Dynamics and Stability of a Within-Host HIV-HBV Co-Infection Model with Time Delays

Ahmed Elaiw^*Abdulaziz S. AlhmadiAatef Hobiny

• King Abdulaziz University, Jeddah, Saudi Arabia

This study introduces a mathematical framework developed to investigate the interactions and progression of HIV-HBV co-infection within a host. The model reflects the biological characteristics of the two viruses: HBV predominantly infects liver cells (hepatocytes), while HIV primarily targets CD4 + T cells and also has the ability to infect hepatocytes. A set of seven nonlinear delay differential equations (DDEs) is developed to represent the dynamic interactions among uninfected and virus-infected hepatocytes, uninfected and HIV-infected CD4 + T cells, as well as circulating HIV and HBV particles. The model incorporates two biologically significant delays: the first reflects the time interval between initial infection and the onset of productive infection in host cells, while the second accounts for the maturation duration of newly produced virions before they become infectious. The model's mathematical consistency is verified by showing that its solutions remain bounded and nonnegative throughout the system's dynamics.Equilibrium points, along with their associated threshold parameters, are identified, and the conditions for their existence and stability are thoroughly derived.The global stability of the equilibria is established through the application of carefully designed Lyapunov functionals in conjunction with Lyapunov-LaSalle asymptotic stability theorem, ensuring a rigorous and comprehensive analysis of the system's long-term behav-1 ior. The theoretical findings are corroborated by numerical simulations. We perform sensitivity analysis of the basic reproduction numbers, R 0 for HIV and R 1 for HBV.The effects of antiviral treatment and time delay on the HIV-HBV co-dynamics are disscussed. The minimum required efficacies for anti-HIV and anti-HBV therapies are determined. When drug effectiveness surpasses these levels, the model predicts the full elimination of both viruses from the host. On the other hand, the length of the time delay interval plays a role similar to that of antiviral treatment. This suggests a potential strategy for developing drug therapies aimed at extending the time delay period. The results of this study highlight the importance of considering time delays in dual viral infection models and support the development of treatments that extend the delay period and improve therapeutic effectiveness.