EXTINCTION AND QUASI-STATIONARITY FOR DISCRETE-TIME, ENDEMIC SIS AND SIR MODELS

Stochastic discrete-time susceptible-infected-susceptible (SIS) and susceptible-infected-recovered (SIR) models of endemic diseases are introduced and analyzed. For the deterministic, mean-field model, the basic reproductive number R-0 determines their global dynamics. If R-0 <= 1, then the frequency of infected individuals asymptotically converges to zero. If R-0 > 1, then the infectious class uniformly persists for all time; conditions for a globally stable, endemic equilibrium are given. In contrast, the infection goes extinct in finite time with a probability of 1 in the stochastic models for all R-0 values. To understand the length of the transient prior to extinction as well as the behavior of the transients, the quasi-stationary distributions and the associated mean time to extinction are analyzed using large deviation methods. When R-0 > 1, these mean times to extinction are shown to increase exponentially with the population size N. Moreover, as N approaches infinity, the quasi-stationary distributions are supported by a compact set bounded away from extinction; sufficient conditions for convergence to a Dirac measure at the endemic equilibrium of the deterministic model are also given. In contrast, when R-0 < 1, the mean times to extinction are bounded above 1/(1-alpha), where alpha < 1 is the geometric rate of decrease of the infection when rare; as N approaches infinity, the quasi-stationary distributions converge to a Dirac measure at the disease-free equilibrium for the deterministic model. For several special cases, explicit formulas for approximating the quasi-stationary distribution and the associated mean extinction are given. These formulas illustrate how for arbitrarily small R-0 values, the mean time to extinction can be arbitrarily large, and how for arbitrarily large R-0 values, the mean time to extinction can be arbitrarily large.