Chaos in Childhood Epidemics

  • W. M. Schaffer
  • L. F. Olsen
Part of the NATO ASI Series book series (NSSB, volume 208)


Whereas childhood infections such as chickenpox evidence effectively periodic dynamics, case reports for diseases such as measles and rubella fluctuate more erratically [1]. Elsewhere [2,3], it has been suggested that the observed fluctuations correspond to low dimensional chaos of the sort which arises in epidemiological models [5] of the SEIR variety, i.e., differential equations, subject to periodic forcing.1 The basis for this assertion is as follows:
  1. 1.

    Reconstructed phase portraits, Poincaré sections, and return maps constructed from actual data are remarkably similar to those obtained from the models (Fig. 1).

  2. 2.

    For measles, both the models and the data yield correlation dimensions of about 25. A comparable value is obtained for the Lyapunov dimension as computed directly from the differential equations.

  3. 3.

    Both the models and the data yield estimated positive Lyapunov exponents of between 0.4 and 0.5 bpy.



Correlation Dimension Childhood Infection Epidemiological Model World City Positive Lyapunov Exponent 
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  1. 1.
    W. P. London and J.A. Yorke. Am. J. EpidemioL 98, 453 (1973).Google Scholar
  2. 2.
    W. M. Schaffer and M. Kot. J. Theor. BioL 112, 403 (1985).MathSciNetCrossRefGoogle Scholar
  3. 3.
    W. M. Schaffer. IMA J. Math. AppL Med Biol. Z 221 (1985).Google Scholar
  4. 4.
    K. Dietz. Lect Notes Biomath. 11, 1, (1976).CrossRefGoogle Scholar
  5. 5.
    L. F. Olsen, G.L. Truty and W. M. Schaffer. Theor. Pop. Biol. 33, 344 (1988).MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    B. Schwartz. Proc. 12th IMACS World Cons, ’88 on Scient Comput (In press).Google Scholar
  7. 7.
    J. L. Aron and I. B. Schwartz. J. Theor. BioL 110, 665 (1984).MathSciNetCrossRefGoogle Scholar
  8. 8.
    W. M. Schaffer, L. F. Olsen, G. L. Truty and S. L. Fulmer. In S. Krasner. The Ubiquity of Chaos AAAS. (1989).Google Scholar

Copyright information

© Plenum Press, New York 1989

Authors and Affiliations

  • W. M. Schaffer
    • 1
  • L. F. Olsen
    • 2
  1. 1.Department of Ecology and Evolutionary BiologyThe University of ArizonaTucsonUSA
  2. 2.Institute of BiochemistryOdense UniversityOdenseDenmark

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