Abstract: This work presents an advanced and detailed analysis of the mechanisms of hepatitis B virus (HBV) propagation in an environment characterized by variability and stochasticity. Based on some biological features of the virus and the assumptions, the corresponding deterministic model is formulated, which takes into consideration the effect of vaccination. This deterministic model is extended to a stochastic framework by considering a new form of disturbance which makes it possible to simulate strong and significant fluctuations. The long-term behaviors of the virus are predicted by using stochastic differential equations with second-order multiplicative $\alpha$-stable jumps. By developing the assumptions and employing the novel theoretical tools, the threshold parameter responsible for ergodicity (persistence) and extinction is provided. The theoretical results of the current study are validated by numerical simulations and parameters estimation is also performed. Moreover, we obtain the following new interesting findings: (a) in each class, the average time depends on the value of $\alpha$; (b) the second-order noise has an inverse effect on the spread of the virus; (c) the shapes of population densities at stationary level quickly changes at certain values of $\alpha$. The last three conclusions can provide a solid research base for further investigation in the field of biological and ecological modeling.

Key words: HBV model, nonlinear perturbation, probabilistic bifurcation, long-run forecast, numerical simulation