Analysis Of Some Mathematical Models For The Spatial Transmission Of West Nile Virus

West Nile virus(WNv)is an infectious disease which is found in temperate and tropical regions of the world.It has kept spreading across the world and still re-mains a threat to wildlife management and public health.As we know,no effective vaccine for the virus is currently available and antibiotics cannot work since a viral pathogen,not bacteria,causes West.Nile disease.There exists no specific treatment for WNv other than supportive therapy for severe cases and use of mosquito repellent becomes the most effective measure.Mathematically,it is necessary to understand the dynamics of WNv transmission.Most early models have only scrutinized the non-spatial dynamical formulation of the imodel.However,spatial diffusion is an important factor to affect the persistence and eradication of infectious diseases like WNv.In this dissertation,we consider mathematical models and utilize theory of partial differential equations to study the characterization of spatial-temporal transmission of WNv in bird-mosquito cycle.To begin with,the background of WNv transmission and history of related work are introduced in Chapter 1.and motivations and contributions are presented.Chapter 2 is devoted to the expanding process in the transmission of WNv.We formulate a free boundary problem with coupled system,which describes the diffusion of birds by a partial differential equation and the move-ment of mosquitoes by an ordinary differential equation.By straightening the free boundaries and the contraction mapping theorem,we prove the global existence and uniqueness of the solution,the comparison principle is also given.The basic reproduction number for the non-spatial epidemic model is defined and a threshold parameter for the corresponding problem with null Dirichlet boundary condition is introduced by considering the associated eigenvalue problems.The risk index associated with the disease in spatial setting is introduced and its properties are discussed.Some sufficient conditions for the WNv to vanish or to spread are giv-en.The sharp threshold with respect,to the expanding capacity is presented.The asymptotic behavior of the solution to the free boundary problem when the spread-ing occurs is considered.It is shown that the initial number of infected populations,the diffusion rate of birds and the length of initial habitat exhibit important im-pacts on the vanishing or spreading of the virus.A spreading-vanishing dichotomy is stated.In order to illustrate the impact of big and small expanding capacity on the transmission of WNv,numerical simulations are presented.In Chapter 3,we deal with a strongly-coupled elliptic system,which describe a West Nile virus model with cross-diffusion in a heterogeneous environment.The basic reproduction number for the corresponding cross-diffusive WNv model in a heterogenous environment is introduced through the next gener-ation infection operator and some related eigenvalue problems.The existence of coexistence states is presented by using a method of upper and lower solution-s.The true positive solutions are obtained by monotone iterative schemes.Our results show that a cross-diffusive WNv model possesses at least one coexistence so-lution if the basic reproduction number is greater than one and the cross-diffusion rates are small enough.Otherwise,the model have no positive solution.Numerical simulations are given to illustrate the influence of cross-diffusion and environmental heterogeneity on the transmission of WNv.Chapter 4 focuses on asymptotic periodicity in a diffusive West Nile virus model in a heterogeneous environment.W'e introduce a threshold pa-rameter for the corresponding diffusive WNv model in a heterogeneous environment by using the standard spectral analysis and the associated eigenvalue problems.It is shown that if a threshold parameter exceeds one,the model admits at least one nontrivial T-periodic solution,whereas if a threshold parameter is less than or equal to onethe model has no nontrivial T-periodic solution.By means of monotone iter-ative schemes,the true solution is obtained and the asymptotic behavior of periodic solutions is presented.To illustrate the effect of the environmental heterogeneity and seasonal periodicity on the outbreaks of WNv,some numerical simulations are presented.In Chapter 5,we are concerned with the spreading-vanishing dichoto-my for a time-periodic diffusive WNv model with moving front.To investi-gate the impact of spatial heterogeneity of environment,temporal periodicity on the persistence and era.dication of the West Nile virus,the free boundary is introduced to represent the moving front of the infected region.The basic reproduction number and the risk index,which depend on spatial heterogeneity,temporal periodicity and spatial diffusion,are introduced by utilizing the next generation infection operator and associated eigenvalue problems.We deal with the T-periodic boundary value problem in half space describe WNv.Sufficient conditions for the spreading and vanishing of the West Nile virus are given.Numerical simulations are presented to illustrate the influence of the expanding capability on the moving front.The last chapter is for epidemiological explanation and future considerations.