Periodic solution and ergodic stationary distribution of two stochastic SIQS epidemic systems

This paper proposes two stochastic SIQS epidemic models with periodic parameters and Markov switching. We first prove that the stochastic non-autonomous periodic system has a nontrivial positive periodic solution by using the Khasminskii's theory. Then the sufficient conditions for extinction of the disease are obtained. Furthermore, we construct suitable stochastic Lyapunov functions with regime switching to prove the existence of ergodic stationary distribution of the stochastic SIQS epidemic model. At last, some rigorous numerical simulations are presented to illustrate our theoretical results.