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Global dynamics of a mathematical model for HTLV-I infection of $CD{4}^{+}T$ cells with delayed CTL response. (English) Zbl 1239.34086
Summary: Human T-cell leukaemia virus type I (HTLV-I) preferentially infects the $CD{4}^{+}$ T cells. The HTLV-I infection causes a strong HTLV-I specific immune response from $CD{8}^{+}$ cytotoxic T cells (CTLs). The persistent cytotoxicity of the CTL is believed to contribute to the development of a progressive neurologic disease, HTLV-I associated myelopathy/tropical spastic paraparesis (HAM/TSP). We investigate the global dynamics of a mathematical model for the CTL response to HTLV-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 ${R}_{0}$ and ${R}_{1}$, basic reproduction numbers for viral infection and for CTL response, respectively. If ${R}_{0}\le 1$, the infection-free equilibrium ${P}_{0}$ is globally asymptotically stable, and the HTLV-I viruses are cleared. If ${R}_{1}\le 1<{R}_{0}$, the asymptomatic-carrier equilibrium ${P}_{1}$ is globally asymptotically stable, and the HTLV-I infection becomes chronic but with no persistent CTL response. If ${R}_{1}>1$, a unique HAM/TSP equilibrium ${P}_{2}$ exists, at which the HTLV-I infection is chronic with 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.
##### MSC:
 34K18 Bifurcation theory of functional differential equations 92C60 Medical epidemiology 37N25 Dynamical systems in biology 34K20 Stability theory of functional-differential equations 34K60 Qualitative investigation and simulation of models