Stochastic Modeling and Analysis of Hepatitis B Virus Dynamics Under Transportation and Lévy Noise

Hepatitis B virus (HBV) remains a global health threat, especially under the influence of human mobility and environmental uncertainty. This paper proposes a stochastic SVAZR epidemic model that incorporates transportation dynamics and Lévy noise to account for both continuous fluctuations and sudden random shocks. The model explicitly includes population movement between two regions and mobility-related transmission. Using Lyapunov functions and stopping time arguments, we prove the existence and uniqueness of a global positive solution. A stochastic basic reproduction number is derived, establishing threshold conditions for disease extinction and persistence. Specifically, the disease dies out almost surely when this threshold is below one, whereas persistence occurs under suitable conditions. Numerical simulations support the theoretical results and illustrate the impact of transportation and Lévy noise on HBV dynamics. Our findings highlight the critical role of mobility and stochastic perturbations in shaping epidemic behavior and offer insights for designing effective disease control strategies.