Mathematical modeling and dynamic analysis of anti-tumor immune response

The competitive interaction of tumor-immune system is very complex. We aim to establish a simple and realistic mathematical model to understand the key factors that impact the outcome of an antitumor response. Based on the principle that lymphocytes undergo two stages of development (namely immature and mature), we develop a new anti-tumor-immune response model and investigate its property and bifurcation. The corresponding sufficient criteria for the asymptotic stabilities of equilibria and the existence of stable periodic oscillations of tumor levels are obtained. Theoretical results indicate that the system orderly undergoes different states with the flow rate of mature immune cells increasing, from the unlimited expansion of tumor, to the stable large tumor-present equilibrium, to the periodic oscillation, to the stable small tumor-present equilibrium, and finally to the stable tumor-free equilibrium, which exhibits a variety of dynamic behaviors. In addition, these dynamic behaviors are in accordance with some phenomena observed clinically, such as tumor dormant, tumor periodic oscillation, immune escape of tumor and so on. Numerical simulations are carried out to verify the results of our theoretical analysis.