Qualitative Dynamics and Hopf Bifurcation in a Hepatitis B Delay Model with Immune Interaction - Sireesha Devi; Ranjith Kumar (Jambura Journal of Biomathematics (JJBM), 2026)
In this research, we build a three-compartment delay differential equation (DDE) model to examine the intra host dynamics of Hepatitis B Virus (HBV) infection. The model includes a biologically based intracellular delay that represents the eclipse phase of infection and a non monotone saturation incidence function that shows how the immune system controls infection dynamics. The suggested formulation takes into account saturation effects that come from immune depletion and limited cellular resources. This is different from standard HBV models that assume bilinear or simple saturation incidence rates. The system is made up of healthy hepatocytes, contaminated hepatocytes, and immune cells that kill cells. The fundamental reproduction number ${R_{0}}$, is calculated and used to find the minimum level of infection that has to be present for it to continue. To study the local stability of the equilibria, we linearize the system and get the transcendental characteristic equation that goes with it. Hopf bifurcation study concerning the delay parameter $τ $ indicates the presence of crucial delay values that distinguish stable steady states from oscillatory dynamics. Sensitivity analysis shows that parameters linked to transmission have the biggest effect on the fundamental reproduction number ${R_{0}}$, while characteristics related to delay mostly affect the system's stability and the appearance of oscillatory dynamics. Numerical simulations, encompassing time-series graphs and bifurcation diagrams, demonstrate that elevating the delay over a crucial threshold can trigger persistent oscillations in infected and immune cell populations, mirroring the recurring fluctuations seen in chronic HBV infection. These findings underscore the significant influence of intracellular delay and immune-regulated infection mechanisms on the long-term dynamics of HBV, offering a theoretical foundation for the advancement of HBV infection models.