Threshold dynamics of a West Nile virus model with impulsive culling and incubation period

<p style='text-indent:20px;'>In this paper, we propose a time-delayed West Nile virus (WNv) model with impulsive culling of mosquitoes. The mathematical difficulty lies in how to choose a suitable phase space and deal with the interaction of delay and impulse. By the recent theory developed in [<xref ref-type="bibr" rid="b3">3</xref>], we define the basic reproduction number <inline-formula><tex-math id="M1">\begin{document}$ \mathcal {R}_0 $\end{document}</tex-math></inline-formula> as the spectral radius of a linear integraloperator and show that <inline-formula><tex-math id="M2">\begin{document}$ \mathcal {R}_0 $\end{document}</tex-math></inline-formula> acts as a threshold parameter determining the persistence of the model. More precisely, it is proved that if <inline-formula><tex-math id="M3">\begin{document}$ \mathcal {R}_0&lt;1 $\end{document}</tex-math></inline-formula>, then the disease-free periodic solution is globally attractive, while if <inline-formula><tex-math id="M4">\begin{document}$ \mathcal {R}_0&gt;1 $\end{document}</tex-math></inline-formula>, then the disease is uniformly persistent.Numerical simulations suggest that culling frequency and culling rate are strongly influenced by the biting rate. We also find that prolonging the length of the incubation period in mosquitoes can reduce the risk of disease spreading.</p>