Model Construction And Study On Complex Dynamic Behavior Of Biological Immune System In Stochastic Switching Environment

Cancer is still the leading cause of death in the world,but little is known about its formation and destruction mechanism.Therefore,understanding the dynamic destruction mechanism and growth law of tumor is of great significance for the prevention and control of tumor.In recent years,scholars at home and abroad have made very rich research results on the research of deterministic biological immune system,but the relevant results of the dynamic behavior of biological immune system in random environment still need to be improved.At present,there are still some deficiencies and difficult problems to be solved.On the one hand,the random switching of tumor living environment in different states makes the dynamic behavior analysis of tumor immune system particularly complex.On the other hand,the tumor immune system in the random switching environment will be affected by various environmental noise,which makes it more difficult to study the growth mechanism and evolution law of tumor cells.In this paper,based on the theory of stochastic differential equations,the dynamic modeling and analysis of tumor immune system are carried out,and the tumor evolution behavior is investigated by studying the dynamic behavior and system response,which provides new theoretical value and research ideas for understanding the specific biological mechanism of tumor evolution and tumor treatment.The main contributions and conclusions of this paper are as follows:1.The dynamic behavior of univariate tumor growth system in random switching environment is studied.Firstly,based on the existing deterministic tumor growth models,we propose a univariate tumor growth model induced by white noise in a random switching environment.The random switching environment is expressed as a multi-state Markov chain,which corresponds to the sudden instantaneous transition between two or more sets of parameters in two or more different environments or states.Then we prove the existence and uniqueness of the positive solution.Then,the threshold conditions of extinction and weak persistence are strictly deduced by using the ergodicity of Markov chain,the properties of limit distribution,Lyapunov function and Ito formula.In addition,we also give a sufficient condition for the random persistence of tumor cells.Finally,the important theoretical results are further illustrated and verified by using Milstein method and Gillespie algorithm for random simulation,and it is found that the existence of random switching environment is conducive to the extinction of cancer.2.The extinction and persistence of Kirschner Panetta(KP)tumor immune model induced by white noise in random switching environment are studied.Kirschner Panetta tumor immune model is a three-dimensional equation that describes the interaction between activated immune system cells,tumor cells and cytokine interleukin-2(IL-2).Firstly,we consider the influence of environmental fluctuation in random switching environment.Assuming that environmental fluctuation mainly stimulates the mortality of immune cells and effector molecules and the growth rate of tumor cells,we introduce multiplicative white noise into Kirschner Panetta system,construct a random Kirschner Panetta tumor immune model induced by white noise in random switching environment,and then establish sufficient conditions for extinction,mean non persistence,weak persistence and random persistence.The critical threshold between weak persistence and extinction is obtained.The conclusion shows that in the case of immune cell monitoring,environmental noise can improve the clearance rate of tumor cells.The study of global dynamics in deterministic and random environments is helpful for clinicians and oncologists.3.The dynamic properties of random Kirschner Panetta tumor immune system induced by Lévy noise in random switching environment are studied.The research shows that only white noise and random switching environment can not explain the huge environmental changes in the actual tumor system.Therefore,we add Lévy noise to the Kirschner Panetta model induced by two noises to construct a random Kirschner Panetta tumor immune system with Markov switching and Lévy noise.Firstly,the existence and uniqueness of the global positive solution of the system are proved.Then,based on the M-matrix analysis method and It(?)s formula,the sufficient conditions for random persistence,extinction,mean non persistence and mean strong persistence of tumor cells are proved mathematically.The stochastic model we constructed is helpful to have a new understanding of the complexity of tumor growth when the tumor microenvironment changes sharply or suddenly.