Stochastic dynamical analysis and optimal control of the SIR epidemic model with transmission rate affected by lévy noise

Considering the influence of vaccination immunization time-delay and the unpredictability of natural environment changes, this study focuses on the dynamical behavior and optimal control of the SIR model with stochastic transmission rate and vaccination immunization time-delay. First, the existence and uniqueness of the global positive solution for the proposed model are proved by building the Lyapunov function. Second, the sufficient conditions of the disease extinction and persistence can be obtained. Furthermore, we apply Pontryagin’s stochastic maximum principle to determine the optimal control strategy for minimizing the spread of the disease. Finally, the numerical simulation verified theoretical results based on the tuberculosis data. From these we can find that the extinction and persistence of the disease are significantly affected by the Lévy noise intensity, the spread of the disease can be controlled by disease prevention and patient management in a short time.