Delay Differential Equations Modelling and Stability of Tumor Growth with Immune Responses

The development of tumors is a very complicated biological process, which is controlled by various interacting factors, such as the proliferation of cells, immune system, nutrients, and treatment. Mathematical and computational modelling is an efficient method to the comprehension of the mechanisms governing tumor dynamics. In the present paper, we construct a delayed differential equation model that is used to explain the interaction between tumor cells and the responses of immune system. Linearization techniques and analysis of characteristic equations are used to analyse the model in terms of the existence of equilibrium points, stability properties of those points and the bifurcation behaviour that may occur. The results demonstrate that time delay can disrupt the otherwise steady equilibria to produce oscillating tumor dynamics, which shows the dramatic effect of the delayed immune response or treatment response on tumor dynamics.