Stationary distribution, extinction and probability density function of a stochastic SEIV epidemic model with general incidence and Ornstein–Uhlenbeck process

Abstract

Considering the great benefit of vaccination and the variability of environmental influence, a stochastic SEIV epidemic model with mean-reversion Ornstein–Uhlenbeck process and general incidence rate is investigated in this paper. First, it is theoretically proved that stochastic model has a unique global solution. Next, by constructing a series of suitable Lyapunov functions, we obtain a sufficient criterion R0s>1 for the existence of stationary distribution which means the disease will last for a long time. Then, the sufficient condition for the extinction of the infectious disease is also derived. Furthermore, an exact expression of probability density function near the quasi-endemic equilibrium is obtained by solving the corresponding four-dimensional matrix equation. Finally, some numerical simulations are carried out to illustrate the theoretical results.