# American Institute of Mathematical Sciences

• Previous Article
Maximum principle for the optimal harvesting problem of a size-stage-structured population model
• DCDS-B Home
• This Issue
• Next Article
Nonconstant positive solutions to the ratio-dependent predator-prey system with prey-taxis in one dimension
doi: 10.3934/dcdsb.2021239
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

## 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
##### References:

show all references

##### 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
 [1] Louis D. Bergsman, James M. Hyman, Carrie A. Manore. A mathematical model for the spread of west nile virus in migratory and resident birds. Mathematical Biosciences & Engineering, 2016, 13 (2) : 401-424. doi: 10.3934/mbe.2015009 [2] Rongsong Liu, Jiangping Shuai, Jianhong Wu, Huaiping Zhu. Modeling spatial spread of west nile virus and impact of directional dispersal of birds. Mathematical Biosciences & Engineering, 2006, 3 (1) : 145-160. doi: 10.3934/mbe.2006.3.145 [3] Abdelrazig K. Tarboush, Jing Ge, Zhigui Lin. Coexistence of a cross-diffusive West Nile virus model in a heterogenous environment. Mathematical Biosciences & Engineering, 2018, 15 (6) : 1479-1494. doi: 10.3934/mbe.2018068 [4] Jing Chen, Jicai Huang, John C. Beier, Robert Stephen Cantrell, Chris Cosner, Douglas O. Fuller, Guoyan Zhang, Shigui Ruan. Modeling and control of local outbreaks of West Nile virus in the United States. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2423-2449. doi: 10.3934/dcdsb.2016054 [5] Hui Cao, Yicang Zhou. The basic reproduction number of discrete SIR and SEIS models with periodic parameters. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 37-56. doi: 10.3934/dcdsb.2013.18.37 [6] Nicolas Bacaër, Xamxinur Abdurahman, Jianli Ye, Pierre Auger. On the basic reproduction number $R_0$ in sexual activity models for HIV/AIDS epidemics: Example from Yunnan, China. Mathematical Biosciences & Engineering, 2007, 4 (4) : 595-607. doi: 10.3934/mbe.2007.4.595 [7] Nitu Kumari, Sumit Kumar, Sandeep Sharma, Fateh Singh, Rana Parshad. Basic reproduction number estimation and forecasting of COVID-19: A case study of India, Brazil and Peru. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021170 [8] Attila Dénes, Gergely Röst. Single species population dynamics in seasonal environment with short reproduction period. Communications on Pure & Applied Analysis, 2021, 20 (2) : 755-762. doi: 10.3934/cpaa.2020288 [9] Ling Xue, Caterina Scoglio. Network-level reproduction number and extinction threshold for vector-borne diseases. Mathematical Biosciences & Engineering, 2015, 12 (3) : 565-584. doi: 10.3934/mbe.2015.12.565 [10] Gerardo Chowell, R. Fuentes, A. Olea, X. Aguilera, H. Nesse, J. M. Hyman. The basic reproduction number $R_0$ and effectiveness of reactive interventions during dengue epidemics: The 2002 dengue outbreak in Easter Island, Chile. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1455-1474. doi: 10.3934/mbe.2013.10.1455 [11] Danfeng Pang, Yanni Xiao, Xiao-Qiang Zhao. A cross-infection model with diffusion and incubation period. Discrete & Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2021316 [12] Zhikun She, Xin Jiang. Threshold dynamics of a general delayed within-host viral infection model with humoral immunity and two modes of virus transmission. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3835-3861. doi: 10.3934/dcdsb.2020259 [13] Zhaohui Yuan, Xingfu Zou. Global threshold dynamics in an HIV virus model with nonlinear infection rate and distributed invasion and production delays. Mathematical Biosciences & Engineering, 2013, 10 (2) : 483-498. doi: 10.3934/mbe.2013.10.483 [14] Tianhui Yang, Lei Zhang. Remarks on basic reproduction ratios for periodic abstract functional differential equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6771-6782. doi: 10.3934/dcdsb.2019166 [15] Luca Bolzoni, Rossella Della Marca, Maria Groppi, Alessandra Gragnani. Dynamics of a metapopulation epidemic model with localized culling. Discrete & Continuous Dynamical Systems - B, 2020, 25 (6) : 2307-2330. doi: 10.3934/dcdsb.2020036 [16] Zhiting Xu, Xiao-Qiang Zhao. A vector-bias malaria model with incubation period and diffusion. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2615-2634. doi: 10.3934/dcdsb.2012.17.2615 [17] Tom Burr, Gerardo Chowell. The reproduction number $R_t$ in structured and nonstructured populations. Mathematical Biosciences & Engineering, 2009, 6 (2) : 239-259. doi: 10.3934/mbe.2009.6.239 [18] Tianhui Yang, Ammar Qarariyah, Qigui Yang. The effect of spatial variables on the basic reproduction ratio for a reaction-diffusion epidemic model. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021170 [19] Cruz Vargas-De-León. Global analysis of a delayed vector-bias model for malaria transmission with incubation period in mosquitoes. Mathematical Biosciences & Engineering, 2012, 9 (1) : 165-174. doi: 10.3934/mbe.2012.9.165 [20] Steve Drekic, Jae-Kyung Woo, Ran Xu. A threshold-based risk process with a waiting period to pay dividends. Journal of Industrial & Management Optimization, 2018, 14 (3) : 1179-1201. doi: 10.3934/jimo.2018005

2020 Impact Factor: 1.327