In this paper, a stochastic epidemic model with logistic growth is discussed. Based on stochastic differential equation theory, stochastic control method, etc., the properties of the solution of the model nearby the epidemic equilibrium of the original deterministic system are investigated, the sufficient conditions to ensure the stability of the disease-free equilibrium of the model are established, and two event-triggered controllers to drive the disease from endemic to extinction are constructed. The related results show that the disease becomes endemic when the transmission coefficient exceeds a certain threshold. Furthermore, when the disease is endemic, we can drive the disease from endemic to extinction by choosing suitable event-triggering gains and control gains. Finally, the effectiveness of the results is illustrated by a numerical example.