Stability Analysis Of Complex Biologcal Dynamics Model

Human T-cell leukaemia/lymphoma virus type I(HTLV-I) is a retrovirus that has been identified as the causative agent of HTLV-I-associated myelopathy/tropical spastic paraparesis(HAM/TSP) and other illnesses. HTLV-I infects primarily CD4(10)T cells and the transmission occurs through direct cell-to-cell contact. HAM/TSP patients harbor higher proviral loads in peripheral blood lymphocytes than asymptomatic carriers. Also, HAM/TSP patients exhibit a remarkably high number of circulating HTLV-I-specific CD8(10)cytotoxic T lymphocytes(CTLs) in the peripheral blood. While CTLs have a protective role by killing the infected cells and lowering the proviral load, a high level of CTLs and their cytotoxicity are believed to be a main cause of the development of HAM/TSP. A mathematical model for HTLV-I infection of CD4(10)T cells that incorporates the CD8(10)cytotoxic T-cell(CTL)response is investigated. Our mathematical analysis reveals that the system can stabilize at a carrier steady-state with persistent viral infection but no CTL response,and at a HAM/TSP steady-state at which both the viral infection and CTL response are persistent. We also establish two threshold parameters0 R and1R, the basic reproduction numbers for viral persistence and for CTL response, respectively. If10R(27), the infection-free equilibrium is globally asymptotically stable, and the HTLV-I viruses are cleared. If1 0R(27)1(27)R,the immune-free equilibrium is globally asymptoti-cally stable, and the HTLV-I infection is chronic but with no persistent CTL response. If 11R(29),a unique HAM/TSP equilibrium exists, and the HTLV-I infection becomes chronic with a persistent CTL response. Our numerical simulations suggest that if 11R(29), an increase of the intracellular delay may stabilize the HAM/TSP equilibrium while the immune delay can destabilize it.