Large population asymptotics for a multitype stochastic SIS epidemic model in randomly switched environment

Abstract

We consider an epidemic SIS model described by a multitype birth-and-death process in a randomly switched environment. That is, the infection and cure rates of the process depend on the state of a finite Markov jump process (the environment), whose transitions also depend on the number of infectives. The total size of the population is constant and equal to some K $\in$ N * , and the number of infectives vanishes almost surely in finite time. We prove that, as K $\rightarrow$ $\infty$, the process composed of the proportions of infectives of each type X^K and the state of the environment $Ξ$^K , converges to a piecewise deterministic Markov process (PDMP) given by a system of randomly switched ODEs. The long term behaviour of this PDMP has been previously investigated by Bena{ï}m and Strickler, and depends only on the sign of the top Lyapunov exponent $Λ$ of the linearised PDMP at 0: if $Λ$ < 0, the proportion of infectives in each group converges to zero, while if $Λ$ > 0, the disease becomes endemic. In this paper, we show that the large population asymptotics of X^K also strongly depend on the sign of $Λ$: if negative, then from fixed initial proportions of infectives the disease disappears in a time of order at most log(K), while if positive, the typical extinction time grows at least as a power of K. We prove that in the situation where the origin is accessible for the linearised PDMP, the mean extinction time of X^K is logarithmically equivalent to K^p * , where p * > 0 is fully characterised. We also investigate the quasi-stationary distribution $μ$^K of (X^K , $Ξ$^K) and show that, when $Λ$ < 0, weak limit points of ($μ$^K), K>0 are supported by the extinction set, while when $Λ$ > 0, limit points belong to the (non empty) set of stationary distributions of the limiting PDMP which do not give mass to the extinction set.