Study On Kinetic Behavior And Drug Control Strategy Of Several Tumor Immune Systems

In recent years,the dynamics of tumor immune system has a made great progress.The tumor immune process is presented in the form of a mathematical model by modeling.The model is analyzed from a mathematical point of view,and the analysis conclusion of the model is provided as a theoretical basis for clinical trials.This method has become a necessary trend.Many mathematicians have applied mathematical methods to analyze the mathematical model of the tumor immune system,but the growth rate of effector cells stimulated by tumor cells in mathematical models has many different forms,so it is necessary to study the development of the process and control strategy of tumors from the mathematical theory level.The main content of this paper consists of three parts are as follows:The first part mainly studies the global state of the Galach's model.Firstly,the positive invariant set D of the corresponding system is given after dimensionless and the model is boundedness after analysis.Secondly,the existence and local stability of the equilibrium point of the system are discussed in the invariant set.The sufficient conditions of the positive equilibrium point,the conditions of local asymptotic stability of the tumor-free equilibrium point and the locality of the positive equilibrium point are given by theoretical analysis.Thirdly,by finding the Dulac function,it is proved that the system has no periodic solution on the positive invariant set,and the global asymptotic stability of the system's positive equilibrium point is obtained.It is obtained that if the positive equilibrium pointP^*exists,it is globally asymptotically stable;if the positive equilibrium point_*P exists,it is the saddle point;if the positive equilibrium point_*P^*exists,it is the saddle node.Finally,use MATLAB numerical simulations to verify the correctness of the results obtained from the numerical aspects,and give a biological explanation for the model,providing the control of drugs for the effector cell input rate in clinical ACI therapy?inherited cellular immunotherapy?to fight tumors.The second part mainly studies the global state of the Kuznetsov's model.The positive invariant set S of the system after dimensionless is obtained,and the sufficient conditions for the existence of the positive equilibrium point of the Kuznetsov's model,the conditions for the local asymptotic stability of the tumor-free equilibrium point and the conditions for the local stability of the positive equilibrium point are given.Compared with the Galach's model,the Kuznetsov's model is more complicated and difficult to study.Based on the previous studies,this paper gives the existence conditions of the equilibrium point of the model and the theoretical analysis of the stability.The theoretical results are completely consistent with numerical analysis results.The third part mainly studies the dynamics of the three-dimensional system of MM form growth rate.This model is based on the three-dimensional tumor immune model proposed by Dong Yueping,whose growth rate of effector cells stimulated by tumor cells is bilinear.Change it to the M-M form,a new improved model was obtained.The analysis of the model is divided into four cases:s₁?28?s₂?28?0,s₁?0?s₂?28?0,s₁?28?0?s₂?0,s₁?0?s₂?0,where the biological meaning of₁s is the input rate of the effector cells,and the biological significance of₂s is the input rate of the helper T cells.The existence and stability conditions of the equilibrium points of each model are given respectively.The numerical simulation diagram of the existence of equilibrium points is given.Each case of the model is made for explanation in meaning.In the study of Model 1 and Model 2,the global state of the equilibrium point of the system is studied respectively.Based on the existing conclusions,the structuring and system theorization of the dynamic analysis of the system is realized.For the research of Model 3,the existing model is improved,the stability problem of the model is discussed in detail,and the theoretical analysis results are obtained.At the same time,the biological meanings of the three types of models are given.The results of theoretical analysis will provide a basis for drug control in ACI therapy of the effector cell's input rate.