Periodic and Chaotic Dynamics in Childhood Infections

  • W. M. Schaffer
  • L. F. Olsen
  • G. L. Truty
  • S. L. Fulmer
  • D. J. Graser
Conference paper
Part of the Springer Series in Synergetics book series (SSSYN, volume 39)


Most biologists, when confronted with a time series, assume that the dynamical possibilities are quite limited. Either a system sits still, presumably at equilibrium, or it may oscillate with a fixed period. Anything else, they will probably tell you, is evidence of noise — observational error or chance perturbations from without. By these criteria, most biological systems, especially at the population level, are extremely noisy. Hence, it is no accident that mathematical biology places a heavy emphasis on stochastic models as well as on statistical techniques designed to extract the “deterministic” component of the signal.


Lyapunov Exponent Correlation Dimension Childhood Disease Chaotic Orbit Chaotic Time Series 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    R.M. Anderson, B.T. Grenfell, R.M. May: J. Hyg. Camb. 93, 587 (1984)CrossRefGoogle Scholar
  2. 2.
    W.M. Schaffer: IMA J. Math. Appl. Med. Biol. 2, 221 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    W.M. Schaffer, M. Kot: J. theor. Biol. 112, 403 (1985)MathSciNetCrossRefGoogle Scholar
  4. 4.
    L.F. Olsen: In Chaos in Biological Systems, ed. by H. Degn, A.V. Holden, L.F. Olsen, NATO ASI Ser., Vol. 138 (Plenum, New York 1987) p. 249;Google Scholar
  5. 4a.
    L.F. Olsen, W.M. Schaffer, G.L. Truty: Theor. Pop. Biol., in pressGoogle Scholar
  6. 5.
    R.M. May, R.M. Anderson: Nature 280, 459 (1979)CrossRefGoogle Scholar
  7. 6.
    R.M. Anderson: In Population Dynamics of Infectious Diseases. Theory and Applications, ed. by R.M. Anderson (Chapman and Hall, New York 1982) p. 1Google Scholar
  8. 7.
    K. Dietz: Lect. Notes Biomath. 11, 1 (1976)Google Scholar
  9. 8.
    P.E.M. Fine, J.A. Clarkson: Int. J. Epidem. 11, 5 (1982)CrossRefGoogle Scholar
  10. 9.
    I.B. Schwartz, H.L. Smith: J. Math. Biol. 18, 233 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 10.
    H. Smith: J. Math. Anal. Appl. 64, 467 (1978)MathSciNetADSzbMATHCrossRefGoogle Scholar
  12. 11.
    J.L. Aron, I.B. Schwartz: J. theor. Biol. 110, 665 (1984)MathSciNetCrossRefGoogle Scholar
  13. 12.
    M. Kot, W.M. Schaffer, G.L. Truty, D.J. Graser, L.F. Olsen: Ecol. Mod. (1988), in pressGoogle Scholar
  14. 13.
    F. Takens: In Dynamical Systems and Turbulence, Warwick, ed. by D.A. Rand, L.S. Young (Springer, New York 1981), p. 366CrossRefGoogle Scholar
  15. 14.
    W.M. Schaffer: In Chaos in Biological Systems, ed. by H. Degn, A.V. Holden, L.F. Olsen, NATO ASI Ser., Vol. 138 (Plenum, New York 1987) p. 233Google Scholar
  16. 15.
    O.E. Rössler: Phys. Lett. 57A, 397 (1976)ADSGoogle Scholar
  17. 16.
    P. Grassberger, I. Procaccia: Physica 9D, 189 (1983)MathSciNetADSGoogle Scholar
  18. 17.
    A. Wolf, J.B. Swift, H.L. Swinney, J.A. Vastano: Physica 16D, 285 (1985)MathSciNetADSGoogle Scholar
  19. 18.
    A.M. Albano, A.I. Mees, G.C. de Guzman, P.E. Rapp: In Chaos in Biological Systems, ed. by H. Degn, A.V. Holden, L.F. Olsen, NATO ASI Ser., Vol. 138 (Plenum, New York 1987) p. 207Google Scholar
  20. 19.
    D.S. Broomhead, G.P. King: Physica 20D, 217 (1986)MathSciNetADSGoogle Scholar
  21. 20.
    A.I. Mees, P.E. Rapp, L.S. Jennings: preprintGoogle Scholar
  22. 21.
    W.M. Schaffer, S.E. Ellner, M. Kot: J. Math. Biol. 24, 479 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 22.
    J.-P. Eckmann, D. Ruelle: Rev. Mod. Phys. 57, 617 (1985)MathSciNetADSCrossRefGoogle Scholar
  24. 23.
    I. Shimada, T. Nagashima: Prog. Theor. Phys. 61, 1606 (1979)MathSciNetADSCrossRefGoogle Scholar
  25. 24.
    G. Bennetin, L. Galgani, A. Giorgil1i, J.-M. Strelcyn: Meccanica 15, 9 (1980)ADSCrossRefGoogle Scholar
  26. 25.
    W.M. Schaffer, G.L. Truty: Dynamical Software: II. User’s Manual and Introduction to Chaotic Systems (Dynamical Systems, Inc., Tucson 1987)Google Scholar
  27. 26.
    C. Nicolis, G. Nicolis: Nature 311, 529 (1984)ADSCrossRefGoogle Scholar
  28. 27.
    P. Grassberger: Nature 323, 609 (1986)ADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1988

Authors and Affiliations

  • W. M. Schaffer
    • 1
  • L. F. Olsen
    • 1
  • G. L. Truty
    • 1
  • S. L. Fulmer
    • 1
  • D. J. Graser
    • 1
  1. 1.Department of Ecology and Evolutionary BiologyThe University of ArizonaTucsonUSA

Personalised recommendations