Chaotic dynamics of a delayed tumor-immune interaction model

Due to the unpredictable growth of tumor cells, the tumor-immune interactive dynamics continues to draw attention from both applied mathematicians and oncologists. Mathematical modeling is a powerful tool to improve our understanding of the complicated biological system for tumor growth. With this goal, we report a mathematical model which describes how tumor cells evolve and survive the brief encounter with the immune system mediated by immune effector cells and host cells which includes discrete time delay. We analyze the basic mathematical properties of the considered model such as positivity of the system and the boundedness of the solutions. By analyzing the distribution of eigenvalues, local stability analysis of the biologically feasible equilibria and the existence of Hopf bifurcation are obtained in which discrete time delay is used as a bifurcation parameter. Based on the normal form theory and center manifold theorem, we obtain explicit expressions to determine the direction of Hopf bifurcation and the stability of Hopf bifurcating periodic solutions. Numerical simulations are carried out to illustrate the rich dynamical behavior of the delayed tumor model. Our model simulations demonstrate that the delayed tumor model exhibits regular and irregular periodic oscillations or chaotic behaviors, which indicate the scenario of long-term tumor relapse.