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There are $S$ individuals who are susceptible to infection, and $I$ who are infectious. $S + I = N$, where $N$ is the total size of the population.

Each infectious transmit the disease to a susceptible after an exponentially distribution time with mean $\lambda$. This results in the susceptible individual becoming an infectious individual ($S -1$ and $I + 1$).

Is it correct to write the rate of change in $S$ as:

$${dS \over dt } = -\lambda SI.$$

I know this is the SIR model. What I am trying to do here is to understand how to generalize the deterministic version in the wiki article to a stochastic version where the waiting time before infection is transmitted is exponentially distributed.

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2 Answers 2

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Call $\Sigma_t$ and $\Phi_t$ the random variables describing the number of susceptible and infectious individuals at time $t$. The hypothesis sustaining the SIR models is, first, that every individual is constantly in contact with every other one and, second, that each contact between an infected individual and a susceptible one causes the transmission of the disease with some given, constant, probability.

Thus, the probability that each susceptible individual becomes infected by a given infected individual during the time interval $(t,t+s)$ is $\lambda s+o(s)$ when $s\to0$, for some parameter $\lambda$ which describes the frequency of the individual encounters times the probability that a given encounter does cause the infection. The probability that each susceptible individual becomes infected during the same time interval is $\lambda s\Phi_t+o(s)$ since $\Phi_t$ encounters may happen to this susceptible individual, independently. This implies that $\Sigma_t$ becomes $\Sigma_t+1$ after a random time which is the minimum of $\Sigma_t\Phi_t$ independent random times, each exponential with parameter $\lambda$. Thus, $\Sigma_t$ becomes $\Sigma_t+1$ after an exponential random time with parameter $\lambda\Sigma_t\Phi_t$. Finally, when $\Sigma_t$ becomes $\Sigma_t+1$, $\Phi_t$ becomes $\Phi_t-1$.

Turning to the expectations $S(t)=\mathbb E(\Sigma_t)$ and $I(t)=\mathbb E(\Phi_t)$, one sees that $\Sigma_t+\Phi_t=N$ is constant, hence $\lambda\Sigma_t\Phi_t=\lambda\Sigma_t(N-\Sigma_t)$, and that $$ S'(t)=-\lambda\mathbb E(\Sigma_t\Phi_t),\qquad I'(t)=\lambda\mathbb E(\Sigma_t\Phi_t). $$ Thus, in full rigor, $$ S'(t)\ne-\lambda S(t)I(t),\qquad I'(t)\ne\lambda S(t)I(t), $$ and in fact, Cauchy-Schwarz inequality $\mathbb E(\Sigma_t^2)\gt\mathbb E(\Sigma_t)^2$ implies that $$ S'(t)\gt-\lambda S(t)I(t),\qquad I'(t)\lt\lambda S(t)I(t). $$ Of course, in the limit $N\to\infty$ of large populations, one can expect that $\Sigma_t\approx S(t)$ and $\Phi_t\approx I(t)$, recovering the usual deterministic approximation.

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Very nice and detailed answer. Thank you so much! –  Legendre Mar 31 '13 at 16:27

If you would like to write that $\dot S=-\lambda SI$, this gives you a description of the underlying contact process (which is time homogeneous Poisson process). If you still prefer to talk about some exponential distributions, then it means that each susceptible stays susceptible for an exponentially distributed time period with the mean $1/(\lambda I)$.

Much more details, in very non-technical language, are given in, e.g., this book.

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Thanks! One quick question: why is it a homogeneous Poisson process when only one contact is enough to turn a susceptible into an infectious individual? –  Legendre Mar 30 '13 at 18:42
@Legendre I do not quite follow your question. We have a number of susceptibles and a number of infectious. Total of $N$ individuals. The contact process is about contacts between these $N$ individuals. –  Artem Mar 30 '13 at 19:02
@Legendre And of course this is not the SIR model since you have no removed individuals. What you consider is usually called just SI model. –  Artem Mar 30 '13 at 19:24
A Poisson process counts the number of events in a given time interval. This number can be more than 1. However, when an infectious make contact with a susceptible, only 1 contact is required to transmit the disease. –  Legendre Mar 30 '13 at 19:48

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