Analysis of a stochastic HBV epidemic model driven by Black–Karasinski process: complete characterization of dynamical behaviour and density function

Abstract In this paper, we propose a complex stochastic hepatitis B virus (HBV) epidemic model with vaccination strategy, where random fluctuations of the transmission dynamics of HBV is driven by Black–Karasinski process, for the first time. It is shown that Black–Karasinski process is a biologically and mathematically reasonable assumption compared with existing stochastic modelling approaches. For the deterministic model, the basic reproduction number $\mathscr{R}_{0}$, possible equilibria and related asymptotic stability are studied. Then for the stochastic model, the existence and global positivity of the solution are proved. We further derive two stochastic critical values $\mathscr{R}_{0}^{S}$ and $\mathscr{R}_{0}^{E}$ related to $\mathscr{R}_{0}$ to characterize the long-term behaviour of HBV, and it turns out that (i) the stochastic model has a stationary distribution if $\mathscr{R}_{0}^{S}>1$; (ii) the infected individuals will go extinct exponentially fast when $\mathscr{R}_{0}^{E}<1$; (iii) $\mathscr{R}_{0}^{S}=\mathscr{R}_{0}^{E}=\mathscr{R}_{0}$ if there is no environmental noise. Our results reveal that random fluctuations introduced will facilitate HBV prevalence. Moreover, by analysing the stable structure of the model, we provide a complete classification and explicit approximation for the local density function of the stationary distribution. Finally, some numerical examples are performed to support our theoretical findings. The techniques and methods of analysis in this paper can be applied to many complex high-dimensional epidemic models motivated by Black–Karasinski process.