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

We consider an epidemic SIS model described by a multitype birth-and-death process in a randomly switching 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∈N∗, and the number of infectives vanishes almost surely in finite time. We prove that, as K→∞, the process composed of the proportions of infectives of each type XK and the state of the environment ΞK, converges to a piecewise deterministic Markov process (PDMP) given by a system of randomly switching ODEs. The long term behaviour of this PDMP has been previously investigated by Benaïm and Strickler, and depends only on the sign of the top Lyapunov exponent Λ of the linearised PDMP at 0: if Λ0, the disease becomes endemic. In this paper, we show that the large population asymptotics of XK 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 XK is logarithmically equivalent to Kp∗, where p∗>0 is fully characterised. We also investigate the quasi-stationary distribution μK of (XK,ΞK) and show that, when Λ0 are supported by the extinction set, while when Λ>0, limit points belong to the (nonempty) set of stationary distributions of the limiting PDMP which do not give mass to the extinction set.