Analysis of a Stochastic HBV Model With Saturated Incidence Rate and Incomplete Immunity

Numerous mathematical models have been developed to study the transmission dynamics of Hepatitis B. However, many existing models overlook two critical characteristics of HBV: (1) both acutely and chronically infected individuals are infectious but exhibit different levels of infectivity and (2) the immunity conferred by vaccination differs from that acquired through recovery. Furthermore, deterministic models often fail to account for the significant impact of random elements on disease dynamics. To address these gaps, this study establishes a stochastic HBV model with a saturated incidence rate and incomplete immunity. The model stratifies the population into five compartments: susceptible individuals, acutely infected individuals, chronically infected individuals, recoverers, and vaccinators. It specifically captures the distinct transmission dynamics of acute and chronic Hepatitis B using saturated incidence rates and incorporates differential immunity between recoverers and vaccinators, aligning more closely with observed epidemiology. Leveraging the stochastic comparison theorem, we derive an extinction threshold for the disease, which defines the basic reproduction number for the stochastic system. Furthermore, by applying Ito^’s formula and constructing a suitable Lyapunov function, we obtain sufficient conditions for the existence of a stationary distribution, indicating disease persistence. Numerical simulations confirm these analytical results. Our findings demonstrate that intervention strategies aimed at reducing the infection rate among susceptible individuals and increasing vaccination coverage for both newborns and susceptible adults are effective in controlling the scale of Hepatitis B. The results also highlight how random noise influences HBV transmission, providing insights for designing effective intervention strategies and advancing the overall control of Hepatitis B.