Dynamical Analysis Of Two Classes Of Immune Response Models

Tumor immunotherapy,especially immune checkpoint inhibitors,has attracted much attention as the latest anti-tumor therapy in modern times.In Chapters 2,3 and 4,a class of tumor-immune model using the immune checkpoint inhibitor,anti-PD-1,will be studied.Firstly,without treatment,we study the global dynamics of the tumor immune interaction model with exponential and logistic tumor growth rate,respectively.Secondly,we consider the influence of time delay on the tumor-immune model with logistic growth rate,and find that the model exhibits Hopf bifurcation,i.e.,time delay may induce periodic oscillations between tumor and T cell.Thirdly,when the immune checkpoint inhibitor anti-PD-1 is used,we investigate the stability of the equilibria and the existence of Hopf bifurcation in the tumor-immune model with exponential growth rate.Finally,we analyze a dynamical model for the innnate immune response to initial pulmonary infections in Chapter 5.It is shown that the model undergoes a sequence of bifurcations including saddlenode bifurcation,subcritical and supercritical Bogdanov-Takens bifurcations,Hopf bifurcation,and degenerate Hopf bifurcation as the parameters vary.The thesis consists of six chapters as follows.In Chapter 1,we review the research background and status of tumor immunotherapy and tumour-immuune interaction models with immune checkpoint in-hibitor anti-PD-1 or anti-PD-L1.We also present the preliminary knowledge that will be used in this thesis.In Chapter 2,we fill several key gaps in the study of the global dynamics of a highly nonlinear tumor-immune model with an immune checkpoint inhibitor proposed by Nikolopoulou et al.(Letters in Biomathematics,5(2018),S137-S159).For this tumour-immune interaction model,it is known that the model has a unique tumour-free equilibrium and at most two tumorous equilibria.We present sufficient and necessary conditions for the global stability of the tumour-free equilibrium or the unique tumorous equilibrium.The global dynamics is obtained by employing a new Dulac function to establish the nonexistence of nontrivial positive periodic orbits.Our analysis shows that we can almost completely classify the global dynamics of the model with two critical values CK0 and CK1(CK0>Ck1)for the carrying capacity CK of tumour cells and one critical value dT0 for the death rate dT of T cells.Specifically,we obtain the following conclusions.(i)When no tumorous equilibrium exists,the tumour-free equilibrium is globally asymptotically stable.(ii)When CK?CK1 and dT>dT0,the unique tumorous equilibrium is globally asymptotically stable.(iii)When CK>CK1,the model exhibits saddle-node bifurcation of tumorous equilibria.In this case,we show that when a unique tumorous equilibrium exists,tumor cells can persist for all positive initial densities,or can be eliminated for some initial densities and persist for other initial densities.When two distinct tumorous equilibria exist,we show that the model exhibits bistable phenomenon,and tumor cells have alternative fates depending on the positive initial densities.(iv)When CK>CK0 and dT=dT0,or dT>dT0,tumor cells will persist for all positive initial densities.In Chapter 3,we investigate the effect of time delay on the interaction between T cells and tumor cells,so a time delay is introduced into the tumor immune model in Chapter 2.For this delay model,we only study the case when it has a unique tumor equilibrium.We obtain the stability of the equilibrium with time delay ?varying,and prove that it undergoes Hopf bifurcation near the equilibrium.In addition,the expression of the direction of Hopf bifurcation and the stability of the bifurcating periodic solution are given.This means that when tumor cells are induced to express PD-L1 and bind to PD-1 expressed by T cells,the complex PD-1-PD-L1 is produced,which can inhibit the activation and proliferation of T cells.This process is affected by time delay,and the density of T cells and tumor cells may coexist in the form of periodic oscillations.In Chapter 4,when the immune checkpoint inhibitor is used,we consider a three-dimensional tumor-immune model with exponential tumor growth rate,and discuss the stability of equilibria and bifurcation analysis of the model with and without treatment,respectively.Specifically,for the model without treatment,there is always a tumor-free equilibrium and at most one tumorous equilibrium.We analyze the stability and types of equilibria.For the model with treatment,there are at most five tumor-free equilibria and two positive equilibria,we analyze the stability of the equilibria,and prove that a family of periodic solutions will be produced by Hopf bifurcation near one of the positive equilibria.Biologically,our analysis shows that there are two thresholds ?A1 and dT1 for an intravenous,continuous injection?A and the death rate dT of T cells.These two thresholds can describe the number and types of tumor-free equilibria as well as the number and stability of positive equilibria.This implies that,due to the addition of anti-PD-1 therapy,the existence and stability of all tumor-free equilibria,positive equilibria,and periodic solutions are dependent on the threshold of continuous intravenous injection.In Chapter 5,we study a dynamical model for the innate immune response to initial pulmonary infections.The model mainly describes the interaction between a pathogen and neutrophilis(also known as polymorphonuclear leukocytes).It has been reported that COVID-19 patients had an increased neutrophil count and a decreased lymphocyte count in the severe phase and neutrophils may contribute to organ damage and mortality.It is shown that the model undergoes a sequence of bifurcations including subcritical and supercritical Bogdanov-Takens bifurcations,Hopf bifurcation,and degenerate Hopf bifurcation as the parameters vary,and the model exhibits rich dynamics such as the existence of multiple coexistent periodic oscillations,homoclinic orbits,bi-stability and tri-stability,etc.In Chapter 6,we conclude the current work and mention the limitations of this thesis and future research prospects.