Stationary distribution extinction and optimal control for the stochastic hepatitis B epidemic model with partial immunity

In this paper, a stochastic model (with random noise transmission) is designed. The model possesses substantial potential to describe the dynamical behavior of the Hepatitis B (HBV) virus and it's control by applying the strategy of vaccinating an offspring. The number of basic reproductive is calculated and proved that the system holds some sharp threshold properties. It is investigated that the model has a bounded, unique and positive solution subject to initial positive data. Furthermore, the stability of the investigated system has been presented by using stochastic Lyapunov functional theory. Stationary distribution and extinction of the infection are examined by providing sufficient conditions. To control the spread of the disease through some external measures, we used optimal control theory and analyzed stochastic as well as deterministic control problems. For further verification of the obtained analytical results, additional graphical solutions have been presented for the ease of understanding. This study may provide a strong theoretical basis for understanding worldwide chronic infectious diseases.