Bulletin of Mathematical Biology

, Volume 67, Issue 3, pp 509–528 | Cite as

Mathematical models for the Aedes aegypti dispersal dynamics: Travelling waves by wing and wind

  • Lucy Tiemi Takahashi
  • Norberto Anibal Maidana
  • Wilson Castro FerreiraJr.
  • Petronio Pulino
  • Hyun Mo Yang
Article

Abstract

Biological invasion is an important area of research in mathematical biology and more so if it concerns species which are vectors for diseases threatening the public health of large populations. That is certainly the case for Aedes aegypti and the dengue epidemics in South America. Without the prospect of an effective and cheap vaccine in the near future, any feasible public policy for controlling the dengue epidemics in tropical climates must necessarily include appropriate strategies for minimizing the mosquito population factor. The present paper discusses some mathematical models designed to describe A. aegypti’s vital and dispersal dynamics, aiming to highlight practical procedures for the minimization of its impact as a dengue vector. A continuous model including diffusion and advection shows the existence of a stable travelling wave in many situations and a numerical study relates the wavefront speed to a few crucial parameters. Strategies for invasion containment and its prediction based on measurable parameters are analysed.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Carr, J., 1981. Applications of Center Manifold Theory. Springer, New York.Google Scholar
  2. Cummings, D.A.T., Irizarry, R.A., Huang, N.E., Endy, T.P., Nisalak, A., Ungchusak, K., Burke, D.S., 2004. Travelling waves in the occurrence of dengue haemorrhagic fever in Thailand. Nature 427, 344–347.CrossRefGoogle Scholar
  3. Ereno, D., 2003. Novas armas contra a dengue. Revista Pesquisa Fapesp 85. http://revistapesquisafapesp.br.
  4. Ermentrout, B., 2002. Simulating, Analyzing, and Animating Dynamical Systems. A Guide to XPPAUT for Researchers and Students. SIAM.Google Scholar
  5. Ferreira, C.P., Yang, H.M., 2003. Estudo dinâmico da população de mosquitos Aedes aegypti, TEMA—Seleta do XXV CNMAC 4.2, SBMAC e FAPESP, São Carlos e São Paulo, pp. 187–196.Google Scholar
  6. Gubler, D.J., 1998. Dengue and dengue hemorrhagic fever. Clin. Microb. Rev. 11, 480–496.Google Scholar
  7. Hagstrom, T, Keller, H.B., 1986. The numerical calculation of travelling wave solutions of nonlinear parabolic equations. SIAM J. Sci. Stat. Comput. 7, 978–985.MathSciNetCrossRefMATHGoogle Scholar
  8. Hartman, P., 1973. Ordinary Differential Equations. Hartman, Baltimore.MATHGoogle Scholar
  9. Heinze, S., Papanicolaou, G., Stevens, A., 2001. Variational principles for propagation speeds in inhomogeneous media. SIAM J. Appl. Math. 621, 129–148.MathSciNetCrossRefGoogle Scholar
  10. Lucia, M., Muratov, C.B., Novaga, M., 2004. Linear vs. nonlinear selection for the propagation speed of the solutions of scalar reaction-diffusion equations invading an unstable equilibrium. Comm. Pure Appl. Math. 57, 616–636.MathSciNetCrossRefMATHGoogle Scholar
  11. Murray, J.D., 1993. Mathematical Biology. Springer, Berlin.MATHGoogle Scholar
  12. Murray, J.D., Stanley, E.A., Brown, D.L., 1986. On the spatial spread of rabies among foxes. Proc. R. Soc. Lond. B 229, 111–150.CrossRefGoogle Scholar
  13. Pauwelussen, J.P., 1981. Nerve impulse propagation in a branching nerve system: a simple model. Physica 4D, 67–88.MathSciNetGoogle Scholar
  14. Potapov, A.B., Lewis, M.A., 2004. Climate and competition: the effect of moving range boundaries on habitat invasibility. Bull. Math. Biol. 66, 975–1008.CrossRefMathSciNetGoogle Scholar
  15. Sandstede, B., 2002. Stability of travelling waves. In: Fiedler, B. (Ed.), Handbook of Dynamical Systems II. Elsevier, Amsterdam, pp. 983–1059.Google Scholar
  16. Segel, L.A. (Ed.), 1980. Mathematical Models in Molecular and Cellular Biology. Cambridge University Press, Cambridge.MATHGoogle Scholar
  17. Shigesada, N., Kawasaki, K., 1997. Biological Invasions: Theory and Practice. Oxford University Press.Google Scholar
  18. Takahashi, L.T., 2004. Modelos Matemáticos de Epidemiologia com Vetores: Simulação da Propagação Urbana e Geográfica da Dengue. Ph.D. Thesis, Univ. Estadual de Campinas—UNICAMP, Campinas, Brazil.Google Scholar
  19. Teixeira, C.F., da S. Augusto, L.G., Morata, T.C., 2003. Hearing health of workers exposed to noise and insecticides. Rev. Saúde Pública 37, 417–423.CrossRefGoogle Scholar
  20. Vasconcelos, P.F.C., Rosa, A.P.A.T., Pinheiro, F.P., Rodrigues, S.G., Rosa, E.S.T., Cruz, A.C.R., Rosa, J.F.S.T., 1999. Aedes aegypti, dengue and re-urbanization of yellow fever in Brazil and other South American Countries—past and present situation and future perspectives. Dengue Bulletin 23, 1–10. http://www.cepis.ops-oms.org/bvsair/e/repindex/repi78/pagina/text/fulltext/vol23.pdf.Google Scholar
  21. Veronesi, R., 1991. Doenças Infecciosas e Parasitárias. Guanabara Koogan, Rio de Janeiro.Google Scholar
  22. Volpert, A.I., Volpert, V.A., 1994. Travelling Wave Solutions of Parabolic Systems. American Mathematical Society, Providence, RI.Google Scholar
  23. Weinberger, F.H., 1982. Long-time behavior of a class of biological models. SIAM J. Math. Anal. 13, 353–396.MATHMathSciNetCrossRefGoogle Scholar
  24. Yang, H.M., Ferreira, C.P., Ternes, S., 2003. Dinâmica populacional do vetor transmissor da dengue, TEMA-Seleta do XXV CNMAC 4.2, SBMAC e FAPESP, São Carlos e São Paulo, pp. 287–296.Google Scholar

Copyright information

© Society for Mathematical Biology 2005

Authors and Affiliations

  • Lucy Tiemi Takahashi
    • 1
  • Norberto Anibal Maidana
    • 1
  • Wilson Castro FerreiraJr.
    • 1
  • Petronio Pulino
    • 1
  • Hyun Mo Yang
    • 1
  1. 1.Departamento de Matemática Aplicada-IMECCUniversidade Estadual de CampinasCampinas SPBrazil

Personalised recommendations