The ergodicity and extinction of stochastically perturbed SIR and SEIR epidemic models with saturated incidence

In this paper, we include stochastic perturbations into SIR and SEIR epidemic models with saturated incidence and investigate their dynamics according to the basic reproduction number R0. The long time behavior of the two stochastic systems is studied. Mainly, we utilize stochastic Lyapunov functions to show under some conditions, the solution has the ergodic property as R0>1, while exponential stability as R0⩽1. At last, we make simulations to conform our analytical results.