Global dynamics analysis of a stochastic SIRS epidemic model with vertical transmission and different periods of immunity

In this work, we analyse a stochastic susceptible-infected-recovered-susceptible (SIRS) epidemic model with vertical transmission and different periods of immunity. This model has a global positive solution. Firstly, we establish sufficient conditions for extinction and persistence in the mean of a disease. Then, we prove the global stability of the system under a suitable condition of perturbation intensity. In the case of the non-autonomous system, we show that there exists at least one positive periodic solution. Finally, some numerical examples are introduced to show the validity of our results.