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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 Name | Affiliation | | 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. |
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