Dynamics and stability of a within-host HIV-HBV co-infection model with time delays

Human immunodeficiency virus (HIV) and hepatitis B virus (HBV) co-infection is common due to their shared transmission routes. Understanding their interaction within host cells is key to improving treatment strategies. Mathematical models are crucial tools for analyzing within-host viral dynamics and informing therapeutic interventions. This study presents a mathematical framework designed to investigate the interactions and progression of HIV-HBV co-infection within a host. The model captures the distinct biological characteristics of the two viruses: HBV primarily infects liver cells (hepatocytes), while HIV targets CD4 + T cells and can also infect hepatocytes. A system of seven non-linear delay differential equations (DDEs) is formulated 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 time delays: the first represents the latency between the 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 non-negative throughout the system's dynamics. Equilibrium points and their associated threshold parameters are identified, with conditions for existence and stability rigorously derived. 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 behavior. The theoretical findings are corroborated by numerical simulations. We conducted a sensitivity analysis of the basic reproduction numbers, R 0 for HIV and R 1 for HBV. The effects of antiviral treatment and time delays on the HIV-HBV co-dynamics are discussed. Minimum efficacy thresholds for anti-HIV and anti-HBV therapies are Determined, and when drug effectiveness surpasses these levels, the model predicts the full elimination of both viruses from the host. Additionally, the length of the time delay interval plays a role similar to that of antiviral treatment, suggesting a potential strategy for developing drug therapies aimed at extending the time delay period. The results of this study highlight the importance of incorporating time delays in models of dual viral infection and support the development of treatment strategies that enhance therapeutic outcomes by extending these delays.