Rich dynamics in a delayed HTLV-I infection model: Stability switch, multiple stable cycles, and torus

Abstract

In this paper, we investigate the impact of time delay in CTL immune response on a HTLV-I infection model. By defining basic reproduction number for viral infection R0 and basic reproduction number for CTL response RCTL, we characterize the model dynamics according to whether these two threshold values are greater than one. Especially, we obtain the global dynamics if R0≤1 or RCTL≤1<R0, as well as infection persistent result when R0>1. However, the model dynamics become much richer when RCTL<1. In this case, we use the time delay as a bifurcation parameter to obtain stability switch result on the positive equilibrium and global bifurcation diagrams for the model system. We also conduct higher-order normal form analysis and apply center manifold theory to classify the rich model dynamics near the double Hopf bifurcation points. Our analysis indicates that time delay in CTL immune response can induce not only Hopf bifurcation and double Hopf bifurcation, but also quasi-periodic orbits (torus) and coexistence of multiple stable periodic solutions.