Spatial Dynamics Of Several Classes Of Reaction–diffusion Epidemic Models

In recent years,a great deal of mathematical models are used to characterize the various infectious diseases.The epidemic model is a tool which has been used to study the mechanisms by which diseases spread,to predict the future course of an outbreak and to evaluate strategies to control an epidemic.Especially,some reaction–diffusion epidemic models have attracted more and more attention and study.Threshold dynamics,traveling wave solutions and asymptotic speed of spread are crucial in the study of the spatial spread of infectious diseases.Threshold dynamics describes the transmission results of the infectious diseases in the long time behavior;traveling waves can give rise to a moving zone of transition from an infective state to a diseases-free state.Furthermore,asymptotic speed of spread just represent the velocity of disease spread.It is well-known that some diffusion–reaction epidemic models with multi-group,multi-strain or multiple parallel infectious stages not only possess complicated structure(the number of equations increasing)of such evolution systems,but also lose the monotonicity(this leads to the lack of the comparison principle).Thus,the study of threshold dynamics,traveling wave solutions and asymptotic speed of spreading for such systems is very challenging.This thesis is devoted to such problems.The main results are as follows:Firstly,we study full information on the existence and nonexistence of traveling waves of a two-group SIR epidemic model with constant recruitment.Our tactic is first to construct an invariant cone of initial functions defined in a bounded domain,and apply a fixed point theorem on this cone,then we prove that when the basic reproduction ratio R0> 1,there exists the minimal wave speed c*> 0 such that for each c?c*,the system admits a nontrivial traveling wave solution with wave speed c by using the Lapunov functional and a limiting argument,and for R0> 1and 0 < c < c*,there exists no nontrivial traveling waves satisfying the system.In addition,when R0? 1,we also show that the system admits no nontrivial traveling waves.Secondly,for infectious diseases that progress through a long infectious period,infectivity or infectiousness can vary significantly in time.Thus,we analyze a diffusive epidemic model with multiple parallel infectious stages.Furthermore,we obtain complete information about the existence and nonexistence of nontrivial traveling wave fronts.Namely,we prove that when the basic reproduction number R0 is larger than one,there exists a critical number c*> 0 such that for each c?c*,the system admits a nontrivial traveling wave front with wave speed c,and for c < c*,the system admits no nontrivial traveling wave fronts.When R0? 1,we show that there exists no nontrivial traveling wave fronts.In addition,we also consider the effects of the diffusion coefficients on the minimal wave speed c*.Thirdly,we incorporate spatial diffusion,the distributed latency of disease and temporal heterogeneity into the multi–group SIR disease model and to investigate the threshold dynamics of the derived model.We introduce the basic reproduction number R0 of the model via a next generation operator and obtain the relationship between R0 and the spectral radius of Poincar(?) map of the associated linear equation.Then,in terms of the comparison arguments and persistence theory,we establish the threshold dynamics of the system.Fourthly,we focus on a time periodic and two–strain SIS epidemic model with latent period.We firstly analyze the threshold dynamics of a single SIS epidemic model.Defining the basic reproduction number of strain i by Ri0 for i = 1,2,we prove that if Ri0?1(i = 1,2),then the disease–free periodic solution is globally attractive;If Ri0> 1,then the infectious disease of strain i is uniformly persistent.Furthermore,the main question that we address is the threshold dynamics of the two–strain SIS epidemic model.Introducing the invasion numbers of the two strains by (?)i0(i = 1,2)which determine the ability of each strain to invade the single–strain equilibrium of the other strain,we show that if Ri0?1(i = 1,2),then the disease–free periodic solution is globally attractive;if Ri0> 1 Rj0(i ? j,i,j = 1,2),then competitive exclusion,where the j-th strain dies out and the i-th strain persists,is a possible outcome and if (?)i0> 1(i = 1,2),then the disease is uniformly persistent.The last section is devoted to considering the asymptotic speed of spread of a time periodic reaction–diffusion SIR epidemic model which lacks the comparison principle.We firstly estimates the uniform boundedness of the solution of the model and then took into account the spreading properties of the corresponding solution of the model in terms of the basic reproduction number R0.More specifically,if R0 1,then the solution of the system converges to the disease-free equilibrium as t ? ? and if R0> 1,the disease is persistent behind the front and extinct ahead the front.