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In this paper, we formulate the intricate process of Human Immunodeficiency Virus (HIV) infection in a mathematical model. In the formulation of the model, the density of CD 4 + T-cells is categorized into healthy and infected classes while the density of free HIV viruses in the human host's blood is considered a separate class. It is reported that when HIV enters a human's body, it infects a large amount of CD 4 + T-cells and unbalance the supply of new cells from the thymus. Therefore, we incorporate variable source term relying on the viral load for the supply of new cells instead of using a fixed value for the supply of these cells. We implement a continuous Galerkin Petrov time discretization scheme, particularly cGP(2)-scheme to visualize the dynamical behavior of the mentioned model and a detailed description of the effects of different physical parameters of interest are depicts graphically and discuss that how the level of healthy, infected CD 4 + T-cells and free HIV particles varies concerning the emerging parameters in the model. In the proposed cGP(2)-method, two unknowns terms can be found by solving a 2 × 2 block system. This method is accurate of order 3 in the whole time interval and shows even superconvergence of order 4 in the discrete-time points. Furthermore, determine the solution of the model utilizing the fourth-order Runge-Kutta (RK4)-method and find out the absolute errors between the solutions obtained from both approaches. Finally, the validity and reliability of the proposed scheme are verified by comparing the numerical and graphical results with the RK-4 method. We examine the accuracy of the cGP(2)-sheme and observe that it produces more accurate results at relatively larger step sizes as in comparison to RK-4 scheme. Correspondence between the findings of cGP(2)-method and RK4-method reveal that the new technique is a promising tool for obtaining approximate solutions to the nonlinear systems of real-world problems. [ABSTRACT FROM AUTHOR]

This paper proposes and analyzes an adaptive immunity HIV infection model. The model describes the interaction between healthy CD 4 + T cells, silent (latent) infected cells, active infected cells, free HIV particles, Cytotoxic T lymphocytes (CTLs) and antibodies. The healthy CD 4 + T cells can be infected when they are contacted by one of the following: (i) free HIV particles, and this is known as virus-to-cell (VTC) transmission (ii) silent infected cells, and we call this mode of infection as silent HIV-infected cell-to-cell (CTC) transmission, and (iii) active infected cells, and we call this mechanism active HIV-infected CTC transmission. The incidence rates of the healthy CD 4 + T cells with free HIV particles, silent infected cells, and active infected cells are given by general functions. Moreover, the production/proliferation and removal/death rates of all compartments are represented by general functions. The model is an improvement of the existing HIV infection models which have neglected the incidence between the silent infected cells and healthy CD 4 + T cells. We first show that the model is well-posed. Then, we show that the model has five equilibria and their existence are governed by five threshold parameters. Under a set of conditions on the general functions and the threshold parameters, we have proven the global asymptotic stability of all equilibria by using Lyapunov's method. We have illustrated the theoretical results via numerical simulations. We have studied the effect of CTC transmission on the dynamical behavior of the system. We have shown that inclusion of CTC transmission decreases the concentration of the healthy CD4 + T cells and increases the concentrations of the infected cells and free HIV particles. [ABSTRACT FROM AUTHOR]