Analysis Of Several Tumor Immune Models With Saturated Inhibition

The kinetic research on the interaction between tumors and immunity began in 1970 and has become a rapidly developing topic.The most basic and most challenging problem in tumor immune research is to understand the interaction between tumor and immune response.Because many factors are involved in the interaction between tumor and immune cells,there are still many unclear aspects in tumor immune response research.An ideal mathematical model can give people an in-depth understanding of the dynamic interaction between tumor cells and the immune system.Therefore,theoretical models can help people to understand tumors and develop new treatment strategies.Firstly,a two-dimensional ordinary differential equation model of the interaction between tumor cells and effector cells with Michaelis-Menten type inhibitory function is established.The positive invariant set of the model is got by proving that the solution of the model under the initial conditions is non-negative and bounded.The dynamic behavior of the model is discussed in the positive invariant set.The existence and local stability of equilibria are discussed.The global dynamic behavior of the model is obtained by constructing Dulac function to exclude the existence of the periodic solution.Meanwhile,it is found that the saddle-node bifurcation may occur,which makes the existence of the tumor cells depending on the initial condition of the model.Finally,the numerical simulation not only verifies the accuracy of the results,but also analyses the effects of the inhibition coefficient of tumor cells to effector cells and the inhibition coefficient of effector cells to tumor cells on the dynamics of the mode.Secondly,considering the time delay in the growth of tumor cells,a tumor immune model with time lag was established.The first,by linearizing at equilibria of the model,the characteristic equations of the model are obtained,and then the local stability of equilibria are obtained.The second,we proved the existence of the Hopf bifurcation and analyzed the nature of the Hopf bifurcation.Meanwhile,we get the expressions of Hopf bifurcation direction and periodic solution stability.Finally,the theoretical results obtained are verified.Lastly,we established a three-dimensional ordinary differential equation model containing tumor cells,effector cells and regulatory T cells(Tregs).The non-negativity and boundedness of the model solution and the existence of the equilibria of the model are discussed.Then,by analyzing the characteristic equation of the model at the equilibria,the Routh-Hurwitz criterion is used to prove the stability of the model at the equilibria.Finally,by proving that the transversal condition is established,the existence of the Hopf branch is proved.At the same time,numerical simulations are used to verify the accuracy of the analysis results.