HIV dynamics in a periodic environment with general transmission rates

In the current study, we present a mathematical model for human immunodeficiency virus type-1 (HIV-1) transmission, incorporating Cytotoxic T-Lymphocyte immune impairment within a seasonal environment. The model divides the infected cell compartment into two sub-compartments: latently infected cells and productively infected cells. Additionally, we consider three possible routes of infection, allowing HIV to spread among susceptible cells via direct contact with the virus, latently infected cells, or productively infected cells. The system is analyzed, and the basic reproduction number is derived using an integral operator. We demonstrate that the HIV-free periodic trajectory is globally asymptotically stable if ${\mathcal{R}}_0<1$, while HIV persists when ${\mathcal{R}}_0>1$. Several numerical simulations are provided to validate the theoretical results.