Tumor-immune Dynamic Model Considering Growth Delay And Saturation Transformation

In this paper,two tumor-immune system dynamic models are established considering the growth delay of tumors and the saturated transformation of resting immune cells.In the first chapter,we introduce the research background of this article,briefly describe the research status of tumor immune models and the main work of this article,and give the main basic theoretical knowledge required.In the second chapter,a three-dimensional nonlinear delay differential system consisting of tumor cells,resting immune cells and hunting immune cells is constructed to study tumor growth delay,tumor immune system competition and the role of chemotherapy.The study finds that the introduction of chemotherapy and tumor growth delay may cause a rich dynamic behavior in the tumor immune competition system.In particular,we prove that the system has a bistable phenomenon,indicating that whether the tumor can be eradicated under certain conditions depends on the initial density of the corresponding cells.The research also shows that when the tumor growth delay exceeds a certain threshold,the coexistence equilibrium point will lose stability,and the system may have a Hopf bifurcation;at the same time,we analyze the direction of Hopf bifurcation and the stability of the periodic solution of the branch through standard and central manifold theory.The above studies indicate that tumor growth lag may be an important mechanism for explaining the short-term oscillation and long-term recurrence of tumors.Finally,through numerical simulation,we further find that the larger the growth delayof the tumor,the larger the amplitude of the branch periodic solution,and the longer the period,indicating that the recurrence period of the tumor is longer,which may help to control the growth of the tumor.In the third chapter,we study a tumor immune system model considering chemotherapy and the saturation transformation of resting immune cells.First of all,we study the stability of the system's extinction equilibrium point,no immune equilibrium point and no tumor equilibrium point and other boundary equilibrium point,and find that different chemotherapy effects and side effects may produce completely different treatment effects.Then we use the saturation conversion coefficient of resting immune cells as the branch parameter to obtain the conditions for the system to generate periodic oscillations through the Hopf bifurcation at the coexistence equilibrium point.Finally,we verify some of the above relevant theoretical research results through some numerical calculations,and further find that with the increase of the saturation conversion coefficient,the density of tumor cells at the positive equilibrium point of the system decreases,and the amplitude of the periodic solution generated by the system through the Hopf bifurcation becomes significantly larger,the period becomes slightly longer,and the minimum value of periodic tumor cells tends to zero,indicating that the increase in saturation transformation coefficient may help to control the growth of the tumor or even eliminate the tumor.The above research conclusions indicate that the saturated transformation of resting immune cells to hunting immune cells may also be an important factor that can not be ignored to explain the short-term oscillation or long-term recurrence of tumor cells.In the fourth chapter,the research conclusions obtained in this paper are briefly summarized,and its biological significance and theoretical value are analyzed.At the same time,issues and directions for further research are pointed out.