Bifurcation Analysis of a Modified Tumor-immune System Interaction Model Involving Time Delay

Abstract

We study stability and Hopf bifurcation analysis of a model that refers to the competition between the immune system and an aggressive host such as a tumor. The model which describes this competition is governed by a reaction-diffusion system including time delay under the Neumann boundary conditions, and is based on Kuznetsov-Taylor's model. Choosing the delay parameter as a bifurcation parameter, we first show that Hopf bifurcation occurs. Second, we determine two properties of the periodic solution, namely its direction and stability, by applying the normal form theory and the center manifold reduction for partial functional differential equations. Furthermore, we discuss the effects of diffusion on the dynamics by analyzing a model with constant coefficients and perform some numerical simulations to support the analytical results. The results show that diffusion has an important effects on the dynamics of a mathematical model.

Stability and Bifurcation of Tumor Immune Model with Time Delay

“…We have noticed that during tumor proliferation, there is usually a period of latency. Many studies have shown that time delays can disrupt the homeostasis of the human immune system and lead to altered stability (Kayan et al, 2017;Banerjee et al, 2008). Therefore, considering the time delay factor in tumor immune models better reflects the objective pattern and has important scientific significance.…”

Mathematical analysis of a cancer model with time-delay in tumor-immune interaction and stimulation processes

“…They observed that uniform activation of helper T -cells can help in ECs stimulation and tumor control. Considering a reaction-diffusion system, including time delay under the Neumann boundary conditions, Kayan et al [ 30 ] modified the model [ 11 ] which described tumor-immune competitions. They studied the Hopf bifurcation analysis and found that the effect of diffusion of tumor-immune interaction can significantly change the dynamics of the model.…”