Bifurcation analysis of a delayed stochastic HBV epidemic model: Cell-to-cell transmission

In the present study, we develop a model for hepatitis B virus (HBV) that integrates cell-to-cell transmission of the virus, CTL immunological response, delay effects, and the effect of fluctuating environmental noise. The inheritance randomness of the system shifting the system from a deterministic to another system consists of stochastic differential equations (SDEs). After model formulation, the dynamical aspects of the model are investigated by using the key threshold quantity R0. If the threshold value R0 is assumed less than unity, then it is proven that the solution trajectories are fluctuating around the disease-free state (DFS) of the associated deterministic model. The fluctuation of the trajectories around the DFS explains the phenomenon of extinction of the stochastic models. It is also proved that the trajectories of the SDE model oscillate around the endemic state of the underlying ODE system when R0s>1 explains the sure persistence theory of the proposed model. One notable finding of the present manuscript is the positive association between the fluctuating amplitude and the intensities of the white noises. This understanding can provide useful control tools for managing the worst dynamic(s) of the infection. The exact value of the parameter for time delay determines the stability of the model under consideration. By analyzing the time-delayed parameter, it was identified that the model is stable by keeping the value of the parameter below the crucial value that causes the Hopf bifurcation. This conclusion is supported by the numerical examples having epidemiological significance. Furthermore, the findings suggest that stochastic environmental disturbance can impact the spread of infectious illnesses. Notably, the high levels of noise can help to slow down the spread of epidemics in the population.