Nonlinear Analysis: Real World Applications

Human T-cell leukaemia virus type I (HTLV-I) preferentially infects the CD4 T cells. The HTLV-I infection causes a strong HTLV-I specific immune response from CD8 cytotoxic T cells (CTLs). The persistent cytotoxicity of the CTL is believed to contribute to the development of a progressive neurologic disease, HTLV-I associatedmyelopathy/tropical spastic paraparesis (HAM/TSP). We investigate the global dynamics of a mathematical model for the CTL response toHTLV-I infection in vivo. To account for a series of immunological events leading to the CTL response, we incorporate a time delay in the response term. Our mathematical analysis establishes that the global dynamics are determined by two threshold parameters R0 and R1, basic reproduction numbers for viral infection and for CTL response, respectively. If R0 ≤ 1, the infection-free equilibrium P0 is globally asymptotically stable, and the HTLV-I viruses are cleared. If R1 ≤ 1 < R0, the asymptomatic-carrier equilibrium P1 is globally asymptotically stable, and the HTLV-I infection becomes chronic but with no persistent CTL response. If R1 > 1, a unique HAM/TSP equilibrium P2 exists, at which the HTLV-I infection is chronicwith a persistent CTL response.We show that the time delay can destabilize the HAM/TSP equilibrium, leading to Hopf bifurcations and stable periodic oscillations. Implications of our results to the pathogenesis of HTLV-I infection and HAM/TSP development are discussed. © 2011 Elsevier Ltd. All rights reserved.