Modelling the Drugs Therapy for HIV Infection with Discrete-Time Delay

A discrete-time-delay differential mathematical model that described HIV infection of CD4+ T cells with drugs therapy is analyzed. The stability of the two equilibria and the existence of Hopf bifurcation at the positive equilibrium are investigated. Using the normal form theory and center manifold argument, the explicit formulas which determine the stability, the direction, and the period of bifurcating periodic solutions are derived. Numerical simulations are carried out to explain the mathematical conclusions. 1. Introduction Recently there has been a substantial effort in the mathematical modelling of virus dynamics [1–8]. These models focus on uninfected target cells, infected cells that are producing virus, and virus. A basic mathematical model describing HIV infection dynamic model is of the following form which has been studied in [5, 9]: In system (1), the following variables are includes: uninfected cells at time (unit is cells ), infected cells at time (unit is cells ), and virus at time (unit is virions ). Parameters , and are the death rates of the uninfected cells, the infected cells, and the virus particles, respectively. is the contact rate between uninfected cells and the virus particles. is the average number of virus particles produced by an infected cell. Reverse transcriptase inhibitors (RTIs) are a class of antiretroviral drugs used to treat HIV infection. RTIs inhibitors work by inhibiting the action of reverse transcriptase. RTIs inhibit the activity of reverse transcriptase, a viral DNA polymerase enzyme that retroviruses need to reproduce. In [10], Srivastava et al. developed a mathematical model for primary infection with RTIs. They subdivided the infected cells class in two subclasses: pre-RT (denoted by ) and post-RT (denoted by ). They assumed that a virus enters a resting T cell, the viral RNA may not be completely reverse transcribed into DNA, the unintegrated virus may decay with time and partial DNA transcripts are labile and degrade quickly [11, 12]. And they also assumed that a fraction of cells in pre-RT class reverts back to uninfected class and the remaining proceeds to post-RT class and becomes productively infected due to presence of RT inhibitors. The model of Srivastava et al. is as follows where is the efficacy of reverse transcriptase inhibitors (RTIs), is the transition rate from pre-RT (i.e., ) infected T cells class to productively post-RT (i.e., ) which is a productively infected class, and is the reverting rate of infected cells to uninfected class due to noncompletion of reverse transcription [11, 12].