The Spatial Spreading Of Mosquitoes And Transmission Of Mosquito-borne Diseases

Infectious diseases caused by insect vectors such as mosquitoes have been a considerable threat to human health.Mosquitoes cause more human suffering than any other organism,over one million people worldwide die from mosquito-borne diseases every year.Since no effective treatment or vaccine is available to treat or prevent those mosquito-borne disease in spite of the sustainable advancement of science and technology as well as the great improvement of medical facilities,it is necessary to understand its distribution and transmission mechanism.In 1911,Dr.Ronald Ross,the British microbiologist built a continuous differential equa-tion model to study the dynamic behavior of the transmission of malaria between mosquitoes and human,which initiated the research on mathematical models for mosquito-borne diseases.Mathematical models can provide a solid and tractable knowledge for the behavior and dynamics of the Aedes aegypi mosquito,which is the yellow fever mosquito carrying and transmitting most of the mosquito-borne diseases such as dengue fever,chickungunya,zika and yellow fever,etc,in order to develop public policies and strategies for control of those diseases.Most of the early models for infectious diseases are considered in a homoge-neous environment with study only on the epidemiological process as time goes on.Recently,it has been commonly realized that spatial diffusion,environmental heterogeneity and periodicity can make a great difference to the transmission dy-namics and regional risks of the model,which corresponds more to the reality.The dissertation focuses on the impact of environmental heterogeneity,periodicity,free boundary as well as periodically evolving domain on the spatial transmission char-acteristics of the Aedes aegypti mosquito.The specific research work is organized as follows.Chapter 1 gives a brief introduction of the background,research status and latest results about our research topic.Chapter 2 deals with the diffusion dynamics of the Aedes aegypti mosquito in a heterogeneious environment.Considering the spatial heterogeneity,we consid-er a reaction-diffusion-advection model with free boundaries to describe the spatial dispersal dynamics of the Aedes aegypti mosquito,where the vector mosquitoes pop-ulation is divided into two life stages:the winged form(adult female mosquitoes)and an aquatic population(including eggs,larvae and pupae).The global exis-tence and uniqueness of the solution are presented by utilizing contraction mapping principle and standard Lp theory with Sobolev imbedding theorem of parabolic sys-tems.Subsequently,the threshold for the model as well as its analytical properties is defined by variational method,and the long time asymptotic behavior of the reaction-diffusion-advection system is discussed.We give the sufficient conditions for the mosquitoes to be eradicated or to spread.Numerical simulations indicate the impact of advection and expanding capacity of the boundary on the mosquitoes'diffusion fronts.In Chapter 3,we consider periodic environment and the free boundary condi-tion to propose the Aedes aegypti transmission model which consists more with the actual diffusion pattern.The spatial-temporal threshold depending on temporal-periodicity and sufficient conditions for the spreading or vanishing of the mosquitoes are investigated by employing classic theories for parabolic systems such as oper-ator theory.By the method of upper and lower solutions as well as the iterative sequences,we finally prove that when spreading occurs,the long time solution of our system oscillates between two positive T-periodic solutions.Chapter 4 is devoted to the diffusive model for Aedes aegypti mosquito on a periodically evolving domain,which reflects the impact of the periodic evolution to the domain boundary depending on natural environment on the asymptotic behavior of the solution.For later analysis,we transform it into a reaction-diffusion system on a fixed domain with time-dependent diffusion term.We present the threshold depending on the domain evolution rate p(t)and analyze its implications by using the semigroup theory and spectral theory.Subsequently,by means of this threshold,we discuss the periodic solution together with its attractivity,and the impact of periodical domain evolution on the spreading or vanishing of the mosquitoes.With the notation(?),theoretical results and numerical simulations imply that the periodical domain evolution has a positive effect on the persistence of the Aedes aegypti mosquito if ?-2(t)<1.Chapter 5 is concerned with the reaction-diffusion model incorporating both the human and Aedes aegypti mosquito for the transmission of dengue virus.The results we establish are based on the biological reproductive index Q0 for the mosquito population and the basic reproduction number R0 for the infected classes with the application of calculus of variations,spectral radius,as well as eigenvalue problems.It is shown that when Q0<1,the Aedes aegypti mosquito will be extinct,as a result dengue can be controlled;while if Q0>1,mosquitoes will persist and the transmission of dengue virus can be estimated by the condition whether R0<1 or R0>1.Numerical simulations consisting with the theoretical analysis indicate that the disease risk is increasing with Q0 and R0,which suggest that it is of necessity to create an appropriate environment for disease prevention and control.Finally,overall results are summarized in Chapter 6,together with some prob-lems for future research.