Abstract
In this paper, we investigate the dynamic behavior and optimal control strategies of a stochastic SIRS (Susceptible-Infectious-Recovered-Susceptible) epidemic model subject to both Markovian switching and Lévy jumps. The model accounts for continuous environmental fluctuations via Brownian motion, finite-state environmental regime transitions via a continuous-time Markov chain, and sudden non-Gaussian environmental perturbations through a compensated Poisson random measure. By constructing suitable stochastic Lyapunov functions, we first establish the existence and uniqueness of a global positive solution. We then derive threshold conditions that dictate either the exponential extinction of the infectious disease or its persistence in the mean. Furthermore, we formulate a stochastic optimal control problem incorporating time-dependent vaccination and treatment interventions to minimize both the burden of infection and the associated operational implementation costs. Using the stochastic Maximum Principle and numerical approximations via an adapted Euler-Maruyama scheme for jump-diffusions, we compute the optimal control paths across various environmental regimes. Numerical simulations confirm our theoretical analysis, illustrating that high-intensity Lévy jumps and regime transitions can suppress disease transmission, while optimal interventions significantly attenuate epidemic peaks under minimal resource expenditure.
