Stability and Hopf Bifurcation Analysis for an Age-Structured Tumor Immune Model with Time Delay

In this paper, we propose and analyze an age-structured tumor immune model with time delay. We divide immune cells into two kinds. One is those whose growth is independent of tumor and the other is those whose growth depends on the simulation of the tumor. For these cells, their physiological ages are considered. A mature time delay t is introduced to the tumor-simulation-dependent immune cells to restrict those cells who participate in the immune response to grow to a minimum physiological age. The existence and stability threshold R is established for the tumor-free equilibrium state. If R < 0, the tumor-free equilibrium state is both locally and globally asymptotically stable. Whereas, when R > 0, the tumor equilibrium state is locally asymptotically stable if t ? [0,t(k=0)) and a Hopf bifurcation occurs when t passes through the threshold t(k=0). This may partly explain the periodic recurrence of some tumors. Finally, theoretical results are verified by some numerical simulations.