A control treatment for a stochastic epidemic model with relapse and Crowly–Martin incidence

In this paper, we study a stochastic epidemic model with relapse, non linear incidence and a random transmission rate. The existence, uniqueness and boundedness of a positive solution are proved for any positive initial value. Using the Lyapunov analysis, we investigate the asymptotic behaviour of the solution. Mainly, we give sufficient conditions for extinction and persistence of the disease. Then, we propose an optimal control for both deterministic and stochastic models. Finally, we give some numerical illustrations to demonstrate our analytical results.