Mathematical Modeling of HIV Treatment with Delayed Drug Response

Human Immunodeficiency Virus (HIV) remains a major public health concern despite significant advances in antiretroviral therapy (ART), particularly dolutegravir-based regimens known for strong viral suppression. However, the therapeutic effect of ART is not instantaneous, as biological and pharmacological processes introduce delays between treatment initiation and observable viral load reduction. Ignoring such delayed drug response may oversimplify infection dynamics and affect the accuracy of predictive models. In this study, a delay differential equation (DDE) model of within-host HIV infection under antiretroviral therapy is developed, incorporating both intracellular infection delay and delayed drug-response effects. The model describes the interaction among healthy CD4+ T cells, infected CD4+ T cells, and free viral particles. Fundamental mathematical properties of the system are established, and the basic reproduction number is derived to characterize infection persistence. Equilibrium points are determined, and stability analysis is performed to identify conditions for viral suppression. The results demonstrate that delayed drug response influences the threshold dynamics of the system and affects the rate of viral load decay and immune recovery. In particular, treatment reduces the effective reproduction number, and viral clearance is achieved when the reproduction number is less than unity. These findings emphasize the importance of incorporating biologically realistic delay mechanisms in mathematical models of HIV treatment dynamics. Aims: The aim of this study is to examine the biological implications of delayed drug response in HIV treatment by developing a mathematical model and analyzing how treatment delay influences viral suppression and CD4+ T-cell dynamics. Model Design: A deterministic compartmental delay differential equation (DDE) model of within-host HIV treatment dynamics. Place and Duration of Study: Department of Mathematics, Federal University of Petroleum Resources, Effurun, Delta State, Nigeria. April 2025 and January 2026. Methodology: A deterministic delay differential equation (DDE) model was formulated to describe within-host HIV dynamics under delayed drug response. The model consists of four state variables representing healthy CD4+ T cells, infected CD4+ T cells, viral load, and drug concentration. Results: The basic reproduction number was derived as R0 = \(\frac{\beta\lambda{N}}{cd}\). Analysis shows that the disease-free equilibrium is locally asymptotically stable when R0 < 1 1and unstable when R0 > 1. The endemic equilibrium exists only when R0 > 1. Numerical simulations using parameter values listed in Table 1 yielded R0 = 1.82in the absence of treatment, indicating persistent infection. Following drug administration, the effective reproduction number decreased to R0 treated = 0.74, resulting in viral suppression and immune recovery. Conclusion: A delay differential equation model of HIV treatment incorporating delayed drug response was developed and analyzed. The study demonstrates that the basic reproduction number governs infection persistence and that treatment efficacy can drive the system below the epidemic threshold. Delayed drug response influences transient viral dynamics but does not alter the fundamental stability condition. These findings provide theoretical insight into optimizing HIV treatment strategies.