Periodic Phenomena and Driving Mechanisms in Transmission of West Nile Virus with Maturation Time

  • Chunhua Shan
  • Guihong Fan
  • Huaiping ZhuEmail author


West Nile virus (WNv) transmission shows both seasonal pattern in every single year and cyclic pattern over years. In this paper we formulate a compartmental model with bird demographics and maturation time of mosquitoes during metamorphosis to study the impact of ambient temperature on the transmission and recurrence of disease. We show that avian birds serve as a reservoir of viruses, whilst maturation time affects disease transmission in sophisticated ways. It turns out that large maturation delay will lead to the extinction of mosquitoes and the disease; small maturation delay will stabilize the epidemic level of the disease; and intermediate maturation delay will cause sustainable oscillations of mosquito population, recurrence of diseases, and even mixed-mode oscillation of diseases with an alternation between oscillations of distinct large and small amplitudes. With bifurcation theory, we prove that temperature can drive the oscillation of mosquito population, which leads recurrence of WNv through the incidence interaction between mosquitoes and hosts, while the biting and transmission process itself will not generate sustained oscillations. Our results provide a sound explanation for understanding interactions between vectors and hosts, and driving mechanisms of periodic phenomena in the transmission of WNv and other mosquito-borne diseases.


West Nile virus Maturation delay Transmission dynamics Hopf bifurcation Mixed-mode oscillations Period-doubling 



Funding was provided by Natural Sciences and Engineering Research Council of Canada (CA), Canadian Institute of Health Research (CIHR) (CA).


  1. 1.
    Abdelrazec, A., Lenhart, S., Zhu, H.: Transmission dynamics of west nile virus in mosquitoes and corvids and non-corvids. J. Math. Biol. 68, 1553–1582 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Arino, J., Wang, L., Wolkowicz, G.S.: An alternative formulation for a delayed logistic equation. J. Theor. Biol. 241, 109–119 (2005)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Berezansky, L., Braverman, E., Idels, L.: Nicholson’s blowflies differential equations revisited: main results and open problems. Appl. Math. Model. 34, 1405–1417 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bowman, C., Gumel, A.B., van Den Driessche, P., Wu, J., Zhu, H.: A mathematical model for assessing control strategies against West Nile virus. Bull. Math. Biol. 67, 1107–1133 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Brust, R.: Weight and develpoment time of different stadia of mosquitoes reared at various constant temperature. Can. Entomol. 99, 986–993 (1967)CrossRefGoogle Scholar
  6. 6.
    Campbell, L.G., Martin, A.A., Lanciotti, R.S., Guble, D.J.: West Nile virus. Lancet Infect. Dis. 2, 519–529 (2002)CrossRefGoogle Scholar
  7. 7.
    Centers for disease control and prevention, 2002. West Nile virus: virus history and distribution. (2002)
  8. 8.
    Centers for disease control and prevention, 2012. West Nile virus update: November 6, (2002)
  9. 9.
    Cooke, K.L., van den Driessche, P., Zou, X.: Interaction of maturation delay and nonlinear birth in population and epidemic models. J. Math. Biol. 39, 332–352 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cruz-Pacheco, G., Esteva, L., Montano-Hirose, J.A., Vargas, C.: Modelling the dynamics of West Nile virus. J. Math. Biol. 67, 1157–1172 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Cruz-Pacheco, G., Esteva, L., Vargas, C.: Seasonality and outbreaks in West Nile virus infection. Bull. Math. Biol. 71, 1378–1393 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Dohm, D.J., Sardelis, M.R., Turell, M.J.: Experimental vertical transmission of West Nile virus by Culex pipiens (Diptera: Culicidae). J. Med. Entomol. 39, 640–644 (2002)CrossRefGoogle Scholar
  13. 13.
    Fan, G., Liu, J., van den Driessche, P., Wu, J., Zhu, H.: A delay differential equations model for the impact of temperature on West Nile virus between birds and mosquitoes. Math. Biosci. 228(2), 119–126 (2010)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Garett-Jones, C.: Prognosis for interruption of malaria transmission through assessment of mosquitoes vectorical capacity. Nature 204, 1173–1175 (1964)CrossRefGoogle Scholar
  15. 15.
    Goddard, L.B., Roth, A.E., Reisen, W.K., Scott, T.W.: Vertical transmission of West Nile virus by three California Culex (Diptera: Culicidae) species. J. Med. Entomol. 40, 743–746 (2003)CrossRefGoogle Scholar
  16. 16.
    Gurney, W., Blythe, S., Nisbet, R.: Nicholson’s blowflies revisited. Nature 287(4), 17–21 (1980)CrossRefGoogle Scholar
  17. 17.
    Hagstrum, D.W., Workman, E.B.: Interaction of temperature and feeding rate in determining the rate of development of larval Culex tarsalis. Ann. Entomol. Soc. Am. 64, 668–671 (1971)CrossRefGoogle Scholar
  18. 18.
    Hale, J.K., Lunel, S.M.: Introduction to Functional Differential Equations. Applied Mathematical Sciences, 99. Springer, New York (1993)CrossRefzbMATHGoogle Scholar
  19. 19.
    Komar, N., Langevin, S., Nemeth, N., Edwards, E., Hettler, D., Davis, B., Bowen, R., Bunning, M.: Experimental infection of North American bird with the New York 1999 strain of West Nile virus. Emerg. Infect. Dis. 9(2003), 311–322 (1999)Google Scholar
  20. 20.
    Laperriere, V., Brugger, K., Rubel, F.: Simulation of the seasonal cycles of bird, equine and human West Nile virus cases. Prev. Vet. Med. 98, 99–110 (2011)CrossRefGoogle Scholar
  21. 21.
    Lassiter, M., Apperson, C., Roe, R.: Juvenile hormone metabolism during the fourth stadium and pupal stage of the south house mosquito Culex quinquefasciatus say. J. Insect Physiol. 41, 869–876 (1995)CrossRefGoogle Scholar
  22. 22.
    Lewis, M., Renclawowicz, J., van den Driessche, P.: Travelling waves and spread rates for a West Nile virus model. Bull. Math. Biol. 68, 3–23 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Lord, C.C., Day, J.F.: Simulation studies of St. Louis encephalitis virus in south Florida. Vector Borne Zoonotic Dis. 1(4), 299–315 (2001)CrossRefGoogle Scholar
  24. 24.
    Maidana, N.A., Yang, H.M.: Spatial spreading of west nile virus described by traveling waves. J. Theor. Biol. 258, 403–417 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Munga, S., Minakawa, N., Zhou, G.: Survivorship of immature stages of Anopheles gambiae s.l. (Diptera: Culicidae) in natural habitats in western kenya highlands. J. Med. Entomol. 44(5), 758–764 (2007)CrossRefGoogle Scholar
  26. 26.
    Sander, E., Yorke, J.: Connecting period-doubling cascades to chaos. Int. J. Bifurc. Chaos Appl. Sci. Eng. 22, 1250022 (2012)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Shu, H., Wang, L., Wu, J.: Global dynamics of nicholson’s blowflies equation revisited: onset and termination of nonlinear oscillations. J. Differ. Equ. 255, 2565–2586 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Simpson, J.E., Hurtado, P.J., Medlock, J., Molaei, G., Andreadis, T.G., Galvani, A.P., Diuk-Wasser, M.A.: Vector host-feeding preference drive transmission of multi-host pathogens: West Nile virus as a model system. Proc. R. Soc. B 279, 924–933 (2012)Google Scholar
  29. 29.
    Smith, H.J.: Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems. Mathematical Surveys and Monographs, vol. 41. American Mathematical Society, Providence, RI (1995)zbMATHGoogle Scholar
  30. 30.
    Thomas, D., Weedermann, M., Billings, L., Hoffacker, J., Washington -Allen, R.A.: When to spray: a time-scale calculus approach to controlling the impact of West Nile virus. Ecol. Soc. 14(2) (2009)Google Scholar
  31. 31.
    Tuno, N., Okeka, W., Minakawa, N., Takagi, M.: Survivorship of Anopheles gambiae sensu stricto (Diptera: Culicidae) larvae in western kenya highland forest. J. Med. Entomol. 42(3), 270–277 (2005)CrossRefGoogle Scholar
  32. 32.
    van den Driessche, P., Watmough, J.: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180, 29–48 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Wan, H., Zhu, H.: A new model with delay for mosquito population dynamics. Math. Biosci. Eng. 11(6), 1395–1410 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Wang, J., Ogden, N.H., Zhu, H.: The impact of weather conditions on Culex pipiens and Culex restuans (Diptera: Culicidae) abundance: a case study in Peel region. J. Med. Entomol. 48(2), 468–475 (2011)CrossRefGoogle Scholar
  35. 35.
    Wei, J., Li, M.Y.: Hopf bifurcation analysis in a delayed nicholson blowflies equation. Nonlinear Anal. 60, 1351–1367 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Wolkowicz, G.S.K., Xia, H.: Global asymptotic behavior of a chemostat model with discrete delays. SIAM J. Appl. Math. 57, 1019–1043 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Wonham, M .J., de Camino Beck, T., Lewis, M .A.: An epidemiological model for West Nile virus: invasion analysis and control applications. Proc. R. Soc. Lond. Ser. B 271, 501–507 (2004)CrossRefGoogle Scholar
  38. 38.
    Wonham, M.J., Lewis, M.A., Renclawowicz, J., van den Driessche, P.: Transmission asumptions generate confflicting prediction in host-vector disease models: a case study in West Nile virus. Ecol. Lett. 9, 706–725 (2006)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsThe University of ToledoToledoUSA
  2. 2.Department of MathematicsColumbus State UniversityColumbusUSA
  3. 3.LAMPS and Department of Mathematics and StatisticsYork UniversityTorontoCanada

Personalised recommendations