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Modeling the effect of activation of CD4$^+$ T cells on HIV dynamics
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 |
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 [
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. |


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