Tumor-immune Dynamics Model Considering Treatment,Distributed Delay And Tumor Mutation

In this article,a series of kinetic models describing the interaction between tumor cells and immune cells in human body are established.The effects of chemotherapy,immunotherapy,distributed delay and tumor mutation on the kinetic behavior of the models are investigated.Through theoretical analysis and numerical simulation,some significant results are obtained,which provide some valuable suggestions for the treatment of cancer.In the first chapter,the background of this study is introduced,and the research status of tumor immune model and the main work of this thesis are summarized.In the second chapter,we consider a kinetic model of the interaction between tumor cells and immune cells in two states(hunting state and resting state).The effects of chemotherapy,immunotherapy and transition time delay between immune cells on the kinetic behavior of the system are analyzed.The existence and stability conditions of equilibria and existence conditions of periodic solution are obtained.The study finds that if chemotherapy is effective enough,tumor cells will be removed.However,if we want to eliminate tumor cells under the condition that immune cells continue to survive,we need to ensure that the chemotherapy effect is better,increase the intensity of immunotherapy.When the intensity of immunotherapy is high and the effect of chemotherapy is low,tumor and immune cells may coexist.At the same time,we also vi find that the introduction of distributed delay may lead to periodic oscillation.Especially,when the mean transition delay is small,the positive equilibrium is a stable focus.When the mean transition delay increases beyond a threshold,the positive equilibrium becomes unstable and produces periodic solutions,which can explain the recurrence of tumors.In the third chapter,a kinetic model considering both the stimulation and destruction effects of tumor cells on immune cell proliferation is studied.First,nonnegativity of solution and dissipative property of system are proved.Then the existence and stability conditions of the boundary equilibria and positive equilibrium are discussed.We find that the tumor can be completely eliminated if the immune cells have a high clearance rate of tumor cells.Especially,when the clearance rate of immune cells and the destruction rate of tumor cells are both large,the equilibria of no tumor and no immune can be stable simultaneously.When the destruction rate of tumor cells is high and the clearance rate of immune cells is low,the non-immune equilibrium and positive equilibrium can be stable simultaneously.Numerical simulation verifies the correctness of theoretical analysis.The above results explain the complex interrelationship between tumor cells and immune system.In the fourth chapter,a kinetic model describing the interaction between two kinds of tumor cells and immune system is established based on the assumption that some innate tumor cells mutate under the pressure of immune cells.First,nonnegativity of solution and dissipative property of system are proved.Then the existence conditions of all kinds of boundary equilibria and positive equilibrium are obtained,and their asymptotic stability is analyzed.We find that the tumor can be eliminated if the immune cells have a high clearance rate of two types of tumor cells.When the mutation rate of the tumors is high and the clearance rate of immune cells to adaptive tumor cells is high,to innate tumor cells is low,tumor cells can also be eliminated.However,if the mutation rate is large,and the clearance rate of immune cells to innate tumor cells is large,to adaptive tumor cells is small,it can only clear the innate tumor cell,but not destroy the mutated tumor cells.Finally,we analyze the influence of thevii stimulation delay of immune cells on the dynamic behavior of the system.It finds that the introduction of the delay may lead to Hopf bifurcation at two equilibria and then to periodic oscillation.In particular,at the equilibrium of mutated tumor-immune cells,when the average stimulation delay is in the middle level,the equilibrium becomes unstable and periodic solutions are produced.When the average stimulation delay is large or small,the equilibrium is locally asymptotically stable.At the positive equilibrium,we find that the equilibrium is an asymptotically stable focus when the average delay is small.When the delay increases beyond a threshold,the positive equilibrium loses its stability and produces periodic solutions by Hopf bifurcation.These results may explain the long-term recurrence of tumors under different conditions.Numerical simulation confirms the theoretical results.In the fifth chapter,the conclusions of this study are summarized briefly,their biological significance and theoretical value are analyzed,and the problems and directions for further study are pointed out.