Dynamical Study Of Tumor-immune System With Nonlinear Interactions

The immune system is one of the important systems to protect the human body from the invasion of foreign bacteria.The immune system has the ability to recognize and eliminate antigens while maintaining the balance and stability of its internal en-vironment.The occurrence of human tumor is closely related to the immune system.In 2004,Dong-Miyazaki-Takeuchi(DMT)constructed a bilinear dynamics model to simulate the interaction between tumor cells,effector cells and helper T cells.In fact,the relationship between the tumor and the immune system is very complex and often highly nonlinear,which requires us to use more realistic nonlinear terms,such us Michaelis-Menten saturated type.This paper is based on DMT model,considering the nonlinear interaction between effector cells and tumor cells,to construct two dynamic model of interaction between tumors and the immune system.This paper is divided into four parts,mainly to study the nonlinear interaction between effector cells and tumor cells.The results can help to better understand the nonlinear between tumor and immune system and its effect on tumor immunodynamics.The first part mainly introduces the background,present situation and develop-ment of tumor immune system,as well as the stability theory of ordinary differential equation.The second part studies the mathematical model of the nonlinear interac-tion between effector cells and tumor cells,construct the nonlinear dynamics model between tumor cell,effector cells and helper T cells,and discuss the number of e-quilibrium points and the stability of the model.Through numerical simulation,the existence of three positive equilibrium points corresponding to the small tumor equi-librium,the intermediate equilibrium and the large tumor equilibrium was verified.The results show that the system is bistable(stable for small tumor equilibrium and large equilibrium)and has stable periodic solution.Through branching analysis of some important parameters,such as the stimulation rate of tumor antigens to helper T cells and inhibition rate of effector cells to tumor cell,it is found that these two parameters play a very important role in tumor suppression and elimination.The third part studies the mathematical model of the nonlinear relationship between tu-mor cells and effector cells in adoptive immunotherapy.In the process of effector cell eliminating tumor cells,immunosuppressive agents are released to suppress the growth of effector cells,thus achieving the effect of immunosuppression.By analyzing the dynamic properties of the mathematical model,we find that the system has sta-ble periodic solution and bistability.We also find that appropriately increasing the activation rate of helper T cell to effector cell is beneficial to tumor elimination.The fourth part summarizes the research results of this dissertation and looks forward to the future research work.