Vaccination strategies in a stochastic $\mathcal{S}\mathcal{I}\mathcal{V}\mathcal{R}$ epidemic model

Effective disease control measures are essential for mitigating epidemic risks. This study introduces a novel stochastic susceptible-infected-vaccinated-recovered $\mathcal{S}\mathcal{I}\mathcal{V}\mathcal{R}$ epidemic model that incorporates white noise in vaccination dynamics. Unlike traditional deterministic models, our stochastic framework accounts for the inherent randomness in real-world disease transmission and the effectiveness of interventions. We rigorously establish the existence and uniqueness of global positive solutions using Lyapunov functions and derive conditions for disease extinction and persistence under stochastic perturbations. A key contribution is the introduction of a stochastic reproduction number $R_0^{\ast }$, which refines classical epidemic thresholds by integrating randomness. Through numerical simulations, we illustrate the impact of stochasticity on disease dynamics, demonstrating that noise can drive disease extinction even in scenarios where deterministic models predict persistence. This study provides a more realistic epidemiological framework for optimizing vaccination strategies under uncertainty, offering significant advances in epidemic modeling and public health policy.