J. Nonl. Mod. Anal., 3 (2021), pp. 301-319.
Published online: 2021-04
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${\it Wolbachia}$ 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.
}, issn = {2562-2862}, doi = {https://doi.org/10.12150/jnma.2021.301}, url = {http://global-sci.org/intro/article_detail/jnma/18792.html} }${\it Wolbachia}$ 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.