790: 16787: 16784: 16781: 16776: 16773: 16768: 16765: 16762: 16760: 16755: 16752: 16747: 16744: 16738: 16737: 16734: 16729: 16726: 16721: 16718: 16715: 16712: 16709: 16706: 16703: 16700: 16697: 16694: 16691: 16688: 16685: 16682: 16680: 16675: 16672: 16667: 16664: 16658: 16657: 16634: 16631: 16628: 16625: 16622: 16619: 16616: 16603: 16600: 16587: 16575: 16572: 16555: 16551: 16546: 16541: 16537: 16530: 16524: 16520: 16516: 16513: 16510: 16505: 16502: 16499: 16495: 16491: 16488: 16466: 16444: 16441: 16438: 16434: 16411: 16407: 16384: 16380: 16359: 16337: 16333: 16306: 16301: 16297: 16290: 16286: 16282: 16279: 16274: 16270: 16266: 16258: 16253: 16249: 16243: 16238: 16235: 16232: 16228: 16200: 16195: 16191: 16183: 16180: 16175: 16170: 16166: 16143: 16140: 16137: 16132: 16128: 16124: 16121: 16118: 16113: 16109: 16083: 16080: 16075: 16071: 16063: 16060: 16057: 16053: 16047: 16042: 16038: 16030: 16027: 16022: 16019: 16014: 16009: 16005: 16001: 15995: 15992: 15987: 15982: 15978: 15959: 15956: 15951: 15943: 15939: 15927: 15915: 15911: 15907: 15903: 15891: 15880: 15879: 15863: 15860: 15855: 15850: 15845: 15838: 15835: 15832: 15828: 15805: 15804: 15781: 15778: 15775: 15771: 15765: 15758: 15755: 15752: 15748: 15732: 15709: 15706: 15703: 15699: 15693: 15686: 15683: 15680: 15676: 15660: 15639: 15634: 15631: 15624: 15621: 15618: 15614: 15608: 15603: 15598: 15595: 15588: 15585: 15582: 15578: 15557: 15553: 15549: 15545: 15541: 15537: 15533: 15525: 15511: 15507: 15484:social physics 15474: 15470: 15467: 15457: 15454: 15448: 15445: 15431: 15409: 15405: 15382: 15378: 15355: 15351: 15339: 15338: 15323: 15320: 15317: 15314: 15311: 15306: 15302: 15296: 15292: 15288: 15285: 15280: 15276: 15272: 15270: 15267: 15264: 15261: 15258: 15253: 15249: 15246: 15243: 15237: 15234: 15229: 15225: 15219: 15215: 15211: 15208: 15203: 15199: 15195: 15193: 15190: 15185: 15181: 15178: 15175: 15169: 15166: 15161: 15157: 15151: 15147: 15143: 15140: 15135: 15131: 15127: 15125: 15109: 15106: 15101: 15100: 15089: 15086: 15083: 15080: 15077: 15074: 15071: 15068: 15065: 15062: 15059: 15056: 15053: 15050: 15047: 15044: 15041: 15038: 15035: 15032: 15029: 15026: 15021: 15018: 15015: 15012: 15008: 15004: 15001: 14998: 14995: 14992: 14985: 14982: 14979: 14974: 14971: 14968: 14965: 14957: 14952: 14948: 14942: 14939: 14914: 14911: 14908: 14905: 14902: 14899: 14896: 14885: 14884: 14869: 14866: 14863: 14860: 14857: 14854: 14851: 14848: 14843: 14840: 14835: 14832: 14829: 14826: 14823: 14820: 14818: 14813: 14810: 14805: 14802: 14796: 14795: 14792: 14787: 14784: 14779: 14776: 14773: 14770: 14767: 14764: 14761: 14758: 14755: 14752: 14749: 14746: 14744: 14739: 14736: 14731: 14728: 14722: 14721: 14704: 14703: 14692: 14689: 14686: 14683: 14680: 14677: 14674: 14671: 14668: 14665: 14662: 14658: 14653: 14650: 14645: 14642: 14639: 14636: 14633: 14630: 14606: 14603: 14602: 14601: 14600: 14599: 14598: 14597: 14581: 14576: 14571: 14566: 14561: 14556: 14551: 14546: 14541: 14536: 14531: 14507: 14504: 14503: 14502: 14501: 14500: 14485: 14482: 14479: 14476: 14473: 14470: 14467: 14465: 14460: 14457: 14452: 14449: 14443: 14442: 14439: 14436: 14433: 14430: 14427: 14424: 14421: 14418: 14415: 14412: 14409: 14407: 14402: 14399: 14394: 14391: 14385: 14384: 14381: 14378: 14375: 14372: 14369: 14366: 14363: 14358: 14354: 14351: 14348: 14342: 14339: 14337: 14332: 14329: 14324: 14321: 14315: 14314: 14311: 14308: 14305: 14300: 14296: 14293: 14290: 14284: 14281: 14278: 14275: 14272: 14270: 14265: 14262: 14257: 14254: 14248: 14247: 14244: 14241: 14238: 14235: 14232: 14229: 14226: 14223: 14220: 14218: 14213: 14210: 14205: 14202: 14196: 14195: 14185: 14184: 14183: 14168: 14163: 14158: 14153: 14148: 14143: 14138: 14133: 14128: 14105: 14102: 14101: 14100: 14099: 14098: 14083: 14080: 14077: 14074: 14071: 14068: 14065: 14062: 14059: 14056: 14053: 14051: 14046: 14043: 14038: 14035: 14029: 14028: 14025: 14022: 14019: 14016: 14013: 14010: 14007: 14002: 13998: 13995: 13992: 13986: 13983: 13981: 13976: 13973: 13968: 13965: 13959: 13958: 13955: 13952: 13949: 13946: 13943: 13940: 13935: 13931: 13928: 13925: 13919: 13916: 13913: 13910: 13908: 13903: 13900: 13895: 13892: 13886: 13885: 13861: 13860: 13859: 13858: 13857: 13856: 13840: 13835: 13830: 13825: 13820: 13815: 13810: 13786: 13785:The SEIS model 13783: 13779: 13778: 13761: 13757: 13753: 13750: 13747: 13744: 13741: 13738: 13733: 13729: 13725: 13722: 13719: 13717: 13712: 13709: 13702: 13698: 13694: 13688: 13687: 13682: 13678: 13674: 13671: 13668: 13665: 13662: 13659: 13654: 13650: 13646: 13643: 13640: 13637: 13634: 13631: 13629: 13624: 13621: 13614: 13610: 13606: 13600: 13599: 13576: 13573: 13570: 13567: 13556: 13555: 13544: 13541: 13538: 13535: 13531: 13527: 13524: 13521: 13518: 13515: 13512: 13509: 13506: 13503: 13500: 13497: 13494: 13491: 13488: 13485: 13482: 13479: 13476: 13473: 13469: 13463: 13460: 13457: 13454: 13450: 13446: 13443: 13440: 13437: 13434: 13431: 13428: 13425: 13422: 13419: 13414: 13410: 13399: 13388: 13385: 13382: 13379: 13376: 13373: 13370: 13367: 13364: 13361: 13358: 13355: 13352: 13349: 13346: 13342: 13338: 13335: 13332: 13329: 13326: 13323: 13320: 13317: 13314: 13311: 13308: 13305: 13302: 13299: 13296: 13293: 13290: 13287: 13284: 13280: 13274: 13271: 13268: 13265: 13261: 13257: 13254: 13251: 13246: 13242: 13227: 13226: 13214: 13210: 13207: 13204: 13201: 13198: 13195: 13192: 13189: 13186: 13183: 13180: 13177: 13174: 13171: 13168: 13165: 13162: 13159: 13156: 13153: 13150: 13147: 13144: 13141: 13138: 13135: 13130: 13126: 13122: 13119: 13116: 13113: 13110: 13107: 13104: 13101: 13098: 13095: 13092: 13089: 13086: 13083: 13079: 13075: 13071: 13067: 13064: 13061: 13058: 13055: 13052: 13049: 13046: 13043: 13040: 13037: 13034: 13031: 13028: 13025: 13022: 13019: 13016: 13013: 13009: 12990: 12989: 12978: 12972: 12969: 12966: 12962: 12954: 12951: 12948: 12944: 12939: 12934: 12930: 12903: 12883: 12880: 12877: 12874: 12871: 12868: 12865: 12862: 12859: 12856: 12845: 12844: 12829: 12826: 12823: 12820: 12817: 12814: 12811: 12808: 12806: 12801: 12798: 12793: 12790: 12784: 12783: 12780: 12777: 12774: 12771: 12768: 12765: 12762: 12759: 12756: 12753: 12750: 12748: 12743: 12740: 12735: 12732: 12726: 12725: 12722: 12719: 12716: 12713: 12710: 12707: 12704: 12699: 12695: 12692: 12689: 12683: 12680: 12678: 12673: 12670: 12665: 12662: 12656: 12655: 12650: 12646: 12643: 12640: 12634: 12631: 12628: 12625: 12622: 12619: 12616: 12613: 12611: 12606: 12603: 12598: 12595: 12589: 12588: 12565: 12545: 12542: 12522: 12500: 12497: 12493: 12472: 12444: 12443:The SEIR model 12441: 12412: 12409: 12408: 12407: 12406: 12405: 12390: 12387: 12384: 12381: 12378: 12375: 12372: 12370: 12365: 12362: 12357: 12354: 12348: 12347: 12344: 12341: 12338: 12335: 12332: 12329: 12324: 12320: 12317: 12314: 12308: 12305: 12303: 12298: 12295: 12290: 12287: 12281: 12280: 12277: 12274: 12271: 12266: 12262: 12259: 12256: 12250: 12247: 12244: 12241: 12238: 12236: 12231: 12228: 12223: 12220: 12214: 12213: 12210: 12207: 12204: 12201: 12198: 12195: 12192: 12189: 12186: 12184: 12179: 12176: 12171: 12168: 12162: 12161: 12105: 12104:The MSIR model 12102: 12088: 12085: 12082: 12079: 12076: 12073: 12070: 12067: 12064: 12061: 12058: 12055: 12052: 12032: 12029: 12026: 12023: 12012: 12011: 12010: 12009: 11998: 11995: 11992: 11989: 11986: 11983: 11978: 11973: 11969: 11965: 11962: 11959: 11956: 11953: 11937: 11936: 11935: 11934: 11919: 11916: 11913: 11910: 11907: 11904: 11898: 11895: 11890: 11887: 11881: 11879: 11876: 11873: 11870: 11867: 11864: 11861: 11858: 11852: 11849: 11844: 11841: 11835: 11833: 11830: 11827: 11824: 11821: 11818: 11815: 11812: 11806: 11803: 11798: 11795: 11789: 11787: 11784: 11781: 11778: 11775: 11772: 11769: 11766: 11763: 11760: 11757: 11754: 11751: 11748: 11745: 11742: 11739: 11736: 11730: 11727: 11722: 11719: 11713: 11711: 11708: 11705: 11702: 11699: 11696: 11693: 11690: 11687: 11684: 11681: 11678: 11672: 11669: 11664: 11661: 11655: 11653: 11628: 11625: 11622: 11619: 11599: 11596: 11593: 11590: 11586: 11582: 11579: 11576: 11573: 11570: 11567: 11547: 11544: 11541: 11538: 11534: 11530: 11527: 11524: 11521: 11518: 11515: 11495: 11492: 11489: 11486: 11482: 11478: 11475: 11472: 11469: 11466: 11463: 11443: 11440: 11437: 11434: 11431: 11428: 11425: 11422: 11419: 11416: 11413: 11410: 11407: 11404: 11401: 11398: 11378: 11375: 11372: 11369: 11366: 11363: 11343: 11340: 11337: 11334: 11331: 11328: 11325: 11322: 11302: 11299: 11296: 11293: 11290: 11287: 11284: 11281: 11278: 11275: 11272: 11269: 11266: 11263: 11260: 11257: 11254: 11251: 11248: 11237: 11236: 11221: 11218: 11215: 11212: 11209: 11206: 11200: 11197: 11192: 11189: 11183: 11181: 11178: 11175: 11172: 11169: 11166: 11163: 11160: 11154: 11151: 11146: 11143: 11137: 11135: 11132: 11129: 11126: 11123: 11120: 11117: 11114: 11108: 11105: 11100: 11097: 11091: 11089: 11086: 11083: 11080: 11077: 11074: 11071: 11068: 11065: 11062: 11059: 11056: 11053: 11050: 11047: 11044: 11041: 11038: 11035: 11032: 11029: 11023: 11020: 11015: 11012: 11006: 11004: 11001: 10998: 10995: 10992: 10989: 10986: 10983: 10980: 10977: 10974: 10971: 10968: 10965: 10962: 10959: 10953: 10950: 10945: 10942: 10936: 10934: 10903: 10900: 10897: 10894: 10891: 10888: 10885: 10882: 10879: 10863: 10860: 10847: 10844: 10839: 10835: 10814: 10794: 10774: 10771: 10766: 10762: 10757: 10753: 10746: 10741: 10737: 10712: 10708: 10702: 10698: 10673: 10669: 10646: 10642: 10616: 10612: 10574: 10570: 10546: 10543: 10540: 10537: 10534: 10531: 10528: 10525: 10522: 10519: 10516: 10496: 10493: 10490: 10487: 10484: 10481: 10478: 10475: 10472: 10469: 10466: 10463: 10460: 10457: 10454: 10451: 10448: 10445: 10442: 10439: 10436: 10416: 10413: 10410: 10407: 10385: 10381: 10358: 10354: 10333: 10330: 10327: 10324: 10321: 10318: 10315: 10312: 10292: 10289: 10286: 10283: 10279: 10275: 10272: 10269: 10266: 10263: 10260: 10240: 10237: 10234: 10231: 10227: 10223: 10220: 10217: 10214: 10211: 10208: 10188: 10185: 10182: 10179: 10176: 10173: 10170: 10167: 10164: 10161: 10158: 10138: 10135: 10132: 10129: 10126: 10123: 10120: 10100: 10097: 10094: 10091: 10088: 10085: 10082: 10079: 10076: 10073: 10070: 10050: 10047: 10044: 10041: 10038: 10019: 10018: 10003: 10000: 9997: 9994: 9991: 9988: 9985: 9979: 9976: 9971: 9968: 9962: 9960: 9957: 9954: 9951: 9948: 9945: 9942: 9939: 9933: 9930: 9925: 9922: 9916: 9914: 9911: 9908: 9905: 9902: 9899: 9896: 9893: 9888: 9884: 9881: 9878: 9875: 9872: 9869: 9863: 9857: 9854: 9849: 9846: 9840: 9838: 9835: 9832: 9829: 9826: 9823: 9820: 9817: 9812: 9808: 9805: 9802: 9799: 9796: 9793: 9787: 9784: 9778: 9775: 9770: 9767: 9761: 9759: 9740: 9739:The SIRV model 9737: 9724: 9721: 9718: 9715: 9712: 9701: 9700: 9685: 9682: 9679: 9676: 9670: 9667: 9662: 9659: 9653: 9651: 9648: 9645: 9642: 9639: 9633: 9630: 9625: 9622: 9616: 9614: 9611: 9608: 9605: 9602: 9599: 9596: 9593: 9588: 9584: 9581: 9578: 9572: 9566: 9563: 9558: 9555: 9549: 9547: 9544: 9539: 9535: 9532: 9529: 9523: 9520: 9514: 9511: 9506: 9503: 9497: 9495: 9456: 9453: 9450: 9430: 9427: 9424: 9404: 9401: 9398: 9389:and mortality 9378: 9375: 9372: 9352: 9349: 9346: 9343: 9340: 9337: 9334: 9331: 9328: 9325: 9322: 9319: 9316: 9313: 9310: 9307: 9304: 9301: 9298: 9295: 9268: 9265: 9262: 9253:and mortality 9242: 9239: 9236: 9216: 9213: 9210: 9190: 9187: 9184: 9181: 9178: 9175: 9172: 9169: 9166: 9163: 9160: 9157: 9154: 9151: 9148: 9145: 9142: 9139: 9136: 9133: 9117: 9116:The SIRD model 9114: 9110: 9109: 9098: 9095: 9092: 9089: 9086: 9083: 9080: 9077: 9074: 9068: 9065: 9060: 9057: 9031: 9028: 9025: 9022: 9019: 8995: 8992: 8989: 8969: 8966: 8963: 8936: 8933: 8930: 8927: 8924: 8921: 8918: 8915: 8912: 8909: 8906: 8886: 8883: 8878: 8874: 8869: 8863: 8859: 8855: 8852: 8832: 8829: 8826: 8823: 8820: 8800: 8797: 8794: 8790: 8786: 8783: 8780: 8777: 8774: 8769: 8765: 8753: 8752: 8735: 8732: 8729: 8725: 8721: 8718: 8715: 8709: 8705: 8699: 8696: 8693: 8690: 8687: 8662: 8659: 8655: 8651: 8648: 8637: 8636: 8621: 8618: 8614: 8608: 8605: 8600: 8597: 8593: 8589: 8586: 8583: 8580: 8577: 8572: 8569: 8566: 8563: 8559: 8555: 8552: 8549: 8544: 8541: 8536: 8534: 8531: 8528: 8525: 8522: 8519: 8516: 8513: 8508: 8505: 8502: 8499: 8495: 8491: 8488: 8485: 8480: 8477: 8472: 8470: 8447: 8444: 8441: 8438: 8435: 8432: 8429: 8414: 8413: 8399: 8395: 8389: 8386: 8381: 8378: 8375: 8372: 8369: 8366: 8363: 8360: 8354: 8351: 8346: 8343: 8326: 8325: 8313: 8310: 8307: 8304: 8301: 8298: 8295: 8292: 8289: 8286: 8283: 8280: 8277: 8274: 8268: 8265: 8260: 8257: 8251: 8245: 8242: 8237: 8234: 8213: 8212: 8197: 8194: 8191: 8186: 8182: 8179: 8176: 8170: 8167: 8165: 8160: 8157: 8152: 8149: 8143: 8142: 8139: 8136: 8133: 8128: 8124: 8121: 8118: 8112: 8109: 8106: 8104: 8099: 8096: 8091: 8088: 8082: 8081: 8039: 8036: 8034: 8031: 8025: 8022: 8021: 8020: 8009: 8003: 8000: 7997: 7993: 7988: 7982: 7979: 7976: 7971: 7968: 7962: 7959: 7932: 7929: 7926: 7922: 7899: 7893: 7889: 7886: 7879: 7876: 7865: 7864: 7853: 7847: 7844: 7841: 7836: 7833: 7827: 7824: 7801: 7798: 7787: 7786: 7775: 7769: 7766: 7763: 7759: 7754: 7751: 7748: 7742: 7739: 7736: 7733: 7730: 7727: 7724: 7720: 7714: 7709: 7705: 7701: 7698: 7695: 7690: 7686: 7682: 7677: 7673: 7669: 7666: 7663: 7660: 7657: 7632: 7628: 7605: 7601: 7580: 7577: 7572: 7568: 7564: 7559: 7555: 7551: 7548: 7545: 7542: 7539: 7522: 7519: 7505: 7501: 7497: 7477: 7457: 7454: 7451: 7431: 7428: 7425: 7422: 7419: 7416: 7396: 7392: 7388: 7368: 7348: 7328: 7308: 7288: 7268: 7256: 7255: 7244: 7241: 7238: 7232: 7229: 7224: 7221: 7207: 7206: 7195: 7192: 7189: 7186: 7183: 7180: 7177: 7171: 7168: 7163: 7160: 7146: 7145: 7134: 7131: 7128: 7125: 7122: 7116: 7113: 7108: 7105: 7079: 7059: 7039: 7019: 6999: 6996: 6993: 6990: 6987: 6984: 6964: 6961: 6958: 6955: 6952: 6949: 6929: 6926: 6923: 6920: 6917: 6914: 6893: 6873: 6870: 6867: 6864: 6861: 6858: 6855: 6852: 6849: 6846: 6843: 6840: 6837: 6834: 6831: 6828: 6817: 6816: 6800: 6795: 6790: 6785: 6780: 6761: 6760: 6747: 6744: 6741: 6738: 6728: 6716: 6713: 6710: 6707: 6697: 6685: 6682: 6679: 6676: 6653: 6650: 6647: 6644: 6624: 6621: 6618: 6615: 6595: 6592: 6589: 6586: 6558: 6555: 6552: 6532: 6529: 6526: 6506: 6503: 6500: 6480: 6477: 6474: 6471: 6468: 6465: 6462: 6459: 6456: 6453: 6450: 6447: 6444: 6441: 6438: 6435: 6432: 6429: 6426: 6423: 6396: 6393: 6390: 6370: 6367: 6364: 6344: 6341: 6338: 6335: 6332: 6329: 6326: 6323: 6320: 6317: 6314: 6311: 6308: 6305: 6302: 6299: 6296: 6293: 6290: 6287: 6271: 6268: 6252: 6248: 6236: 6235: 6224: 6220: 6215: 6211: 6208: 6203: 6199: 6194: 6188: 6185: 6180: 6176: 6172: 6169: 6164: 6160: 6155: 6149: 6146: 6141: 6136: 6132: 6129: 6126: 6119: 6115: 6105: 6102: 6099: 6096: 6093: 6090: 6087: 6084: 6081: 6078: 6075: 6072: 6069: 6066: 6063: 6060: 6057: 6052: 6049: 6046: 6042: 6038: 6035: 6032: 6029: 6026: 6023: 6020: 6017: 6014: 6011: 6006: 6002: 5991: 5979: 5975: 5972: 5969: 5966: 5961: 5958: 5952: 5948: 5938: 5935: 5932: 5929: 5926: 5923: 5920: 5917: 5914: 5911: 5908: 5905: 5902: 5899: 5896: 5893: 5890: 5885: 5882: 5879: 5875: 5871: 5868: 5865: 5860: 5856: 5841: 5840: 5829: 5823: 5820: 5817: 5813: 5808: 5803: 5799: 5780: 5779: 5768: 5764: 5760: 5757: 5754: 5751: 5746: 5743: 5737: 5733: 5729: 5725: 5722: 5719: 5716: 5713: 5710: 5707: 5704: 5701: 5698: 5695: 5692: 5689: 5686: 5682: 5667: 5666: 5651: 5648: 5645: 5642: 5639: 5636: 5633: 5631: 5626: 5623: 5618: 5615: 5609: 5608: 5605: 5602: 5599: 5596: 5593: 5590: 5585: 5581: 5578: 5575: 5569: 5566: 5564: 5559: 5556: 5551: 5548: 5542: 5541: 5536: 5532: 5529: 5526: 5520: 5517: 5514: 5511: 5508: 5505: 5502: 5500: 5495: 5492: 5487: 5484: 5478: 5477: 5454: 5434: 5422: 5419: 5404: 5400: 5395: 5391: 5388: 5385: 5365: 5362: 5357: 5353: 5330: 5326: 5322: 5317: 5313: 5288: 5285: 5280: 5276: 5255: 5252: 5247: 5243: 5220: 5216: 5193: 5189: 5185: 5182: 5177: 5173: 5150: 5146: 5142: 5139: 5134: 5130: 5106: 5102: 5079: 5075: 5047: 5025: 5021: 4991: 4988: 4983: 4979: 4958: 4955: 4950: 4946: 4925: 4922: 4917: 4913: 4889: 4885: 4861: 4857: 4830: 4826: 4822: 4801: 4797: 4793: 4770: 4766: 4745: 4723: 4702: 4699: 4678: 4658: 4638: 4618: 4615: 4610: 4606: 4585: 4582: 4577: 4573: 4549: 4546: 4542: 4538: 4535: 4532: 4529: 4524: 4520: 4497: 4494: 4490: 4486: 4483: 4480: 4477: 4472: 4468: 4445: 4441: 4430: 4429: 4428: 4427: 4413: 4408: 4404: 4398: 4393: 4389: 4382: 4377: 4373: 4359: 4358: 4354: 4353: 4352: 4351: 4336: 4332: 4326: 4322: 4316: 4311: 4307: 4293: 4292: 4288: 4287: 4286: 4285: 4270: 4266: 4260: 4256: 4250: 4245: 4241: 4227: 4226: 4210: 4207: 4194: 4191: 4188: 4168: 4165: 4162: 4159: 4139: 4136: 4133: 4130: 4110: 4107: 4104: 4101: 4081: 4078: 4075: 4072: 4052: 4049: 4046: 4043: 3986: 3982: 3959: 3955: 3932: 3928: 3904: 3884: 3864: 3844: 3841: 3838: 3834: 3830: 3827: 3807: 3804: 3801: 3798: 3795: 3792: 3772: 3769: 3766: 3763: 3760: 3757: 3737: 3734: 3730: 3726: 3706: 3686: 3660: 3657: 3652: 3649: 3646: 3643: 3640: 3636: 3615: 3595: 3592: 3589: 3586: 3575: 3574: 3557: 3553: 3549: 3545: 3540: 3536: 3532: 3529: 3524: 3519: 3515: 3509: 3506: 3501: 3498: 3496: 3494: 3491: 3488: 3485: 3482: 3481: 3478: 3475: 3472: 3469: 3466: 3463: 3460: 3457: 3454: 3451: 3448: 3445: 3443: 3441: 3438: 3435: 3432: 3429: 3428: 3425: 3422: 3419: 3416: 3413: 3410: 3407: 3404: 3401: 3398: 3395: 3392: 3389: 3387: 3385: 3382: 3379: 3376: 3373: 3372: 3367: 3364: 3361: 3358: 3355: 3351: 3347: 3344: 3341: 3338: 3335: 3332: 3330: 3328: 3325: 3322: 3319: 3316: 3315: 3289: 3286: 3281: 3277: 3256: 3253: 3250: 3247: 3244: 3241: 3238: 3218: 3215: 3212: 3209: 3205: 3199: 3195: 3191: 3188: 3183: 3179: 3175: 3172: 3169: 3166: 3163: 3160: 3155: 3151: 3147: 3144: 3140: 3136: 3131: 3127: 3104: 3100: 3079: 3076: 3073: 3068: 3064: 3060: 3055: 3031: 3027: 3015: 3014: 3003: 3000: 2997: 2994: 2991: 2986: 2982: 2977: 2974: 2971: 2968: 2965: 2962: 2959: 2956: 2953: 2950: 2947: 2944: 2941: 2938: 2935: 2932: 2929: 2926: 2923: 2920: 2917: 2914: 2911: 2908: 2905: 2902: 2899: 2896: 2893: 2888: 2883: 2880: 2877: 2874: 2869: 2864: 2861: 2858: 2855: 2850: 2845: 2831: 2830: 2819: 2814: 2810: 2807: 2801: 2798: 2794: 2788: 2783: 2779: 2775: 2770: 2763: 2759: 2751: 2747: 2743: 2735: 2730: 2726: 2720: 2717: 2712: 2709: 2695: 2694: 2683: 2680: 2677: 2674: 2671: 2668: 2665: 2662: 2659: 2656: 2653: 2650: 2647: 2644: 2641: 2636: 2624: 2613: 2610: 2607: 2602: 2597: 2594: 2591: 2588: 2583: 2578: 2575: 2572: 2569: 2566: 2563: 2558: 2546: 2535: 2532: 2529: 2526: 2523: 2520: 2517: 2514: 2511: 2506: 2481: 2478: 2475: 2472: 2469: 2466: 2463: 2460: 2455: 2440: 2437: 2433: 2432: 2421: 2416: 2413: 2408: 2405: 2402: 2379: 2364: 2363: 2352: 2349: 2346: 2343: 2340: 2325: 2322: 2317: 2316: 2305: 2302: 2299: 2296: 2293: 2290: 2284: 2281: 2276: 2273: 2256: 2255: 2244: 2241: 2238: 2235: 2232: 2229: 2226: 2223: 2218: 2214: 2199: 2198: 2187: 2184: 2181: 2178: 2175: 2172: 2166: 2163: 2158: 2155: 2138: 2137: 2126: 2123: 2120: 2117: 2114: 2111: 2108: 2105: 2100: 2096: 2081: 2080: 2069: 2066: 2063: 2059: 2055: 2052: 2047: 2044: 2037: 2033: 2028: 2024: 2018: 2015: 2010: 2007: 1978: 1975: 1970: 1966: 1954: 1953: 1942: 1939: 1934: 1929: 1925: 1921: 1918: 1915: 1910: 1906: 1902: 1898: 1892: 1888: 1882: 1878: 1874: 1871: 1868: 1862: 1859: 1854: 1850: 1846: 1843: 1838: 1834: 1830: 1827: 1824: 1819: 1815: 1788: 1787: 1774: 1769: 1765: 1761: 1756: 1752: 1748: 1743: 1739: 1735: 1731: 1725: 1721: 1717: 1712: 1708: 1704: 1701: 1698: 1693: 1689: 1665: 1662: 1659: 1656: 1636: 1632: 1628: 1625: 1622: 1619: 1616: 1611: 1607: 1586: 1582: 1578: 1575: 1572: 1569: 1566: 1561: 1557: 1536: 1532: 1528: 1525: 1522: 1519: 1516: 1511: 1507: 1486: 1483: 1480: 1477: 1457: 1454: 1451: 1448: 1437: 1436: 1425: 1420: 1416: 1412: 1409: 1406: 1403: 1400: 1397: 1394: 1391: 1388: 1385: 1382: 1377: 1373: 1369: 1365: 1361: 1358: 1355: 1352: 1349: 1346: 1343: 1340: 1337: 1308: 1303: 1299: 1294: 1288: 1284: 1257: 1254: 1250: 1246: 1241: 1237: 1214: 1211: 1207: 1203: 1198: 1194: 1180:the so-called 1178: 1177: 1166: 1161: 1158: 1153: 1148: 1144: 1117: 1106: 1105: 1094: 1091: 1088: 1080: 1077: 1074: 1071: 1068: 1065: 1062: 1059: 1056: 1053: 1050: 1047: 1044: 1041: 1038: 1024: 1023: 1012: 1009: 1006: 1000: 997: 992: 989: 983: 977: 974: 969: 966: 960: 954: 951: 946: 943: 891: 871: 851: 831: 812: 811: 799: 793: 790: 787: 784: 778: 775: 770: 767: 761: 759: 756: 753: 750: 747: 744: 741: 738: 735: 729: 726: 721: 718: 712: 710: 707: 704: 701: 698: 695: 692: 686: 683: 678: 675: 669: 667: 664: 628: 625: 608: 605: 602: 571: 567: 563: 560: 557: 537: 517: 514: 477: 473: 468: 464: 461: 441: 421: 399: 395: 390: 386: 383: 380: 377: 374: 371: 368: 364: 360: 357: 353: 349: 346: 343: 323: 320: 258: 254: 233: 230: 227: 224: 150: 149: 144:ecovered" or " 130: 120: 101: 98: 46:nfectious, or 15: 9: 6: 4: 3: 2: 30836: 30825: 30822: 30820: 30817: 30815: 30812: 30810: 30807: 30806: 30804: 30789: 30786: 30784: 30781: 30779: 30776: 30774: 30771: 30769: 30766: 30764: 30761: 30760: 30758: 30754: 30748: 30745: 30743: 30740: 30738: 30737:Crime mapping 30735: 30733: 30730: 30728: 30725: 30723: 30720: 30718: 30715: 30714: 30712: 30708: 30702: 30699: 30697: 30694: 30692: 30689: 30688: 30686: 30682: 30676: 30673: 30671: 30668: 30666: 30663: 30661: 30658: 30656: 30653: 30651: 30648: 30646: 30645:Climate model 30643: 30641: 30638: 30636: 30633: 30632: 30630: 30628:Environmental 30626: 30620: 30617: 30615: 30612: 30610: 30607: 30605: 30602: 30600: 30597: 30595: 30592: 30590: 30587: 30586: 30584: 30582: 30578: 30574: 30567: 30562: 30560: 30555: 30553: 30548: 30547: 30544: 30530: 30527: 30526: 30525: 30522: 30520: 30517: 30515: 30512: 30510: 30507: 30505: 30502: 30499: 30497: 30494: 30492: 30489: 30488: 30486: 30482: 30476: 30473: 30471: 30468: 30466: 30463: 30461: 30458: 30455: 30453: 30450: 30448: 30445: 30443: 30440: 30438: 30435: 30433: 30430: 30428: 30425: 30423: 30420: 30418: 30415: 30413: 30410: 30408: 30405: 30401: 30398: 30397: 30395: 30393: 30390: 30388: 30385: 30384: 30382: 30380: 30374: 30363: 30361: 30358: 30356: 30353: 30351: 30348: 30346: 30343: 30341: 30338: 30336: 30335:Sterilization 30333: 30331: 30328: 30326: 30323: 30321: 30318: 30316: 30313: 30309: 30306: 30304: 30303:Surgical mask 30301: 30299: 30296: 30295: 30294: 30291: 30289: 30286: 30284: 30281: 30277: 30274: 30272: 30269: 30266: 30263: 30262: 30261: 30260:Public health 30258: 30256: 30253: 30249: 30246: 30245: 30244: 30241: 30239: 30236: 30232: 30229: 30228: 30227: 30224: 30220: 30217: 30215: 30212: 30210: 30207: 30206: 30205: 30202: 30200: 30197: 30195: 30192: 30190: 30187: 30185: 30182: 30180: 30177: 30176: 30174: 30172: 30166: 30158: 30155: 30153: 30150: 30148: 30145: 30143: 30140: 30138: 30135: 30134: 30133: 30130: 30128: 30125: 30123: 30120: 30118: 30115: 30113: 30112:Phage therapy 30110: 30108: 30105: 30101: 30098: 30097: 30096: 30095:Immunotherapy 30093: 30091: 30088: 30086: 30083: 30081: 30078: 30076: 30073: 30071: 30068: 30066: 30063: 30059: 30056: 30055: 30054: 30053:Antimicrobial 30051: 30047: 30044: 30043: 30042: 30039: 30037: 30034: 30030: 30027: 30026: 30025: 30022: 30021: 30019: 30015: 30012: 30010: 30002: 29995: 29993: 29990: 29988: 29985: 29983: 29980: 29978: 29975: 29973: 29970: 29967: 29965: 29962: 29960: 29957: 29954: 29952: 29949: 29946: 29944:Genitourinary 29943: 29939: 29936: 29935: 29934: 29931: 29927: 29924: 29922: 29919: 29918: 29916: 29915: 29913: 29907: 29897: 29894: 29892: 29889: 29887: 29884: 29882: 29879: 29877: 29874: 29872: 29869: 29867: 29864: 29862: 29859: 29857: 29854: 29852: 29849: 29847: 29844: 29842: 29839: 29837: 29834: 29830: 29827: 29825: 29822: 29821: 29820: 29817: 29815: 29812: 29810: 29807: 29806: 29804: 29802:in population 29798: 29792: 29789: 29787: 29784: 29782: 29779: 29777: 29774: 29772: 29769: 29767: 29766:Herd immunity 29764: 29762: 29759: 29757: 29754: 29752: 29749: 29747: 29744: 29742: 29739: 29737: 29734: 29732: 29729: 29728: 29726: 29724: 29720: 29702: 29699: 29698: 29696: 29692: 29685: 29684: 29682: 29678: 29671: 29670: 29668: 29664: 29658: 29655: 29654: 29652: 29650:Genitourinary 29648: 29642: 29638: 29636: 29633: 29631: 29628: 29626: 29623: 29622: 29620: 29616: 29610: 29607: 29605: 29602: 29600: 29597: 29593: 29592:Contamination 29590: 29589: 29588: 29585: 29584: 29582: 29578: 29570: 29567: 29565: 29562: 29561: 29560:Vector-borne 29559: 29555: 29552: 29550: 29547: 29546: 29544: 29540: 29537: 29534: 29532: 29529: 29527: 29524: 29523: 29521: 29519: 29516: 29515: 29513: 29507: 29501: 29498: 29496: 29493: 29489: 29486: 29485: 29484: 29481: 29479: 29476: 29475: 29473: 29469: 29466: 29462: 29456: 29453: 29452: 29450: 29444: 29438: 29435: 29431: 29428: 29424: 29421: 29418: 29415: 29414: 29413: 29410: 29409: 29407: 29403: 29400: 29398: 29395: 29394: 29392: 29390: 29387: 29386: 29384: 29380: 29376: 29370: 29367: 29365: 29362: 29360: 29357: 29355: 29352: 29351: 29349: 29347: 29346:Cross-species 29343: 29340: 29336: 29329: 29326: 29323: 29320: 29318: 29315: 29313: 29310: 29307: 29306: 29304: 29300: 29297: 29293: 29287: 29286:Window period 29284: 29282: 29279: 29277: 29274: 29272: 29269: 29267: 29264: 29262: 29259: 29257: 29254: 29252: 29251:Latent period 29249: 29247: 29244: 29242: 29239: 29237: 29234: 29232: 29229: 29227: 29224: 29222: 29219: 29217: 29214: 29213: 29211: 29205: 29202: 29200: 29196: 29186: 29183: 29181: 29178: 29176: 29173: 29171: 29168: 29166: 29163: 29161: 29158: 29154: 29151: 29150: 29148: 29146: 29143: 29141: 29138: 29136: 29133: 29131: 29130:Deforestation 29128: 29126: 29123: 29119: 29116: 29114: 29111: 29110: 29108: 29106: 29103: 29100: 29097: 29095: 29092: 29091: 29089: 29087: 29083: 29075:Stress levels 29074: 29071: 29067: 29064: 29063: 29061: 29058: 29055: 29052: 29049: 29046: 29045: 29044: 29041: 29039: 29036: 29034: 29031: 29029: 29026: 29022: 29019: 29017: 29014: 29013: 29012: 29009: 29007: 29004: 29002: 28999: 28997: 28994: 28992: 28989: 28988: 28986: 28984: 28980: 28974: 28971: 28967: 28964: 28962: 28959: 28957: 28954: 28953: 28952: 28949: 28945: 28942: 28940: 28937: 28935: 28932: 28930: 28927: 28926: 28925: 28922: 28920: 28917: 28913: 28910: 28909: 28908: 28907:Pathogenicity 28905: 28901: 28898: 28897: 28896: 28893: 28891: 28888: 28886: 28883: 28882: 28880: 28878: 28874: 28871: 28867: 28862: 28858: 28850: 28845: 28843: 28838: 28836: 28831: 28830: 28827: 28819: 28815: 28814: 28809: 28805: 28803: 28800: 28799: 28789: 28785: 28781: 28775: 28771: 28767: 28760: 28759: 28754: 28749: 28745: 28739: 28735: 28730: 28726: 28720: 28716: 28711: 28707: 28705:0-19-854040-X 28701: 28697: 28693: 28689: 28688: 28675: 28671: 28667: 28663: 28658: 28653: 28648: 28643: 28639: 28635: 28631: 28627: 28623: 28616: 28608: 28604: 28600: 28596: 28592: 28588: 28584: 28580: 28575: 28570: 28566: 28562: 28555: 28547: 28543: 28539: 28535: 28531: 28527: 28523: 28519: 28515: 28511: 28506: 28501: 28497: 28493: 28486: 28478: 28474: 28470: 28466: 28462: 28458: 28454: 28450: 28443: 28435: 28431: 28427: 28423: 28419: 28415: 28411: 28407: 28400: 28384: 28383:Games Journal 28380: 28373: 28365: 28363:0-412-21610-8 28359: 28355: 28351: 28344: 28336: 28334:0-387-98902-1 28330: 28326: 28319: 28317: 28315: 28306: 28302: 28297: 28292: 28288: 28284: 28280: 28276: 28272: 28265: 28257: 28253: 28249: 28245: 28241: 28237: 28234:(8): 815–21. 28233: 28229: 28222: 28213: 28208: 28204: 28200: 28196: 28189: 28181: 28177: 28172: 28167: 28163: 28159: 28155: 28151: 28147: 28143: 28139: 28132: 28124: 28120: 28115: 28110: 28106: 28102: 28098: 28094: 28090: 28083: 28075: 28071: 28067: 28063: 28058: 28053: 28049: 28045: 28042:(4): 365–82. 28041: 28037: 28030: 28017: 28011: 28007: 28003: 27999: 27995: 27988: 27980: 27974: 27970: 27969: 27961: 27954: 27945: 27937: 27933: 27929: 27925: 27921: 27917: 27913: 27909: 27905: 27901: 27897: 27890: 27888: 27879: 27875: 27870: 27865: 27861: 27857: 27853: 27849: 27845: 27841: 27837: 27830: 27828: 27819: 27815: 27811: 27807: 27803: 27799: 27794: 27789: 27785: 27781: 27777: 27773: 27769: 27762: 27754: 27750: 27745: 27740: 27735: 27730: 27726: 27722: 27718: 27711: 27703: 27699: 27695: 27691: 27687: 27683: 27679: 27675: 27671: 27667: 27663: 27656: 27649: 27645: 27640: 27632: 27628: 27623: 27618: 27613: 27608: 27604: 27600: 27595: 27590: 27586: 27582: 27578: 27571: 27569: 27560: 27556: 27552: 27548: 27543: 27538: 27534: 27530: 27526: 27522: 27517: 27512: 27508: 27504: 27500: 27493: 27491: 27481: 27476: 27469: 27461: 27457: 27452: 27447: 27442: 27437: 27433: 27429: 27424: 27419: 27415: 27411: 27407: 27400: 27392: 27388: 27384: 27380: 27376: 27372: 27368: 27364: 27360: 27356: 27349: 27340: 27335: 27331: 27327: 27323: 27316: 27314: 27304: 27299: 27294: 27289: 27285: 27281: 27277: 27270: 27268: 27266: 27264: 27255: 27251: 27246: 27241: 27236: 27231: 27227: 27223: 27219: 27215: 27211: 27204: 27202: 27200: 27192: 27186: 27178: 27174: 27169: 27164: 27160: 27156: 27152: 27148: 27144: 27140: 27136: 27129: 27118: 27112: 27104: 27100: 27095: 27090: 27086: 27082: 27078: 27074: 27071:(2): 191504. 27070: 27066: 27062: 27055: 27047: 27045:3-540-19465-7 27041: 27037: 27033: 27029: 27022: 27015: 27009: 27004: 26999: 26995: 26988: 26980: 26974: 26970: 26969: 26964: 26958: 26950: 26946: 26942: 26938: 26931: 26923: 26919: 26915: 26911: 26906: 26901: 26897: 26893: 26889: 26885: 26881: 26877: 26873: 26866: 26864: 26862: 26860: 26851: 26847: 26842: 26837: 26833: 26829: 26825: 26821: 26817: 26810: 26808: 26806: 26804: 26796: 26795:Section 2.1.3 26792: 26788: 26783: 26778: 26774: 26770: 26766: 26762: 26758: 26751: 26744: 26740: 26736: 26731: 26726: 26722: 26718: 26714: 26710: 26706: 26699: 26691: 26689:3-540-56526-4 26685: 26681: 26674: 26666: 26659: 26651: 26649:0-19-856585-2 26645: 26641: 26634: 26626: 26624:0-85264-231-8 26620: 26616: 26609: 26607: 26598: 26594: 26590: 26586: 26582: 26578: 26574: 26570: 26563: 26555: 26551: 26546: 26541: 26536: 26531: 26527: 26523: 26519: 26512: 26503: 26498: 26494: 26490: 26486: 26479: 26470: 26465: 26458: 26456: 26446: 26441: 26436: 26431: 26427: 26423: 26419: 26415: 26411: 26404: 26402: 26393: 26389: 26384: 26379: 26375: 26371: 26367: 26363: 26359: 26352: 26350: 26348: 26346: 26344: 26328: 26321: 26314: 26312: 26310: 26301: 26297: 26293: 26289: 26285: 26281: 26276: 26271: 26267: 26263: 26256: 26254: 26252: 26243: 26239: 26234: 26229: 26225: 26221: 26217: 26213: 26209: 26205: 26201: 26194: 26186: 26182: 26178: 26174: 26170: 26164: 26160: 26156: 26152: 26148: 26141: 26139: 26129: 26124: 26120: 26116: 26112: 26108: 26104: 26097: 26095: 26093: 26091: 26081: 26076: 26072: 26068: 26064: 26060: 26056: 26049: 26040: 26035: 26031: 26027: 26023: 26019: 26015: 26008: 25999: 25994: 25990: 25986: 25982: 25978: 25974: 25967: 25963: 25953: 25950: 25948: 25945: 25943: 25940: 25938: 25935: 25933: 25930: 25928: 25925: 25923: 25920: 25918: 25915: 25914: 25908: 25906: 25902: 25897: 25895: 25882: 25879: 25876: 25873: 25872: 25871: 25863: 25854: 25840: 25818: 25814: 25810: 25800: 25796: 25792: 25787: 25783: 25772: 25770: 25752: 25748: 25727: 25707: 25704: 25699: 25695: 25672: 25669: 25656: 25653: 25650: 25622: 25619: 25616: 25591: 25587: 25583: 25577: 25574: 25571: 25566: 25562: 25553: 25535: 25531: 25527: 25524: 25521: 25518: 25513: 25509: 25488: 25485: 25480: 25476: 25466: 25450: 25446: 25425: 25417: 25401: 25396: 25388: 25384: 25374: 25370: 25363: 25356: 25348: 25344: 25339: 25335: 25332: 25325: 25321: 25316: 25312: 25303: 25299: 25294: 25288: 25284: 25276: 25272: 25267: 25263: 25260: 25254: 25249: 25244: 25236: 25232: 25222: 25218: 25211: 25203: 25200: 25196: 25172: 25168: 25145: 25141: 25132: 25128: 25112: 25107: 25103: 25097: 25093: 25087: 25083: 25079: 25071: 25067: 25063: 25058: 25054: 25047: 25044: 25041: 25038: 25033: 25029: 25006: 25002: 24989: 24986: 24969: 24965: 24942: 24938: 24929: 24912: 24908: 24899: 24898: 24897: 24889: 24870: 24866: 24859: 24856: 24836: 24833: 24824: 24816: 24812: 24805: 24802: 24797: 24792: 24788: 24782: 24779: 24774: 24765: 24757: 24753: 24746: 24743: 24740: 24731: 24723: 24719: 24712: 24709: 24687: 24683: 24673: 24659: 24656: 24651: 24647: 24624: 24621: 24608: 24605: 24602: 24574: 24571: 24568: 24559: 24543: 24537: 24533: 24529: 24526: 24520: 24517: 24495: 24492: 24488: 24484: 24479: 24475: 24454: 24449: 24440: 24436: 24429: 24426: 24420: 24414: 24411: 24400: 24392: 24388: 24381: 24378: 24375: 24352: 24347: 24338: 24334: 24327: 24324: 24315: 24307: 24303: 24299: 24293: 24290: 24282: 24274: 24270: 24266: 24243: 24240: 24236: 24232: 24224: 24220: 24213: 24210: 24207: 24198: 24190: 24186: 24179: 24176: 24173: 24164: 24156: 24152: 24145: 24142: 24120: 24117: 24113: 24105: 24097: 24093: 24089: 24084: 24080: 24076: 24071: 24067: 24059: 24051: 24047: 24043: 24037: 24029: 24025: 24004: 23982: 23978: 23963: 23950: 23943: 23939: 23931: 23925: 23922: 23916: 23913: 23891: 23887: 23879: 23878:doubling time 23863: 23854: 23838: 23834: 23812: 23792: 23786: 23783: 23775: 23769: 23766: 23763: 23757: 23754: 23734: 23711: 23705: 23697: 23687: 23674: 23665: 23662: 23659: 23650: 23647: 23644: 23634: 23630: 23626: 23623: 23617: 23609: 23606: 23602: 23598: 23592: 23589: 23584: 23580: 23557: 23551: 23548: 23545: 23540: 23537: 23530: 23525: 23522: 23519: 23513: 23508: 23505: 23483: 23477: 23472: 23463: 23459: 23455: 23450: 23444: 23439: 23436: 23428: 23425: 23424: 23411: 23408: 23399: 23384: 23368: 23353: 23323: 23320: 23316: 23293: 23290: 23286: 23277: 23261: 23253: 23237: 23230:. Basically, 23215: 23212: 23208: 23204: 23201: 23198: 23189: 23172: 23169: 23166: 23163: 23160: 23157: 23154: 23150: 23146: 23140: 23132: 23128: 23124: 23119: 23115: 23111: 23106: 23102: 23098: 23093: 23089: 23082: 23077: 23045: 23021: 23006: 23002: 22988: 22983: 22980: 22976: 22950: 22926: 22911: 22907: 22893: 22888: 22885: 22881: 22872: 22870: 22857: 22848: 22835: 22825: 22798: 22793: 22789: 22785: 22769: 22764: 22760: 22756: 22740: 22736: 22702: 22698: 22694: 22678: 22674: 22665: 22652: 22629: 22624: 22620: 22611: 22598: 22575: 22570: 22566: 22557: 22544: 22510: 22506: 22497: 22484: 22461: 22431: 22428: 22425: 22422: 22419: 22416: 22413: 22407: 22375: 22372: 22369: 22366: 22363: 22360: 22357: 22351: 22348: 22335: 22332: 22326: 22323: 22320: 22314: 22311: 22308: 22305: 22299: 22296: 22283: 22280: 22274: 22271: 22268: 22262: 22259: 22256: 22253: 22250: 22244: 22241: 22228: 22225: 22222: 22219: 22216: 22213: 22210: 22207: 22204: 22201: 22195: 22192: 22183: 22173: 22159: 22139: 22119: 22099: 22079: 22071: 22070: 22065: 22064: 22059: 22055: 22051: 22041: 22038: 22022: 22018: 22009: 21990: 21986: 21963: 21959: 21951:To calculate 21935: 21925: 21916: 21908: 21904: 21900: 21897: 21894: 21888: 21880: 21876: 21872: 21866: 21858: 21854: 21839: 21828: 21825: 21819: 21813: 21810: 21807: 21804: 21799: 21795: 21787: 21766: 21757: 21749: 21745: 21741: 21738: 21735: 21729: 21721: 21717: 21713: 21707: 21699: 21695: 21680: 21669: 21666: 21661: 21657: 21649: 21648: 21647: 21631: 21627: 21618: 21614: 21598: 21595: 21592: 21572: 21569: 21564: 21560: 21539: 21536: 21531: 21527: 21506: 21503: 21498: 21494: 21471: 21467: 21441: 21437: 21433: 21430: 21427: 21424: 21421: 21416: 21412: 21408: 21403: 21399: 21392: 21389: 21383: 21372: 21368: 21357: 21351: 21346: 21342: 21318: 21310: 21306: 21296: 21282: 21260: 21257: 21235: 21215: 21194: 21174: 21166: 21148: 21145: 21140: 21134: 21128: 21125: 21120: 21116: 21107: 21102: 21086: 21082: 21078: 21056: 21053: 21050: 21046: 21023: 21019: 20998: 20976: 20973: 20969: 20948: 20928: 20908: 20888: 20868: 20846: 20843: 20839: 20818: 20813: 20807: 20802: 20795: 20790: 20784: 20779: 20776: 20756: 20749: 20733: 20713: 20704: 20691: 20685: 20679: 20676: 20671: 20667: 20646: 20639: 20635: 20631: 20613: 20609: 20586: 20582: 20567: 20564: 20547: 20541: 20538: 20535: 20532: 20526: 20518: 20514: 20507: 20501: 20495: 20492: 20489: 20482: 20481: 20480: 20463: 20459: 20455: 20452: 20445: 20439: 20434: 20429: 20425: 20421: 20417: 20413: 20410: 20407: 20404: 20398: 20390: 20386: 20378: 20377: 20376: 20359: 20356: 20351: 20346: 20343: 20337: 20331: 20325: 20320: 20316: 20312: 20308: 20304: 20301: 20295: 20289: 20282: 20278: 20272: 20268: 20264: 20261: 20254: 20253: 20252: 20235: 20229: 20206: 20200: 20174: 20171: 20168: 20162: 20156: 20150: 20147: 20144: 20138: 20135: 20132: 20126: 20121: 20113: 20107: 20104: 20101: 20095: 20090: 20078: 20077: 20076: 20075:one obtains: 20059: 20056: 20053: 20047: 20044: 20038: 20035: 20032: 20026: 20023: 20017: 20014: 20011: 20005: 20002: 19996: 19993: 19990: 19984: 19975: 19961: 19958: 19955: 19952: 19949: 19946: 19943: 19940: 19934: 19926: 19922: 19898: 19895: 19890: 19883: 19880: 19877: 19871: 19865: 19857: 19853: 19849: 19843: 19840: 19837: 19831: 19825: 19817: 19813: 19809: 19803: 19800: 19797: 19791: 19785: 19777: 19773: 19768: 19760: 19756: 19750: 19746: 19742: 19736: 19733: 19730: 19724: 19717: 19716: 19715: 19698: 19695: 19689: 19686: 19683: 19677: 19674: 19668: 19665: 19662: 19656: 19649: 19648: 19647: 19644: 19627: 19624: 19619: 19615: 19611: 19608: 19602: 19577: 19573: 19569: 19562: 19559: 19554: 19550: 19543: 19537: 19534: 19529: 19525: 19521: 19518: 19512: 19505: 19501: 19495: 19491: 19487: 19478: 19475: 19472: 19466: 19463: 19460: 19457: 19454: 19448: 19441: 19440: 19439: 19419: 19416: 19413: 19407: 19401: 19395: 19392: 19386: 19383: 19380: 19374: 19368: 19362: 19359: 19356: 19350: 19347: 19344: 19338: 19333: 19325: 19319: 19316: 19313: 19307: 19302: 19290: 19289: 19271: 19268: 19265: 19259: 19253: 19247: 19244: 19238: 19235: 19232: 19226: 19220: 19214: 19211: 19205: 19201: 19197: 19191: 19188: 19183: 19179: 19172: 19166: 19163: 19158: 19154: 19150: 19147: 19141: 19133: 19129: 19123: 19119: 19112: 19109: 19106: 19100: 19097: 19091: 19088: 19085: 19079: 19074: 19066: 19060: 19057: 19054: 19048: 19043: 19031: 19030: 19013: 19009: 19005: 18998: 18995: 18990: 18986: 18979: 18973: 18970: 18965: 18961: 18957: 18954: 18948: 18941: 18937: 18931: 18927: 18920: 18917: 18914: 18908: 18905: 18899: 18896: 18893: 18887: 18881: 18875: 18872: 18869: 18863: 18860: 18857: 18851: 18846: 18838: 18832: 18829: 18826: 18820: 18815: 18803: 18802: 18801: 18799: 18780: 18777: 18772: 18768: 18744: 18741: 18734: 18731: 18728: 18722: 18715: 18711: 18705: 18701: 18697: 18691: 18685: 18678: 18677: 18662: 18659: 18652: 18649: 18646: 18640: 18633: 18629: 18623: 18619: 18615: 18609: 18603: 18596: 18595: 18580: 18577: 18570: 18567: 18564: 18558: 18551: 18547: 18541: 18537: 18533: 18527: 18521: 18514: 18513: 18512: 18495: 18492: 18489: 18483: 18480: 18474: 18471: 18468: 18462: 18459: 18453: 18450: 18447: 18441: 18412: 18409: 18406: 18403: 18396: 18388: 18384: 18378: 18373: 18369: 18363: 18359: 18351: 18350: 18349: 18327: 18324: 18321: 18315: 18312: 18309: 18306: 18303: 18299: 18290: 18287: 18284: 18280: 18273: 18270: 18267: 18261: 18258: 18254: 18249: 18246: 18243: 18237: 18229: 18225: 18217: 18216: 18215: 18211: 18187: 18184: 18181: 18178: 18175: 18172: 18169: 18166: 18163: 18157: 18149: 18145: 18141: 18135: 18132: 18129: 18121: 18117: 18113: 18107: 18104: 18096: 18092: 18088: 18082: 18078: 18075: 18072: 18069: 18066: 18064: 18056: 18053: 18048: 18045: 18035: 18032: 18029: 18026: 18023: 18020: 18017: 18014: 18011: 18005: 17997: 17993: 17989: 17983: 17977: 17974: 17971: 17965: 17957: 17953: 17949: 17943: 17939: 17936: 17931: 17928: 17923: 17920: 17917: 17914: 17911: 17908: 17905: 17902: 17900: 17892: 17889: 17884: 17881: 17867: 17866: 17865: 17838: 17834: 17830: 17824: 17818: 17815: 17810: 17807: 17803: 17799: 17796: 17793: 17790: 17783: 17782: 17781: 17760: 17757: 17754: 17751: 17748: 17745: 17742: 17739: 17736: 17733: 17731: 17723: 17720: 17715: 17712: 17702: 17697: 17694: 17689: 17686: 17683: 17680: 17677: 17674: 17671: 17668: 17662: 17659: 17656: 17650: 17647: 17644: 17642: 17634: 17631: 17626: 17623: 17609: 17608: 17607: 17599: 17581: 17576: 17572: 17568: 17562: 17556: 17549: 17548: 17547: 17530: 17527: 17521: 17514: 17511: 17506: 17500: 17494: 17491: 17488: 17481: 17480: 17479: 17470: 17468: 17444: 17440: 17436: 17431: 17428: 17425: 17420: 17416: 17412: 17409: 17402: 17401: 17400: 17383: 17379: 17374: 17370: 17367: 17361: 17358: 17355: 17347: 17343: 17338: 17334: 17331: 17322: 17319: 17316: 17308: 17304: 17299: 17293: 17289: 17286: 17283: 17280: 17276: 17269: 17263: 17260: 17254: 17248: 17244: 17235: 17229: 17218: 17215: 17209: 17203: 17199: 17196: 17193: 17187: 17184: 17181: 17173: 17169: 17161: 17160: 17144: 17140: 17137: 17133: 17129: 17126: 17123: 17119: 17115: 17111: 17107: 17104: 17101: 17098: 17095: 17091: 17084: 17078: 17075: 17069: 17063: 17059: 17050: 17044: 17033: 17030: 17024: 17021: 17018: 17010: 17006: 16998: 16997: 16996: 16982: 16962: 16939: 16936: 16933: 16930: 16924: 16918: 16910: 16904: 16892: 16891: 16890: 16876: 16849: 16846: 16843: 16840: 16837: 16834: 16831: 16829: 16821: 16818: 16813: 16810: 16800: 16794: 16791: 16788: 16782: 16779: 16774: 16771: 16766: 16763: 16761: 16753: 16750: 16745: 16742: 16732: 16727: 16724: 16719: 16716: 16713: 16710: 16707: 16701: 16698: 16695: 16689: 16686: 16683: 16681: 16673: 16670: 16665: 16662: 16648: 16647: 16646: 16629: 16626: 16623: 16617: 16614: 16599: 16585: 16571: 16568: 16553: 16549: 16544: 16539: 16535: 16528: 16522: 16518: 16514: 16511: 16508: 16503: 16500: 16497: 16493: 16489: 16486: 16478: 16464: 16442: 16439: 16436: 16432: 16409: 16405: 16382: 16378: 16370:. Parameters 16357: 16335: 16331: 16322: 16304: 16299: 16295: 16288: 16280: 16277: 16272: 16268: 16256: 16251: 16247: 16241: 16236: 16233: 16230: 16226: 16217: 16214: 16198: 16193: 16189: 16181: 16178: 16173: 16168: 16164: 16155: 16138: 16135: 16130: 16126: 16119: 16116: 16111: 16107: 16098: 16095: 16081: 16078: 16073: 16069: 16061: 16058: 16055: 16051: 16045: 16040: 16036: 16028: 16025: 16020: 16017: 16012: 16007: 16003: 15999: 15993: 15990: 15985: 15980: 15976: 15964: 15955: 15947: 15925: 15889: 15843: 15813: 15812: 15811: 15810: 15802: 15733: 15730: 15661: 15658: 15632: 15629: 15596: 15593: 15563: 15562: 15561: 15509: 15505: 15495: 15491: 15489: 15488:social stress 15485: 15481: 15466: 15464: 15453: 15444: 15429: 15407: 15403: 15380: 15376: 15353: 15349: 15321: 15318: 15315: 15312: 15309: 15304: 15294: 15290: 15286: 15283: 15278: 15265: 15262: 15259: 15256: 15251: 15247: 15244: 15241: 15235: 15232: 15227: 15217: 15213: 15209: 15206: 15201: 15188: 15183: 15179: 15176: 15173: 15167: 15164: 15159: 15149: 15145: 15141: 15138: 15133: 15116: 15115: 15114: 15105: 15087: 15081: 15078: 15075: 15069: 15066: 15063: 15060: 15057: 15048: 15042: 15039: 15033: 15027: 15016: 15010: 14999: 14996: 14993: 14990: 14983: 14980: 14977: 14969: 14963: 14955: 14950: 14946: 14940: 14937: 14928: 14927: 14926: 14912: 14909: 14906: 14903: 14900: 14897: 14894: 14867: 14861: 14858: 14855: 14849: 14846: 14841: 14838: 14830: 14824: 14821: 14819: 14811: 14808: 14803: 14800: 14790: 14785: 14782: 14774: 14768: 14765: 14762: 14759: 14756: 14753: 14750: 14747: 14745: 14737: 14734: 14729: 14726: 14712: 14711: 14710: 14707: 14687: 14681: 14678: 14672: 14669: 14666: 14660: 14656: 14651: 14648: 14640: 14634: 14631: 14628: 14621: 14620: 14619: 14616: 14611: 14517: 14516: 14515: 14514: 14513: 14512: 14511: 14483: 14480: 14477: 14474: 14471: 14468: 14466: 14458: 14455: 14450: 14447: 14437: 14431: 14428: 14425: 14419: 14416: 14413: 14410: 14408: 14400: 14397: 14392: 14389: 14379: 14373: 14370: 14367: 14361: 14356: 14352: 14349: 14346: 14340: 14338: 14330: 14327: 14322: 14319: 14309: 14306: 14303: 14298: 14294: 14291: 14288: 14282: 14279: 14276: 14273: 14271: 14263: 14260: 14255: 14252: 14242: 14239: 14236: 14233: 14230: 14227: 14221: 14219: 14211: 14208: 14203: 14200: 14186: 14115: 14114: 14113: 14112: 14111: 14110: 14109: 14081: 14075: 14072: 14069: 14063: 14060: 14057: 14054: 14052: 14044: 14041: 14036: 14033: 14023: 14017: 14014: 14011: 14005: 14000: 13996: 13993: 13990: 13984: 13982: 13974: 13971: 13966: 13963: 13953: 13950: 13947: 13944: 13941: 13938: 13933: 13929: 13926: 13923: 13917: 13911: 13909: 13901: 13898: 13893: 13890: 13876: 13875: 13874: 13873: 13872: 13870: 13866: 13796: 13795: 13794: 13793: 13792: 13791: 13790: 13782: 13759: 13755: 13748: 13745: 13742: 13736: 13731: 13727: 13723: 13720: 13718: 13710: 13707: 13700: 13696: 13692: 13680: 13676: 13669: 13666: 13663: 13657: 13652: 13648: 13641: 13635: 13632: 13630: 13622: 13619: 13612: 13608: 13604: 13590: 13589: 13588: 13571: 13565: 13542: 13539: 13536: 13533: 13529: 13522: 13516: 13513: 13507: 13501: 13498: 13492: 13486: 13483: 13477: 13471: 13467: 13458: 13452: 13441: 13438: 13432: 13426: 13423: 13420: 13417: 13412: 13408: 13400: 13386: 13380: 13377: 13374: 13371: 13368: 13365: 13362: 13356: 13353: 13350: 13347: 13344: 13340: 13333: 13327: 13324: 13318: 13312: 13309: 13303: 13297: 13294: 13288: 13282: 13278: 13269: 13263: 13252: 13249: 13244: 13240: 13232: 13231: 13230: 13212: 13208: 13205: 13202: 13199: 13196: 13193: 13190: 13187: 13184: 13181: 13178: 13175: 13172: 13169: 13166: 13163: 13160: 13157: 13154: 13151: 13148: 13145: 13142: 13139: 13136: 13133: 13128: 13120: 13117: 13114: 13108: 13102: 13099: 13096: 13093: 13090: 13087: 13084: 13077: 13073: 13069: 13062: 13056: 13053: 13047: 13041: 13038: 13032: 13026: 13023: 13017: 13011: 13007: 12999: 12998: 12997: 12995: 12976: 12970: 12967: 12964: 12960: 12952: 12949: 12946: 12942: 12937: 12932: 12928: 12920: 12919: 12918: 12915: 12901: 12881: 12878: 12875: 12872: 12869: 12866: 12863: 12860: 12857: 12854: 12827: 12824: 12821: 12818: 12815: 12812: 12809: 12807: 12799: 12796: 12791: 12788: 12778: 12772: 12769: 12766: 12760: 12757: 12754: 12751: 12749: 12741: 12738: 12733: 12730: 12720: 12714: 12711: 12708: 12702: 12697: 12693: 12690: 12687: 12681: 12679: 12671: 12668: 12663: 12660: 12648: 12644: 12641: 12638: 12632: 12629: 12626: 12623: 12620: 12617: 12614: 12612: 12604: 12601: 12596: 12593: 12579: 12578: 12577: 12563: 12543: 12540: 12498: 12495: 12491: 12470: 12457: 12453: 12451: 12440: 12438: 12433: 12431: 12430:typhoid fever 12427: 12422: 12418: 12411:Carrier state 12388: 12385: 12382: 12379: 12376: 12373: 12371: 12363: 12360: 12355: 12352: 12342: 12339: 12336: 12333: 12330: 12327: 12322: 12318: 12315: 12312: 12306: 12304: 12296: 12293: 12288: 12285: 12275: 12272: 12269: 12264: 12260: 12257: 12254: 12248: 12245: 12242: 12239: 12237: 12229: 12226: 12221: 12218: 12208: 12205: 12202: 12199: 12196: 12193: 12187: 12185: 12177: 12174: 12169: 12166: 12152: 12151: 12150: 12149: 12148: 12146: 12142: 12138: 12129: 12125: 12123: 12119: 12115: 12111: 12101: 12083: 12077: 12071: 12065: 12062: 12056: 12050: 12027: 12021: 11996: 11993: 11987: 11981: 11976: 11971: 11967: 11963: 11957: 11951: 11944: 11943: 11942: 11941: 11940: 11917: 11911: 11905: 11902: 11896: 11893: 11888: 11885: 11874: 11871: 11865: 11859: 11856: 11850: 11847: 11842: 11839: 11828: 11825: 11819: 11813: 11810: 11804: 11801: 11796: 11793: 11782: 11779: 11770: 11764: 11761: 11755: 11749: 11743: 11740: 11737: 11734: 11728: 11725: 11720: 11717: 11706: 11703: 11697: 11691: 11688: 11685: 11682: 11679: 11676: 11670: 11667: 11662: 11659: 11644: 11643: 11642: 11641: 11640: 11623: 11617: 11594: 11588: 11584: 11577: 11571: 11568: 11565: 11542: 11536: 11532: 11525: 11519: 11516: 11513: 11490: 11484: 11480: 11473: 11467: 11464: 11461: 11441: 11438: 11432: 11426: 11423: 11417: 11411: 11408: 11402: 11396: 11376: 11373: 11367: 11361: 11341: 11338: 11335: 11332: 11326: 11320: 11297: 11291: 11288: 11282: 11276: 11273: 11267: 11261: 11258: 11252: 11246: 11219: 11213: 11207: 11204: 11198: 11195: 11190: 11187: 11176: 11173: 11167: 11161: 11158: 11152: 11149: 11144: 11141: 11130: 11127: 11121: 11115: 11112: 11106: 11103: 11098: 11095: 11084: 11081: 11075: 11069: 11066: 11063: 11057: 11051: 11048: 11045: 11042: 11036: 11030: 11027: 11021: 11018: 11013: 11010: 10999: 10996: 10990: 10984: 10981: 10978: 10975: 10969: 10963: 10960: 10957: 10951: 10948: 10943: 10940: 10925: 10924: 10919: 10915: 10901: 10898: 10895: 10892: 10889: 10886: 10883: 10880: 10877: 10869: 10859: 10845: 10842: 10837: 10833: 10812: 10792: 10772: 10769: 10764: 10760: 10755: 10751: 10744: 10739: 10735: 10710: 10706: 10700: 10696: 10671: 10667: 10644: 10640: 10632: 10614: 10610: 10600: 10595: 10591: 10572: 10568: 10558: 10544: 10541: 10535: 10529: 10526: 10520: 10514: 10494: 10491: 10485: 10479: 10476: 10470: 10464: 10461: 10455: 10449: 10446: 10440: 10434: 10411: 10405: 10379: 10356: 10352: 10331: 10328: 10325: 10322: 10319: 10316: 10313: 10310: 10287: 10281: 10277: 10270: 10264: 10261: 10258: 10235: 10229: 10225: 10218: 10212: 10209: 10206: 10186: 10183: 10177: 10171: 10168: 10162: 10156: 10136: 10133: 10130: 10124: 10118: 10098: 10092: 10089: 10086: 10080: 10074: 10068: 10048: 10045: 10042: 10039: 10036: 10023: 10001: 9998: 9992: 9986: 9983: 9977: 9974: 9969: 9966: 9955: 9952: 9946: 9940: 9937: 9931: 9928: 9923: 9920: 9909: 9906: 9900: 9894: 9891: 9886: 9882: 9879: 9873: 9867: 9861: 9855: 9852: 9847: 9844: 9833: 9830: 9824: 9818: 9815: 9810: 9806: 9803: 9797: 9791: 9785: 9782: 9776: 9773: 9768: 9765: 9750: 9749: 9748: 9746: 9736: 9722: 9719: 9716: 9713: 9710: 9683: 9680: 9677: 9674: 9668: 9665: 9660: 9657: 9646: 9643: 9640: 9637: 9631: 9628: 9623: 9620: 9609: 9606: 9603: 9600: 9597: 9594: 9591: 9586: 9582: 9579: 9576: 9570: 9564: 9561: 9556: 9553: 9542: 9537: 9533: 9530: 9527: 9521: 9518: 9512: 9509: 9504: 9501: 9486: 9485: 9484: 9482: 9478: 9474: 9454: 9451: 9448: 9428: 9425: 9422: 9402: 9399: 9396: 9376: 9373: 9370: 9350: 9347: 9341: 9335: 9332: 9329: 9326: 9320: 9314: 9311: 9308: 9305: 9299: 9293: 9284: 9266: 9263: 9260: 9240: 9237: 9234: 9214: 9211: 9208: 9188: 9185: 9179: 9173: 9170: 9167: 9164: 9158: 9152: 9149: 9146: 9143: 9137: 9131: 9122: 9113: 9096: 9090: 9087: 9084: 9078: 9075: 9072: 9066: 9063: 9058: 9055: 9045: 9044: 9043: 9029: 9026: 9023: 9020: 9017: 9009: 8993: 8990: 8987: 8967: 8964: 8961: 8952: 8948: 8931: 8925: 8922: 8919: 8916: 8910: 8904: 8884: 8881: 8876: 8872: 8867: 8857: 8853: 8850: 8830: 8827: 8824: 8821: 8818: 8798: 8792: 8788: 8784: 8781: 8778: 8772: 8763: 8733: 8730: 8727: 8723: 8719: 8716: 8713: 8703: 8697: 8691: 8685: 8678: 8677: 8676: 8660: 8657: 8653: 8649: 8646: 8619: 8616: 8612: 8606: 8603: 8598: 8595: 8591: 8587: 8581: 8575: 8567: 8561: 8550: 8547: 8542: 8539: 8529: 8526: 8523: 8517: 8511: 8503: 8497: 8486: 8483: 8478: 8475: 8461: 8460: 8459: 8445: 8442: 8436: 8430: 8419: 8397: 8393: 8387: 8384: 8379: 8376: 8370: 8367: 8364: 8358: 8352: 8349: 8344: 8341: 8331: 8330: 8329: 8311: 8308: 8302: 8296: 8293: 8287: 8281: 8275: 8272: 8266: 8263: 8258: 8255: 8249: 8243: 8240: 8235: 8232: 8222: 8221: 8220: 8218: 8195: 8192: 8189: 8184: 8180: 8177: 8174: 8168: 8166: 8158: 8155: 8150: 8147: 8137: 8134: 8131: 8126: 8122: 8119: 8116: 8110: 8107: 8105: 8097: 8094: 8089: 8086: 8072: 8071: 8070: 8063: 8059: 8057: 8053: 8044: 8038:The SIS model 8030: 8007: 8001: 7998: 7995: 7991: 7986: 7980: 7977: 7974: 7969: 7966: 7960: 7957: 7950: 7949: 7948: 7930: 7927: 7924: 7920: 7897: 7891: 7887: 7884: 7877: 7874: 7851: 7845: 7842: 7839: 7834: 7831: 7825: 7822: 7815: 7814: 7813: 7799: 7796: 7773: 7767: 7764: 7761: 7757: 7752: 7749: 7746: 7740: 7734: 7731: 7728: 7722: 7718: 7707: 7703: 7699: 7688: 7684: 7680: 7675: 7671: 7658: 7648: 7647: 7646: 7630: 7626: 7603: 7599: 7570: 7566: 7562: 7557: 7553: 7540: 7527: 7518: 7503: 7499: 7495: 7475: 7455: 7452: 7449: 7429: 7426: 7423: 7420: 7417: 7414: 7394: 7390: 7386: 7366: 7346: 7326: 7306: 7286: 7266: 7242: 7239: 7236: 7230: 7227: 7222: 7219: 7209: 7208: 7193: 7190: 7187: 7184: 7181: 7178: 7175: 7169: 7166: 7161: 7158: 7148: 7147: 7132: 7129: 7126: 7123: 7120: 7114: 7111: 7106: 7103: 7093: 7092: 7091: 7077: 7057: 7037: 7017: 6994: 6991: 6988: 6982: 6959: 6956: 6953: 6947: 6924: 6921: 6918: 6912: 6891: 6868: 6862: 6859: 6853: 6847: 6844: 6838: 6832: 6829: 6826: 6766: 6765: 6764: 6742: 6736: 6729: 6711: 6705: 6698: 6680: 6674: 6667: 6666: 6665: 6648: 6642: 6619: 6613: 6590: 6584: 6572: 6556: 6553: 6550: 6530: 6527: 6524: 6504: 6501: 6498: 6478: 6475: 6469: 6463: 6460: 6457: 6454: 6448: 6442: 6439: 6436: 6433: 6427: 6421: 6412: 6394: 6391: 6388: 6368: 6365: 6362: 6342: 6339: 6333: 6327: 6324: 6321: 6318: 6312: 6306: 6303: 6300: 6297: 6291: 6285: 6276: 6270:The SIR model 6267: 6250: 6246: 6222: 6218: 6213: 6209: 6206: 6201: 6197: 6192: 6186: 6183: 6178: 6174: 6170: 6167: 6162: 6158: 6153: 6147: 6144: 6139: 6134: 6130: 6127: 6124: 6117: 6113: 6103: 6094: 6088: 6085: 6079: 6073: 6070: 6064: 6058: 6044: 6033: 6030: 6024: 6018: 6015: 6012: 6009: 6004: 6000: 5992: 5977: 5973: 5970: 5967: 5964: 5959: 5950: 5946: 5936: 5927: 5921: 5918: 5912: 5906: 5903: 5897: 5891: 5877: 5866: 5863: 5858: 5854: 5846: 5845: 5844: 5827: 5821: 5818: 5815: 5811: 5806: 5801: 5797: 5789: 5788: 5787: 5785: 5766: 5762: 5758: 5755: 5752: 5749: 5744: 5735: 5731: 5727: 5720: 5714: 5711: 5705: 5699: 5696: 5690: 5684: 5680: 5672: 5671: 5670: 5649: 5646: 5643: 5640: 5637: 5634: 5632: 5624: 5621: 5616: 5613: 5603: 5600: 5597: 5594: 5591: 5588: 5583: 5579: 5576: 5573: 5567: 5565: 5557: 5554: 5549: 5546: 5534: 5530: 5527: 5524: 5518: 5515: 5512: 5509: 5503: 5501: 5493: 5490: 5485: 5482: 5468: 5467: 5466: 5432: 5418: 5402: 5398: 5393: 5389: 5386: 5383: 5363: 5360: 5355: 5351: 5328: 5324: 5320: 5315: 5311: 5302: 5301:herd immunity 5286: 5283: 5278: 5274: 5253: 5250: 5245: 5241: 5218: 5214: 5191: 5187: 5183: 5180: 5175: 5171: 5148: 5144: 5140: 5137: 5132: 5128: 5104: 5100: 5077: 5073: 5063: 5061: 5045: 5023: 5019: 5009: 5003: 4989: 4986: 4981: 4977: 4956: 4953: 4948: 4944: 4923: 4920: 4915: 4911: 4887: 4883: 4859: 4855: 4846: 4844: 4828: 4824: 4820: 4799: 4795: 4791: 4768: 4764: 4743: 4735: 4721: 4700: 4697: 4676: 4656: 4636: 4616: 4613: 4608: 4604: 4583: 4580: 4575: 4571: 4547: 4544: 4540: 4536: 4533: 4530: 4527: 4522: 4518: 4495: 4492: 4488: 4484: 4481: 4478: 4475: 4470: 4466: 4443: 4439: 4411: 4406: 4402: 4396: 4391: 4387: 4380: 4375: 4371: 4363: 4362: 4361: 4360: 4356: 4355: 4334: 4330: 4324: 4320: 4314: 4309: 4305: 4297: 4296: 4295: 4294: 4290: 4289: 4268: 4264: 4258: 4254: 4248: 4243: 4239: 4231: 4230: 4229: 4228: 4224: 4220: 4219: 4218: 4216: 4206: 4192: 4189: 4186: 4163: 4157: 4134: 4128: 4105: 4099: 4076: 4070: 4047: 4041: 4033: 4028: 4026: 4022: 4018: 4010: 4006: 4002: 3980: 3953: 3926: 3916: 3902: 3882: 3862: 3842: 3839: 3836: 3832: 3828: 3825: 3805: 3802: 3799: 3796: 3793: 3790: 3770: 3767: 3764: 3761: 3758: 3755: 3735: 3732: 3728: 3724: 3704: 3684: 3675: 3672: 3658: 3655: 3647: 3641: 3638: 3634: 3613: 3590: 3584: 3555: 3551: 3547: 3538: 3534: 3527: 3522: 3517: 3513: 3507: 3504: 3499: 3497: 3489: 3483: 3473: 3467: 3464: 3461: 3455: 3449: 3446: 3444: 3436: 3430: 3420: 3414: 3411: 3405: 3399: 3396: 3393: 3390: 3388: 3380: 3374: 3362: 3356: 3353: 3349: 3342: 3336: 3333: 3331: 3323: 3317: 3306: 3305: 3304: 3301: 3287: 3284: 3275: 3251: 3248: 3242: 3236: 3213: 3207: 3203: 3193: 3189: 3181: 3177: 3167: 3161: 3158: 3149: 3142: 3138: 3134: 3129: 3125: 3098: 3077: 3074: 3066: 3062: 3029: 3025: 3001: 2998: 2995: 2992: 2989: 2984: 2980: 2975: 2966: 2960: 2957: 2951: 2945: 2942: 2936: 2930: 2927: 2924: 2921: 2915: 2909: 2903: 2894: 2881: 2875: 2862: 2856: 2836: 2835: 2834: 2817: 2812: 2808: 2805: 2799: 2796: 2792: 2781: 2777: 2761: 2757: 2749: 2745: 2741: 2733: 2728: 2724: 2718: 2715: 2710: 2707: 2700: 2699: 2698: 2678: 2672: 2669: 2666: 2663: 2657: 2651: 2648: 2642: 2625: 2608: 2595: 2589: 2576: 2573: 2570: 2564: 2547: 2533: 2527: 2521: 2518: 2512: 2495: 2494: 2493: 2476: 2470: 2467: 2461: 2436: 2419: 2414: 2411: 2406: 2403: 2400: 2393: 2392: 2391: 2377: 2369: 2350: 2347: 2344: 2341: 2338: 2331: 2330: 2329: 2321: 2303: 2300: 2297: 2291: 2282: 2279: 2274: 2271: 2261: 2260: 2259: 2242: 2239: 2236: 2230: 2224: 2221: 2216: 2212: 2204: 2203: 2202: 2185: 2182: 2179: 2173: 2164: 2161: 2156: 2153: 2143: 2142: 2141: 2124: 2121: 2118: 2112: 2106: 2103: 2098: 2094: 2086: 2085: 2084: 2067: 2064: 2061: 2057: 2053: 2050: 2045: 2042: 2035: 2031: 2026: 2022: 2016: 2013: 2008: 2005: 1995: 1994: 1993: 1990: 1976: 1973: 1968: 1964: 1940: 1927: 1923: 1919: 1916: 1908: 1904: 1900: 1896: 1890: 1886: 1880: 1876: 1872: 1866: 1860: 1857: 1852: 1848: 1844: 1841: 1832: 1828: 1825: 1822: 1813: 1805: 1804: 1803: 1801: 1793: 1767: 1763: 1759: 1750: 1741: 1737: 1733: 1729: 1723: 1719: 1715: 1706: 1702: 1699: 1696: 1687: 1679: 1678: 1677: 1663: 1654: 1634: 1630: 1617: 1614: 1605: 1584: 1580: 1567: 1564: 1555: 1534: 1530: 1523: 1517: 1514: 1509: 1505: 1481: 1475: 1452: 1446: 1423: 1418: 1414: 1404: 1398: 1395: 1389: 1383: 1375: 1371: 1367: 1363: 1356: 1350: 1347: 1341: 1335: 1328: 1327: 1326: 1324: 1319: 1306: 1301: 1297: 1292: 1286: 1282: 1273: 1255: 1252: 1248: 1244: 1239: 1235: 1212: 1209: 1205: 1201: 1196: 1192: 1183: 1164: 1159: 1156: 1151: 1146: 1142: 1134: 1133: 1132: 1129: 1115: 1092: 1089: 1086: 1078: 1072: 1066: 1063: 1057: 1051: 1048: 1042: 1036: 1029: 1028: 1027: 1010: 1007: 1004: 998: 995: 990: 987: 981: 975: 972: 967: 964: 958: 952: 949: 944: 941: 931: 930: 929: 927: 922: 920: 916: 912: 908: 903: 889: 869: 849: 829: 818:The SIR model 816: 791: 788: 785: 782: 776: 773: 768: 765: 754: 751: 748: 745: 742: 739: 736: 733: 727: 724: 719: 716: 705: 702: 699: 696: 693: 690: 684: 681: 676: 673: 662: 654: 653: 652: 650: 646: 638: 633: 624: 622: 606: 603: 600: 591: 589: 585: 569: 565: 561: 558: 555: 535: 515: 512: 504: 500: 495: 493: 475: 471: 466: 462: 459: 439: 419: 397: 393: 388: 384: 381: 378: 375: 372: 369: 366: 362: 355: 351: 347: 341: 333: 329: 315: 311: 303: 299: 297: 293: 289: 285: 281: 276: 274: 256: 252: 228: 222: 214: 210: 206: 202: 198: 194: 190: 186: 182: 178: 169: 165: 163: 159: 155: 147: 143: 138: 134: 131: 128: 124: 121: 118: 114: 111: 110: 109: 107: 100:The SIR model 97: 95: 91: 87: 82: 80: 75: 73: 70:in 1956. The 69: 66:in 1927, and 65: 61: 57: 52: 49: 45: 41: 37: 33: 29: 25: 21: 30809:Epidemiology 30243:Notification 30214:Hand washing 30209:Food hygiene 30194:Disinfection 30090:Immunization 30041:Anthelmintic 30029:prophylactic 29917:Respiratory 29841:Hyperendemic 29791:WAIFW matrix 29750: 29446:Environment- 29199:Transmission 29175:Urbanization 28973:Host tropism 28869:Determinants 28855:Concepts in 28811: 28757: 28733: 28714: 28695: 28629: 28625: 28615: 28564: 28560: 28554: 28495: 28491: 28485: 28452: 28448: 28442: 28412:(1): 48–70. 28409: 28405: 28399: 28387:. Retrieved 28382: 28372: 28353: 28343: 28324: 28278: 28274: 28264: 28231: 28227: 28221: 28202: 28198: 28188: 28145: 28141: 28131: 28096: 28092: 28082: 28039: 28035: 28029: 28019:, retrieved 27997: 27987: 27967: 27960: 27948:J.H. Jones, 27944: 27906:(1): 29–48. 27903: 27899: 27843: 27839: 27775: 27771: 27761: 27724: 27720: 27710: 27669: 27665: 27655: 27639: 27587:(1): 22497. 27584: 27580: 27506: 27502: 27468: 27413: 27409: 27399: 27358: 27354: 27348: 27329: 27325: 27283: 27279: 27217: 27213: 27190: 27185: 27142: 27138: 27128: 27111: 27068: 27064: 27054: 27027: 27021: 26993: 26987: 26966: 26957: 26940: 26936: 26930: 26879: 26875: 26823: 26819: 26794: 26767:(1): 35–55. 26764: 26760: 26750: 26742: 26712: 26708: 26698: 26679: 26673: 26658: 26639: 26633: 26614: 26572: 26568: 26562: 26525: 26521: 26511: 26492: 26488: 26478: 26469:2003.05666v1 26417: 26413: 26365: 26361: 26330:. Retrieved 26326: 26265: 26261: 26207: 26203: 26193: 26150: 26110: 26106: 26062: 26058: 26048: 26021: 26017: 26007: 25980: 25976: 25966: 25898: 25891: 25869: 25860: 25851: 25773: 25552:simple model 25467: 25414:The largest 24993: 24895: 24674: 24560: 23969: 23966:Simple model 23855: 23693: 23429: 23426: 23401: 23370: 23339: 23190: 22873: 22850: 22828: 22826: 22645: 22591: 22537: 22477: 22174: 22068: 22067: 22062: 22061: 22053: 22047: 22039: 21950: 21613:steady state 21302: 21275:. Note that 21103: 20747: 20705: 20629: 20573: 20565: 20562: 20478: 20374: 20192: 19976: 19913: 19713: 19645: 19594: 19437: 18759: 18432: 18347: 18209: 18206: 17862: 17779: 17605: 17596: 17545: 17476: 17463: 17398: 16954: 16868: 16605: 16577: 16569: 16479: 16323: 16218: 16215: 16156: 16099: 16096: 15965: 15961: 15948: 15881: 15808: 15806: 15800: 15728: 15656: 15531: 15476: 15459: 15450: 15340: 15111: 15102: 14886: 14708: 14705: 14612: 14608: 14509: 14107: 13868: 13864: 13862: 13788: 13780: 13557: 13228: 12993: 12991: 12916: 12846: 12462: 12449: 12446: 12434: 12417:tuberculosis 12414: 12144: 12140: 12136: 12134: 12107: 12013: 11938: 11238: 10867: 10865: 10559: 10028: 9744: 9742: 9702: 9480: 9476: 9472: 9470: 9111: 9007: 8953: 8949: 8754: 8638: 8415: 8327: 8216: 8214: 8068: 8049: 8027: 7866: 7788: 7528: 7524: 7257: 6818: 6762: 6606:; infected, 6576: 6237: 5842: 5781: 5668: 5424: 5064: 5004: 4847: 4736: 4431: 4212: 4029: 3917: 3676: 3673: 3576: 3302: 3016: 2832: 2696: 2442: 2434: 2365: 2327: 2318: 2257: 2200: 2139: 2082: 1991: 1955: 1789: 1438: 1320: 1271: 1179: 1130: 1107: 1025: 923: 904: 821: 642: 592: 502: 498: 496: 331: 327: 325: 308: 291: 279: 277: 272: 212: 208: 204: 200: 196: 192: 188: 184: 180: 176: 174: 151: 145: 141: 136: 132: 126: 122: 116: 112: 105: 103: 83: 76: 53: 47: 43: 42:usceptible, 39: 35: 31: 27: 19: 18: 30496:Eradication 30407:Biosecurity 30132:Vaccination 30107:Inoculation 30085:Drug safety 30080:Combination 30007:and Control 29955:Soft tissue 29909:Anatomical 29876:Seasonality 29856:Mesoendemic 29836:Holoendemic 29829:Farr's laws 29741:Attack rate 29539:Animal bite 29535:Insect bite 29471:Respiratory 29098:Air quality 29086:Environment 28996:Comorbidity 28912:Attack rate 28895:Infectivity 28813:3Blue1Brown 28530:2117/126209 28389:24 November 28350:Anderson RM 27416:(1): 5576. 27280:Mathematics 27145:: 147–149. 26743:Section 4.1 26569:SIAM Review 26268:: 184–194. 25917:Attack rate 22058:SEIR‌ model 21617:fixed point 20251:must hold: 15463:Erdos Renyi 14615:common cold 12426:Mary Mallon 10594:Riemann sum 9227:, recovery 8052:common cold 4025:Mathematica 919:Ronald Ross 30803:Categories 30581:Biological 30379:infections 30320:Sanitation 30288:Quarantine 30147:resistance 30065:Antiseptic 30046:Ascaricide 30036:Antifungal 30024:Antibiotic 30005:Prevention 29938:Intestinal 29871:Prevalence 29851:Inequality 29800:Occurrence 29599:Breastmilk 29483:Bioaerosol 29455:Sapronosis 29430:Horizontal 29302:Endogenous 29281:Viral load 29241:Index case 28021:2022-11-08 27950:Notes on R 27594:2106.08966 27516:2006.10490 27509:: 110077. 27423:2003.13967 27220:(2): 386. 26495:(2): e14. 26210:(3): 105. 26185:0070.15101 25958:References 25416:eigenvalue 23727:over time 22871:such that 15486:models of 10602:real-time 8420:, so that 3044:satisfies 926:non-linear 148:esistant". 30491:Discovery 30417:Disease X 30377:Emerging 30325:Screening 30226:Isolation 30070:Antiviral 29896:Twindemic 29846:Incidence 29723:Modelling 29686:Perinatal 29618:Cutaneous 29509:Linked to 29419:Perinatal 29338:Exogenous 29330:formation 29072:Pregnancy 28929:Endotoxin 28924:Virulence 28788:239899862 28674:226242440 28057:1874/8051 27920:0025-5564 27860:1742-5689 27802:1432-1416 27793:1874/8051 27727:: 64870. 27702:253314121 27694:0218-1274 27559:219792089 27254:233589998 27191:SIR model 27159:2452-0918 26922:251484056 26820:ACS Omega 26392:225555567 26275:1403.2160 26224:1742-6316 25749:τ 25670:− 25617:τ 25588:τ 25578:⁡ 25532:τ 25477:τ 25345:τ 25333:− 25322:τ 25300:τ 25273:τ 25261:− 25104:τ 25094:τ 25068:τ 25055:τ 24988:Isolation 24939:τ 24909:τ 24860:⁡ 24806:⁡ 24789:∫ 24783:τ 24747:⁡ 24713:⁡ 24622:− 24569:τ 24544:τ 24530:⁡ 24496:τ 24450:τ 24430:⁡ 24382:⁡ 24348:τ 24328:⁡ 24241:τ 24214:⁡ 24180:⁡ 24146:⁡ 24085:τ 24005:τ 23926:⁡ 23770:⁡ 23666:γ 23660:μ 23651:κ 23645:μ 23635:∗ 23627:β 23624:κ 23607:− 23593:ρ 23552:μ 23546:γ 23541:κ 23538:− 23526:κ 23520:μ 23464:∗ 23456:β 23291:− 23213:− 23155:μ 23147:λ 23133:∗ 23120:∗ 23107:∗ 23094:∗ 23078:∗ 23032:∂ 23022:∗ 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25285:R 25277:E 25268:/ 25264:1 25255:( 25250:= 25245:) 25237:I 25233:n 25223:E 25219:n 25212:( 25204:t 25201:d 25197:d 25187:, 25173:I 25169:n 25146:E 25142:n 25113:. 25108:I 25098:E 25088:2 25084:K 25080:+ 25077:) 25072:I 25064:+ 25059:E 25051:( 25048:K 25045:+ 25042:1 25039:= 25034:0 25030:R 25007:0 25003:R 24970:0 24966:R 24943:I 24927:. 24913:E 24876:) 24871:0 24867:R 24863:( 24837:t 24834:d 24831:) 24828:) 24825:t 24822:( 24817:0 24813:R 24809:( 24798:t 24793:0 24780:1 24775:+ 24772:) 24769:) 24766:0 24763:( 24758:E 24754:n 24750:( 24741:= 24738:) 24735:) 24732:t 24729:( 24724:E 24720:n 24716:( 24688:0 24684:R 24657:= 24652:0 24648:R 24625:1 24617:d 24606:= 24603:K 24582:d 24575:5 24572:= 24538:0 24534:R 24521:= 24518:K 24493:K 24489:e 24485:= 24480:0 24476:R 24455:. 24446:) 24441:0 24437:R 24433:( 24421:= 24415:t 24412:d 24407:) 24404:) 24401:t 24398:( 24393:E 24389:n 24385:( 24376:d 24353:. 24344:) 24339:0 24335:R 24331:( 24319:) 24316:t 24313:( 24308:E 24304:n 24300:= 24294:t 24291:d 24286:) 24283:t 24280:( 24275:E 24271:n 24267:d 24244:. 24237:/ 24233:t 24230:) 24225:0 24221:R 24217:( 24208:+ 24205:) 24202:) 24199:0 24196:( 24191:E 24187:n 24183:( 24174:= 24171:) 24168:) 24165:t 24162:( 24157:E 24153:n 24149:( 24121:t 24118:K 24114:e 24109:) 24106:0 24103:( 24098:E 24094:n 24090:= 24081:/ 24077:t 24072:0 24068:R 24063:) 24060:0 24057:( 24052:E 24048:n 24044:= 24041:) 24038:t 24035:( 24030:E 24026:n 23983:0 23979:R 23951:. 23944:d 23940:T 23935:) 23932:2 23929:( 23917:= 23914:K 23892:d 23888:T 23864:K 23839:0 23835:R 23813:N 23793:. 23787:t 23784:d 23779:) 23776:N 23773:( 23764:d 23755:K 23735:t 23715:) 23712:t 23709:( 23706:N 23675:. 23669:) 23663:+ 23657:( 23654:) 23648:+ 23642:( 23631:S 23618:= 23615:) 23610:1 23603:V 23599:F 23596:( 23590:= 23585:0 23581:R 23558:) 23549:+ 23531:0 23523:+ 23514:( 23509:= 23506:V 23484:) 23478:0 23473:0 23460:S 23451:0 23445:( 23440:= 23437:F 23412:. 23409:j 23385:i 23380:x 23354:j 23349:x 23324:j 23321:i 23317:G 23294:1 23287:V 23262:V 23238:F 23216:1 23209:V 23205:F 23202:= 23199:G 23176:) 23173:0 23170:, 23167:0 23164:, 23161:0 23158:, 23151:/ 23144:( 23141:= 23138:) 23129:R 23125:, 23116:I 23112:, 23103:E 23099:, 23090:S 23086:( 23083:= 23073:x 23046:j 23041:x 23027:) 23017:x 23012:( 23007:i 23003:V 22989:= 22984:j 22981:i 22977:V 22951:j 22946:x 22932:) 22922:x 22917:( 22912:i 22908:F 22894:= 22889:j 22886:i 22882:F 22858:V 22836:F 22812:) 22808:x 22804:( 22799:+ 22794:i 22790:V 22783:) 22779:x 22775:( 22765:i 22761:V 22757:= 22754:) 22750:x 22746:( 22741:i 22737:V 22716:) 22712:x 22708:( 22703:i 22699:V 22692:) 22688:x 22684:( 22679:i 22675:F 22653:i 22625:i 22621:V 22599:i 22576:+ 22571:i 22567:V 22545:i 22524:) 22520:x 22516:( 22511:i 22507:F 22485:i 22462:i 22457:x 22435:) 22432:R 22429:, 22426:I 22423:, 22420:E 22417:, 22414:S 22411:( 22408:= 22404:x 22376:. 22373:R 22364:I 22358:= 22349:R 22336:, 22333:I 22330:) 22324:+ 22318:( 22312:E 22306:= 22297:I 22284:, 22281:E 22278:) 22272:+ 22266:( 22260:I 22257:S 22251:= 22242:E 22229:, 22226:I 22223:S 22214:S 22202:= 22193:S 22184:{ 22069:I 22063:E 22054:E 22023:0 22019:R 21991:0 21987:R 21964:0 21960:R 21936:. 21926:= 21923:) 21920:) 21917:t 21914:( 21909:n 21905:C 21901:, 21895:, 21892:) 21889:t 21886:( 21881:2 21877:C 21873:, 21870:) 21867:t 21864:( 21859:1 21855:C 21851:( 21840:t 21829:0 21823:) 21820:0 21817:( 21814:I 21811:, 21808:1 21800:0 21796:R 21767:= 21764:) 21761:) 21758:t 21755:( 21750:n 21746:C 21742:, 21736:, 21733:) 21730:t 21727:( 21722:2 21718:C 21714:, 21711:) 21708:t 21705:( 21700:1 21696:C 21692:( 21681:t 21670:1 21662:0 21658:R 21632:0 21628:R 21599:0 21596:= 21593:I 21573:R 21570:= 21565:3 21561:C 21540:I 21537:= 21532:2 21528:C 21507:S 21504:= 21499:1 21495:C 21472:i 21468:C 21447:) 21442:n 21438:C 21434:, 21431:. 21428:. 21425:. 21422:, 21417:2 21413:C 21409:, 21404:1 21400:C 21396:( 21393:f 21390:= 21384:t 21380:d 21373:i 21369:C 21364:d 21358:= 21347:i 21343:C 21319:n 21283:G 21261:f 21258:m 21236:f 21228:× 21216:m 21195:m 21175:f 21149:f 21146:m 21141:= 21138:) 21135:G 21132:( 21126:= 21121:0 21117:R 21087:a 21079:G 21057:1 21054:+ 21051:a 21024:a 20999:a 20977:j 20974:i 20970:g 20949:m 20929:f 20909:i 20889:j 20869:i 20847:j 20844:i 20840:g 20819:, 20814:) 20808:0 20803:m 20796:f 20791:0 20785:( 20780:= 20777:G 20757:G 20734:m 20714:f 20692:. 20689:) 20686:G 20683:( 20677:= 20672:0 20668:R 20647:G 20614:0 20610:R 20587:0 20583:R 20548:. 20545:) 20542:0 20539:, 20536:0 20533:, 20530:) 20527:a 20524:( 20515:n 20511:( 20508:= 20505:) 20502:a 20499:( 20496:S 20493:F 20490:D 20464:, 20460:) 20456:q 20453:d 20449:) 20446:q 20443:( 20435:a 20430:0 20418:( 20408:C 20405:= 20402:) 20399:a 20396:( 20387:n 20360:a 20357:d 20352:) 20347:q 20344:d 20341:) 20338:q 20335:( 20326:a 20321:0 20309:( 20299:) 20296:a 20293:( 20283:M 20279:a 20273:0 20265:= 20262:1 20239:) 20236:a 20233:( 20210:) 20207:. 20204:( 20178:) 20175:t 20172:, 20169:a 20166:( 20163:n 20160:) 20157:a 20154:( 20145:= 20142:) 20139:a 20136:, 20133:t 20130:( 20127:n 20122:a 20114:+ 20111:) 20108:a 20105:, 20102:t 20099:( 20096:n 20091:t 20063:) 20060:a 20057:, 20054:t 20051:( 20048:r 20045:+ 20042:) 20039:a 20036:, 20033:t 20030:( 20027:i 20024:+ 20021:) 20018:a 20015:, 20012:t 20009:( 20006:s 20003:= 20000:) 19997:a 19994:, 19991:t 19988:( 19985:n 19962:r 19959:, 19956:i 19953:, 19950:s 19947:= 19944:j 19941:, 19938:) 19935:a 19932:( 19927:j 19899:a 19896:d 19891:) 19887:) 19884:t 19881:, 19878:a 19875:( 19872:r 19869:) 19866:a 19863:( 19858:r 19850:+ 19847:) 19844:t 19841:, 19838:a 19835:( 19832:i 19829:) 19826:a 19823:( 19818:i 19810:+ 19807:) 19804:t 19801:, 19798:a 19795:( 19792:s 19789:) 19786:a 19783:( 19778:s 19769:( 19761:M 19757:a 19751:0 19743:= 19740:) 19737:0 19734:, 19731:t 19728:( 19725:s 19699:0 19696:= 19693:) 19690:0 19687:, 19684:t 19681:( 19678:r 19675:= 19672:) 19669:0 19666:, 19663:t 19660:( 19657:i 19631:) 19628:t 19625:; 19620:1 19616:a 19612:, 19609:a 19606:( 19603:k 19578:1 19574:a 19570:d 19566:) 19563:t 19560:, 19555:1 19551:a 19547:( 19544:i 19541:) 19538:t 19535:; 19530:1 19526:a 19522:, 19519:a 19516:( 19513:k 19506:M 19502:a 19496:0 19488:= 19485:) 19482:) 19476:, 19470:( 19467:i 19464:, 19461:t 19458:, 19455:a 19452:( 19449:F 19423:) 19420:t 19417:, 19414:a 19411:( 19408:i 19405:) 19402:a 19399:( 19393:+ 19390:) 19387:t 19384:, 19381:a 19378:( 19375:r 19372:) 19369:a 19366:( 19357:= 19354:) 19351:a 19348:, 19345:t 19342:( 19339:r 19334:a 19326:+ 19323:) 19320:a 19317:, 19314:t 19311:( 19308:r 19303:t 19275:) 19272:t 19269:, 19266:a 19263:( 19260:i 19257:) 19254:a 19251:( 19242:) 19239:t 19236:, 19233:a 19230:( 19227:i 19224:) 19221:a 19218:( 19206:1 19202:a 19198:d 19195:) 19192:t 19189:, 19184:1 19180:a 19176:( 19173:i 19170:) 19167:t 19164:; 19159:1 19155:a 19151:, 19148:a 19145:( 19142:k 19134:M 19130:a 19124:0 19116:) 19113:t 19110:, 19107:a 19104:( 19101:s 19098:= 19095:) 19092:a 19089:, 19086:t 19083:( 19080:i 19075:a 19067:+ 19064:) 19061:a 19058:, 19055:t 19052:( 19049:i 19044:t 19014:1 19010:a 19006:d 19002:) 18999:t 18996:, 18991:1 18987:a 18983:( 18980:i 18977:) 18974:t 18971:; 18966:1 18962:a 18958:, 18955:a 18952:( 18949:k 18942:M 18938:a 18932:0 18924:) 18921:t 18918:, 18915:a 18912:( 18909:s 18903:) 18900:t 18897:, 18894:a 18891:( 18888:s 18885:) 18882:a 18879:( 18870:= 18867:) 18864:a 18861:, 18858:t 18855:( 18852:s 18847:a 18839:+ 18836:) 18833:a 18830:, 18827:t 18824:( 18821:s 18816:t 18781:+ 18773:M 18769:a 18745:a 18742:d 18738:) 18735:a 18732:, 18729:t 18726:( 18723:r 18716:M 18712:a 18706:0 18698:= 18695:) 18692:t 18689:( 18686:R 18663:a 18660:d 18656:) 18653:a 18650:, 18647:t 18644:( 18641:i 18634:M 18630:a 18624:0 18616:= 18613:) 18610:t 18607:( 18604:I 18581:a 18578:d 18574:) 18571:a 18568:, 18565:t 18562:( 18559:s 18552:M 18548:a 18542:0 18534:= 18531:) 18528:t 18525:( 18522:S 18499:) 18496:a 18493:, 18490:t 18487:( 18484:r 18481:, 18478:) 18475:a 18472:, 18469:t 18466:( 18463:i 18460:, 18457:) 18454:a 18451:, 18448:t 18445:( 18442:s 18413:1 18407:t 18404:d 18400:) 18397:t 18394:( 18385:S 18379:T 18374:0 18364:0 18360:R 18331:) 18328:T 18325:, 18322:t 18319:( 18316:D 18313:O 18310:M 18300:E 18291:T 18281:E 18277:) 18274:p 18268:1 18265:( 18259:1 18255:p 18247:1 18244:= 18241:) 18238:t 18235:( 18226:S 18210:I 18185:, 18182:2 18179:, 18176:1 18173:, 18170:0 18167:= 18164:n 18158:, 18155:) 18146:T 18142:n 18139:( 18136:S 18133:p 18130:+ 18127:) 18118:T 18114:n 18111:( 18108:V 18105:= 18102:) 18097:+ 18093:T 18089:n 18086:( 18083:V 18079:, 18076:V 18067:= 18057:t 18054:d 18049:V 18046:d 18033:, 18030:2 18027:, 18024:1 18021:, 18018:0 18015:= 18012:n 18006:, 18003:) 17994:T 17990:n 17987:( 17984:S 17981:) 17978:p 17972:1 17969:( 17966:= 17963:) 17958:+ 17954:T 17950:n 17947:( 17944:S 17940:, 17937:S 17932:N 17929:I 17918:S 17909:N 17903:= 17893:t 17890:d 17885:S 17882:d 17839:0 17835:R 17831:1 17825:) 17811:+ 17808:1 17804:( 17797:1 17791:P 17761:V 17752:S 17746:+ 17743:P 17740:N 17734:= 17724:t 17721:d 17716:V 17713:d 17703:S 17698:N 17695:I 17684:S 17675:S 17666:) 17663:P 17657:1 17654:( 17651:N 17645:= 17635:t 17632:d 17627:S 17624:d 17582:, 17573:P 17566:) 17563:0 17560:( 17557:P 17525:) 17522:I 17519:( 17512:P 17507:, 17504:) 17501:I 17498:( 17495:P 17492:= 17489:P 17445:0 17441:R 17437:1 17429:1 17426:= 17417:P 17410:P 17384:. 17380:) 17375:) 17371:1 17365:) 17362:P 17356:1 17353:( 17348:0 17344:R 17339:( 17335:N 17332:, 17326:) 17323:P 17317:1 17314:( 17309:0 17305:R 17300:N 17294:( 17290:= 17287:E 17284:E 17281:= 17277:) 17273:) 17270:t 17267:( 17264:I 17261:, 17258:) 17255:t 17252:( 17249:S 17245:( 17236:+ 17230:t 17219:0 17213:) 17210:0 17207:( 17204:I 17200:, 17197:1 17191:) 17188:P 17182:1 17179:( 17174:0 17170:R 17145:) 17141:0 17138:, 17134:) 17130:P 17124:1 17120:( 17116:N 17112:( 17108:= 17105:E 17102:F 17099:D 17096:= 17092:) 17088:) 17085:t 17082:( 17079:I 17076:, 17073:) 17070:t 17067:( 17064:S 17060:( 17051:+ 17045:t 17034:1 17028:) 17025:P 17019:1 17016:( 17011:0 17007:R 16983:I 16963:S 16940:, 16937:P 16934:N 16931:= 16928:) 16925:t 16922:( 16919:V 16911:+ 16905:t 16877:V 16850:V 16841:P 16838:N 16832:= 16822:t 16819:d 16814:V 16811:d 16801:I 16798:) 16792:+ 16786:( 16780:S 16775:N 16772:I 16764:= 16754:t 16751:d 16746:I 16743:d 16733:S 16728:N 16725:I 16714:S 16705:) 16702:P 16696:1 16693:( 16690:N 16684:= 16674:t 16671:d 16666:S 16663:d 16633:) 16630:1 16627:, 16624:0 16621:( 16615:P 16586:V 16554:) 16550:t 16545:2 16540:o 16529:( 16523:2 16519:h 16515:c 16512:e 16509:s 16504:x 16501:a 16498:m 16494:I 16490:= 16487:I 16465:I 16443:x 16440:a 16437:m 16433:I 16410:o 16406:R 16383:o 16358:S 16336:o 16332:S 16305:2 16300:o 16296:R 16289:2 16285:) 16281:1 16273:o 16269:R 16265:( 16257:2 16252:o 16248:S 16242:= 16237:x 16234:a 16231:m 16227:I 16199:N 16194:o 16190:S 16174:= 16169:o 16165:R 16142:) 16139:1 16131:o 16127:R 16123:( 16117:= 16112:o 16082:0 16079:= 16074:2 16070:I 16062:x 16059:a 16056:m 16052:I 16046:2 16041:o 16029:2 16026:3 16021:+ 16018:I 16013:2 16008:o 15994:t 15991:d 15986:I 15981:2 15977:d 15862:I 15859:2 15849:I 15844:+ 15837:. 15834:. 15831:. 15827:S 15780:n 15777:g 15774:i 15770:S 15757:h 15754:x 15751:e 15747:S 15708:h 15705:x 15702:e 15698:S 15685:s 15682:e 15679:r 15675:S 15638:I 15633:2 15630:+ 15623:s 15620:e 15617:r 15613:S 15602:I 15597:2 15594:+ 15587:n 15584:g 15581:i 15577:S 15510:t 15506:R 15408:R 15404:D 15381:I 15377:D 15354:S 15350:D 15322:, 15319:I 15313:+ 15310:R 15305:2 15295:R 15291:D 15287:= 15284:R 15279:t 15266:, 15263:I 15252:N 15248:S 15245:I 15236:+ 15233:I 15228:2 15218:I 15214:D 15210:= 15207:I 15202:t 15189:, 15184:N 15180:S 15177:I 15165:S 15160:2 15150:S 15146:D 15142:= 15139:S 15134:t 15088:, 15085:) 15082:0 15079:, 15076:N 15073:( 15070:= 15067:E 15064:F 15061:D 15058:= 15055:) 15052:) 15049:t 15046:( 15043:I 15040:, 15037:) 15034:t 15031:( 15028:S 15025:( 15017:+ 15011:t 15000:1 14994:t 14991:d 14981:+ 14973:) 14970:t 14967:( 14956:T 14951:0 14941:T 14938:1 14913:I 14907:S 14901:N 14898:= 14895:R 14868:I 14865:) 14859:+ 14853:( 14847:S 14842:N 14839:I 14834:) 14831:t 14828:( 14822:= 14812:t 14809:d 14804:I 14801:d 14791:S 14786:N 14783:I 14778:) 14775:t 14772:( 14763:S 14754:N 14748:= 14738:t 14735:d 14730:S 14727:d 14691:) 14688:t 14685:( 14679:= 14676:) 14673:T 14670:+ 14667:t 14664:( 14657:, 14652:N 14649:I 14644:) 14641:t 14638:( 14632:= 14629:F 14580:S 14570:R 14560:I 14550:E 14540:S 14530:M 14484:R 14475:I 14469:= 14459:t 14456:d 14451:R 14448:d 14438:I 14435:) 14429:+ 14423:( 14417:E 14411:= 14401:t 14398:d 14393:I 14390:d 14380:E 14377:) 14371:+ 14365:( 14357:N 14353:I 14350:S 14341:= 14331:t 14328:d 14323:E 14320:d 14310:S 14299:N 14295:I 14292:S 14280:M 14274:= 14264:t 14261:d 14256:S 14253:d 14243:M 14234:M 14222:= 14212:t 14209:d 14204:M 14201:d 14167:R 14157:I 14147:E 14137:S 14127:M 14082:I 14079:) 14073:+ 14067:( 14061:E 14055:= 14045:t 14042:d 14037:I 14034:d 14024:E 14021:) 14015:+ 14009:( 14001:N 13997:I 13994:S 13985:= 13975:t 13972:d 13967:E 13964:d 13954:I 13948:+ 13945:S 13934:N 13930:I 13927:S 13912:= 13902:t 13899:d 13894:S 13891:d 13869:t 13867:( 13865:S 13839:S 13829:I 13819:E 13809:S 13760:1 13756:I 13752:) 13746:+ 13740:( 13732:1 13728:E 13724:a 13721:= 13711:t 13708:d 13701:1 13697:I 13693:d 13681:1 13677:E 13673:) 13670:a 13667:+ 13661:( 13653:1 13649:I 13645:) 13642:t 13639:( 13633:= 13623:t 13620:d 13613:1 13609:E 13605:d 13575:) 13572:t 13569:( 13543:. 13540:E 13537:E 13534:= 13530:) 13526:) 13523:t 13520:( 13517:R 13514:, 13511:) 13508:t 13505:( 13502:I 13499:, 13496:) 13493:t 13490:( 13487:E 13484:, 13481:) 13478:t 13475:( 13472:S 13468:( 13459:+ 13453:t 13442:0 13436:) 13433:0 13430:( 13427:I 13424:, 13421:1 13413:0 13409:R 13387:, 13384:) 13381:0 13378:, 13375:0 13372:, 13369:0 13366:, 13363:N 13360:( 13357:= 13354:E 13351:F 13348:D 13345:= 13341:) 13337:) 13334:t 13331:( 13328:R 13325:, 13322:) 13319:t 13316:( 13313:I 13310:, 13307:) 13304:t 13301:( 13298:E 13295:, 13292:) 13289:t 13286:( 13283:S 13279:( 13270:+ 13264:t 13253:1 13245:0 13241:R 13213:} 13209:N 13206:= 13203:R 13200:+ 13197:I 13194:+ 13191:E 13188:+ 13185:S 13182:, 13179:0 13173:R 13170:, 13167:0 13161:I 13158:, 13155:0 13149:E 13146:, 13143:0 13137:S 13134:: 13129:4 13125:] 13121:N 13118:, 13115:0 13112:[ 13106:) 13103:R 13100:, 13097:I 13094:, 13091:E 13088:, 13085:S 13082:( 13078:{ 13070:) 13066:) 13063:0 13060:( 13057:R 13054:, 13051:) 13048:0 13045:( 13042:I 13039:, 13036:) 13033:0 13030:( 13027:E 13024:, 13021:) 13018:0 13015:( 13012:S 13008:( 12994:N 12977:. 12968:+ 12953:a 12950:+ 12943:a 12938:= 12933:0 12929:R 12902:N 12882:, 12879:N 12876:= 12873:R 12870:+ 12867:I 12864:+ 12861:E 12858:+ 12855:S 12828:. 12825:R 12816:I 12810:= 12800:t 12797:d 12792:R 12789:d 12779:I 12776:) 12770:+ 12764:( 12758:E 12755:a 12752:= 12742:t 12739:d 12734:I 12731:d 12721:E 12718:) 12715:a 12712:+ 12706:( 12698:N 12694:S 12691:I 12682:= 12672:t 12669:d 12664:E 12661:d 12649:N 12645:S 12642:I 12630:S 12621:N 12615:= 12605:t 12602:d 12597:S 12594:d 12564:N 12541:N 12499:1 12492:a 12471:a 12450:E 12389:R 12380:I 12374:= 12364:t 12361:d 12356:R 12353:d 12343:I 12334:I 12323:N 12319:I 12316:S 12307:= 12297:t 12294:d 12289:I 12286:d 12276:S 12265:N 12261:I 12258:S 12246:M 12240:= 12230:t 12227:d 12222:S 12219:d 12209:M 12200:M 12188:= 12178:t 12175:d 12170:M 12167:d 12145:. 12143:) 12141:t 12139:( 12137:M 12087:) 12081:( 12078:I 12075:) 12069:( 12066:S 12063:= 12060:) 12054:( 12051:j 12031:) 12025:( 12022:I 11994:d 11991:) 11985:( 11982:a 11977:t 11972:0 11964:= 11961:) 11958:t 11955:( 11918:S 11915:) 11909:( 11906:q 11903:= 11894:d 11889:D 11886:d 11875:, 11872:S 11869:) 11863:( 11860:b 11857:= 11848:d 11843:V 11840:d 11829:, 11826:I 11823:) 11817:( 11814:k 11811:= 11802:d 11797:R 11794:d 11783:, 11780:I 11777:] 11774:) 11768:( 11765:q 11762:+ 11759:) 11753:( 11750:k 11747:[ 11741:I 11738:S 11735:= 11726:d 11721:I 11718:d 11707:, 11704:S 11701:) 11695:( 11692:b 11686:I 11683:S 11677:= 11668:d 11663:S 11660:d 11627:) 11624:t 11621:( 11618:a 11598:) 11595:t 11592:( 11589:a 11585:/ 11581:) 11578:t 11575:( 11569:= 11566:q 11546:) 11543:t 11540:( 11537:a 11533:/ 11529:) 11526:t 11523:( 11520:v 11517:= 11514:b 11494:) 11491:t 11488:( 11485:a 11481:/ 11477:) 11474:t 11471:( 11465:= 11462:k 11442:0 11439:= 11436:) 11433:0 11430:( 11427:D 11424:= 11421:) 11418:0 11415:( 11412:V 11409:= 11406:) 11403:0 11400:( 11397:R 11374:= 11371:) 11368:0 11365:( 11362:I 11336:1 11333:= 11330:) 11327:0 11324:( 11321:S 11301:) 11298:t 11295:( 11289:, 11286:) 11283:t 11280:( 11274:, 11271:) 11268:t 11265:( 11262:v 11259:, 11256:) 11253:t 11250:( 11247:a 11220:I 11217:) 11214:t 11211:( 11205:= 11199:t 11196:d 11191:D 11188:d 11177:, 11174:S 11171:) 11168:t 11165:( 11162:v 11159:= 11153:t 11150:d 11145:V 11142:d 11131:, 11128:I 11125:) 11122:t 11119:( 11113:= 11107:t 11104:d 11099:R 11096:d 11085:, 11082:I 11079:) 11076:t 11073:( 11064:I 11061:) 11058:t 11055:( 11046:I 11043:S 11040:) 11037:t 11034:( 11031:a 11028:= 11022:t 11019:d 11014:I 11011:d 11000:, 10997:S 10994:) 10991:t 10988:( 10985:v 10979:I 10976:S 10973:) 10970:t 10967:( 10964:a 10958:= 10952:t 10949:d 10944:S 10941:d 10902:D 10899:, 10896:V 10893:, 10890:R 10887:, 10884:I 10881:, 10878:S 10846:1 10838:e 10834:R 10813:N 10793:S 10773:1 10765:N 10761:S 10756:t 10752:R 10745:= 10740:e 10736:R 10711:0 10707:R 10701:t 10697:R 10672:t 10668:R 10645:0 10641:R 10615:t 10611:R 10573:t 10569:R 10545:0 10542:= 10539:) 10536:0 10533:( 10530:V 10527:= 10524:) 10521:0 10518:( 10515:R 10495:N 10492:= 10489:) 10486:0 10483:( 10480:V 10477:+ 10474:) 10471:0 10468:( 10465:R 10462:+ 10459:) 10456:0 10453:( 10450:I 10447:+ 10444:) 10441:0 10438:( 10435:S 10415:) 10412:0 10409:( 10406:S 10380:S 10357:c 10353:b 10329:2 10323:1 10317:b 10314:+ 10311:k 10291:) 10288:t 10285:( 10278:/ 10274:) 10271:t 10268:( 10265:v 10262:= 10259:b 10239:) 10236:t 10233:( 10226:/ 10222:) 10219:t 10216:( 10210:= 10207:k 10187:0 10184:= 10181:) 10178:0 10175:( 10172:V 10169:= 10166:) 10163:0 10160:( 10157:R 10137:N 10131:= 10128:) 10125:0 10122:( 10119:I 10099:N 10096:) 10087:1 10084:( 10081:= 10078:) 10075:0 10072:( 10069:S 10049:v 10046:, 10040:, 10002:, 9999:S 9996:) 9993:t 9990:( 9987:v 9984:= 9978:t 9975:d 9970:V 9967:d 9956:, 9953:I 9950:) 9947:t 9944:( 9938:= 9932:t 9929:d 9924:R 9921:d 9910:, 9907:I 9904:) 9901:t 9898:( 9887:N 9883:S 9880:I 9877:) 9874:t 9871:( 9862:= 9856:t 9853:d 9848:I 9845:d 9834:, 9831:S 9828:) 9825:t 9822:( 9819:v 9811:N 9807:S 9804:I 9801:) 9798:t 9795:( 9783:= 9777:t 9774:d 9769:S 9766:d 9720:, 9714:, 9684:, 9681:I 9675:= 9669:t 9666:d 9661:D 9658:d 9647:, 9644:I 9638:= 9632:t 9629:d 9624:R 9621:d 9610:, 9607:I 9598:I 9587:N 9583:S 9580:I 9571:= 9565:t 9562:d 9557:I 9554:d 9543:, 9538:N 9534:S 9531:I 9519:= 9513:t 9510:d 9505:S 9502:d 9452:= 9426:= 9400:= 9374:= 9351:0 9348:= 9345:) 9342:0 9339:( 9336:R 9333:, 9330:3 9327:= 9324:) 9321:0 9318:( 9315:I 9312:, 9306:= 9303:) 9300:0 9297:( 9294:S 9264:= 9238:= 9212:= 9189:0 9186:= 9183:) 9180:0 9177:( 9174:R 9171:, 9168:3 9165:= 9162:) 9159:0 9156:( 9153:I 9150:, 9144:= 9141:) 9138:0 9135:( 9132:S 9097:. 9094:) 9091:I 9085:N 9082:( 9076:I 9067:t 9064:d 9059:I 9056:d 9030:I 9024:N 9021:= 9018:S 8994:0 8991:= 8988:R 8968:0 8965:= 8935:) 8932:t 8929:( 8926:I 8920:N 8917:= 8914:) 8911:t 8908:( 8905:S 8885:1 8877:0 8873:I 8868:/ 8858:I 8854:= 8851:V 8822:= 8799:N 8796:) 8789:/ 8779:1 8776:( 8773:= 8764:I 8751:. 8734:t 8724:e 8720:V 8717:+ 8714:1 8704:I 8698:= 8695:) 8692:t 8689:( 8686:I 8661:1 8654:y 8650:= 8647:I 8620:. 8617:N 8613:) 8596:1 8592:( 8588:= 8585:) 8582:t 8579:( 8576:I 8568:+ 8562:t 8551:1 8530:, 8527:0 8524:= 8521:) 8518:t 8515:( 8512:I 8504:+ 8498:t 8487:1 8446:0 8440:) 8437:0 8434:( 8431:I 8412:, 8398:2 8394:I 8388:N 8377:I 8374:) 8362:( 8359:= 8353:t 8350:d 8345:I 8342:d 8324:. 8312:N 8309:= 8306:) 8303:t 8300:( 8297:I 8294:+ 8291:) 8288:t 8285:( 8282:S 8276:0 8273:= 8267:t 8264:d 8259:I 8256:d 8250:+ 8244:t 8241:d 8236:S 8233:d 8217:N 8196:I 8185:N 8181:I 8178:S 8169:= 8159:t 8156:d 8151:I 8148:d 8138:I 8132:+ 8127:N 8123:I 8120:S 8108:= 8098:t 8095:d 8090:S 8087:d 8008:. 8002:v 7999:+ 7992:1 7978:+ 7970:N 7961:= 7958:I 7931:v 7928:+ 7921:1 7898:, 7892:N 7888:I 7878:= 7852:. 7843:+ 7835:N 7826:= 7823:S 7800:N 7774:, 7765:+ 7758:1 7753:= 7750:x 7747:d 7741:x 7738:) 7732:+ 7726:( 7719:e 7708:0 7700:= 7697:] 7694:) 7689:S 7685:T 7676:L 7672:T 7668:( 7662:[ 7656:E 7631:S 7627:T 7604:L 7600:T 7579:] 7576:) 7571:S 7567:T 7558:L 7554:T 7550:( 7544:[ 7538:E 7500:/ 7496:1 7456:I 7453:S 7430:S 7424:= 7421:S 7418:b 7415:a 7395:N 7391:/ 7387:S 7367:b 7347:b 7327:a 7287:b 7267:a 7243:I 7237:= 7231:t 7228:d 7223:R 7220:d 7194:I 7185:I 7182:S 7176:= 7170:t 7167:d 7162:I 7159:d 7133:I 7130:S 7121:= 7115:t 7112:d 7107:S 7104:d 6998:) 6995:0 6992:= 6989:t 6986:( 6983:R 6963:) 6960:0 6957:= 6954:t 6951:( 6948:I 6928:) 6925:0 6922:= 6919:t 6916:( 6913:S 6892:N 6872:) 6869:t 6866:( 6863:R 6860:+ 6857:) 6854:t 6851:( 6848:I 6845:+ 6842:) 6839:t 6836:( 6833:S 6830:= 6827:N 6799:R 6789:I 6779:S 6746:) 6743:t 6740:( 6737:R 6715:) 6712:t 6709:( 6706:I 6684:) 6681:t 6678:( 6675:S 6652:) 6649:t 6646:( 6643:R 6623:) 6620:t 6617:( 6614:I 6594:) 6591:t 6588:( 6585:S 6554:= 6528:= 6502:= 6479:0 6476:= 6473:) 6470:0 6467:( 6464:R 6461:, 6458:3 6455:= 6452:) 6449:0 6446:( 6443:I 6440:, 6434:= 6431:) 6428:0 6425:( 6422:S 6392:= 6366:= 6343:0 6340:= 6337:) 6334:0 6331:( 6328:R 6325:, 6322:3 6319:= 6316:) 6313:0 6310:( 6307:I 6304:, 6298:= 6295:) 6292:0 6289:( 6286:S 6251:0 6247:R 6223:. 6219:) 6214:) 6210:1 6202:0 6198:R 6193:( 6179:, 6175:) 6171:1 6163:0 6159:R 6154:( 6140:, 6128:+ 6118:( 6114:= 6104:= 6101:) 6098:) 6095:t 6092:( 6089:R 6086:, 6083:) 6080:t 6077:( 6074:I 6071:, 6068:) 6065:t 6062:( 6059:S 6056:( 6045:t 6034:0 6028:) 6025:0 6022:( 6019:I 6016:, 6013:1 6005:0 6001:R 5978:) 5974:0 5971:, 5968:0 5965:, 5951:( 5947:= 5937:= 5934:) 5931:) 5928:t 5925:( 5922:R 5919:, 5916:) 5913:t 5910:( 5907:I 5904:, 5901:) 5898:t 5895:( 5892:S 5889:( 5878:t 5867:1 5859:0 5855:R 5828:, 5819:+ 5807:= 5802:0 5798:R 5767:. 5763:) 5759:0 5756:, 5753:0 5750:, 5736:( 5732:= 5728:) 5724:) 5721:t 5718:( 5715:R 5712:, 5709:) 5706:t 5703:( 5700:I 5697:, 5694:) 5691:t 5688:( 5685:S 5681:( 5650:R 5641:I 5635:= 5625:t 5622:d 5617:R 5614:d 5604:I 5595:I 5584:N 5580:S 5577:I 5568:= 5558:t 5555:d 5550:I 5547:d 5535:N 5531:S 5528:I 5516:S 5504:= 5494:t 5491:d 5486:S 5483:d 5403:0 5399:R 5394:/ 5390:N 5387:= 5384:S 5364:1 5361:= 5356:E 5352:R 5329:0 5325:R 5321:= 5316:t 5312:R 5287:1 5279:e 5275:R 5254:1 5246:e 5242:R 5219:e 5215:R 5192:0 5188:R 5181:= 5176:t 5172:R 5149:0 5145:R 5138:= 5133:t 5129:R 5105:t 5101:R 5078:0 5074:R 5024:t 5020:R 4990:1 4982:e 4978:R 4957:1 4954:= 4949:e 4945:R 4924:1 4916:e 4912:R 4888:e 4884:R 4860:t 4856:R 4825:/ 4821:1 4796:/ 4792:1 4769:t 4765:R 4701:I 4698:S 4677:S 4657:I 4617:2 4614:= 4609:0 4605:R 4584:2 4581:= 4576:0 4572:R 4548:1 4541:y 4537:a 4534:d 4528:= 4523:0 4496:1 4489:y 4485:a 4482:d 4476:= 4471:0 4444:0 4440:R 4412:N 4407:t 4397:S 4392:t 4381:= 4376:e 4372:R 4335:t 4325:t 4315:= 4310:t 4306:R 4269:0 4259:0 4249:= 4244:0 4240:R 4225:: 4193:0 4187:t 4167:) 4164:0 4161:( 4158:I 4138:) 4135:0 4132:( 4129:S 4109:) 4106:0 4103:( 4100:R 4080:) 4077:0 4074:( 4071:I 4051:) 4048:0 4045:( 4042:S 4015:W 3981:R 3954:I 3927:S 3883:R 3843:= 3837:d 3833:/ 3829:R 3826:d 3806:t 3803:d 3800:I 3794:= 3771:t 3768:d 3765:I 3762:= 3756:d 3733:d 3729:/ 3725:d 3705:I 3685:I 3659:u 3656:= 3651:) 3648:t 3645:( 3635:e 3614:t 3594:) 3591:t 3588:( 3552:t 3548:d 3544:) 3535:t 3531:( 3528:I 3523:t 3518:0 3508:N 3500:= 3493:) 3490:t 3487:( 3477:) 3474:t 3471:( 3462:+ 3459:) 3456:0 3453:( 3450:R 3447:= 3440:) 3437:t 3434:( 3431:R 3424:) 3421:t 3418:( 3415:R 3409:) 3406:t 3403:( 3400:S 3394:N 3391:= 3384:) 3381:t 3378:( 3375:I 3366:) 3363:t 3360:( 3350:e 3346:) 3343:0 3340:( 3337:S 3334:= 3327:) 3324:t 3321:( 3318:S 3288:0 3285:= 3276:I 3255:) 3252:0 3246:) 3243:0 3240:( 3237:S 3217:) 3214:0 3211:( 3208:S 3204:/ 3194:S 3190:= 3187:( 3178:/ 3174:) 3171:) 3168:0 3165:( 3162:R 3150:R 3146:( 3139:e 3135:= 3130:T 3126:u 3099:R 3078:0 3075:= 3072:) 3067:T 3063:u 3059:( 3054:I 3030:T 3026:u 3002:, 2999:1 2993:u 2985:T 2981:u 2976:, 2973:) 2970:) 2967:0 2964:( 2961:R 2958:, 2955:) 2952:0 2949:( 2946:S 2940:) 2937:0 2934:( 2931:R 2925:N 2922:, 2919:) 2916:0 2913:( 2910:S 2907:( 2904:= 2901:) 2898:) 2895:1 2892:( 2887:R 2882:, 2879:) 2876:1 2873:( 2868:I 2863:, 2860:) 2857:1 2854:( 2849:S 2844:( 2818:, 2809:N 2800:= 2793:, 2787:) 2778:u 2774:( 2769:I 2758:u 2746:u 2742:d 2734:1 2729:u 2716:N 2711:= 2708:t 2682:) 2679:u 2676:( 2661:) 2658:0 2655:( 2652:R 2649:= 2646:) 2643:u 2640:( 2635:R 2612:) 2609:u 2606:( 2601:S 2593:) 2590:u 2587:( 2582:R 2574:N 2571:= 2568:) 2565:u 2562:( 2557:I 2534:u 2531:) 2528:0 2525:( 2522:S 2519:= 2516:) 2513:u 2510:( 2505:S 2480:) 2477:t 2474:( 2471:S 2468:= 2465:) 2462:u 2459:( 2454:S 2420:. 2415:N 2412:I 2404:= 2401:F 2378:N 2351:, 2348:I 2342:= 2339:F 2304:, 2301:0 2295:) 2292:0 2289:( 2283:t 2280:d 2275:I 2272:d 2243:, 2240:N 2234:) 2231:0 2228:( 2225:S 2217:0 2213:R 2186:, 2183:0 2177:) 2174:0 2171:( 2165:t 2162:d 2157:I 2154:d 2125:, 2122:N 2116:) 2113:0 2110:( 2107:S 2099:0 2095:R 2068:, 2065:I 2058:) 2054:1 2046:N 2043:S 2036:0 2032:R 2027:( 2023:= 2017:t 2014:d 2009:I 2006:d 1977:0 1974:= 1969:0 1965:s 1941:. 1938:) 1933:) 1928:0 1924:r 1917:1 1914:( 1909:0 1905:R 1897:e 1891:0 1887:R 1881:0 1877:s 1870:( 1867:W 1861:1 1853:0 1849:R 1842:= 1833:r 1826:1 1823:= 1814:s 1798:W 1773:) 1768:0 1764:r 1751:r 1747:( 1742:0 1738:R 1730:e 1724:0 1720:s 1716:= 1707:r 1700:1 1697:= 1688:s 1664:, 1655:t 1635:N 1631:/ 1627:) 1621:( 1618:R 1615:= 1606:r 1585:N 1581:/ 1577:) 1571:( 1568:S 1565:= 1556:s 1535:N 1531:/ 1527:) 1524:0 1521:( 1518:S 1515:= 1510:0 1506:s 1485:) 1482:0 1479:( 1476:R 1456:) 1453:0 1450:( 1447:S 1424:, 1419:N 1415:/ 1411:) 1408:) 1405:0 1402:( 1399:R 1393:) 1390:t 1387:( 1384:R 1381:( 1376:0 1372:R 1364:e 1360:) 1357:0 1354:( 1351:S 1348:= 1345:) 1342:t 1339:( 1336:S 1307:. 1302:c 1298:T 1293:/ 1287:r 1283:T 1256:1 1245:= 1240:r 1236:T 1213:1 1202:= 1197:c 1193:T 1165:, 1152:= 1147:0 1143:R 1116:N 1093:, 1090:N 1087:= 1079:= 1076:) 1073:t 1070:( 1067:R 1064:+ 1061:) 1058:t 1055:( 1052:I 1049:+ 1046:) 1043:t 1040:( 1037:S 1011:, 1008:0 1005:= 999:t 996:d 991:R 988:d 982:+ 976:t 973:d 968:I 965:d 959:+ 953:t 950:d 945:S 942:d 890:N 870:R 850:I 830:S 792:, 789:I 783:= 777:t 774:d 769:R 766:d 755:, 752:I 743:S 740:I 734:= 728:t 725:d 720:I 717:d 706:, 703:S 700:I 691:= 685:t 682:d 677:S 674:d 663:{ 607:0 604:= 570:D 566:/ 562:1 559:= 536:D 516:I 503:R 499:I 476:2 472:N 467:/ 463:I 460:S 420:N 398:2 394:N 389:/ 385:I 382:S 373:= 370:t 367:d 363:/ 359:) 356:N 352:/ 348:S 345:( 342:d 332:I 328:S 292:t 280:t 273:t 257:0 253:R 232:) 229:0 226:( 223:R 213:t 211:( 209:R 205:t 203:( 201:I 197:t 195:( 193:S 189:t 185:R 181:I 177:S 146:r 142:r 137:r 133:R 127:i 123:I 117:s 113:S 48:R 44:I 40:S 36:R 32:I 28:S

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