Permanence and Extinction for the Stochastic SIR Epidemic Model

TL;DR

This paper investigates the stochastic SIR epidemic model with general incidence functions and multiple noise sources, establishing conditions for disease extinction or persistence and analyzing convergence rates.

Contribution

It provides a comprehensive analysis of extinction and permanence conditions for a stochastic SIR model with multi-noise perturbations, including convergence rate results.

Findings

Derived necessary and sufficient conditions for epidemic extinction.

Established criteria for epidemic permanence.

Demonstrated convergence rates of solutions through numerical examples.

Abstract

The aim of this paper is to study the stochastic SIR equation with general incidence functional responses and in which both natural death rates and the incidence rate are perturbed by white noises. We derive a sufficient and almost necessary condition for the extinction and permanence for an epidemic system with multi noises \begin{equation*} \begin{cases} dS(t)=\big[a_1-b_1S(t)-I(t)f(S(t),I(t))\big]dt + \sigma_1 S(t) dB_1(t) -I(t)g(S(t),I(t))dB_3(t),\\ dI(t)=\big[-b_2I(t) + I(t)f(S(t),I(t))\big]dt + \sigma_2I(t) dB_2(t) + I(t)g(S(t),I(t))dB_3(t). \end{cases} \end{equation*} Moreover, the rate of all convergences of the solution are also established. A number of numerical examples are given to illustrate our results