Dynamical analysis of a stochastic epidemic HBV model with log-normal Ornstein–Uhlenbeck process and vertical transmission term

Considering the transmission rate perturbed by log-normal Ornstein–Uhlenbeck process, we develop a stochastic HBV model with vertical transmission term. For higher-dimensional deterministic system, the local asymptotic stability of the endemic equilibrium is given by proving the global stability of the corresponding linearized system. For stochastic system, the existence of stationary distribution is obtained by constructing several suitable Lyapunov functions and using the ergodicity of the Ornstein–Uhlenbeck process and the critical value corresponding to the basic reproduction number for determined system is derived, which means the persistence of the disease. And sufficient conditions for disease extinction are given. Furthermore, by solving five-dimensional Fokker–Planck equation, the exact expression of the probability density function near the quasi-equilibrium is provided to reveal the statistical properties. In the end, numerical simulations illustrate our theoretical results and exhibit the trends of the critical values for persistence and extinction of diseases along with the change of noise intensity and reversion speed.