Dynamical Behavior of a Stochastic Epidemiological Model: Stationary Distribution and Extinction of a SIRS Model with Stochastic Perturbations

This paper deals with a new epidemiological model of SIRS with stochastic perturbations. The primary objective is to establish the existence of a unique non-negative nonlocal solution. Using the basic reproduction number $\mathscr{R}_0$ derived from the associated deterministic model, we demonstrate the existence of a stationary distribution in the stochastic model. In addition, we study the fluctuation of the unique solution of the deterministic problem around the disease-free equilibrium under certain conditions. In particular, we reveal scenarios where random effects induce disease extinction, contrary to the persistence predicted by the deterministic model. The theoretical insights are complemented by numerical simulations, which provide further validation of our findings.