Stability and Bifurcation of the Immune Model of Chronic Myeloid Leukemia with Time Delay

In order to investigate the effects of the differentiation delay in chronic myeloid leukemia stem cells on the dynamic behaviors among chronic myeloid leukemia stem cells, chronic myeloid leukemia mature cells and effector T cells, a delayed immune model of chronic myeloid leukemia with the treatment of interferon-alpha and imatinib is established. Through the stability analysis of the modified time-delay model, using normal form and central manifold theory, the critical conditions and properties of Hopf bifurcation are obtained. In addition, the conditions for the control of chronic myeloid leukemia by interferon-alpha and imatinib combined with effector T cells are given. For example, under the intervention of drugs, adjusting the time delay of differentiation can make the number of stem cells and mature cells of chronic myeloid leukemia show a stable and non-increasing state, or transition from large amplitude periodic oscillation state to small periodic oscillation state. Finally, numerical simulations verify the obtained theories.