Qualitative Study of a Nonlinear Stochastic SIRS Epidemic System

This article examines the qualitative dynamics of an SIRS epidemic model with saturated incidence in random environments. First, we show that the global positive solutions of the stochastic model belongs to a bounded positively invariant set with probability one. Then we obtain the threshold between persistence and extinction of diseases. Furthermore, we establish certain asymptotic results regarding large time behavior. The results show that the introduction of noise in the deterministic SIRS model can change the properties of the epidemic model significantly.