A regime-switching SIR epidemic model with a ratio-dependent incidence rate and degenerate diffusion

In this paper, we present a regime-switching SIR epidemic model with a ratio-dependent incidence rate and degenerate diffusion. We utilize the Markov semigroup theory to obtain the existence of a unique stable stationary distribution. We prove that the densities of the distributions of the solutions can converge in L(1) to an invariant density under certain condition. Moreover, the sufficient conditions for the extinction of the disease, which means the disease will die out with probability one, are given in two cases. Meanwhile, we obtain a threshold parameter which can be utilized in identifying the stochastic extinction and persistence of the disease. Some numerical simulations are given to illustrate the analytical results.