Dynamical behavior of stochastic SIRS model with two different incidence rates and Markovian switching

In this paper, we discuss SIRS models with two different incidence rates and Markovian switching. First, we consider that the parameters are perturbed by random environment modulated by Markovian switching. The segment method is used to prove that the model has a unique solution and the estimate of the solution is provided. The threshold values for determining extinction or persistence in mean of diseases are presented by theoretical analysis and some inequalities techniques. Furthermore, some results reveal that stochastic disturbances can suppress the disease outbreak. Because of regime switching, the diseases will be extinct (or persistent) although they might be persistent (or extinct) in some certain environments. Then, the model in which incidence rate functions are perturbed by random environment is also discussed and the values to judge the disease extinction are obtained. At last, a few examples are set to illustrate these interesting phenomena, and their simulations have been carried out to verify our theoretical outcomes.