Spatiotemporal Dynamics Of Time-periodic Reaction-diffusion Epidemic Models

The time periodic reaction-diffusion epidemic models play a powerful role in understanding the evolution of spatial distribution of epidemics with seasonality,the spatiotemporal dynamics of which emerges as one of the front topics in evolu-tion equations.Besides the essential difficulties from the non-monotone coupling structure and non-automomy of S-I type epidemic models,other realistic factors also leads to more challengies in analyzing the spatial dynamics.When it comes to analysing the models,new difficulties also result from the introduction of other realistic factors,except for the coupling of non-autonomy and non-monotonicity.The current thesis is devoted to the geographic spread dynamics of time periodic reaction-diffusion S-I type epidemic models posed on unbounded domain and the threshold dynamics on bounded domain.The details are as follows:Chapter 2 aims to probe the existence of periodic traveling waves for a time periodic reaction-diffusion Kermack-Mc Kendrick epidemic model with nonlocal in-teraction derived by the mobility of individuals during the latent period of the dis-ease.Firstly,when the basic reproduction number R₀of the corresponding spatial homogeneous system is larger than 1,the critical value of wave speed c^*ensuring the existence of traveling waves is defined,and the existence of time periodic trav-eling waves with speed c>c^*is further obtained.Secondly,there is no non-trivial periodic traveling wave when the basic reproduction number R₀<1.Moreover,the influences of seasonality and nonlocal effects on c^*are discussed by numerical simulations,which reveal the significant role of seasonal fluctuation and incubation period in disease prevalence.Chapter 3 is concerned with the propagation dynamics of time periodic reaction-diffusion S-I epidemic model with Logistic demographic structure.Firstly,the ex-istence of asymptotic speeds of spread for the model system is explored by using dynamical system method and persistence theory.In this case,the difficulties from Logistic growth（cause a trivial equilibrium）and non-autonomy of the system need to be overcome.Secondly,the existence of weak periodic traveling wave solution-s is proved by constructing appropriate upper and lower solutions combined with the limit arguments,and the asymptotic boundary conditions are verified by using the results on the asymptotic speeds of spread,and then the existence of persistent periodic traveling wave solutions is obtained.Finally,the existence of the critical periodic traveling wave solution is proved based on the method of taking the limit of supercritical wave speed sequence and the existence results of supercritical waves.In chapter 4,we study the asymptotic speeds of spread and periodic traveling wave for the time periodic reaction-diffusion S-I epidemic model with treatment.Compared with the traditional S-I endemic model,the dimension of the model is increased due to treatment intervention,which brings new difficulties in investigating the dynamics of model system.Firstly,the characterization of wave speed c^*is given by analyzing linearized systems（two-dimension）.The existence of asymptotic speeds of spread is obtained by combining the periodic principal eigenvalue problem,comparison method and persistence theory of two-dimensional infected systems.Secondly,by constructing the suitable super-and sub-solutions for the truncation problem corresponding to the traveling wave systems,we obtain the existence of periodic solutions via the fixed point theorem twice.Then,the existence of periodic traveling wave solutions is proved by regularity estimation and limit arguments.The aforementioned conclusions imply that the asymptotic speeds of spread coincides with the minimum wave speed of periodic traveling waves.Finally,the effects of treatment ratio on the asymptotic speed of spread c^*and the ability of disease expansion were discussed numerically.In Chapter 5,the existence of periodic traveling wave solution with critical wave speed for a time periodic reaction-diffusion S-I epidemic model without demograph-ic structure is investigated.Although we discuss the existence of critical periodic traveling wave in chapter 3 via the limit method in terms of the supercritical speed sequence associated with periodic traveling wave solutions,it is invalid for the cur-rent model system.The essential difficulties arise from the asymptotic boundary properties and the non-autonomy of the system.We adopt the approach of super-and sub-solutions and fixed point theorem to the truncated problem associated with the traveling wave system combined with limiting arguments to obtain the existence result.Here the construction of the super-and sub-solutions is quite different from the supercritical speed case.In Chapter 6,we incorporate the spatial and temporal heterogeneity,extrin-sic incubation period of dengue virus and crowding effect of host population in-to dengue fever transmission,and propose a time-periodic and nonlocal delayed reaction-diffusion model.By overcoming the difficulties caused by high dimension-ality and complexity of spatiotemporal environment,we establish the threshold dy-namics of the model system in terms of the basic reproduction number R₀.The disease will go extinction if R₀<1,while the disease will be uniformly persistent if R₀>1.Furthermore,by virtue of numerical calculations for R₀,the effects of external incubation period,spatial heterogeneity,seasonal variation and crowding effect on the disease transmission are analyzed,respectively.