The Impact Factors Of The Spatio-temporal Transmission Of Dengue Fever And Mathematical Analysis Of The Related Models

De.ngue fever is one of the most destructive vector-borne diseases,and the World Health Organization declared it as "the fastest-spreading insect-borne infec-tious disease worldwide" in 2014.Although medical level is increasing and sanitary conditions are improving nowadays,the outbreak of dengue fever causes tens of millions of people infected and tens of thousands of deaths around the world every year,which seriously threatens public health.The infection mechanism of dengue fever is more complex,so that there haven't been still particularly safe and effective vaccines until now.Therefore,in order to prevent and control dengue fever and other mosquito-borne diseases,the investigations on their transmission mechanism have been one of the important research topics in the current epidemiology and ecological mathematics.The research on the mathematical models for mosquito-borne infectious dis-eases can be traced back to the malaria model built by Dr.Ross,a British microbi-ologist and Nobel Prize winner,in 1911.However,the models designed specially for dengue fever have been constructed since the 1960s and 70s.In recent decades,a variety of dengue fever models have been successively built and studied according to the complex transmission mechanism.These results not only enrich the dynamical theory of infectious diseases,but also help people gain an insight into the propa-gation process of dengue fever,furthermore,provide the theoretical basis for the effective prevention and control of mosquito-borne diseases including dengue fever.Considering that most existing dengue models,associated with fewer external factors,are characterized by the ordinary differential equations,We will focus on the availability of medical resources,the migration of infected human,media,coverage and others.Specific research arrangement is as follows:Chapter 1 reviews the elementary knowledge and research advances on dengue fever,as well as introduces the studying background of this paper.Chapter 2 is devoted to a dengue fever model incorporating the impacts of lim-ited medical resources and self-protection.The hospital bed number and utilization of bed nets are used to represent the availability of medical resources and the effica-cy of self-protection,respectively.Some properties of a threshold,named the basic reproduction number,are obtained by the spectral radius of next infection opera-tor and the relevant eigenvalue problem.We explore the stabilities of disease free equilibrium and endemic one by combining this threshold with Lyapunov function,iteration sequences,Sobolev imbedding theorem and so on.This chapter discovers the significant impacts of medical resources and self-protection on the transmission of dengue fever.Chapter 3 introduces the cross-diffusion into the reaction-diffusion model of dengue fever in a heterogenous environment,which emphasizes the influence be-tween human and mosquitoes in the diffusion.We focus on the coexistence solutions of the corresponding strongly-coupled elliptic system.Firstly,the sufficient condi-tions on the existence and non-existence of coexistence solutions are proposed by utilizing the spectral analysis,variational method,variable substitution,construc-tion of upper-lower solutions and so on.Subsequently,combining the theoretical results with numerical simulation and epidemiologic explanation,we show that the value of the basic reproduction number and the extent of cross-diffusion have great influences on whether the virus carried in human and that in mosquitoes can co-exist.In Chapter 4,we construct a dengue fever model with free boundaries,in which the index of media coverage is incorporated.This research aims to explore the im-pacts of human activities,including the migration capability of infected persons and the effectiveness of media coverage,on the transmission dynamics of dengue fever.By using the ways of variable substitution,Lp theory,contraction mapping principle and Schauder estimate,we obtain the existence and uniqueness of the global clas-sic solution.Next,the explicit expression of the basic reproduction number——risk index,related to the spatial-temporal variables and media index,is exhibited by employing semigroup theory,eigenvalue problem and variational method.With the help of this risk index,together with the comparison principle and strong max-imal principle,we discuss the sufficient conditions oi the vanishing or spreading of dengue fever,as well as the long-time asymptotic profile of the solution when spreading occurs.In last section,we give the corresponding epidemiologic explana-tion due to these analytical findings,and further reveal that human activities play an important role on the transmission of dengue fever.Chapter 5 is concerned with a dengue fever model with the nonlocal incidence rate and free boundaries.The nonlinear contact rate in an integral form with a ker-nel function is introduced to the model,which characterizes the interaction between the susceptible persons at the current location and the infected mosquitoes in the whole area.The corresponding risk index not only relates to the free boundaries,but also includes the nonlocal term.Besides,we discuss the impact of nonlocality on the basic reproduction number.In the analysis process about the model's dynam-ics,some techniques,such as energy integral,comparison principle and eigenvalue problem,are applied to investigate the various conditions on whether the dengue virus can be controlled.We also draw some conclusions about the complex impact of free boundary condition and nonlocality on the spread of dengue virus through the epidemic explanations.The whole work of this paper is summarized in Chapter 6.We also make some further plan about the future research.