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This paper develops a mathematical model to investigate the Human Immunodeficiency Virus (HIV) infection dynamics. The model includes two transmission modes (cell-to-cell and cell-free), two adaptive immune responses (cytotoxic T-lymphocyte (CTL) and antibody), a saturated CTL immune response, and latent HIV infection. The existence and local stability of equilibria are fully characterized by four reproduction numbers. Through sensitivity analyses, we assess the partial rank correlation coefficients of these reproduction numbers and identify that the infection rate via cell-to-cell transmission, the number of new viruses produced by each infected cell during its life cycle, the clearance rate of free virions, and immune parameters have the greatest impact on the reproduction numbers. Additionally, we compare the effects of immune stimulation and cell-to-cell spread on the model's dynamics. The findings highlight the significance of adaptive immune responses in increasing the population of uninfected cells and reducing the numbers of latent cells, infected cells, and viruses. Furthermore, cell-to-cell transmission is identified as a facilitator of HIV transmission. The analytical and numerical results presented in this study contribute to a better understanding of HIV dynamics and can potentially aid in improving HIV management strategies. [ABSTRACT FROM AUTHOR]

The dynamics of a model describing the human immunodeficiency virus (HIV) infection with cytotoxic T-lymphocyte (CTL), antibodies and two saturated rates is investigated and studied in this paper. The model includes five nonlinear differential equations describing the evolution of uninfected cells, infected ones, free HIV viruses, CTL immune response and antibodies. This model includes also two treatments that represent the efficiency of drug treatment in inhibiting viral production and preventing new infections. Existence, positivity and boundedness of solutions are given. Existence of the optimal control pair is established and the Pontryagin's maximum principle is used to find an optimal treatment strategy that maximizes the number of uninfected CD4+ T cells as well as cytotoxic T-lymphocyte and antibody immune responses. Finally, the optimality system is derived and solved numerically. Results show that administrating good therapy maximizes the amount of healthy CD4+ T cells and deceases considerably the viral load and the infected cells. [ABSTRACT FROM AUTHOR]