Threshold Dynamics of Stochastic SIS Epidemic Models with Logistic Recruitment Rate

This study explores an SIS epidemic model that integrates logistic growth and a saturated incidence rate among susceptible individuals. The primary objective is to examine the long-term behavior of the stochastic system. For this purpose, a key parameter, [Formula: see text], is introduced to differentiate the system dynamics into two cases. In scenarios where [Formula: see text], the model predicts the eventual extinction of the disease. In contrast, for [Formula: see text], we show an ergodic stationary distribution. Numerical simulations validate our theoretical conditions for extinction, persistence, and the existence of a stationary distribution.