Several Classes Of Dynamic Modeling,stability And Periodic Oscillations Describing The Inflammatory Response In The Lesion Area Of Kawasaki Disease

In this thesis,several classes of dynamic models were constructed and studied for problems related to the inflammatory response in the lesion area of Kawasaki disease.Using Lyapunov stability theory,Lyapunov-LaSalle invariance principle,permanence theory,Hopf bifurcation theory,center manifold theorem and normal form method,coincidence degree theory,etc.,the dynamics properties of several classes of nonlinear Kawasaki disease dynamic models are studied,such as the local and global asymptotic stability of the equilibria,the permanence of the model,the existence of Hopf bifurcation and periodic solutions(periodic oscillations),etc.Specifically,the following aspects of the study are included:In Chapter 3,for a class of nonlinear ordinary differential equation dynamic models of Kawasaki disease,sufficient conditions for the global asymptotic stability of the inflammatory factors-free equilibrium and the inflammatory factors-existent equilibrium are established by constructing suitable Lyapunov functions and analyzing their derivatives in detail and using the classical Lyapunov-LaSalle invariance principle.In Chapter 4,based on some new biological findings,a class of nonlinear Kawasaki disease differential equation dynamic model with time delay is constructed to describe the interaction between important cytokines and endothelial cells in Kawasaki disease lesion areas.This model exhibits forward/backward bifurcation.By analyzing the corresponding characteristic equations in detail,the local stability of the inflammatory factors-free equilibrium and the inflammatory factorsexistent equilibria are established.Interestingly,the time delay does not affect the local stability of the inflammatory factors-free equilibrium;however,the time delay as the bifurcation parameter may change the local stability of the inflammatory factors-existent equilibrium,and stability switches as well as Hopf bifurcation may occur within certain parameter ranges.Further,the properties of Hopf bifurcation are investigated by using the center manifold theorem and normal form method.By skillfully constructing Lyapunov functionals and combining Barbalat's lemma and Lyapunov-LaSalle invariance principle,we establish some sufficient conditions for the global stability of the inflammatory factors-free equilibrium and the inflammatory factors-existent equilibrium.In addition,the permanence of the model is obtained,and some explicit expressions for the ultimate lower bound of any positive solutions of the model are obtained.In Chapter 5,considering that Kawasaki disease is influenced by environmental factors,a class of time delay differential equation dynamic model for nonautonomous Kawasaki disease is constructed based on Chapter 4.For the general nonautonomous Kawasaki disease model,the permanence of the model is obtained by analyzing the properties of the solutions and the range of time delays very precisely.Inflammatory factor extinction and global attractivity of the model are obtained by constructing appropriate Lyapunoval functionals combined with Barbalat's lemma and Fluctuation lemma.For the corresponding periodic Kawasaki disease model,the continuation theorem in the coincidence degree theory is used to establish sufficient conditions for the existence of positive periodic solutions of the model.In addition,three methods are applied to some classic biological mathematical periodic models,and several types of explicit sufficient conditions are obtained.The main results obtained largely improve or extend some of the existing results in the literatures.In Chapter 6,considering that the normal proliferation of endothelial cells in the focal area of Kawasaki disease is portrayed by logistic growth and it is more reasonable to divide the endothelial cells into normal endothelial cells and injured endothelial cells,a class of nonlinear Kawasaki disease ordinary differential equation dynamic model with logistic growth is constructed to describe the vascular injury in the focal area of Kawasaki disease.This model exhibits forward/backward bifurcation.The local stability of vascular injury-free equilibrium and vascular-injury equilibrium are obtained by analyzing their characteristic equations.The global stability of the equilibria of the model are established by constructing appropriate Lyapunov functions combined with the Lyapunov-LaSalle invariance principle.Interestingly,the parameter(endothelial cell growth factor promotes the growth rate of endothelial cell proliferation)as a bifurcation parameter can cause Hopf bifurcation under certain conditions,and vascular-injury equilibrium is always unstable when this parameter is larger.The results suggest that the control of vascular injury in Kawasaki disease requires control of the size of the basic reproduction number and also the growth rate of endothelial cell proliferation promoted by endothelial cell growth factor.The main innovations of this thesis are summarized as:1.For a class of nonlinear ordinary differential equation dynamic model of Kawasaki disease,sufficient conditions for the global asymptotic stability of the inflammatory factors-free equilibrium and the inflammatory factors-existent equilibrium of the model are established for both forward and backward bifurcation scenarios are established.The local asymptotic stability results in the literature are extended.2.Based on the work in Chapter 3,a class of nonlinear time delay differential equation dynamic model for Kawasaki disease is constructed and its global dynamics are investigated.It is found that the model exhibits forward/backward bifurcation,and the time delay as the bifurcation parameter can change the stability of the inflammatory factors-existent equilibrium(stability switches)and cause Hopf bifurcation within certain parameter ranges,which reveals the complexity of the inflammatory response in the Kawasaki disease focal area.The permanence of the model is obtained,and some explicit expressions for the ultimate lower bounds of any positive solution of the model are obtained,which can be used to estimate the inflammatory factor concentration in the focal area of Kawasaki disease.3.Based on the work in Chapter 4,a class of nonautonomous time delay differential equation dynamic models for Kawasaki disease is constructed and their global dynamics are studied.More general permanence results are obtained,which can be applied to the permanence studies of some virus models and extend and improve the permanence results in Chapter 4.For the corresponding periodic system,the upper and lower bounds of the corresponding operator equations are constructed skillfully,and then the continuation theorem in the coincidence degree theory is used to establish two types of explicit and sufficient conditions for the existence of the positive periodic solutions of the model.In addition,three methods are extracted and applied to some biological mathematical periodic models,and the results obtained largely improve and extend some results in the existing literatures.4.A class of nonlinear Kawasaki disease ordinary differential equation dynamic model is constructed to describe the vascular injury in Kawasaki disease.It is found that the model exhibits forward/backward bifurcation,and the model-related parameters can cause Hopf bifurcation.The global dynamics of the model axe established,including the global stability of the equilibria and the permanence of the model is obtained(some explicit estimate of the ultimate lower bounds of any positive solution of the model is obtained).A theoretical control strategy for vascular damage in Kawasaki disease is given.