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

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.