Stochastically asymptotically stability of the multi-group SEIR and SIR models with random perturbation

In this paper, we discuss the multi-group SEIR and SIR model with random perturbation, allowing random fluctuation around the endemic equilibrium of deterministic SEIR and SIR models. We prove that the endemic equilibriums of the two models are both stochastic asymptotically stable. In addition, the stability condition is obtained by the construction of the Lyapunov function and graph theory. Finally, numerical simulations are presented to illustrate our mathematical findings.