Dynamic Effects of Two-Time Delays on a Model for Tumor Growth

Aims and Objective: In order to understand the dynamic mechanisms of tumor growth and make a contribution to develop anti-cancer treatment strategies, a mathematical model for tumor growth with two-time delays is proposed in this article. Materials and Methods: First, the relationships among host cells, tumor cells and effector cells, and the biological meaning of two-time delays are explained. Moreover, the system stability is discussed by analyzing the characteristic equation of the model. In addition, the existence and properties of oscillatory dynamic are also researched by using normative theory and central manifold method. Finally, the numerical simulations are performed to further illustrate and support the theoretical results. Results: Both two-time delays in the model can affect the dynamics of tumor growth. Meanwhile, the system can experience a Hopf bifurcation when the delay crosses a series of critical values. Further, a clear formula is deduced to determine the Hopf bifurcation and the direction of stability of the periodic solution. Finally, these results are verified by using numerical simulation. Conclusion: The results demonstrated that the time from identifying tumor cells to making the appropriate response for the immune system and the time needed for competition between host cells and tumor cells for natural resources and living space is significant for tumor growth. These findings in this paper may help us better understand the behaviors of tumors and develop better anti-cancer treatment strategies.