This paper focuses on the long time dynamics for a class stochastic SEI model with standard incidence and infectivity in incubation period. Firstly, we investigate a unique global positive solution almost surely for any positive initial value. Secondly, we obtain a unique stationary measure and the extinction condition of the epidemic based on the technique of Lyapunov function and inequalities. Thirdly, we explore the asymptotic behavior of the solutions around equilibriums of the corresponding deterministic model from different aspects. Finally, we establish some numerical simulations to illustrate the main presented results.