American Institute of Mathematical Sciences

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October  2016, 21(8): 2423-2449. doi: 10.3934/dcdsb.2016054

Modeling and control of local outbreaks of West Nile virus in the United States

Jing Chen 1, , Jicai Huang 2, , John C. Beier 3, , Robert Stephen Cantrell 4, , Chris Cosner 1, , Douglas O. Fuller 5, , Guoyan Zhang 6, and  Shigui Ruan 1,

1. 

Department of Mathematics, University of Miami, Coral Gables, FL 33146, United States, United States, United States
2. 

School of Mathematics and Statistics, Central China Normal University, Wuhan, Hubei 430079
3. 

Department of Public Health Sciences, University of Miami Miller School of Medicine, Miami, FL 33136, United States
4. 

Department of Mathematics, University of Miami, Department of Mathematics, University of Miami, Coral Gables, FL 33146, United States
5. 

Department of Geography and Regional Studies, University of Miami, Coral Gables, FL 33146, United States
6. 

Florida Department of Health, Miami-Dade County, Epidemiology, Disease Control and Immunizations Services, 8600 NW 17th Street, Suite 200, Miami, FL 33126, United States

Received  April 2016 Revised  June 2016 Published  September 2016

West Nile virus (WNV) was first detected in the United States (U.S.) during an outbreak in New York City in 1999 with 62 human cases including seven deaths. In 2001, the first human case in Florida was identified, and in Texas and California it was 2002 and 2004, respectively. WNV has now been spread to almost all states in the US. In 2015, the Center for Disease Control and Prevention (CDC) reported 2,175 human cases, including 146 deaths, from 45 states. WNV is maintained in a cycle between mosquitoes and animal hosts in which birds are the predominant and preferred reservoirs while most mammals, including humans, are considered dead-end hosts, as they do not appear to develop high enough titers of WNV in the blood to infect mosquitoes. In this article, we propose a deterministic model by including interactions among mosquitoes, birds, and humans to study the local transmission dynamics of WNV. To validate the model, it is used to simulate the WNV human data of infected cases and accumulative deaths from 1999 to 2013 in the states of New York, Florida, Texas, and California as reported to the CDC. These simulations demonstrate that the epidemic of WNV in New York, Texas, and California (and thus in the U.S.) has not reached its equilibrium yet and may be expected to get worse if the current control strategies are not enhanced. Mathematical and numerical analyses of the model are carried out to understand the transmission dynamics of WNV and explore effective control measures for the local outbreaks of the disease. Our studies suggest that the larval mosquito control measure should be taken as early as possible in a season to control the mosquito population size and the adult mosquito control measure is necessary to prevent the transmission of WNV from mosquitoes to birds and humans.
Keywords: sensitive analysis, transmission dynamics., West Nile virus, mathematical modeling, basic reproduction number.
Mathematics Subject Classification: Primary: 37N25, 92D25; Secondary: 92D3.
Citation: 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
References:
[1]

J. F. Anderson, T. G. Andreadis, C. R. Vossbrinck, S. Tirrell, E. Wakem, A. Garmendia and H. J. Van Kruiningen, Isolation of West Nile virus from mosquitoes, crows, and a cooper's hawk in Connecticut, Science, 286 (1999), 2331-2333. doi: 10.1126/science.286.5448.2331.

[2]

W. Bajwa, M. O'Connor, B. E. Slavinski and Z. Shah, Comprehensive Mosquito Surveillance and Control Plan 2012, New York City Department of Health and Mental Hygiene, New York, 2012. Available from: http://www1.nyc.gov/assets/doh/downloads/pdf/wnv/wnvplan2012.pdf.

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C. G. Blackmore, L. M. Stark, W. C. Jeter, R. L. Oliveri, R. G. Brooks, L. A. Conti and S. T. Wiersma, Surveillance results from the first West Nile virus transmission season in Florida, 2001, The American Journal of Tropical Medicine and Hygiene, 69 (2003), 141-150.

[4]

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, Bulletin of Mathematical Biology, 67 (2005), 1107-1133. doi: 10.1016/j.bulm.2005.01.002.

[5]

T. Briese, X. Y. Jia, C. Huang, L. J. Grady and W. I. Lipkin, Identification of a Kunjin/West Nile like flavivirus in brains of patients with New York encephalitis, Lancet, 354 (1999), 1261-1262. doi: 10.1016/S0140-6736(99)04576-6.

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R. B. Clapp, M. K. Klimkiewicz and A. G. Futcher, Longevity records of North American birds: Columbidae through paridae, Journal of Field Ornithology, 54 (1983) (2), 123-137.

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C. Castillo-Chavez and B. Song, Dynamical models of Tuberculosis and their applications, Mathematical Biosciences and Engineering , 1 (2004), 361-404. doi: 10.3934/mbe.2004.1.361.

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G. Cruz-Pacheco, L. Esteva, J. A. Montaño-Hirose and C. Vargas, Modelling the dynamics of West Nile virus, Bulletin of Mathematical Biology, 67 (2005), 1157-1172. doi: 10.1016/j.bulm.2004.11.008.

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O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations, Journal of Mathematical Biology, 28 (1990), 365-382. doi: 10.1007/BF00178324.

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O. Diekmann, J. A. P. Heesterbeek and M. G. Roberts, The construction of nextgeneration matrices for compartmental epidemic models, Journal of the Royal Society Interface, 7 (2009), rsif20090386.

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M. Eidson, L. Kramer, W. Stone, Y. Hagiwara, K. Schmit and New York State West Nile Virus Avian Surveillance Team, Dead bird surveillance as an early warning system for West Nile virus, Emerging Infectious Diseases, 7 (2001), 631-635. doi: 10.3201/eid0704.017405.

[16]

G. L. Hamer, U. D. Kitron, T. L. Goldberg, J. D. Brawn, S. R. Loss, M. O. Ruiz, D. B. Hayes and E. D. Walker, Host selection by Culex pipiens mosquitoes and West Nile virus amplification, The American Journal of Tropical Medicine and Hygiene, 80 (2009) (2), 268-278.

[17]

C. G. Hayes, West Nile Virus, in The Arboviruses: Epidemiology and Ecolog (eds. T.P. Monath), CRC, Boca Raton, FL, 1989, pp. 59-88.

[18]

E. B. Hayes and D. J. Gubler, West Nile Virus: Epidemiology and clinical features of an emerging epidemic in the United States, Annual Review of Medicine, 57 (2006), 181-194. doi: 10.1146/annurev.med.57.121304.131418.

[19]

A. M. Kilpatrick, P. Daszak, M. J. Jones, P. P. Marra and L. D. Kramer, Host heterogeneity dominates West Nile virus transmission, Proceedings of the Royal Society of London B: Biological Sciences, 273 (2006), 2327-2333. doi: 10.1098/rspb.2006.3575.

[20]

N. Komar, West Nile virus: Epidemiology and ecology in North America, Advances in Virus Research, 61 (2003), 185-234.

[21]

R. S. Lacciotti, J. T. Roehrig, V. Deubel, J. Smith and M. Parker et al., Origin of the West Nile virus responsible for an outbreak of encephalitis in the northeastern United States, Science, 286 (1999), 2333-2337.

[22]

V. Laperriere, K. Brugger and F. Rubel, Simulation of the seasonal cycles of bird, equine and human West Nile virus cases, Preventive Veterinary Medicine, 98 (2011), 99-110. doi: 10.1016/j.prevetmed.2010.10.013.

[23]

M. A. Lewis, J. Renclawowicz and P. van den Driessche, Traveling waves and spread rates for a West Nile Virus model, Bulletin of Mathematical Biology, 68 (2006), 3-23. doi: 10.1007/s11538-005-9018-z.

[24]

R. Liu, J. Shuai, J. Wu and H. Zhu, Modeling spatial spread of West Nile virus and impact of directional dispersal of birds, Mathematical Biosciences and Engineering, 3 (2006), 145-160.

[25]

K. Magori, W. I. Bajwa, S. Bowden and J. M. Drake, Decelerating spread of West Nile Virus by percolation in a heterogeneous urban landscape, PLoS Computational Biology, 7 (2011), e1002104, 13pp. doi: 10.1371/journal.pcbi.1002104.

[26]

N. A. Maidana and H. M. Yang, Spatial spreading of West Nile Virus described by traveling wave, Journal of Theoretical Biology, 258 (2009), 403-417. doi: 10.1016/j.jtbi.2008.12.032.

[27]

A. A. Marfin and D. J. Gubler, West Nile encephalitis: An emerging disease in the United States, Clinical Infectious Diseases, 33 (2001), 1712-1719. doi: 10.1086/322700.

[28]

K. O. Murray, E. Mertens and P. Després, West Nile virus and its emergence in the United States of America, Veterinary Research, 41 (2010), p67. doi: 10.1051/vetres/2010039.

[29]

M. S. Nolan, J. Schuermann and K. O. Murray, West Nile virus infection among humans, Texas, USA, 2002-2011, Emerging Infectious Diseases, 19 (2013), 137-139. doi: 10.3201/eid1901.121135.

[30]

D. R. O'Leary, A. A. Marfin, S. P. Montgomery, A. M. Kipp, J. A. Lehman, B. J. Biggerstaff, V. L. Elko, P. D. Collins, J. E. Jones and G. L. Campbell, The epidemic of West Nile virus in the United States, Vector-Borne and Zoonotic Diseases, 4 (2004), 61-70.

[31]

O. G. Pybus, M. A. Suchard, P. Lemey, F. J. Bernardin, A. Rambaut, F. W. Crawford, R. R. Gray, N. Arinaminpathy, S. L. Stramer, M. P. Busch and E. L. Delwart, Unifying the spatial epidemiology and molecular evolution of emerging epidemics, Proceedings of the National Academy of Sciences, 109 (2012), 15066-15071. doi: 10.1073/pnas.1206598109.

[32]

W. Reisen, H. Lothrop, R. Chiles, M. Madon, C. Cossen, L. Woods, S. Husted, V. Kramer and J. Edman, West nile virus in california, Emerging Infectious Diseases, 10 (2004), 1369-1378. doi: 10.3201/eid1008.040077.

[33]

F. Rubel, K. Brugger, M. Hantel, S. Chvala-Mannsberger, T. Bakonyi, H. Weissenbock and N. Nowotny, Explaining Usutu virus dynamics in Austria: Model development and calibration, Preventive Veterinary Medicine, 85 (2008), 166-186. doi: 10.1016/j.prevetmed.2008.01.006.

[34]

J. E. Simpson, P. J. Hurtado, J. Medlock, G. Molaei, T. G. Andreadis, A. P. Galvani and M. A. Diuk-Wasser, Vector host-feeding preferences drive transmission of multi-host pathogens: West Nile virus as a model system, Proceedings of the Royal Society of London B: Biological Sciences, 279 (2008) (1730), 925-933. doi: 10.1098/rspb.2011.1282.

[35]

K. E. Steele, M. J. Linn, R. J. Schoepp, N. Komar, T. W. Geisbert, R. M. Manduca, P. R. Calle, B. L. Raphael, T. L. Clippinger, T. Larsen, J. Smith, R. S. Lanciotti, N. A. Panella and T. S. Mc Namara, Pathology of fatal West Nile virus infections in native and exotic birds during the 1999 outbreak in New York City, NY, Veterinary Pathology, 37 (2000), 208-224.

[36]

D. M. Thomas and B. Urena, A model describing the evolution of West Nile-like encephalitis in New York City, Mathematical and Computer Modelling, 34 (2001), 771-781. doi: 10.1016/S0895-7177(01)00098-X.

[37]

, United States Census Bureau, 2013 Historical Population Data,, last updated Sept. 25, (2013). 

[38]

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

[39]

H. Wan and H. Zhu, The backward bifurcation in compartmental models for West Nile virus, Mathematical Biosciences, 227 (2010), 20-28. doi: 10.1016/j.mbs.2010.05.006.

[40]

M. J. Wonham, T. de Camino-Beck and M. A. Lewis, An epidemiological model for West Nile virus: Invasion analysis and control applications, Proceedings of the Royal Society of London B: Biological Sciences, 271 (2004), 501-507. doi: 10.1098/rspb.2003.2608.

[41]

M. J. Wonham, M. A. Lewis, J. Renclawowicz and P. van den Driessche, Transmission assumptions generate conflicting predictions in host-vector disease models: A case study in West Nile virus, Ecology Letters, 9 (2006), 706-725. doi: 10.1111/j.1461-0248.2006.00912.x.

[42]

, World Health Organization: West Nile virus, 2015. Available from: http://www.who.int/mediacentre/factsheets/fs354/en/.

[43]

J. Zhang, Z. Jin, G. Q. Sun, X. D. Sun and S. Ruan, Modeling seasonal rabies epidemic in China, Bulletin of Mathematical Biology, 74 (2012), 1226-1251. doi: 10.1007/s11538-012-9720-6.

show all references

References:
[1]

J. F. Anderson, T. G. Andreadis, C. R. Vossbrinck, S. Tirrell, E. Wakem, A. Garmendia and H. J. Van Kruiningen, Isolation of West Nile virus from mosquitoes, crows, and a cooper's hawk in Connecticut, Science, 286 (1999), 2331-2333. doi: 10.1126/science.286.5448.2331.

[2]

W. Bajwa, M. O'Connor, B. E. Slavinski and Z. Shah, Comprehensive Mosquito Surveillance and Control Plan 2012, New York City Department of Health and Mental Hygiene, New York, 2012. Available from: http://www1.nyc.gov/assets/doh/downloads/pdf/wnv/wnvplan2012.pdf.

[3]

C. G. Blackmore, L. M. Stark, W. C. Jeter, R. L. Oliveri, R. G. Brooks, L. A. Conti and S. T. Wiersma, Surveillance results from the first West Nile virus transmission season in Florida, 2001, The American Journal of Tropical Medicine and Hygiene, 69 (2003), 141-150.

[4]

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, Bulletin of Mathematical Biology, 67 (2005), 1107-1133. doi: 10.1016/j.bulm.2005.01.002.

[5]

T. Briese, X. Y. Jia, C. Huang, L. J. Grady and W. I. Lipkin, Identification of a Kunjin/West Nile like flavivirus in brains of patients with New York encephalitis, Lancet, 354 (1999), 1261-1262. doi: 10.1016/S0140-6736(99)04576-6.

[6]

Centers for Disease Control and Prevention (CDC), West Nile Virus in the United States: Guidelines for Surveillance, Prevention, and Control, June 4, 2013. Available from: http://www.cdc.gov/westnile/resources/pdfs/wnvGuidelines.pdf.

[7]

Centers for Disease Control and Prevention (CDC), West Nile Virus, March 26, 2014. Available from: http://www.cdc.gov/westnile/index.html.

[8]

J. Chen, L. Zou, Z. Jin and S. Ruan, Modeling the geographic spread of rabies in China, PLoS Neglected Tropical Diseases, 9 (2015), e0003772. doi: 10.1371/journal.pntd.0003772.

[9]

T. M. Colpitts, M. J. Conway, R. R. Montgomery and E. Fikrig, West Nile Virus: Biology, transmission, and human infection, Clinical Microbiology Reviews, 25 (2012), 635-648. doi: 10.1128/CMR.00045-12.

[10]

R. B. Clapp, M. K. Klimkiewicz and A. G. Futcher, Longevity records of North American birds: Columbidae through paridae, Journal of Field Ornithology, 54 (1983) (2), 123-137.

[11]

C. Castillo-Chavez and B. Song, Dynamical models of Tuberculosis and their applications, Mathematical Biosciences and Engineering , 1 (2004), 361-404. doi: 10.3934/mbe.2004.1.361.

[12]

G. Cruz-Pacheco, L. Esteva, J. A. Montaño-Hirose and C. Vargas, Modelling the dynamics of West Nile virus, Bulletin of Mathematical Biology, 67 (2005), 1157-1172. doi: 10.1016/j.bulm.2004.11.008.

[13]

O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations, Journal of Mathematical Biology, 28 (1990), 365-382. doi: 10.1007/BF00178324.

[14]

O. Diekmann, J. A. P. Heesterbeek and M. G. Roberts, The construction of nextgeneration matrices for compartmental epidemic models, Journal of the Royal Society Interface, 7 (2009), rsif20090386.

[15]

M. Eidson, L. Kramer, W. Stone, Y. Hagiwara, K. Schmit and New York State West Nile Virus Avian Surveillance Team, Dead bird surveillance as an early warning system for West Nile virus, Emerging Infectious Diseases, 7 (2001), 631-635. doi: 10.3201/eid0704.017405.

[16]

G. L. Hamer, U. D. Kitron, T. L. Goldberg, J. D. Brawn, S. R. Loss, M. O. Ruiz, D. B. Hayes and E. D. Walker, Host selection by Culex pipiens mosquitoes and West Nile virus amplification, The American Journal of Tropical Medicine and Hygiene, 80 (2009) (2), 268-278.

[17]

C. G. Hayes, West Nile Virus, in The Arboviruses: Epidemiology and Ecolog (eds. T.P. Monath), CRC, Boca Raton, FL, 1989, pp. 59-88.

[18]

E. B. Hayes and D. J. Gubler, West Nile Virus: Epidemiology and clinical features of an emerging epidemic in the United States, Annual Review of Medicine, 57 (2006), 181-194. doi: 10.1146/annurev.med.57.121304.131418.

[19]

A. M. Kilpatrick, P. Daszak, M. J. Jones, P. P. Marra and L. D. Kramer, Host heterogeneity dominates West Nile virus transmission, Proceedings of the Royal Society of London B: Biological Sciences, 273 (2006), 2327-2333. doi: 10.1098/rspb.2006.3575.

[20]

N. Komar, West Nile virus: Epidemiology and ecology in North America, Advances in Virus Research, 61 (2003), 185-234.

[21]

R. S. Lacciotti, J. T. Roehrig, V. Deubel, J. Smith and M. Parker et al., Origin of the West Nile virus responsible for an outbreak of encephalitis in the northeastern United States, Science, 286 (1999), 2333-2337.

[22]

V. Laperriere, K. Brugger and F. Rubel, Simulation of the seasonal cycles of bird, equine and human West Nile virus cases, Preventive Veterinary Medicine, 98 (2011), 99-110. doi: 10.1016/j.prevetmed.2010.10.013.

[23]

M. A. Lewis, J. Renclawowicz and P. van den Driessche, Traveling waves and spread rates for a West Nile Virus model, Bulletin of Mathematical Biology, 68 (2006), 3-23. doi: 10.1007/s11538-005-9018-z.

[24]

R. Liu, J. Shuai, J. Wu and H. Zhu, Modeling spatial spread of West Nile virus and impact of directional dispersal of birds, Mathematical Biosciences and Engineering, 3 (2006), 145-160.

[25]

K. Magori, W. I. Bajwa, S. Bowden and J. M. Drake, Decelerating spread of West Nile Virus by percolation in a heterogeneous urban landscape, PLoS Computational Biology, 7 (2011), e1002104, 13pp. doi: 10.1371/journal.pcbi.1002104.

[26]

N. A. Maidana and H. M. Yang, Spatial spreading of West Nile Virus described by traveling wave, Journal of Theoretical Biology, 258 (2009), 403-417. doi: 10.1016/j.jtbi.2008.12.032.

[27]

A. A. Marfin and D. J. Gubler, West Nile encephalitis: An emerging disease in the United States, Clinical Infectious Diseases, 33 (2001), 1712-1719. doi: 10.1086/322700.

[28]

K. O. Murray, E. Mertens and P. Després, West Nile virus and its emergence in the United States of America, Veterinary Research, 41 (2010), p67. doi: 10.1051/vetres/2010039.

[29]

M. S. Nolan, J. Schuermann and K. O. Murray, West Nile virus infection among humans, Texas, USA, 2002-2011, Emerging Infectious Diseases, 19 (2013), 137-139. doi: 10.3201/eid1901.121135.

[30]

D. R. O'Leary, A. A. Marfin, S. P. Montgomery, A. M. Kipp, J. A. Lehman, B. J. Biggerstaff, V. L. Elko, P. D. Collins, J. E. Jones and G. L. Campbell, The epidemic of West Nile virus in the United States, Vector-Borne and Zoonotic Diseases, 4 (2004), 61-70.

[31]

O. G. Pybus, M. A. Suchard, P. Lemey, F. J. Bernardin, A. Rambaut, F. W. Crawford, R. R. Gray, N. Arinaminpathy, S. L. Stramer, M. P. Busch and E. L. Delwart, Unifying the spatial epidemiology and molecular evolution of emerging epidemics, Proceedings of the National Academy of Sciences, 109 (2012), 15066-15071. doi: 10.1073/pnas.1206598109.

[32]

W. Reisen, H. Lothrop, R. Chiles, M. Madon, C. Cossen, L. Woods, S. Husted, V. Kramer and J. Edman, West nile virus in california, Emerging Infectious Diseases, 10 (2004), 1369-1378. doi: 10.3201/eid1008.040077.

[33]

F. Rubel, K. Brugger, M. Hantel, S. Chvala-Mannsberger, T. Bakonyi, H. Weissenbock and N. Nowotny, Explaining Usutu virus dynamics in Austria: Model development and calibration, Preventive Veterinary Medicine, 85 (2008), 166-186. doi: 10.1016/j.prevetmed.2008.01.006.

[34]

J. E. Simpson, P. J. Hurtado, J. Medlock, G. Molaei, T. G. Andreadis, A. P. Galvani and M. A. Diuk-Wasser, Vector host-feeding preferences drive transmission of multi-host pathogens: West Nile virus as a model system, Proceedings of the Royal Society of London B: Biological Sciences, 279 (2008) (1730), 925-933. doi: 10.1098/rspb.2011.1282.

[35]

K. E. Steele, M. J. Linn, R. J. Schoepp, N. Komar, T. W. Geisbert, R. M. Manduca, P. R. Calle, B. L. Raphael, T. L. Clippinger, T. Larsen, J. Smith, R. S. Lanciotti, N. A. Panella and T. S. Mc Namara, Pathology of fatal West Nile virus infections in native and exotic birds during the 1999 outbreak in New York City, NY, Veterinary Pathology, 37 (2000), 208-224.

[36]

D. M. Thomas and B. Urena, A model describing the evolution of West Nile-like encephalitis in New York City, Mathematical and Computer Modelling, 34 (2001), 771-781. doi: 10.1016/S0895-7177(01)00098-X.

[37]

, United States Census Bureau, 2013 Historical Population Data,, last updated Sept. 25, (2013). 

[38]

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

[39]

H. Wan and H. Zhu, The backward bifurcation in compartmental models for West Nile virus, Mathematical Biosciences, 227 (2010), 20-28. doi: 10.1016/j.mbs.2010.05.006.

[40]

M. J. Wonham, T. de Camino-Beck and M. A. Lewis, An epidemiological model for West Nile virus: Invasion analysis and control applications, Proceedings of the Royal Society of London B: Biological Sciences, 271 (2004), 501-507. doi: 10.1098/rspb.2003.2608.

[41]

M. J. Wonham, M. A. Lewis, J. Renclawowicz and P. van den Driessche, Transmission assumptions generate conflicting predictions in host-vector disease models: A case study in West Nile virus, Ecology Letters, 9 (2006), 706-725. doi: 10.1111/j.1461-0248.2006.00912.x.

[42]

, World Health Organization: West Nile virus, 2015. Available from: http://www.who.int/mediacentre/factsheets/fs354/en/.

[43]

J. Zhang, Z. Jin, G. Q. Sun, X. D. Sun and S. Ruan, Modeling seasonal rabies epidemic in China, Bulletin of Mathematical Biology, 74 (2012), 1226-1251. doi: 10.1007/s11538-012-9720-6.

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