Hopf bifurcation for a discontinuous HTLV-1 model

Developing accurate mathematical models for host immune response in immunosuppressive diseases such as HIV and HTLV-1 are essential to achieve an optimal drug therapy regime. Since for HTLV-1 specific CTL response typically occurs after a time lag, we consider a discontinuous response function to better describe this lagged response during the early stage of the infectious, thus the system of HTLV-1 model will be a discontinuous system. For analyzing the dynamic of the system we use Filippov theory and find conditions in which the Filippov system undergoes a Hopf bifurcation. The Hopf bifurcation help us to find stable and unstable periodic oscillations and can be used to predict whether the CTL response can return to a steady state condition. Also, Hopf bifurcation in sliding mode is investigated. In this case the solutions will remain in the hyper-surface of discontinuity and as a consequence the disease cannot progress, at least for a long time. Finally we use numerical simulations to demonstrate the results by example. nema