Nonlinear Analysis: Real World Applications 17 (2014) 147–160 Contents lists available at ScienceDirect Nonlinear Analysis: Real World Applications journal homepage: www.elsevier.com/locate/nonrwa Backward bifurcation in a mathematical model for HIV infection in vivo with anti-retroviral treatment Michael Y. Li a , Liancheng Wang b,∗ a Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, T6G 2G1 Canada b Department of Mathematical and Statistical Sciences, Kennesaw State University, 1000 Chastain Rd., #1601, Kennesaw, GA 30144, USA article info Article history: Received 15 November 2012 abstract Anti-retroviral treatments (ART) such as HAART have been used to control the replication of HIV virus in HIV-positive patients. In this paper, we study an in-host model of HIV infection with ART and carry out mathematical analysis of the global dynamics and bifurcations of the model in different parameter regimes. Among our discoveries is a parameter region for which backward bifurcation can occur. Biologically, the catastrophic behaviors associated with backward bifurcations may explain the sudden rebound of HIV viral load when ART is stopped, and possibly provide an explanation for the viral blips during ART suppression of HIV. © 2013 Elsevier Ltd. All rights reserved. 1. Introduction Human immunodeficiency virus (HIV) type I is an RNA virus that preferentially targets the CD4 + helper T cells. After entry into a target cell, the viral RNA is reverse transcribed into viral DNA using host genetic materials and viral reverse transcriptase. The viral DNA is transported to the cell nucleus and integrated into the host genome through the action of viral integrase and stays latent. Upon antigenic stimulation, the viral DNA can be transcribed into new viral RNA and viral proteins such as reverse transcriptase, integrase and protease. The viral protease is needed in this stage to cut the long polypeptide chain into individual enzyme components to complete the translation of viral proteins. At this stage, the target cell is called productively infected. The viral RNA genome and viral proteins will be assembled and enveloped to become mature virus near the cell membrane. Mature viruses bud out of the host cell and get released into the plasma to infect new target cells. Viral budding will terminate an infected CD4 + T cells. In addition, HIV infection can lead to increased rate of apoptosis of a target cell, and an infected target cell can be recognized and lysed by the CD8 cytotoxic T cells. These factors combine to progressively reduce the number of CD4 + T cells in the body and leave the body increasingly susceptible to opportunistic infections. Anti-retroviral drugs are designed to block the action of various HIV viral proteins and hence interrupt the viral replication cycle. Reverse transcriptase inhibitor (RTI) based drugs can block the reverse transcription of viral DNA and prevent the host cell from becoming productively infected. Protease inhibitor (PI) based drugs aim to block the action of protease and thus prevent the production of mature viruses. Due to the extremely high rate of mutation of HIV, standard ART regimens combine several drugs from both RTI and PI family to avoid emergence of drug resistance. Mathematical models have been developed to describe the HIV infection dynamics and effects of ART [1–13]. These in-host models are useful for exploring possible mechanisms and outcomes of the viral infection process [1,5,3], and for estimating key parameter values such as virion clearance rate, life span of infected cells, and average viral generation time in vivo [2]. Findings from in-host modeling can be used to guide development of efficient antiviral drug therapies [4]. ∗ Corresponding author. Tel.: +1 678 797 2139; fax: +1 770 423 6629. E-mail addresses: mli@math.ualberta.ca (M.Y. Li), lwang5@kennesaw.edu (L. Wang). 1468-1218/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.nonrwa.2013.11.002