Dynamical Behavior of a stochastic SIRS epidemic model

TL;DR

This paper analyzes a stochastic SIRS epidemic model with telegraph noise, revealing how environmental switching influences disease persistence, stability, and long-term behavior.

Contribution

It introduces a stochastic SIRS model with telegraph noise, providing conditions for disease persistence and stability, and characterizes the long-term dynamics based on a computed threshold.

Findings

Disease persistence depends on the threshold λ.

System can tend towards endemic or disease-free states.

Long-term behavior is characterized by omega-limit sets.

Abstract

In this paper we study the Kernack - MacKendrick model under telegraph noise. The telegraph noise switches at random between two SIRS models. We give out conditions for the persistence of the disease and the stability of a disease free equilibrium. We show that the asymptotic behavior highly depends on the value of a threshold $\lambda$ which is calculated from the intensities of switching between environmental states, the total size of the population as well as the parameters of both SIRS systems. According to the value of $\lambda$, the system can globally tend towards an endemic case or a disease free case. The aim of this work is also to describe completely the omega-limit set of all positive solutions to the model. Moreover, the attraction of the omega-limit set and the stationary distribution of solutions will be pointed out.