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

TL;DR

This paper analyzes the large population behavior of a multitype stochastic SIS epidemic model in a randomly switched environment, showing how the epidemic's extinction or persistence depends on the sign of a Lyapunov exponent and characterizing the asymptotic extinction times.

Contribution

It establishes the asymptotic behavior of the epidemic model as population size grows, linking extinction times and distributions to the sign of the Lyapunov exponent of the linearized PDMP.

Findings

Extinction time scales as log(K) when Lyapunov exponent is negative.

When Lyapunov exponent is positive, extinction time grows at least as a power of K.

Limit distributions depend on the sign of the Lyapunov exponent, with support on extinction set or stationary distributions of the PDMP.

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 $\Xi$^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{\"i}m and Strickler, and depends only on the sign of the top Lyapunov exponent $\Lambda$ of the linearised PDMP at 0: if $\Lambda$ < 0, the proportion of infectives in each group converges to zero, while if $\Lambda$ > 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 $\Lambda$: 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 $\mu$^K of (X^K , $\Xi$^K) and show that, when $\Lambda$ < 0, weak limit points of ($\mu$^K), K>0 are supported by the extinction set, while when $\Lambda$ > 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.