A mathematical model for the dynamics of HIV-1 during the typical course of infection

The level of HIV in blood, referred to as virus load is recognized to be the single best predictor of HIV disease progression (and death) in HIV-1 infected individuals. CD4(+) T cells are the principal target and site of replication of HIV-1 in vivo, where a vast majority of these cells normally exist in a quiescent resting state. However, the replication cycle of HIV can only be completed in activated, but not resting CD4(+) T cells since HIV provirus, which is synthesized in the cytoplasm fails to enter the nucleus and integrate in resting cells. The provirus retained in the resting cell cytoplasm is eventually destroyed. This paper presents a new mathematical model, based upon these and other critical factors, for predicting the number of cell-free HIV in the blood during typical course of HIV infection. The model exhibits two steady states, a trivial steady state (virus-free) and an endemically-infected steady state (with virus and virus-infected CD4(+) T cells). A linearized stability analysis was carried out to determine the stability of these states. A competitive Gauss-Seidel-like implicit finite-difference method was developed and used for the solution of the resulting system of first-order, non-linear, initial-value problem(IVP). Unlike some well-known methods in the literature. this easy-to-implement method gives chaos-free numerical results when large time-steps are used in the numerical simulations. Numerical simulations suggest that (i) during primary HIV infection, the anti-HIV cytotoxic T cell activity plays a crucial role in suppressing virus load and the loss of CD4(+) T cells, (ii) at the final stages of HIV disease, which is commonly associated with severe CD4 lymphopenia, cells other than CD4(+) T cells may be responsible for HIV production and the final surge of viremia.