Analysis Of Several Tumor-Immune Models With Bilinear Stimulation Terms

Cancer is increasingly invading people's lives,the high incidence of cancer not only increase the pain of patients but also bring heavy economic burden to patients' families and society.According to the study of biologist,cancer is caused by tumor(especially malignant tumors).Tumor immunotherapy is an important method to study tumor therapy.In order to study the process of interaction between tumor cells and immune system in human's body,three types of tumor immune models with bilinear stimulation terms were established in this paper.The research is mainly conducted from the following three aspects:Firstly,we propose a simple tumor-immune system model,in which we considered two kinds of effects of the tumor cells(i.e.bilinear stimulation and saturation inhibition)on the effector cells.We discuss the existence and the local asymptotic stability of tumor equilibrium and tumor free equilibrium with the help of stability theory.The sufficient conditions when the saddle node bifurcation is generated are found by applying the center manifold theorem.At the same time,the correctness of the theoretical results is verified further by numerical simulations and the complex global dynamic behavior of the model is shown.Secondly,a modified mathematical model of established model representing tumor-immune interaction with two discrete time delay is proposed.Non-Negativity and boundedness of the solutions of the model has been proven,the sufficient conditions for the existence of equilibrium are also discussed.We analyze the stability of equilibrium in great detail by applying the qualitative theory of delay differential equation and Hopf bifurcation theory.By choosing two discrete time delay ?1,?2 as the bifurcation parameter,we establish the sufficient condition for existence of Hopf bifurcation.It is found that the model will switch between stable behavior and unstable behavior as the parameter values changing,and periodic solutions will be generated under the condition of the tumor equilibrium is unstable.Thirdly,an ordinary differential equation model of effector cells,tumor cells and helper cells is established on the basis of assuming that helper cells grow in Logistic model when there are no effector cells and assuming that there's a mutual inhibition between helper cells and tumor cells.Based on the qualitative and stability theory of differential equations and the bifurcation theory,the existence of the equilibrium point of the model is analyzed concretely,and the local asymptotic stability of each equilibrium point and saddle-node bifurcation are discussed.At the same time,numerical simulations are obtained to validate our analytical findings,and the complex global dynamic state of the model is presented accurately and intuitively through the three-dimensional phase diagram.This theoretical study has a guiding effect on the treatment of tumor.