Dynamical behaviors of an HTLV-I infection model with intracellular delay and immune activation delay

This paper investigates the dynamics of an HTLV-I infection model with intracellular delay and immune activation delay. The primary objective of the study is to consider the effect of the time delay on the stability of the infected equilibrium. Two sharp threshold parameters $\Re_{0}$ and $\Re_{1}$ are identified as the basic reproduction number for viral infection and for CTLs response, respectively, which determine the long time behaviors of the viral infection. In particular, our mathematical analysis reveals that a Hopf bifurcation occurs when immune activation delay passes through a critical value. Using the normal form theory and center manifold arguments, the explicit formulae which determine the stability, the direction, and the period of bifurcating periodic solutions are derived. Numerical simulations are given to support the theoretical results.