A stochastic differential equation SIS model on network under Markovian switching

TL;DR

This paper analyzes a stochastic SIS epidemic model on networks with Markovian regime-switching, establishing conditions for extinction and permanence, and exploring the system's long-term behavior and invariant measures.

Contribution

It introduces a novel stochastic SIS model with Markovian switching on networks, providing new theoretical results on extinction, permanence, and ergodicity.

Findings

Conditions for almost sure extinction identified

Criteria for stochastic permanence established

Existence of invariant measures proven

Abstract

We study a stochastic SIS (susceptible-infected-susceptible) epidemic dynamics on network, under the effect of a Markovian regime-switching. We first prove the existence of a unique global positive solution, and find a positive invariant set for the system. Then, we find sufficient conditions for a.s. extinction and stochastic permanence, showing also their relation with the stationary probability distribution of the Markov chain that governs the switching and with the network topology. We provide an asymptotic lower bound for the time average of the sample-path solution under the conditions ensuring stochastic permanence. From this bound, we are able to prove the existence of an invariant probability measure if the condition of stochastic permanence holds. Under a different condition, we prove the positive recurrence and the ergodicity of the regime-switching diffusion.