# American Institute of Mathematical Sciences

2016, 13(2): 401-424. doi: 10.3934/mbe.2015009

## A mathematical model for the spread of west nile virus in migratory and resident birds

 1 Department of Mathematics, Tulane University, New Orleans, LA 70118, United States, United States, United States

Received  January 2015 Revised  November 2015 Published  December 2015

We develop a mathematical model for transmission of West Nile virus (WNV) that incorporates resident and migratory host avian populations and a mosquito vector population. We provide a detailed analysis of the model's basic reproductive number and demonstrate how the exposed infected, but not infectious, state for the bird population can be approximated by a reduced model. We use the model to investigate the interplay of WNV in both resident and migratory bird hosts. The resident host parameters correspond to the American Crow (Corvus brachyrhynchos), a competent host with a high death rate due to disease, and migratory host parameters to the American Robin (Turdus migratorius), a competent host with low WNV death rates. We find that yearly seasonal outbreaks depend primarily on the number of susceptible migrant birds entering the local population each season. We observe that the early growth rates of seasonal outbreaks is more influenced by the the migratory population than the resident bird population. This implies that although the death of highly competent resident birds, such as American Crows, are good indicators for the presence of the virus, these species have less impact on the basic reproductive number than the competent migratory birds with low death rates, such as the American Robins. The disease forecasts are most sensitive to the assumptions about the feeding preferences of North American mosquito vectors and the effect of the virus on the hosts. Increased research on the these factors would allow for better estimates of these important model parameters, which would improve the quality of future WNV forecasts.
Citation: 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
##### References:
 [1] A. Abdelrazec, S. Lenhart and H. Zhu, Transmission dynamics of west nile virus in mosquitoes and corvids and non-corvids, Journal of mathematical biology, 68 (2014), 1553-1582. doi: 10.1007/s00285-013-0677-3. [2] L. Arriola and J. M. Hyman, Sensitivity analysis for uncertainty quantification in mathematical models, in Mathematical and Statistical Estimation Approaches in Epidemiology, Springer, 2009, 195-247. doi: 10.1007/978-90-481-2313-1_10. [3] L. M. Arriola and J. M. Hyman, Being sensitive to uncertainty, Computing in Science & Engineering, 9 (2007), 10-20. doi: 10.1109/MCSE.2007.27. [4] T. A. Beveroth, M. P. Ward, R. L. Lampman, A. M. Ringia and R. J. Novak, Changes in seroprevalence of west nile virus across illinois in free-ranging birds from 2001 through 2004, The American journal of tropical medicine and hygiene, 74 (2006), 174-179. [5] D. B. Botkin and R. S. Miller, Mortality rates and survival of birds, American Naturalist, 108 (1974), 181-192. doi: 10.1086/282898. [6] C. Bowman, A. 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. [7] C. A. Bradley, S. E. J. Gibbs and S. Altizer, Urban land use predicts west nile virus exposure in songbirds, Ecological Applications, 18 (2008), 1083-1092. doi: 10.1890/07-0822.1. [8] S. Chatterjee, S. Pal and J. Chattopadhyay, Role of migratory birds under environmental fluctuation: a mathematical study, Journal of Biological Systems, 16 (2008), 81-106. doi: 10.1142/S0218339008002423. [9] N. Chitnis, J. Cushing and J. Hyman, Bifurcation analysis of a mathematical model for malaria transmission, SIAM Journal on Applied Mathematics, 67 (2006), 24-45. doi: 10.1137/050638941. [10] N. Chitnis, J. M. Hyman and J. M. Cushing, Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model, Bulletin of mathematical biology, 70 (2008), 1272-1296. doi: 10.1007/s11538-008-9299-0. [11] N. Chitnis, J. M. Hyman and C. A. Manore, Modelling vertical transmission in vector-borne diseases with applications to Rift Valley fever, Journal of Biological Dynamics, 7 (2013), 11-40. doi: 10.1080/17513758.2012.733427. [12] D. Chowell-Puente, P. Delgado, D. Pérez, C. H. S. Tapia, F. Sánchez and D. Murillo, The Impact of Mosquito-Bird Interaction on the Spread of West Nile Virus to Human Populations, Department of Biometrics, Cornell University, Technical Report Series. [13] L. Colton, B. J. Biggerstaff, A. Johnson and R. S. Nasci, Quantification of west nile virus in vector mosquito saliva, Journal of the American Mosquito Control Association, 21 (2005), 49-53. [14] G. Cruz-Pacheco, L. Esteva, J. Montaø-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. [15] G. Cruz-Pacheco, L. Esteva and C. Vargas, Multi-species interactions in west nile virus infection, Journal of Biological Dynamics, 6 (2012), 281-298. doi: 10.1080/17513758.2011.571721. [16] G. Cruz-Pacheco, L. Esteva and C. Vargas, Seasonality and outbreaks in west nile virus infection, Bulletin of mathematical biology, 71 (2009), 1378-1393. doi: 10.1007/s11538-009-9406-x. [17] B. Durand, G. Balança, T. Baldet and V. Chevalier, A metapopulation model to simulate west nile virus circulation in western africa, southern europe and the mediterranean basin, Veterinary research, 41. [18] D. S. Farner, Age groups and longevity in the american robin: Comments, further discussion, and certain revisions, The Wilson Bulletin, 68-81. [19] C. for Disease Control and Prevention, Statistics, surveillance, and control archive, http://www.cdc.gov/ncidod/dvbid/westnile/surv&control_archive.htm, 2012. [20] C. for Disease Control and Prevention, West nile virus clinical description, http://www.cdc.gov/ncidod/dvbid/westnile/clinicians/, 2012. [21] C. for Disease Control and Prevention, West nile virus questions and answers, http://www.cdc.gov/ncidod/dvbid/westnile/qa/pesticides.htm, 2012. [22] L. B. Goddard, A. E. Roth, W. K. Reisen and T. W. Scott, Vertical transmission of west nile virus by three california culex (diptera: Culicidae) species, Journal of medical entomology, 40 (2003), 743-746. doi: 10.1603/0022-2585-40.6.743. [23] J. Heffernan, R. Smith and L. Wahl, Perspectives on the basic reproductive ratio, Journal of the Royal Society Interface, 2 (2005), 281-293. doi: 10.1098/rsif.2005.0042. [24] A. M. Kilpatrick, A. A. Chmura, D. W. Gibbons, R. C. Fleischer, P. P. Marra and P. Daszak, Predicting the global spread of h5n1 avian influenza, Proceedings of the National Academy of Sciences, 103 (2006), 19368-19373. doi: 10.1073/pnas.0609227103. [25] A. M. Kilpatrick, L. D. Kramer, M. J. Jones, P. P. Marra and P. Daszak, West nile virus epidemics in north america are driven by shifts in mosquito feeding behavior, PLoS Biol, 4 (2006), e82. doi: 10.1371/journal.pbio.0040082. [26] N. Komar, West nile virus: epidemiology and ecology in north america, Advances in virus research, 61 (2003), 185-234. [27] J. L. Kwan, S. Kluh and W. K. Reisen, Antecedent avian immunity limits tangential transmission of west nile virus to humans, PLoS ONE, 7 (2012), e34127. doi: 10.1371/journal.pone.0034127. [28] J. Mackenzie, D. Gubler and L. Petersen, Emerging flaviviruses: The spread and resurgence of japanese encephalitis, west nile and dengue viruses, Nature medicine, 10 (2004), S98-S109. doi: 10.1038/nm1144. [29] C. A. Manore, J. K. Davis, R. C. Christofferson, D. M. Wesson, J. M. Hyman and C. N. Mores, Towards an early warning system for forecasting human west nile virus incidence, PLoS currents, 6 2014. doi: 10.1371/currents.outbreaks.f0b3978230599a56830ce30cb9ce0500. [30] C. A. Manore, K. S. Hickmann, S. Xu, H. J. Wearing and J. M. Hyman, Comparing dengue and chikungunya emergence and endemic transmission in A. aegypti and A. albopictus, Journal of theoretical biology, 356 (2014), 174-191. doi: 10.1016/j.jtbi.2014.04.033. [31] R. G. McLean, S. R. Ubico, D. E. Docherty, W. R. Hansen, L. Sileo and T. S. McNamara, West nile virus transmission and ecology in birds, Annals of the New York Academy of Sciences, 951 (2001), 54-57. doi: 10.1111/j.1749-6632.2001.tb02684.x. [32] S. Moore, C. Manore, V. Bokil, E. Borer and P. Hosseini, Spatiotemporal model of barley and cereal yellow dwarf virus transmission dynamics with seasonality and plant competition, Bulletin of Mathematical Biology, 73 (2011), 2707-2730. doi: 10.1007/s11538-011-9654-4. [33] F. Morneau, C. Lépine, R. Décarie, M.-A. Villard and J.-L. DesGranges, Reproduction of american robin (turdus migratorius) in a suburban environment, Landscape and urban planning, 32 (1995), 55-62. [34] A. T. Peterson, D. A. Vieglais and J. K. Andreasen, Migratory birds modeled as critical transport agents for west nile virus in north america, Vector-Borne and Zoonotic Diseases, 3 (2003), 27-37. doi: 10.1089/153036603765627433. [35] Z. Qiu, Dynamics of an epidemic model with host migration, Applied Mathematics and Computation, 218 (2011), 4614-4625. doi: 10.1016/j.amc.2011.10.045. [36] W. K. Reisen, Y. Fang, H. D. Lothrop, V. M. Martinez, J. Wilson, P. O'Connor, R. Carney, B. Cahoon-Young, M. Shafii and A. C. Brault, Overwintering of west nile virus in southern california, Journal of medical entomology, 43 (2006), 344-355. doi: 10.1093/jmedent/43.2.344. [37] W. K. Reisen, M. M. Milby and R. P. Meyer, Population dynamics of adult culex mosquitoes (diptera: Culicidae) along the kern river, kern county, california, in 1990, Journal of medical entomology, 29 (1992), 531-543. doi: 10.1093/jmedent/29.3.531. [38] R. Rosà, G. Marini, L. Bolzoni, M. Neteler, M. Metz, L. Delucchi, E. A. Chadwick, L. Balbo, A. Mosca, M. Giacobini et al., Early warning of west nile virus mosquito vector: Climate and land use models successfully explain phenology and abundance of culex pipiens mosquitoes in north-western italy, Parasites & vectors, 7 (2014), p269. [39] M. R. Sardelis, M. J. Turell, D. J. Dohm and M. L. O'Guinn, Vector competence of selected north american culex and coquillettidia mosquitoes for west nile virus, Emerging infectious diseases, 7 (2001), p1018. [40] 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 B: Biological Sciences, 279 (2012), 925-933. doi: 10.1098/rspb.2011.1282. [41] J. P. Swaddle and S. E. Calos, Increased avian diversity is associated with lower incidence of human west nile infection: Observation of the dilution effect, PloS one, 3 (2008), e2488. doi: 10.1371/journal.pone.0002488. [42] D. 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. [43] S. Tiawsirisup, K. B. Platt, R. B. Evans and W. A. Rowley, Susceptibility of ochlerotatus trivittatus (coq.), aedes albopictus (skuse), and culex pipiens (l.) to west nile virus infection, Vector-Borne & Zoonotic Diseases, 4 (2004), 190-197. [44] S. Tiawsirisup, K. B. Platt, R. B. Evans and W. A. Rowley, A comparision of west nile virus transmission by ochlerotatus trivittatus (coq.), culex pipiens (l.), and aedes albopictus (skuse), Vector-Borne & Zoonotic Diseases, 5 (2005), 40-47. [45] M. J. Turell, M. R. Sardelis, D. J. Dohm and M. L. O'Guinn, Potential for north american mosquitoes to transmit west nile virus, American Journal of Tropical Medicine and Hygiene, 62 (2000), 413-414. [46] R. Unnasch, T. Sprenger, C. Katholi, E. Cupp, G. Hill and T. Unnasch, A dynamic transmission model of eastern equine encephalitis virus, Ecological modelling, 192 (2006), 425-440. doi: 10.1016/j.ecolmodel.2005.07.011. [47] 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. [48] E. B. Vinogradova, Culex pipiens pipiens mosquitoes: taxonomy, distribution, ecology, physiology, genetics, applied importance and control, Pensoft Publishers, 2000. [49] T. P. Weber and N. I. Stilianakis, Ecologic immunology of avian influenza (h5n1) in migratory birds, Emerging infectious diseases, 13 (2007), p1139. doi: 10.3201/eid1308.070319. [50] M. J. Wonham, M. A. Lewis, J. Rencławowicz 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. [51] M. Wonham, T. de Camino-Beck and M. Lewis, An epidemiological model for west nile virus: Invasion analysis and control applications, Proceedings of the Royal Society of London. Series B: Biological Sciences, 271 (2004), 501-507. doi: 10.1098/rspb.2003.2608.

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##### References:
 [1] A. Abdelrazec, S. Lenhart and H. Zhu, Transmission dynamics of west nile virus in mosquitoes and corvids and non-corvids, Journal of mathematical biology, 68 (2014), 1553-1582. doi: 10.1007/s00285-013-0677-3. [2] L. Arriola and J. M. Hyman, Sensitivity analysis for uncertainty quantification in mathematical models, in Mathematical and Statistical Estimation Approaches in Epidemiology, Springer, 2009, 195-247. doi: 10.1007/978-90-481-2313-1_10. [3] L. M. Arriola and J. M. Hyman, Being sensitive to uncertainty, Computing in Science & Engineering, 9 (2007), 10-20. doi: 10.1109/MCSE.2007.27. [4] T. A. Beveroth, M. P. Ward, R. L. Lampman, A. M. Ringia and R. J. Novak, Changes in seroprevalence of west nile virus across illinois in free-ranging birds from 2001 through 2004, The American journal of tropical medicine and hygiene, 74 (2006), 174-179. [5] D. B. Botkin and R. S. Miller, Mortality rates and survival of birds, American Naturalist, 108 (1974), 181-192. doi: 10.1086/282898. [6] C. Bowman, A. 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. [7] C. A. Bradley, S. E. J. Gibbs and S. Altizer, Urban land use predicts west nile virus exposure in songbirds, Ecological Applications, 18 (2008), 1083-1092. doi: 10.1890/07-0822.1. [8] S. Chatterjee, S. Pal and J. Chattopadhyay, Role of migratory birds under environmental fluctuation: a mathematical study, Journal of Biological Systems, 16 (2008), 81-106. doi: 10.1142/S0218339008002423. [9] N. Chitnis, J. Cushing and J. Hyman, Bifurcation analysis of a mathematical model for malaria transmission, SIAM Journal on Applied Mathematics, 67 (2006), 24-45. doi: 10.1137/050638941. [10] N. Chitnis, J. M. Hyman and J. M. Cushing, Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model, Bulletin of mathematical biology, 70 (2008), 1272-1296. doi: 10.1007/s11538-008-9299-0. [11] N. Chitnis, J. M. Hyman and C. A. Manore, Modelling vertical transmission in vector-borne diseases with applications to Rift Valley fever, Journal of Biological Dynamics, 7 (2013), 11-40. doi: 10.1080/17513758.2012.733427. [12] D. Chowell-Puente, P. Delgado, D. Pérez, C. H. S. Tapia, F. Sánchez and D. Murillo, The Impact of Mosquito-Bird Interaction on the Spread of West Nile Virus to Human Populations, Department of Biometrics, Cornell University, Technical Report Series. [13] L. Colton, B. J. Biggerstaff, A. Johnson and R. S. Nasci, Quantification of west nile virus in vector mosquito saliva, Journal of the American Mosquito Control Association, 21 (2005), 49-53. [14] G. Cruz-Pacheco, L. Esteva, J. Montaø-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. [15] G. Cruz-Pacheco, L. Esteva and C. Vargas, Multi-species interactions in west nile virus infection, Journal of Biological Dynamics, 6 (2012), 281-298. doi: 10.1080/17513758.2011.571721. [16] G. Cruz-Pacheco, L. Esteva and C. Vargas, Seasonality and outbreaks in west nile virus infection, Bulletin of mathematical biology, 71 (2009), 1378-1393. doi: 10.1007/s11538-009-9406-x. [17] B. Durand, G. Balança, T. Baldet and V. Chevalier, A metapopulation model to simulate west nile virus circulation in western africa, southern europe and the mediterranean basin, Veterinary research, 41. [18] D. S. Farner, Age groups and longevity in the american robin: Comments, further discussion, and certain revisions, The Wilson Bulletin, 68-81. [19] C. for Disease Control and Prevention, Statistics, surveillance, and control archive, http://www.cdc.gov/ncidod/dvbid/westnile/surv&control_archive.htm, 2012. [20] C. for Disease Control and Prevention, West nile virus clinical description, http://www.cdc.gov/ncidod/dvbid/westnile/clinicians/, 2012. [21] C. for Disease Control and Prevention, West nile virus questions and answers, http://www.cdc.gov/ncidod/dvbid/westnile/qa/pesticides.htm, 2012. [22] L. B. Goddard, A. E. Roth, W. K. Reisen and T. W. Scott, Vertical transmission of west nile virus by three california culex (diptera: Culicidae) species, Journal of medical entomology, 40 (2003), 743-746. doi: 10.1603/0022-2585-40.6.743. [23] J. Heffernan, R. Smith and L. Wahl, Perspectives on the basic reproductive ratio, Journal of the Royal Society Interface, 2 (2005), 281-293. doi: 10.1098/rsif.2005.0042. [24] A. M. Kilpatrick, A. A. Chmura, D. W. Gibbons, R. C. Fleischer, P. P. Marra and P. Daszak, Predicting the global spread of h5n1 avian influenza, Proceedings of the National Academy of Sciences, 103 (2006), 19368-19373. doi: 10.1073/pnas.0609227103. [25] A. M. Kilpatrick, L. D. Kramer, M. J. Jones, P. P. Marra and P. Daszak, West nile virus epidemics in north america are driven by shifts in mosquito feeding behavior, PLoS Biol, 4 (2006), e82. doi: 10.1371/journal.pbio.0040082. [26] N. Komar, West nile virus: epidemiology and ecology in north america, Advances in virus research, 61 (2003), 185-234. [27] J. L. Kwan, S. Kluh and W. K. Reisen, Antecedent avian immunity limits tangential transmission of west nile virus to humans, PLoS ONE, 7 (2012), e34127. doi: 10.1371/journal.pone.0034127. [28] J. Mackenzie, D. Gubler and L. Petersen, Emerging flaviviruses: The spread and resurgence of japanese encephalitis, west nile and dengue viruses, Nature medicine, 10 (2004), S98-S109. doi: 10.1038/nm1144. [29] C. A. Manore, J. K. Davis, R. C. Christofferson, D. M. Wesson, J. M. Hyman and C. N. Mores, Towards an early warning system for forecasting human west nile virus incidence, PLoS currents, 6 2014. doi: 10.1371/currents.outbreaks.f0b3978230599a56830ce30cb9ce0500. [30] C. A. Manore, K. S. Hickmann, S. Xu, H. J. Wearing and J. M. Hyman, Comparing dengue and chikungunya emergence and endemic transmission in A. aegypti and A. albopictus, Journal of theoretical biology, 356 (2014), 174-191. doi: 10.1016/j.jtbi.2014.04.033. [31] R. G. McLean, S. R. Ubico, D. E. Docherty, W. R. Hansen, L. Sileo and T. S. McNamara, West nile virus transmission and ecology in birds, Annals of the New York Academy of Sciences, 951 (2001), 54-57. doi: 10.1111/j.1749-6632.2001.tb02684.x. [32] S. Moore, C. Manore, V. Bokil, E. Borer and P. Hosseini, Spatiotemporal model of barley and cereal yellow dwarf virus transmission dynamics with seasonality and plant competition, Bulletin of Mathematical Biology, 73 (2011), 2707-2730. doi: 10.1007/s11538-011-9654-4. [33] F. Morneau, C. Lépine, R. Décarie, M.-A. Villard and J.-L. DesGranges, Reproduction of american robin (turdus migratorius) in a suburban environment, Landscape and urban planning, 32 (1995), 55-62. [34] A. T. Peterson, D. A. Vieglais and J. K. Andreasen, Migratory birds modeled as critical transport agents for west nile virus in north america, Vector-Borne and Zoonotic Diseases, 3 (2003), 27-37. doi: 10.1089/153036603765627433. [35] Z. Qiu, Dynamics of an epidemic model with host migration, Applied Mathematics and Computation, 218 (2011), 4614-4625. doi: 10.1016/j.amc.2011.10.045. [36] W. K. Reisen, Y. Fang, H. D. Lothrop, V. M. Martinez, J. Wilson, P. O'Connor, R. Carney, B. Cahoon-Young, M. Shafii and A. C. Brault, Overwintering of west nile virus in southern california, Journal of medical entomology, 43 (2006), 344-355. doi: 10.1093/jmedent/43.2.344. [37] W. K. Reisen, M. M. Milby and R. P. Meyer, Population dynamics of adult culex mosquitoes (diptera: Culicidae) along the kern river, kern county, california, in 1990, Journal of medical entomology, 29 (1992), 531-543. doi: 10.1093/jmedent/29.3.531. [38] R. Rosà, G. Marini, L. Bolzoni, M. Neteler, M. Metz, L. Delucchi, E. A. Chadwick, L. Balbo, A. Mosca, M. Giacobini et al., Early warning of west nile virus mosquito vector: Climate and land use models successfully explain phenology and abundance of culex pipiens mosquitoes in north-western italy, Parasites & vectors, 7 (2014), p269. [39] M. R. Sardelis, M. J. Turell, D. J. Dohm and M. L. O'Guinn, Vector competence of selected north american culex and coquillettidia mosquitoes for west nile virus, Emerging infectious diseases, 7 (2001), p1018. [40] 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 B: Biological Sciences, 279 (2012), 925-933. doi: 10.1098/rspb.2011.1282. [41] J. P. Swaddle and S. E. Calos, Increased avian diversity is associated with lower incidence of human west nile infection: Observation of the dilution effect, PloS one, 3 (2008), e2488. doi: 10.1371/journal.pone.0002488. [42] D. 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. [43] S. Tiawsirisup, K. B. Platt, R. B. Evans and W. A. Rowley, Susceptibility of ochlerotatus trivittatus (coq.), aedes albopictus (skuse), and culex pipiens (l.) to west nile virus infection, Vector-Borne & Zoonotic Diseases, 4 (2004), 190-197. [44] S. Tiawsirisup, K. B. Platt, R. B. Evans and W. A. Rowley, A comparision of west nile virus transmission by ochlerotatus trivittatus (coq.), culex pipiens (l.), and aedes albopictus (skuse), Vector-Borne & Zoonotic Diseases, 5 (2005), 40-47. [45] M. J. Turell, M. R. Sardelis, D. J. Dohm and M. L. O'Guinn, Potential for north american mosquitoes to transmit west nile virus, American Journal of Tropical Medicine and Hygiene, 62 (2000), 413-414. [46] R. Unnasch, T. Sprenger, C. Katholi, E. Cupp, G. Hill and T. Unnasch, A dynamic transmission model of eastern equine encephalitis virus, Ecological modelling, 192 (2006), 425-440. doi: 10.1016/j.ecolmodel.2005.07.011. [47] 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. [48] E. B. Vinogradova, Culex pipiens pipiens mosquitoes: taxonomy, distribution, ecology, physiology, genetics, applied importance and control, Pensoft Publishers, 2000. [49] T. P. Weber and N. I. Stilianakis, Ecologic immunology of avian influenza (h5n1) in migratory birds, Emerging infectious diseases, 13 (2007), p1139. doi: 10.3201/eid1308.070319. [50] M. J. Wonham, M. A. Lewis, J. Rencławowicz 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. [51] M. Wonham, T. de Camino-Beck and M. Lewis, An epidemiological model for west nile virus: Invasion analysis and control applications, Proceedings of the Royal Society of London. Series B: Biological Sciences, 271 (2004), 501-507. doi: 10.1098/rspb.2003.2608.
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