Stationary distribution of a stochastic SEIR model with infectivity in the incubation period and homestead-isolation on the susceptible under regime switching.

Abstract

This paper is concerned with a stochastic SEIR model with infectivity in the incubation period and homestead-isolation on the susceptible, which is perturbed by white and colour noises. The model has a unique stationary distribution, which reflects the persistence of epidemics over a long period. Using the Has-minskii theorem and constructing stochastic Lyapunov functions with regime switching, we derive an important condition $R_0^s.$ Comparing the expression for $R_0$ and $R_0^s,$ we can see that if there is no environmental noise, then $R_0^s=R_0.$ It ensures the asymptotic stability of the positive equilibrium $E^{\ast }$ of the corresponding deterministic system.