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Modeling the Transmission of West Nile Virus with {\it Wolbachia} in a Heterogeneous Environment
  
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KeyWord:West Nile virus model; {\it Wolbachia}; Basic reproduction number; Stability
Author NameAffiliation
Mengli Liu School of Mathematics and Statistics, Anhui Normal University, Wuhu, Anhui 241000, China 
Min Zhu School of Mathematics and Statistics, Anhui Normal University, Wuhu, Anhui 241000, China 
Xiaofei Song School of Mathematics and Statistics, Anhui Normal University, Wuhu, Anhui 241000, China 
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Abstract:
      {\it Wolbachia} are maternally transmitted endosymbiotic bacteria. To investigate the effect of {\it Wolbachia} on the spreading and vanishing of West Nile virus, we construct a reaction-diffusion model associated with the {\it Wolbachia} parameter in a heterogeneous environment, which has nonlinear infectious disease parameters. Based on the spectral radius of next infection operator and the related eigenvalue problem, we present a corresponding explicit expression describing the basic reproduction number. Furthermore, utilizing this number, we not only give out the stability of disease-free equilibrium, but also analyze the uniqueness and globally asymptotic behavior of endemic equilibrium. Our theoretical results and numerical simulations indicate that only if {\it Wolbachia} reach a certain magnitude in mosquitoes, it can be effective in the control of West Nile virus.