Spatial spreading of West Nile virus described by traveling waves.

*(English)*Zbl 1405.92260Summary: In this work, we propose a spatial model to analyze the West Nile virus (WNV) propagation across the USA, from east to west. WNV is an arthropod-borne flavivirus that appeared for the first time in New York City in the summer of 1999 and then spread prolifically among birds. Mammals, such as humans and horses, do not develop sufficiently high bloodstream titers to play a significant role in the transmission, which is the reason to consider the mosquito-bird cycle. The model aims to study this propagation based on a system of partial differential reaction-diffusion equations taking the mosquito and the avian populations into account. Diffusion and advection movements are allowed for both populations, being greater in the avian than in the mosquito population. The traveling wave solutions of the model are studied to determine the speed of disease dissemination. This wave speed is obtained as a function of the model’s parameters, in order to assess the control strategies. The propagation of WNV from New York City to California state is established as a consequence of the diffusion and advection movements of birds. Mosquito movements do not play an important role in the disease dissemination, while bird advection becomes an important factor for lower mosquito biting rates.

##### MSC:

92D30 | Epidemiology |

35C07 | Traveling wave solutions |

35Q92 | PDEs in connection with biology, chemistry and other natural sciences |

##### Keywords:

West Nile virus; reaction-diffusion equation; traveling waves; wave speed; sensitivity analysis
PDF
BibTeX
XML
Cite

\textit{N. A. Maidana} and \textit{H. M. Yang}, J. Theor. Biol. 258, No. 3, 403--417 (2009; Zbl 1405.92260)

Full Text:
DOI

##### References:

[1] | Bowman, C.; Gumel, A.B.; Van den Driessche, P.; Wu, J.; Zhu, H., A mathematical model for assessing control strategies against west nile virus, Bulletin of mathematical biology, 67, 1107-1133, (2005) · Zbl 1334.92392 |

[2] | Campbell, L.G.; Martin, A.A.; Lanciotti, R.S.; Gubler, D.J., West nile virus, The lancet infectious diseases, 2, 519-529, (2002) |

[3] | Cruz-Pacheco, G.; Esteva, L.; Montaño-Hirose, J.A.; Vargas, C., Modelling the dynamics of the west nile virus, Bulletin of mathematical biology, 67, 1157-1172, (2005) · Zbl 1334.92397 |

[4] | DeBiasi, R.L.; Tyler, K., West nile virus meningoencephalitis, Nature clinical practice neurology, 3, 5, 264-275, (2006) |

[5] | Dohm, D.J.; Sardelis, M.R.; Turell, J., Experimental vertical transmission of west nile virus by culex pippiens (diptera: culicidae), Journal of medical entomology, 39, 4, 640-644, (2002) |

[6] | Erni, B.; Liecheti, F.; Bruderer, B., Stopover strategies in passarine bird migration: a simulation study, Journal of theoretical biology, 219, 479-493, (2002) |

[7] | Fisher, R.A., The wave of advance of advantageous genes, Annals of eugenics, 7, 355-369, (1937) · JFM 63.1111.04 |

[8] | FlexPDE 5.0, copyright 2005, PDE solution inc. |

[9] | Goddard, L.B.; Roth, A.E.; Reisen, W.K.; Scott, T.W., Vector competence of California mosquitoes for west nile virus, Emerging infectious diseases, 8, 12, 1385-1391, (2002) |

[10] | Harrington, L.C.; Scott, T.W.; Lerdthusnee, K.; Coleman, R.C.; Costero, A.; Clarck, G.G.; Jones, J.J.; Kitthawee, S.; Yapong, P.K.; Sithiprasasna, R.; Edman, J.D., Dispersal of the dengue vector aedes aegypti within and between rural communities, American journal of tropical medicine and hygiene, 72, 2, 209-220, (2005) |

[11] | Hayes, G.G., West nile virus fever, (), 59-88 |

[12] | Hayes, E.B.; Komar, N.; Nasci, R.S.; Montgomery, S.P.; O’Leary, D.R.; Campbell, G.L., Epidemiology and transmission dynamics of west nile virus disease, Emerging infectious diseases, 11, 8, 1167-1173, (2005) |

[13] | Källen, A.; Acuari, P.; Murray, J.D., A simple model for the spatial spread and control of rabies, Journal of theoretical biology, 116, 377-393, (1985) |

[14] | Kenkre, V.M.; Parmenter, R.R.; Peixoto, I.D.; Sadasiv, L., A theoretical framework for the analysis of the west nile virus epidemic, Computer and mathematics with applications, 42, 313-324, (2005) · Zbl 1080.92057 |

[15] | Komar, N.; Langevin, S.; Hinten, S.; Nemeth, N.; Edwars, E.; Hettler, D.; Davis, B.; Bowen, R.; Bunning, M., Experimental infection of north American birds with the New York 1999 strain of west nile virus, Emerging infectious diseases, 9, 3, 311-322, (2003) |

[16] | Lewis, M.; Renclawowicz, J.; Van den Driessche, P., Travelling waves and spread rate for a west nile virus model, Bulletin of mathematical biology, 68, 3-23, (2006) · Zbl 1334.92414 |

[17] | Maidana, N.A.; Ferreira, W.C., The geographic spread of “el mal de las caderas” in capybaras (hydrochaeris hydrochaeris), Bulletin of mathematical biology, 70, 4, 1216-1234, (2008) · Zbl 1142.92039 |

[18] | Murray, J.D.; Stanley, F.R.S.; Brown, D.L., On the spatial spread of rabies among foxes, Proceedings of the royal society of London B, 229, 111-150, (1986) |

[19] | Murray, J.D.; Seward, W.L., On the spatial spread of rabies among foxes with immunity, Journal of theoretical biology, 156, 327-348, (1992) |

[20] | Murray, J.D., Mathematical biology, (2002), Springer Berlin |

[21] | New York City Department of Health and Mental Hygiene, 2004. Comprehensive mosquito surveillance and control plan. \(\langle\)http://www.nyc.gov/html/doh/downloads/pdf/wnv/wnvplan2004.pdf⟩. |

[22] | Okubo, A., Diffusion-type models for Avian range expansion, (), 1038-1049 |

[23] | Rappole, J.H.; Derrickson, S.R.; Hubálek, Z., Migratory birds and spread of west nile virus in the western hemisphere, Emerging infectious diseases, 6, 4, 319-328, (2000) |

[24] | Reed, K.D.; Meece, J.K.; Henkel, J.S.; Shukla, S.K., Bird migration and emerging zoonoses: west nile virus, lyme disease, influenza A and enteropathogens, Clinical medicine and research, 1, 1, 5-12, (2002) |

[25] | Sandstede, B., Stability of traveling waves, (), 983-1059 |

[26] | Stewart, P.A., Migration of blue jays in eastern north America, North American bird bander, 7, 3, 107-112, (1982) |

[27] | Turell, M.J.; O’Guinn, M.L.; Dohm, D.J.; Jones, J.W., Vector competence of north American mosquitoes (diptera: culicidade) for west nile virus, Journal of medical entomology, 38, 2, 130-134, (2001) |

[28] | Volpert, A.I.; Volpert, V.A., Traveling waves solutions of parabolic system, (1994), American Mathematical Society Providence, RI · Zbl 0835.35048 |

[29] | Wonham, M.J.; De-camino-Beck, T.; Lewis, M.A., An epidemiological model for west nile virus: invasion analysis and control applications, Proceedings of the royal society of London B, 271, 501-507, (2004) |

[30] | Work, T.H.; Hurlbut, H.S.; Taylor, R.M., Indigenous wild birds of the nile delta as potential west nile virus circulating reservoir, American journal of tropical medicine and hygiene, 4, 872-888, (1955) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.