Predator-Prey Dynamics in the Interaction of HIV Virus with CD4+T Cells

This study analyzes the interaction dynamics between the human immunodeficiency virus (HIV) and CD4+T cells using a predator–prey mathematical model, in which HIV is represented as the predator and CD4+T cells as the prey. The model aims to describe the long-term behavior of the immune system when challenged by the virus. Analytical results show that the system has two equilibrium points: a disease-free equilibrium E1 and an endemic equilibrium E2, whose explicit forms are derived in closed form. Stability analyses of both the disease-free and endemic states are conducted through system linearization, Jacobian matrix formulation, and application of the Routh–Hurwitz criteria. The disease-free state is found to be locally asymptotically stable when the viral elimination rate by the immune system exceeds a specific threshold determined by the balance between viral infection and CD4+T cell production, indicating that under certain conditions the immune system can suppress the virus naturally. The endemic state, representing chronic infection, is stable when the combined effects of viral replication and immune response surpass the rate at which healthy CD4+T cells are lost, implying that the virus can persist within the host. Numerical simulations in Python, using parameter values from previous studies, confirm the coexistence of the virus and host cells under specific conditions. The findings emphasize the influence of viral replication and immune response rates on system stability, offering insights into how HIV can maintain chronic infection without completely depleting CD4+T cells.