In this paper, we proposed and studied a stochastic SEI (Susceptible-Exposed-Infectious) epidemic model with logistic growth and nonlinear perturbation. By constructing a series of suitable stochastic Lyapunov functions, we derived sufficient conditions for the existence of a unique ergodic stationary distribution. Furthermore, we obtained criteria which ensured the disease approached extinction at an exponential rate. At the same time, under these criteria, the distribution of susceptible individuals converged weakly to a unique invariant probability measure. The numerical simulations supported the theoretical results.