Stationary distribution and extinction of SIR model with nonlinear incident rate under Markovian switching

Taking into account of both white and colored noises, a stochastic epidemic model with nonlinear incident rate under regime switching is formulated. Based on this model, we investigate the dynamic behaviors such as ergodicity and extinction of the SIR model with Beddington-DeAngelis incidence rate and Markov switching. First, we study the existence of the unique positive solution of system (1.3). Secondly, by using Lyapunov functions, we prove that the system has a ergodic stationary distribution under certain sufficient conditions. Then, we obtain the conditions for extinction. Finally, numerical simulations are employed to illustrate our theoretical analysis.