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Stochastic discrete-time SIS and SIR models of endemic diseases areintroduced and analyzed. For the deterministic, mean-field model, the basicreproductive number $R_0$ determines their global dynamics. If $R_0\le 1$, thenthe frequency of infected individuals asymptotically converges to zero. If$R_0>1$, then the infectious class uniformly persists for all time; conditionsfor a globally stable, endemic equilibrium are given. In contrast, theinfection goes extinct in finite time with probability one in the stochasticmodels for all $R_0$ values. To understand the length of the transient prior toextinction as well as the behavior of the transients, the quasi-stationarydistributions and the associated mean time to extinction are analyzed usinglarge deviation methods. When $R_0>1$, these mean times to extinction are shownto increase exponentially with the population size $N$. Moreover, as $N$approaches $\infty$, the quasi-stationary distributions are supported by acompact set bounded away from extinction; sufficient conditions for convergenceto a Dirac measure at the endemic equilibrium of the deterministic model arealso given. In contrast, when $R_0<1$, the mean times to extinction are boundedabove $1/(1-\alpha)$ where $\alpha<1$ is the geometric rate of decrease of theinfection when rare; as $N$ approaches $\infty$, the quasi-stationarydistributions converge to a Dirac measure at the disease-free equilibrium forthe deterministic model. For several special cases, explicit formulas forapproximating the quasi-stationary distribution and the associated meanextinction are given. These formulas illustrate how for arbitrarily small $R_0$values, the mean time to extinction can be arbitrarily large, and how forarbitrarily large $R_0$ values, the mean time to extinction can be arbitrarilylarge.

Extinction and quasi-stationarity for discrete-time, endemic SIS and SIR models