# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021239
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## Threshold dynamics of a West Nile virus model with impulsive culling and incubation period

 School of Mathematics and Statistics, Xidian University, , Xi'an, Shaanxi 710126, China

* Corresponding author: Zhenguo Bai

Received  May 2021 Revised  July 2021 Early access October 2021

Fund Project: This research was supported by the NSF of China (No. 11971369), the NSF of Shaanxi Province of China (No. 2019JM-241) and the Fundamental Research Funds for the Central Universities (No. JB210711)

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 [3], we define the basic reproduction number $\mathcal {R}_0$ as the spectral radius of a linear integraloperator and show that $\mathcal {R}_0$ acts as a threshold parameter determining the persistence of the model. More precisely, it is proved that if $\mathcal {R}_0<1$, then the disease-free periodic solution is globally attractive, while if $\mathcal {R}_0>1$, 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.

Citation: Yaxin Han, Zhenguo Bai. Threshold dynamics of a West Nile virus model with impulsive culling and incubation period. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021239
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##### References:
Comparison of the long-term behavior of infectious mosquitoes and birds in different scenarios: culling and without culling.
Sensitivity analysis of $\mathcal{R}_0$. PRCCs represents the sensitivity index of $\mathcal {R}_0$
The curve of $\mathcal{R}_0$ with respect to $\tau$ for different culling interval
">Figure 4.  The contour plots of $\mathcal {R}_0$ with respect to $T$ and $p$ with different biting rate $\beta$ equal to (a) 0.03, (b) 0.05, (c) 0.07. Other parameters are chosen as in Figure 1
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