American Institute of Mathematical Sciences

Advanced
August  2022, 27(8): 4515-4529. doi: 10.3934/dcdsb.2021239

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

Yaxin Han and  Zhenguo Bai ,

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

* Corresponding author: Zhenguo Bai

Received  May 2021 Revised  July 2021 Published  August 2022 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.

Keywords: West Nile virus, basic reproduction number, extrinsic incubation period, threshold dynamics, culling.
Mathematics Subject Classification: Primary: 92D30, 37N25; Secondary: 34K13.
Citation: Yaxin Han, Zhenguo Bai. Threshold dynamics of a West Nile virus model with impulsive culling and incubation period. Discrete and Continuous Dynamical Systems - B, 2022, 27 (8) : 4515-4529. doi: 10.3934/dcdsb.2021239
References:
[1]

S. Ai, J. Li and J. Lu, Mosquito-stage-structured malaria models and their global dynamics, SIAM J. Appl. Math., 72 (2012), 1213-1237.  doi: 10.1137/110860318.

[2]

N. Bacaër and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality, J. Math. Biol., 53 (2006), 421-436.  doi: 10.1007/s00285-006-0015-0.

[3]

Z. Bai and X.-Q. Zhao, Basic reproduction ratios for periodic and time-delayed compartmental models with impulses, J. Math. Biol., 80 (2020), 1095-1117.  doi: 10.1007/s00285-019-01452-2.

[4]

G. Ballinger and X. Liu, Existence, uniqueness and boundedness results for impulsive delay differential equations, Appl. Anal., 74 (2000), 71-93.  doi: 10.1080/00036810008840804.

[5]

K. W. Blayneh, A. B. Gumel, S. Lenhart and T. Clayton, Backward bifurcation and optimal control in transmission dynamics of West Nile virus, Bull. Math. Biol., 72 (2010), 1006-1028.  doi: 10.1007/s11538-009-9480-0.

[6]

C. Bowman, A. B. Gumel, P. van den Driessche, J. Wu and H. Zhu, A mathematical model for assessing control strategies against West Nile virus, Bull. Math. Biol., 67 (2005), 1107-1133.  doi: 10.1016/j.bulm.2005.01.002.

[7]

G. Fan, J. Liu, P. van den Driessche, J. Wu and H. Zhu, The impact of maturation delay of mosquitoes on the transmission of West Nile virus, Math. Biosci., 228 (2010), 119-126.  doi: 10.1016/j.mbs.2010.08.010.

[8]

S. Gao, L. Chen and Z. Teng, Impulsive vaccination of an SEIRS model with time delay and varying total population size, Bull. Math. Biol., 69 (2007), 731-745. 

[9]

S. A. Gourley, R. Liu and J. Wu, Eradicating vector-borne diseases via age-structured culling, J. Math. Biol., 54 (2007), 309-335.  doi: 10.1007/s00285-006-0050-x.

[10]

X. Hu, Y. Liu and J. Wu, Culling structured hosts to eradicate vector-borne diseases, Math. Biosci. Eng., 6 (2009), 301-319.  doi: 10.3934/mbe.2009.6.301.

[11]

J. Jiang and Z. Qiu, The complete classification for dynamics in a nine-dimensional West Nile virus model, SIAM J. Appl. Math., 69 (2009), 1205-1227.  doi: 10.1137/070709438.

[12]

J. Li, Simple stage-structured models for wild and transgenic mosquito populations, J. Difference Equ. Appl., 15 (2009), 327-347.  doi: 10.1080/10236190802566491.

[13]

X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40.  doi: 10.1002/cpa.20154.

[14]

X. Liu, X. Shen and Y. Zhang, A comparison principle and stability for large-scale impulsive delay differential systems, ANZIAM J., 47 (2005), 203-235.  doi: 10.1017/S1446181100009998.

[15]

Y. Lou and X.-Q. Zhao, A theoretical approach to understanding population dynamics with seasonal developmental durations, J. Nonlinear Sci., 27 (2017), 573-603.  doi: 10.1007/s00332-016-9344-3.

[16]

D. Nash, F. Mostashari and A. Fine, The outbreak of West Nile virus infection in the New York City area in 1999, N. Engl. J. Med., 344 (2001), 1807-1814.  doi: 10.1056/NEJM200106143442401.

[17]

S. Ruan, D. Xiao and J. C. Beier, On the delayed Ross-Macdonald model for malaria transmission, Bull. Math. Biol., 70 (2008), 1098-1114.  doi: 10.1007/s11538-007-9292-z.

[18]

O. S. Sisodiya, O. P. Misra and J. Dhar, Dynamics of cholera epidemics with impulsive vaccination and disinfection, Math. Biosci., 298 (2018), 46-57.  doi: 10.1016/j.mbs.2018.02.001.

[19]

H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity, SIAM J. Appl. Math., 70 (2009), 188-211.  doi: 10.1137/080732870.

[20]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.

[21]

F.-B. Wang, R. Wu and X.-Q. Zhao, A West Nile virus transmission model with periodic incubation periods, SIAM J. Appl. Dyn. Syst., 18 (2019), 1498-1535.  doi: 10.1137/18M1236162.

[22]

X. Wang and X.-Q. Zhao, A periodic vector-bias malaria model with incubation period, SIAM J. Appl. Math., 77 (2017), 181-201.  doi: 10.1137/15M1046277.

[23]

M. J. Wonham, T. de-Camino-Beck and M. A. Lewis, An epidemiological model for West Nile virus: Invasion analysis and control applications, Proc. R. Soc. Lond. B., 271 (2004), 501-507.  doi: 10.1098/rspb.2003.2608.

[24]

X. Xu, Y. Xiao and R. A. Cheke, Models of impulsive culling of mosquitoes to interrupt transmission of West Nile virus to birds, Appl. Math. Model., 39 (2015), 3549-3568.  doi: 10.1016/j.apm.2014.10.072.

[25]

Z. Yang, C. Huang and X. Zou, Effect of impulsive controls in a model system for age-structured population over a patchy environment, J. Math. Biol., 76 (2018), 1387-1419.  doi: 10.1007/s00285-017-1172-z.

[26]

T. Zhang and X.-Q. Zhao, Mathematical modeling for schistosomiasis with seasonal influence: A case study in Hubei, China, SIAM J. Appl. Dyn. Syst., 19 (2020), 1438-1471.  doi: 10.1137/19M1280259.

[27]

X.-Q. Zhao, Basic reproduction ratios for periodic compartmental models with time delay, J. Dyn. Diff. Equat., 29 (2017), 67-82.  doi: 10.1007/s10884-015-9425-2.

[28]

W. Zhou, Y. Xiao and R. A. Cheke, A threshold policy to interrupt transmission of West Nile Virus to birds, Appl. Math. Model., 40 (2016), 8794-8809.  doi: 10.1016/j.apm.2016.05.040.

[29]

W. Zhou, Y. Xiao and J. M. Heffernan, A threshold policy to curb WNV transmission to birds with seasonality, Nonlinear Anal. Real World Appl., 59 (2021), 24pp. doi: 10.1016/j.nonrwa.2020.103273.

show all references

References:
[1]

S. Ai, J. Li and J. Lu, Mosquito-stage-structured malaria models and their global dynamics, SIAM J. Appl. Math., 72 (2012), 1213-1237.  doi: 10.1137/110860318.

[2]

N. Bacaër and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality, J. Math. Biol., 53 (2006), 421-436.  doi: 10.1007/s00285-006-0015-0.

[3]

Z. Bai and X.-Q. Zhao, Basic reproduction ratios for periodic and time-delayed compartmental models with impulses, J. Math. Biol., 80 (2020), 1095-1117.  doi: 10.1007/s00285-019-01452-2.

[4]

G. Ballinger and X. Liu, Existence, uniqueness and boundedness results for impulsive delay differential equations, Appl. Anal., 74 (2000), 71-93.  doi: 10.1080/00036810008840804.

[5]

K. W. Blayneh, A. B. Gumel, S. Lenhart and T. Clayton, Backward bifurcation and optimal control in transmission dynamics of West Nile virus, Bull. Math. Biol., 72 (2010), 1006-1028.  doi: 10.1007/s11538-009-9480-0.

[6]

C. Bowman, A. B. Gumel, P. van den Driessche, J. Wu and H. Zhu, A mathematical model for assessing control strategies against West Nile virus, Bull. Math. Biol., 67 (2005), 1107-1133.  doi: 10.1016/j.bulm.2005.01.002.

[7]

G. Fan, J. Liu, P. van den Driessche, J. Wu and H. Zhu, The impact of maturation delay of mosquitoes on the transmission of West Nile virus, Math. Biosci., 228 (2010), 119-126.  doi: 10.1016/j.mbs.2010.08.010.

[8]

S. Gao, L. Chen and Z. Teng, Impulsive vaccination of an SEIRS model with time delay and varying total population size, Bull. Math. Biol., 69 (2007), 731-745. 

[9]

S. A. Gourley, R. Liu and J. Wu, Eradicating vector-borne diseases via age-structured culling, J. Math. Biol., 54 (2007), 309-335.  doi: 10.1007/s00285-006-0050-x.

[10]

X. Hu, Y. Liu and J. Wu, Culling structured hosts to eradicate vector-borne diseases, Math. Biosci. Eng., 6 (2009), 301-319.  doi: 10.3934/mbe.2009.6.301.

[11]

J. Jiang and Z. Qiu, The complete classification for dynamics in a nine-dimensional West Nile virus model, SIAM J. Appl. Math., 69 (2009), 1205-1227.  doi: 10.1137/070709438.

[12]

J. Li, Simple stage-structured models for wild and transgenic mosquito populations, J. Difference Equ. Appl., 15 (2009), 327-347.  doi: 10.1080/10236190802566491.

[13]

X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40.  doi: 10.1002/cpa.20154.

[14]

X. Liu, X. Shen and Y. Zhang, A comparison principle and stability for large-scale impulsive delay differential systems, ANZIAM J., 47 (2005), 203-235.  doi: 10.1017/S1446181100009998.

[15]

Y. Lou and X.-Q. Zhao, A theoretical approach to understanding population dynamics with seasonal developmental durations, J. Nonlinear Sci., 27 (2017), 573-603.  doi: 10.1007/s00332-016-9344-3.

[16]

D. Nash, F. Mostashari and A. Fine, The outbreak of West Nile virus infection in the New York City area in 1999, N. Engl. J. Med., 344 (2001), 1807-1814.  doi: 10.1056/NEJM200106143442401.

[17]

S. Ruan, D. Xiao and J. C. Beier, On the delayed Ross-Macdonald model for malaria transmission, Bull. Math. Biol., 70 (2008), 1098-1114.  doi: 10.1007/s11538-007-9292-z.

[18]

O. S. Sisodiya, O. P. Misra and J. Dhar, Dynamics of cholera epidemics with impulsive vaccination and disinfection, Math. Biosci., 298 (2018), 46-57.  doi: 10.1016/j.mbs.2018.02.001.

[19]

H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity, SIAM J. Appl. Math., 70 (2009), 188-211.  doi: 10.1137/080732870.

[20]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.

[21]

F.-B. Wang, R. Wu and X.-Q. Zhao, A West Nile virus transmission model with periodic incubation periods, SIAM J. Appl. Dyn. Syst., 18 (2019), 1498-1535.  doi: 10.1137/18M1236162.

[22]

X. Wang and X.-Q. Zhao, A periodic vector-bias malaria model with incubation period, SIAM J. Appl. Math., 77 (2017), 181-201.  doi: 10.1137/15M1046277.

[23]

M. J. Wonham, T. de-Camino-Beck and M. A. Lewis, An epidemiological model for West Nile virus: Invasion analysis and control applications, Proc. R. Soc. Lond. B., 271 (2004), 501-507.  doi: 10.1098/rspb.2003.2608.

[24]

X. Xu, Y. Xiao and R. A. Cheke, Models of impulsive culling of mosquitoes to interrupt transmission of West Nile virus to birds, Appl. Math. Model., 39 (2015), 3549-3568.  doi: 10.1016/j.apm.2014.10.072.

[25]

Z. Yang, C. Huang and X. Zou, Effect of impulsive controls in a model system for age-structured population over a patchy environment, J. Math. Biol., 76 (2018), 1387-1419.  doi: 10.1007/s00285-017-1172-z.

[26]

T. Zhang and X.-Q. Zhao, Mathematical modeling for schistosomiasis with seasonal influence: A case study in Hubei, China, SIAM J. Appl. Dyn. Syst., 19 (2020), 1438-1471.  doi: 10.1137/19M1280259.

[27]

X.-Q. Zhao, Basic reproduction ratios for periodic compartmental models with time delay, J. Dyn. Diff. Equat., 29 (2017), 67-82.  doi: 10.1007/s10884-015-9425-2.

[28]

W. Zhou, Y. Xiao and R. A. Cheke, A threshold policy to interrupt transmission of West Nile Virus to birds, Appl. Math. Model., 40 (2016), 8794-8809.  doi: 10.1016/j.apm.2016.05.040.

[29]

W. Zhou, Y. Xiao and J. M. Heffernan, A threshold policy to curb WNV transmission to birds with seasonality, Nonlinear Anal. Real World Appl., 59 (2021), 24pp. doi: 10.1016/j.nonrwa.2020.103273.

Figure 1.  Comparison of the long-term behavior of infectious mosquitoes and birds in different scenarios: culling and without culling.
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Sensitivity analysis of $ \mathcal{R}_0 $. PRCCs represents the sensitivity index of $ \mathcal {R}_0 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  The curve of $ \mathcal{R}_0 $ with respect to $ \tau $ for different culling interval
Figure Options

Download full-size image

Download as PowerPoint slide

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
Figure Options

Download full-size image

Download as PowerPoint slide

[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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and Continuous Dynamical Systems - B, 2022, 27 (6) : 3005-3017. 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 and Management Optimization, 2018, 14 (3) : 1179-1201. doi: 10.3934/jimo.2018005

2021 Impact Factor: 1.497

Tools

Metrics

  • PDF downloads (396)
  • HTML views (341)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy