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Low Dimensional Strange Attractors in Epidemics of Childhood Diseases in Copenhagen, Denmark

  • Lars Folke Olsen
Part of the NATO ASI Series book series (NSSA, volume 138)

Abstract

The monthly reported cases of six childhood diseases in Copenhagen, Denmark over periods of 30–40 years were analysed by three different non-linear techniques: (1) Calculation of the correlation dimension, (2) calculation of the Lyapunov exponents from apparently 1D maps derived from Poincaré sections of the flows and (3) calculation of the maximal Lyapunov exponents from the flows. Combining the results from these three types of analysis leads to the following conclusions: three of the diseases (measles, mumps and rubella) evolve with chaotic dynamics, one disease (chicken pox) evolves with dynamics corresponding to a noise-perturbed damped oscillation and the remaining two diseases (pertussis and scarlet fever) evolve with dynamics that are completely obscured by random fluctuations.

Keywords

Lyapunov Exponent Chaotic Dynamic Chaotic Attractor Reporting Rate Childhood Disease 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    J.L. Aron and I.B. Schwartz, J. Theor. Biol. 110:665-679 (1984)Google Scholar
  2. [2]
    N.T.J. Bailey, “The Mathematical Theory of Infectious Diseases and its Applications”, Charles Griffin & Co., London (1975).Google Scholar
  3. [3]
    M.S. Bartlett, “Stochastic Population Models in Ecology and Epidemiology”, Methuen & Co., London (1960).Google Scholar
  4. [4]
    K. Dietz, Lect. Notes Biomath. 11: 1 - 15 (1976).CrossRefGoogle Scholar
  5. [5]
    P. Grassberger and I. Procaccia, Physica 9D: 189 - 208 (1983).Google Scholar
  6. [6]
    W.P. London and J.A. Yorke, Am. J. Epidemiol. 98: 453 - 468 (1973).PubMedGoogle Scholar
  7. [7]
    Medical Report for the Kingdom of Denmark 1927-1968, The National Health Service of Denmark, Copenhagen.Google Scholar
  8. [8]
    L.F. Olsen, in “Proceedings of Workshop on Dynamical Systems and Environmental Models”, Wartburg Castle, Eisenach, GDR, March 16-21, 1986, Akademie Verlag, Berlin (in press).Google Scholar
  9. [9]
    W.M. Schaffer, IMA J.Math. Appl. Med. & Biol. 2: 221 - 252 (1985).CrossRefGoogle Scholar
  10. [10]
    W.M. Schaffer, this volume (1987).Google Scholar
  11. [11]
    W.M. Schaffer and M. Kot, J. Theor. Biol. 112: 403 - 427 (1985).PubMedCrossRefGoogle Scholar
  12. [12]
    F. Takens, Lect. Notes Math. 898: 366 - 381 (1981).CrossRefGoogle Scholar
  13. [13]
    A. Wolf, J.B. Swift, H.L. Swinney and J.A. Vastano, Physica 16D: 285 - 317 (1985).Google Scholar

Copyright information

© Springer Science+Business Media New York 1987

Authors and Affiliations

  • Lars Folke Olsen
    • 1
  1. 1.Institute of BiochemistryOdense UniversityOdense MDenmark

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