Spatial Propagation Dynamics Of Non-cooperative Epidemic Models

Due to further globalization,humans are traveling to different parts of the world more frequently.Such movements increase the risk of the spread of pathogens and threaten global public health security.The spatial propagation theory of epidemic models is helpful to quantitatively understand the important factors affecting the spreading of disease and has attracted extensive attention.The spread of many diseases is described by non-cooperative systems,and different non-cooperative systems may have significantly different dynamics which correspond to different natural phenomena.Although there exist many results on the spreading theory of infectious disease models,on the one hand,much attention is dedicated to special traveling wave solutions,on the other hand,different epidemic models admit quite different dynamics,which leads to the lack of universal theory.The propagation theory of many important epidemic models,especially the asymptotic spreading to initial value problems,deserves further study.This thesis aims to investigate the spatial propagation dynamics of several non-cooperative epidemic models with multi-species or multi-groups.Firstly,we study the asymptotic spreading in a within-host viral infection model which describes the spatial expansion speeds of viruses and infected cells within an infected host.We first establish the boundedness of solutions to the Cauchy problem via local Lp-estimates and dual arguments.Then the spreading speed is obtained when the basic reproduction number of the corresponding kinetic system is larger than one.More precisely,the upper bounds of the spreading speed are given by constructing suitable upper solutions,while the lower bounds of the spreading speed are obtained by using the strongly continuous semigroup theory and introducing an auxiliary equation with nonlocal delay.Moreover,we prove that when the diffusion rates are the same,the solution locally uniformly converges to the positive steady state by the corresponding entire solutions and the Lyapunov like argument.When the basic reproduction number of the corresponding kinetic system is less than one,the virus dies out uniformly.Finally,we present some numerical examples to illustrate our main results,explore the possible extension and discuss the biological relevance of these results.Secondly,we investigate the spreading speed and invasion waves of infected species in the non-monotone vector-borne disease models with three species.We determine the spreading speed and minimal wave speed when the basic reproduction number of the corresponding kinetic system is larger than one.To estimate the lower bounds of spreading speed,we introduce the principal eigenvalues of a weak coupled elliptic system as well as the corresponding homogeneous Dirichlet problem and use the pull-back method and the uniform persistence argument.The upper bounds of spreading speed are obtained by constructing an auxiliary system and applying the comparison principle.We also show that solutions converge locally uniformly to the positive steady state by employing two auxiliary monotone systems.Moreover,it is proven that the spreading speed is the minimal wave speed of traveling waves.In particular,the uniqueness and monotonicity of traveling waves are obtained.When the basic reproduction number of the corresponding kinetic system is not larger than one,it is shown that solutions approach to the disease-free steady state uniformly and there are no traveling waves.Finally,numerical simulations are presented to illustrate the analytical results.Thirdly,we study the spreading properties and traveling waves that describe the co-invasion of three species to the new habitat for the same epidemic model as in the second part.Applying the uniform persistence idea from dynamical systems and the properties of the corresponding entire solutions,and combining the auxiliary monotone systems and the comparison principle,we investigate the propagation phenomena in two different cases,in which the susceptible individuals admit the faster and slower ability of spreading,respectively.It turns out that in the former case,the susceptible species may spread faster than the infected species under appropriate conditions,which leads to the spreading of multiple fronts at different speeds;while in the latter case,three species almost spread at the same speed.We further reduce the three-component traveling wave problem to a two-component traveling wave problem of a competitive system by utilizing the idea of force waves and obtain the existence and nonexistence of the co-invasive traveling wave solutions connecting zero to endemic equilibrium.We also conduct numerical simulations to illustrate our theoretical results and the possible extension.Finally,we discuss the spreading properties of the initial value problem for two Polio models with spatial diffusion.Considering the full interaction between the adult and the child classes,we obtain a unique propagation threshold of both groups,from which we can find the factors that determine the spreading ability of the virus.When the interaction between two groups is omitted,it is possible to observe different spreading-vanishing phenomena of the two groups.When the disease spreads successfully in one group,we estimate the spreading speeds of this group and obtain the uniform convergence of the other one.When the disease spreads successfully in both groups,we also estimate the spreading speeds of different groups,which might reveal the nontrivial interaction between the two groups from the view of asymptotic spreading.Finally,some numerical simulations are utilized to discuss and explain our main results.