Stochastic Analysis of Double-Strain Epidemics with Vaccination and L\'evy Noise

This paper introduces a novel stochastic SVIR epidemic model to investigate the dynamics of two co-circulating pathogen strains in a vaccinated population. The model incorporates both Gaussian white noise and Lévy jump processes to capture a spectrum of random disturbances, ranging from continuous environmental fluctuations to abrupt, large-scale events such as outbreaks or intervention shocks. We rigorously establish the existence, uniqueness, and positivity of a global solution using Lyapunov functions and stopping time theory. Stochastic threshold conditions, Formula: see text and Formula: see text, are derived to determine the extinction or persistence of the infections. Our analysis shows that when Formula: see text 1, the infections persist endemically. Numerical simulations validate these theoretical results and reveal that Lévy noise induces more pronounced fluctuations compared to Gaussian noise. Sensitivity analysis indicates that transmission rates are key drivers of both infection prevalence and stochastic variability. This study provides a rigorous analytical framework and valuable insights for understanding and managing concurrent epidemics in complex and uncertain environments, highlighting the critical role of stochasticity in shaping disease outcomes.