ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.12(2011) No.4,pp.503-512 A Delay Differential Equation Model of HIV with Therapy and Cure Rate Khalid Hattaf ∗ , Noura Yousfi Laboratory Analysis, Modeling and Simulation, Department of Mathematics and Computer Science, Faculty of Sciences Ben Msik, University Hassan II, Mohammedia, P.O Box 7955 Sidi Othman, Casablanca, Morocco (Received 18 May 2011 , accepted 29 October 2011) Abstract: The aim of this work is to investigate a new mathematical model with delay that describes HIV infection of CD4 + T-cells during therapy. A novel feature is that both therapy and the delay are incorporated into the model. We prove that the infection will die out if the basic reproduction number R 0 < 1 while the HIV infection may become endemic if R 0 > 1. Stability analysis of both endemic and disease free steady states are also studied. Finally, we give some numerical simulations to illustrate our results. Keywords: HIV infection; time delay; stability 1 Introduction Human immunodeficiency virus (HIV) is a lentivirus (a member of the retrovirus family) that causes acquired immunod- eficiency syndrome (AIDS), a condition in humans in which the immune system begins to fail, leading to life-threatening opportunistic infections. Infection with HIV occurs by the transfer of blood, semen, vaginal fluid, pre-ejaculate, or breast milk. Within these bodily fluids, HIV is present as both free virus particles and virus within infected immune cells. The four major routes of transmission are unsafe sex, contaminated needles, breast milk, and transmission from an in- fected mother to her baby at birth (Vertical transmission). Screening of blood products for HIV has largely eliminated transmission through blood transfusions or infected blood products in the developed world. There are some antiretroviral (ARV) drugs available nowadays which help the immune system in preventing the infection due to HIV even though it is not possible to cure it. Reverse Transcriptase Inhibitors (RTIs), is one of the chemotherapies which opposes the conversion of RNA of the virus to DNA (reverse transcription), so that the viral population will be minimal and on the other-hand the CD4 + count remains higher and the host can survive. Another one is the Protease Inhibitors (PIs) prevents the production of viruses from the actively infected CD4 + T-cells. In the literature, many Mathematical models used ordinary differential equations (ODEs) have been developed in order to understand the dynamics of HIV infection [10–13]. To account for the time between viral entry into a target cell and the production of new virus particles, Herz et al. [5] used a fixed, discrete, delay to model the intracellular phase of the viral life-cycle, defined as the time between infection of a cell and production of new virus particles, and showed that the incorporation of a delay changed the estimated value of the half-life of free virus. This model was analyzed by Li and Shu in [6]. Mittler et al. [7] examined a related model but assumed that the intracellular delay, rather than being discrete, was continuous and varied according to a gamma distribution. In [5, 7], the authors assumed the drug to be 100% effective in order to completely block viral production. It was shown by Nelson et al. [8], using a model with a discrete delay and constant target cell density, that when drug efficacy is less than 100%, as may be the case in vivo, the predicted rate of decline in plasma virus concentration depends on three factors: the death rate of virus producing cells, the efficacy of therapy and the length of the delay. Nelson and Perelson [9] generalized this model by supposing that the delay varies according to a probability distribution. The authors used two drugs (RTIs and PIs). It was shown that when the drug efficacy is less than perfect the estimated value of the loss rate of productively infected T cell is increased when the data is fitted with delay models compared to the values estimated with a non-delay model. All these studies cited above are based on HIV models which omitted the cure of infected cells. A part of these infected cells return to the uninfected state by loss of all covalently closed circular DNA (cccDNA) from their nucleus at a certain rate per infected cell [18]. In addition, * Corresponding author. E-mail address: k.hattaf@yahoo.fr Copyright c ⃝World Academic Press, World Academic Union IJNS.2011.12.31/568