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Global dynamics of a mathematical model for HTLV-I infection of CD4 + T cells with delayed CTL response. (English) Zbl 1239.34086
Summary: 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 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 1, the infection-free equilibrium P 0 is globally asymptotically stable, and the HTLV-I viruses are cleared. If R 1 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:
34K18Bifurcation theory of functional differential equations
92C60Medical epidemiology
37N25Dynamical systems in biology
34K20Stability theory of functional-differential equations
34K60Qualitative investigation and simulation of models
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