Global Dynamics And Travelling Wave Solutions For Diffusive Viral Infection Dynamical Models

The main purpose of this thesis is to study global dynamics of diffusive viral infection dynamical models,which includes the computations of basic reproduc-tion number,persistence,travelling wave solutions,Turing instability,and so on.This thesis mainly applies mathematical theories and methods such as,Lyapunov stability theory and LaSalle invariance principle of functional differential equation-s,analytic semigroups theory and comparison arguments for parabolic equations,Sobolev embedded theorem,the strong maximal principle,Schauder's fixed point theorem,and so on.The main innovations of this thesis are:1.The effects of nonlocal time delay,nonlocal dispersal,and time periodic non-local time delay are innovatively introduced into diffusive viral infection dynamical models.Several new types of partial differential equations dynamical models are established to describe virus transmission.2.Existence result of travelling wave solutions for viral infection dynamical model with nonlocal time delay is established by constructing an invariant cone and employing Schauder's fixed point theorem.3.For a nonlocal dispersal viral infection dynamical model,existence of trav-elling wave solutions and its boundary asymptotic behavior are investigated by constructing suitable Lyapunov functions and employing Lebegue dominated con-vergence theorem.The main difficulties are that the semiflow generated by the model does not have the order-preserving property and the solutions lack of regu-larity.4.For a new viral infection dynamical model with spatial heterogeneity,dis-crete delay,spatial non-locality,and temporal heterogeneity,the mathematical d-ifficulty in proving the existence of the positive infection periodic solution is that we cannot directly verify the solution semiflow N is ?-condensing or N is convex?-contracting(0??? 1).To overcome it,we construct an equivalent norm,and then show that the solution maps of model are ?-contracting.Theoretical analyses and simulations for the spatially homogeneous model demonstrate rich dynamic-s,including the occurrence of Hopf bifurcation,Turing instability,and spatially inhomogeneous pattern formations.The details are as follows:In Chapter 3,a mathematical model for viral infection dynamics with absorp-tion effect and chemotaxis is proposed.By adopting permanence theoretical tech-niques for partial functional differential equations,singular perturbation theory,and characteristic equation analysis,threshold dynamics,sufficient conditions for exis-tence of travelling wave solutions,and necessary conditions for Turing instability at the infection steady state can be obtained.In Chapter 4,a new viral infection dynamical model is developed to make use-ful contributions to understanding caspase-1-mediated pyroptosis by inflammatory cytokine,by incorporating spatial heterogeneity,discrete delay,and spatial non-locality.By adopting Schauder's fixed point theorem,we investigate existence of travelling wave solutions.We successfully construct the super-solutions and sub-solutions.In Chapter 5,based on Chapter 4,a class of non-cooperative reaction-diffusion viral infection dynamical model is considered.We successfully establish the general result of travelling wave solutions.In Chapter 6,existence of travelling wave solutions is investigated for a nonlocal dispersal viral infection dynamical model.The main difficulties are that the semi-flow generated by the model does not have the order-preserving property and the solutions lack of regularity.The boundary asymptotic behavior of travelling wave solutions is obtained by constructing suitable Lyapunov functions and employing Lebesgue dominated convergence theorem.In Chapter 7,the well-posedness of solutions,basic reproduction number,threshold dynamics and Turing instability of viral infection dynamical model with spatial heterogeneity,discrete delay,spatial non-locality,and temporal heterogene-ity are investigated.We are the first time to derive necessary conditions for the occurence of Turing instability for high-dimensional system,which is constituted by four equations.