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

Abstract

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 [Formula: see text] 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 [Formula: see text] is established for the tumor-free equilibrium state. If [Formula: see text], the tumor-free equilibrium state is both locally and globally asymptotically stable. Whereas, when [Formula: see text], the tumor equilibrium state is locally asymptotically stable if [Formula: see text] and a Hopf bifurcation occurs when [Formula: see text] passes through the threshold [Formula: see text]. This may partly explain the periodic recurrence of some tumors. Finally, theoretical results are verified by some numerical simulations.