Stability and Bifurcation of Tumor Immune Model with Time Delay

In this paper, we investigate the effect of time delay on the stability of the tumor immune system using theoretical calculations and numerical simulations. Since it takes a certain time for immune cells to recognize tumor cells to make an appropriate response, a model of tumor-immune system interaction with time delay is established by considering time delay in this process. The four equilibrium points are solved by simplifying the model using Taylor expansion with a small time delay. Then the stability of each equilibrium point of the system under a small time delay is determined by calculating the characteristic roots of each equilibrium point with numerical simulation software. The results show that the system has a bistability phenomenon. The saddle point and stable node are not affected by the delay, while only the stability of the stable foci changes with the time delay with Hopf bifurcation. This study can help determine the optimal time for tumor treatment and provide a reference for analyzing tumor status and treatment.