Threshold dynamics of a stochastic general SIRS epidemic model with migration

In this study, a stochastic SIRS epidemic model that features constant immigration and general incidence rate is investigated. Our findings show that the dynamical behaviors of the stochastic system can be predicted using the stochastic threshold $ R_0^S $. If $ R_0^S < 1 $, the disease will become extinct with certainty, given additional conditions. Conversely, if $ R_0^S > 1 $, the disease has the potential to persist. Moreover, the necessary conditions for the existence of the stationary distribution of positive solution in the event of disease persistence is determined. Our theoretical findings are validated through numerical simulations.