Sharp Threshold Dynamics for a Stochastic SIRS Model with General Nonlinear Incidence

Abstract

This paper studies the dynamics of a stochastic SIRS epidemic model with a general nonlinear incidence rate. The model incorporates environmental noise through white noise perturbations. We first establish basic properties of the total population, including boundedness, recurrence, and the existence of a stationary distribution. Then, under suitable hypotheses on the incidence function, we derive conditions for disease extinction and persistence. The theoretical results are illustrated by a corollary providing explicit lower bounds for the time averages of the infected, recovered, and susceptible compartments. Numerical simulations are used to confirm the theoretical threshold dynamics and demonstrate the practical applicability of the model.