This paper constructs and analyzes a stochastic SAIRV epidemic model that integrates environmental viral load impacts and multi-path transmission dynamics—encompassing both asymptomatic and symptomatic carriers. To accurately capture the complexity of disease spread, Lévy processes are employed to model continuous environmental fluctuations and abrupt population-level disturbances, addressing both gradual and sudden stochastic variations. Theoretically, we first establish the existence and uniqueness of a globally positive solution, ensuring the mathematical well-posedness of the model. We then derive sufficient conditions for the extinction and persistence of the disease, emphasizing the critical roles of multi-route transmission, nonlinear recovery mechanisms, and stochastic perturbations in driving these outcomes. Numerical simulations validate these theoretical findings, and an optimal control strategy, developed using the stochastic Pontryagin maximum principle, is proposed to evaluate the impact of environmental interventions on disease spread. Collectively, these results deepen the understanding of stochastic epidemic evolution and provide a robust theoretical foundation for designing effective control measures.
Li et al. (Mon,) studied this question.