On invariant measures and the asymptotic behavior of a stochastic delayed SIRS epidemic model

TL;DR

This paper analyzes a stochastic delayed SIRS epidemic model with general incidence, establishing existence of solutions, invariant measures, and asymptotic behavior depending on the basic reproduction number, supported by numerical validation.

Contribution

It introduces a stochastic epidemic model with delay and general incidence, proving existence, uniqueness of solutions, and invariant measures, and analyzing long-term behavior based on R0.

Findings

Existence and uniqueness of global positive solutions.

Existence of invariant measures via Krylov-Bogoliubov method.

Asymptotic behavior differs for R0<1 and R0>1, with validation through simulations.

Abstract

In this paper, we consider a stochastic epidemic model with time delay and general incidence rate. We first prove the existence and uniqueness of the global positive solution. By using the Krylov-Bogoliubov method, we obtain the existence of invariant measures. Furthermore , we study a special case where the incidence rate is bilinear with distributed time delay. When the basic reproduction number R0 < 1, the analysis of the asymptotic behavior around the disease-free equilibrium E0 is provided while when R0 > 1, we prove that the invariant measure is unique and ergodic. The numerical simulations also validate our analytical results.