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The backward bifurcation in compartmental models for West Nile virus. (English) Zbl 1194.92067

Summary: In all of the West Nile virus (WNV) compartmental models in the literature, the basic reproduction number serves as a crucial control threshold for the eradication of the virus. However, our study suggests that backward bifurcation is a common property shared by the available compartmental models with a logistic type of growth for the population of host birds. There exists a subthreshold condition for the outbreak of the virus due to the existence of backward bifurcations.

We first review and give a comparison study of the four available compartmental models for the virus, and focus on the analysis of the model proposed by G. Cruz-Pacheco et al. [Bull. Math. Biol. 67, No. 6, 1157 ff (2005)] to explore the backward bifurcation in the model. Our comparison study suggests that the mosquito population dynamics itself cannot explain the occurrence of the backward bifurcation, it is the higher mortality rate of the avian host due to the infection that determines the existence of backward bifurcations.

93A30Mathematical modelling of systems
37N25Dynamical systems in biology
34D99Stability theory of ODE
[1]Baqar, S.; Hayes, C. G.; Murphy, J. R.; Watts, D. M.: Vertical transmission of west nile virus by culex and aedes species mosquitoes, Am. J. Trop. med. Hyg. 48, 757 (1993)
[2]Bowman, C.; Gumel, A. B.; Wu, J.; Den Driessche, P. Van; Zhu, H.: A mathematical model for assessing control strategies against west nile virus, Bull. math. Biol. 67, No. 5, 1107 (2005)
[3]Campbell, L. G.; Martin, A. A.; Lanciotti, R. S.; Gubler, D. J.: West nile virus, Lancet infect. Dis. 2, 519 (2002)
[4]Cruz-Pacheco, Gustavo; Esteva, Lourdes: Juan antonio montaño-hirose and cristobal vargas, modelling the dynamics of west nile virus, Bull. math. Biol. 67, No. 6, 1157 (2005)
[5]Diekmann, O.; Heesterbeek, J. A. P.; Metz, J. A. J.: On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations, J. math. Biol. 28, 365 (1990) · Zbl 0726.92018 · doi:10.1007/BF00178324
[6]Hayes, C. G.: West nile virus: uganda, 1937, to New York city, 1999, Ann. NY acad. Sci. 951, 25 (2001)
[7]Jiang, J.; Qiu, Z.; Wu, J.; Zhu, H.: Threshold conditions for west nile virus outbreaks, Bull. math. Biol. 71, No. 3, 627 (2009) · Zbl 1163.92036 · doi:10.1007/s11538-008-9374-6
[8]Kenkre, V. M.; Parmenter, R. R.; Peixoto, I. D.; Sadasiv, L.: A theoretical framework for the analysis of the west nile virus epidemic, Math. comput. Model. 42, No. 3 – 4, 313 (2005) · Zbl 1080.92057 · doi:10.1016/j.mcm.2004.08.012
[9]Komar, N.; Langevin, S.; Hinten, S.; Nemeth, N.; Edwards, 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, Emerg. infect. Dis. 9, No. 3, 311 (2003)
[10]Lanciotti, R. S.; Rohering, J. T.; Deubel, V.; Smith, J.; Parker, M.; Steele, K.; Crise, B.; Volpe, K. E.; Crabtree, M. B.; Scherret, J. H.; Hall, R. A.; Mackenzie, J. S.; Cropp, C. B.; Panigrahy, B.; Ostlund, E.; Schmitt, B.; Malkinson, M.; Banet, C.; Weissman, J.; Komar, N.; Savage, H. M.; Stone, W.; Mcnamara, T.; Gubler, D. J.: Origin of the west nile virus responsible for an outbreak of encephalitis in the northeastern united states, Science 286, No. 5448, 2333 (2003)
[11]Lewis, M.; Renclawowicz, J.; Den Driessche, P. Van: Traveling waves and spread rates for a west nile virus model, Bull. math. Biol. 68, No. 1, 3 (2006)
[12]Lewis, M.; Renclawowicz, J.; Den Driessche, P. Van; Wonham, M.: A comparison of continuous and discrete-time west nile virus models, Bull. math. Biol. 68, No. 3, 491 (2006)
[13]Liu, R.; Shuai, J.; Wu, J.; Zhu, H.: Modeling spatial spread of west nile virus and impact of directional dispersal of birds, Math. biosci. Eng. 3, No. 1, 145 (2006) · Zbl 1089.92047
[14]Lord, C. C.; Day, J. F.: Simulation studies of St. Louis encephalitis virus in south florida, Vector borne zoonotic diseases 1, No. 4, 299 (2001)
[15]Lord, C. C.; Day, J. F.: Simulation studies of St. Louis encephalitis and west nile viruses: the impact of bird mortality, Vector borne zoonotic diseases 1, No. 4, 317 (2001)
[16]Swayne, D. E.; Beck, J. R.; Zaki, S.: Pathogenicity of west nile virus for turkeys, Avian diseases 44, 932 (2000)
[17]Smithburn, K. C.; Hughes, T. P.; Burke, A. W.; Paul, J. H.: A neurotropic virus isolated from the blood of a native of uganda, Am. J. Trop. med. 20, 471 (1940)
[18]Thomas, D. M.; Urena, B.: A model describing the evolution of west nile-like encephalitis in New York city, Math. comput. Model. 34, 771 (2001) · Zbl 0999.92025 · doi:10.1016/S0895-7177(01)00098-X
[19]Den Driessche, P. Van; Watmough, J.: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. biosci. 180, 29 (2002) · Zbl 1015.92036 · doi:10.1016/S0025-5564(02)00108-6
[20]Wonham, M. J.; De-Camino-Beck, T.; Lewis, M.: An epidemiological model for west nile virus: invasion analysis and control applications, Proc. roy. Soc. London, series B 1538, 501 (2004)
[21]Centers for Disease Control and Prevention, West Nile virus: fact sheet, 2003. lt;http://www.cdc.gov/ncidod/dvbid/westnile/wnvfactSheet.htmgt;.
[22]West Nile Virus Monitor – West Nile Virus Surveillance Information. West Nile Virus – Public Health Agency of Canada. lt;http://www.phac-aspc.gc.ca/wn-no/index-e.htmlgt;.