Threshold dynamics and ergodicity of an SIRS epidemic model with Markovian switching

TL;DR

This paper analyzes a stochastic SIRS epidemic model with Markovian switching, establishing threshold conditions for disease eradication or persistence, and generalizing ergodic analysis methods to broader systems.

Contribution

It introduces a generalized method for analyzing ergodicity in population systems, applicable regardless of environmental regimes or system dimension.

Findings

Disease eradication if R0<1 almost surely

Existence of stationary distribution when R0>1

Convergence to stationary measure under mild conditions

Abstract

This paper studies the spread dynamics of a stochastic SIRS epidemic model with nonlinear incidence and varying population size, which is formulated as a piecewise deterministic Markov process. A threshold dynamic determined by the basic reproduction number $\mathcal{R}_{0}$ is established: the disease can be eradicated almost surely if $\mathcal{R}_{0}<1$, while the disease persists almost surely if $\mathcal{R}_{0}>1$. The existing method for analyzing ergodic behavior of population systems has been generalized. The modified method weakens the required conditions and has no limitations for both the number of environmental regimes and the dimension of the considered system. When $\mathcal{R}_{0}>1$, the existence of a stationary probability measure is obtained. Furthermore, with the modified method, the global attractivity of the $\Omega$-limit set of the system and the convergence in total variation to the stationary measure are both demonstrated under a mild extra condition.