A mathematical model of HTLV-I infection with two time delays

In this paper, we include two time delays in a mathematical model for the CD8+ cytotoxic T lymphocytes (CTLs) response to the Human T-cell leukaemia virus type I (HTLV-I) infection, where one is the intracellular infection delay and the other is the immune delay to account for a series of immunological events leading to the CTL response. We show that the global dynamics of the model system are determined by two threshold values R0, the corresponding reproductive number of a viral infection, and R1, the corresponding reproductive number of a CTL response, respectively. If R0 < 1, the infection-free equilibrium is globally asymptotically stable, and the HTLV-I viruses are cleared. If R1<1<R0, the immune-free equilibrium is globally asymptotically stable, and the HTLV-I infection is chronic but with no persistent CTL response. If 1<R1, a unique HAM/TSP equilibrium exists, and the HTLV-I infection becomes chronic with a persistent CTL response. Moreover, we show that the immune delay can destabilize the HAM/TSP equilibrium, leading to Hopf bifurcations. Our numerical simulations suggest that if 1<R1, an increase of the intracellular delay may stabilize the HAM/TSP equilibrium while the immune delay can destabilize it. If both delays increase, the stability of the HAM/TSP equilibrium may generate rich dynamics combining the 'stabilizing' effects from the intracellular delay with those 'destabilizing' influences from immune delay.