Dynamics Of Three Classes Of Reaction-Diffusion Infection Disease Models And Age-Structure Parasitism Predation Population Model

In recent years,the dynamic model of reaction-diffusion infectious disease and the pop-ulation dynamic model with age structure have been widely concerned by experts and schol-ars at home and abroad.In this paper,we propose three kinds of reaction-diffusion infectious disease models and one parasitism-mutualism-predation population model with age structure.Based on the basic theories of reaction diffusion equation,the theory of uniform continuous survival of dynamical systems,and the basic theories of functional differential equations,we mainly study the the dynamic behavior of the model.The main research contents are summarized as follows:1.In the first chapter,we first introduce the infectious diseases caused by viruses in recent years and the social and economic harm caused by them.Meanwhile,we introduce the importance of multiple relationships between populations for disease transmission.Based on these facts,we give the research background and significance of this paper.Then,we give the research status of the virus infection model with humoral immunity,the multi-group infectious disease model,the vector host infectious disease model with seasonal influence,and the population model with multiple relationships,respectively.Through the introduction and summary of the research status,we give the starting point and main research purpose of this paper in the third part.2.In the second chapter,we set up a class of reaction-diffusion virus propagation model with humoral immune,where we use the nonlinear function to depict infection rates between virus and cells,cells and cells.In heterogeneous environment,we define the virus infec-tion basic reproductive number₀,and we define the basic reproductive number₁with antibody response in the homogeneous environment.Based on the two basic reproductive numbers,using the Lyapunov function,La Salle invariable principle,we focus on research-ing the threshold dynamics of the virus transmission in two environment.And the threshold dynamics of the virus transmission in the absence of antibody response and in the presence of antibody response are obtained.The dynamic behavior of viral infection threshold under antibody response can only be obtained in the case of homogeneous space,which fully illus-trates the difficulty of antibody response and spatial heterogeneity in model analysis.Finally,numerical simulations are used to verify our results and conjecture.3.In the third chapter,considering the differences between individuals,heterogeneous environments and immune loss rate,we establish the mutli-group SIR and mutli-group SEIRS reaction-diffusion infectious diseases model,respectively.Here,we mainly use the nonlinear function to describe the spread of disease between susceptible individuals and infected indi-viduals.Because the SIR model can be regarded as a special case of SEIRS model,in this chapter we mainly introduce the results of SEIRS model.By the next generation operator method,we define the basic reproductive number₀of the model.Moreover,we obtain the global asymptotic stability of the disease-free steady-state under the condition of diffusion coefficient or permanent immunity.Using the uniform persistence theory,we obtain the per-sistence of the system and the existence of the endemic steady-state of the disease.Finally,the global asymptotic stability of the endemic equilibrium is obtained under the condition of homogeneous space and heterogeneous diffusion.4.In the fourth chapter,considering that the virus takes some time to develop in both the host and the vector,and the time is affected by seasonal temperatures,we propose a reaction-diffusion vector-host epidemic model with general incidence and seasonal development peri-od,where we mainly use the nonlinear function to describe the infection rate between infected vector and the susceptible host,or between infected host and the susceptible vector.Through the next generation of infection operator method,we define the basic regeneration number₀.Based on₀,we establish the threshold dynamics of the disease,including the global attractivity of the disease-free periodic steady-state solution,the uniformly persistence of the disease,and the existence of the endemic periodic steady-state solution.We obtain that global of the disease-free periodic steady-state solution when the basic reproduction number₀<1.Furthermore,the uniform persistence of the disease and the positive periodic endemic steady-state are obtained when₀>1.By introducing the seasonal development periodic delay and the nonlinear incidence function,the theoretical analysis of this paper is very difficult.5.In the fifth chapter,considering the parasitism-mutualism-predation among cuckoos,crows and cats,we propose a parasitism-mutualism-predation model with stage-structure,where the nonlinear Holling-type II and Beddington-De Angelis-type functional responses are used to describe the relationships among them.This chapter mainly discusses the ex-istence of positive equilibrium of the three subsystems and the existence of the coexistence equilibrium.Using Krein-Rutman theorem and other mathematical analysis methods,we get the global stability of the trivial equilibrium.Furthermore,the uniformly strong persistence of subsystems and the uniformly weak persistence of the whole system are discussed.Be-cause of the introduction of the age structure and mature time delay,the derivation of the model and the theoretical analysis are quite difficult.Finally,the effects of stage-structure on the dynamic of crows,cuckoos and cats are discussed by using numerical models.