A Nonlinear Caputo Fractional HIV-1 Model with CTL Saturation and Antiretroviral Intervention: Stability and Approximation Methods

This study introduces a fractional-order HIV-1 infection model formulated with the Caputo derivative, combining the effects of antiretroviral therapy and a saturating cytotoxic T lymphocyte (CTL) immune response. The fractional formulation allows the inclusion of memory-dependent viral dynamics and the saturation function provides a biologically consistent representation of immune regulation. The basic reproduction number R<sub>0</sub> is derived and used to determine the threshold behavior of the system. A theoretical analysis is carried out to prove existence and uniqueness of solutions and to investigate both local and global stability of the disease-free equilibrium. The endemic equilibrium is also derived to offer a deeper understanding of long-term infection dynamics. To obtain approximate solutions for the nonlinear system the semi analytical methods, the Differential Transform Method (DTM), Adomian Decomposition Method (ADM), and Homotopy Perturbation Method (HPM), are applied. A fractional predictor–corrector scheme is applied and compared with the semi analytical solutions. Numerical experiments show the influence of the fractional order, therapeutic effectiveness and immune parameters on disease evolution. The results indicate that decreasing the fractional order significantly alters the transient dynamics of viral load and CD4+ T-cell populations, demonstrating that fractional-order modeling provides a more flexible and realistic framework for describing HIV-1 dynamics and evaluating treatment effects.