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Epidemiology is a field for evaluating the sources of illness and clinical effects in communities. Therefore, individuals are viewed globally in order to investigate the dynamics and the control of an infectious disease. However, any governmental spending on a pharmaceutical control policy in times of epidemics can lead to a state budget imbalance. For this matter, we study in this paper stochastic stability and optimization for a treatment strategy, where the infectious disease is characterized by the relapse aspect. We provided a stochastic threshold for extinction and persistence in mean. Then, we established a priori estimates of the optimal control, where we obtained sufficient and necessary conditions for the considered treatment problem of the stochastic epidemic model by applying the maximum condition on the Hamiltonian function and the Ekeland principle. We provide a characterization of stochastic optimal control and sensitivity analysis to discuss its impact on the basic reproduction number with numerical illustrations for the system to carry out our theoretical investigations.

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