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The Role of Mathematical Models in Explaining Recurrent Outbreaks of Infectious Childhood Diseases

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Mathematical Epidemiology

Part of the book series: Lecture Notes in Mathematics ((LNMBIOS,volume 1945))

Infectious childhood diseases such as measles are characterized by recurrent outbreaks. Mathematicians have long used models in an effort to better understand and predict these recurrent outbreak patterns. This paper summarizes and comments upon those efforts, providing a historical outline of childhood disease models that have been developed since the start of the twentieth century. This paper also discusses the influence of data analysis techniques, such as spectral analysis, on the understanding and modelling of childhood disease dynamics.

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References

  1. P. Bonanni. Demographic impact of vaccination: a review. Vaccine, 17:S120–S125, 1998

    Article  Google Scholar 

  2. CDC. Progress toward Poliomyelitis eradication–Nigeria, January 2003–March 2004. MMWR Weekly, 53(16):343–346, 2004

    Google Scholar 

  3. F. Fenner, D.A. Henderson, I. Arita, Z. Jezek, and I.D. Ladnyi. Smallpox and Its Eradication. World Health Organization, Geneva, 1988

    Google Scholar 

  4. F.E. Andre. Vaccinology: past achievements, present roadblocks and future promises. Vaccine, 21:593–595, 2003

    Article  Google Scholar 

  5. World Health Organization. Fact sheet # 286, March 2005

    Google Scholar 

  6. E.J. Gangarosa, A.M. Galazka, C.R. Wolfe, L.M. Phillips, R.E. Gangarosa, E. Miller, and R.T. Chen. Impact of anti-vaccine movements on pertussis control: the untold story. Lancet, 351:356–361, 1998

    Article  Google Scholar 

  7. J.P. Baker. The pertussis controversy in Great Britain, 1974–1986. Vaccine, 21: 4003–4010, 2003

    Article  Google Scholar 

  8. V.A. Jansen, N. Stollenwerk, H.J. Jensen, M.E. Ramsay, W.J. Edmunds, and C.J. Rhodes. Measles outbreaks in a population with declining vaccine uptake. Science, 301:804, 2003

    Google Scholar 

  9. B.T. Grenfell and J. Harwood. (Meta)-Population dynamics of infectious diseases. Trends Ecol. Evolut., 12:395–399, 1997

    Article  Google Scholar 

  10. D.J.D. Earn, P. Rohani, and B.T. Grenfell. Persistence, chaos and synchrony in ecology and epidemiology. Proc. R. Soc. Lond. B, 265:7–10, 1998

    Article  Google Scholar 

  11. D.J.D. Earn, P. Rohani, B.M. Bolker, and B.T. Grenfell. A simple model for complex dynamical transitions in epidemics. Science, 287:667–670, 2000

    Article  Google Scholar 

  12. C.T. Bauch and D.J.D. Earn. Transients and attractors in epidemics. Proc. R. Soc. Lond. B, 270:1573–1578, 2003

    Article  Google Scholar 

  13. P.E.M. Fine. The interval between successive cases of an infectious disease. Am. J. Epidemiol., 158:1039–1047, 2003

    Article  Google Scholar 

  14. M.B. Priestley. Spectral Analysis and Time Series. Academic, London, 1981

    MATH  Google Scholar 

  15. R.M. Anderson, B.T. Grenfell, and R.M. May. Oscillatory fluctuations in the incidence of infectious disease and the impact of vaccination: Time series analysis. J. Hyg. Camb., 93:587–608, 1984

    Article  Google Scholar 

  16. S. Hodder and E. Mortimer. Epidemiology of pertussis and reactions to pertussis vaccine. Epidem. Rev., 14:243–267, 1992

    Google Scholar 

  17. H.R. Babad, D.J. Nokes, N.J. Gay, E. Miller, P. Morgan-Capner, and R.M. Anderson. Predicting the impact of measles vaccination in England and Wales: model validation and analysis of policy options. Epidemiol. Infect., 114:319–344, 1995

    Article  Google Scholar 

  18. New Zealand Ministry of Health. Modelling measles. predicting and preventing measles epidemics in New Zealand: application of a mathematical model, 1998

    Google Scholar 

  19. R.M. Anderson and R.M. May. Infectious Diseases of Humans. Oxford University Press, Oxford, 1991

    Google Scholar 

  20. O. Diekmann and J.A.P. Heesterbeek. Mathematical epidemiology of infectious diseases. Wiley, New York, 2000

    Google Scholar 

  21. H.E. Soper. The interpretation of periodicity in disease prevalence. J. R. Stat. Soc., 92:34–73, 1929

    Article  Google Scholar 

  22. M.S. Bartlett. Deterministic and stochastic models for recurrent epidemics. Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, volume 4, pp. 81–108, 1956

    Google Scholar 

  23. D. Green. Self-oscillation for epidemic models. Math. Biosci., 38:91–111, 1978

    Article  MATH  MathSciNet  Google Scholar 

  24. H.L. Smith. Periodic solutions for a class of epidemic equations. Math. Anal. Appl., 64:467–79, 1978

    Article  MATH  MathSciNet  Google Scholar 

  25. D. Schenzle. An age-structure model of pre- and post-vaccination measles transmission. IMA J. Math. Appl. Med. Biol., 1:169–191, 1984

    Article  MATH  MathSciNet  Google Scholar 

  26. H.W. Hethcote. Optimal ages of vaccination for measles. Math. Biosci., 89:29–52, 1988

    Article  MATH  MathSciNet  Google Scholar 

  27. C. Castillo-Chavez, H.W. Hethcote, V. Andreaen, S.A. Levin, and W.M. Liu. Epidemiological models with age structure, proportionate mixing, and cross-immunity. J. Math. Biol., 27:233–258, 1989

    Article  MATH  MathSciNet  Google Scholar 

  28. M.J. Keeling and B.T. Grenfell. Disease extinction and community size: modeling the persistence of measles. Science, 275:65–67, 1997

    Article  Google Scholar 

  29. D. Greenhalgh. Analytic results on the stability of age-structured epidemic models. IMA J. Math. Appl. Med. Biol., 4:109–144, 1987

    Article  MATH  MathSciNet  Google Scholar 

  30. H.R. Thieme. Persistence under relaxed point-dissipativity (with an application to an endemic model). SIAM J. Math. Anal., 24:407–435, 1993

    Article  MATH  MathSciNet  Google Scholar 

  31. H.W. Hethcote and P. van den Driessche. An SIS epidemic model with variable population size and a delay. J. Math. Biol., 34:177–194, 1996

    Article  Google Scholar 

  32. H.W. Hethcote. An age-structured model for pertussis transmission. Math. Biosci., 145:89–136, 1997

    Article  MATH  MathSciNet  Google Scholar 

  33. H.W. Hethcote. Oscillations in an endemic model for pertussis. Can. Appl. Math. Q., 6:61–88, 1998

    MATH  MathSciNet  Google Scholar 

  34. H.W. Hethcote and P. van den Driessche. Two SIS epidemiologic models with delays. J. Math. Biol., 40:3–26, 2000

    Article  MATH  MathSciNet  Google Scholar 

  35. F. Brauer. Models for the spread of universally fatal diseases. J. Math. Biol., 28: 451–462, 1990

    Article  MATH  MathSciNet  Google Scholar 

  36. Z. Feng and H.R. Thieme. Endemic models with arbitrarily distributed periods of infection II: fast disease dynamics and permanent recovery. SIAM J. Appl. Math., 61:983–1012, 2000

    Article  MATH  MathSciNet  Google Scholar 

  37. B.T. Grenfell, O.N. Bjornstad, and B.F. Finkenstadt. Dynamics of measles epidemics: scaling noise, determinism, and predictability with the TSIR model. Ecol. Monogr., 72:185–202, 2002

    Article  Google Scholar 

  38. W. London and J.A. Yorke. Recurrent outbreaks of measles, chickenpox and mumps. I. Seasonal variation in contact rates. Am. J. Epidem., 98(6):469–482, 1973

    Google Scholar 

  39. J.A. Yorke, N. Nathanson, G. Pianigiani, and J. Martin. Seasonality and the requirements for perpetuation and eradication of viruses in populations. Am. J. Epidem., 109:103–123, 1979

    Google Scholar 

  40. I.B. Schwartz and H.L. Smith. Infinite subharmonic bifurcation in an SEIR epidemic model. J. Math. Biol., 18:233–253, 1983

    Article  MATH  MathSciNet  Google Scholar 

  41. K. Dietz and D. Schenzle. Mathematical Models for Infectious Disease Statistics, pages 167–204. Springer, Berlin Heidelebrg New York, 1985

    Google Scholar 

  42. H.W. Hethcote and S.A. Levin. Periodicity in epidemic models. In S.A. Levin, T.G. Hallam, and L.J. Gross, editors, Biomathematics, volume 18, pages 193–211. Springer, Berlin Heidelberg New York, 1989

    Google Scholar 

  43. O. Diekmann, H. Metz, and H. Heesterbeek. The legacy of Kermack and McKendrick. In D. Mollison, editor, Epidemic models: their structure and relation to data, pages 95–118. Cambridge University Press, Cambridge, 1995

    Google Scholar 

  44. H.W. Hethcote. The mathematics of infectious diseases. SIAM Rev., 42:599–653, 2000

    Article  MATH  MathSciNet  Google Scholar 

  45. F. Brauer and P. van den Driessche. Some directions for mathematical epidemiology. Fields Inst. Commun., 36:95–112, 2003

    Google Scholar 

  46. D. Bernoulli. Essai d’une nouvelle analyse de la mortalite causee par la pette verole. Mem. Math. Phys. Acad. R. Sci. Paris, pages 1–45, 1766.

    Google Scholar 

  47. K. Dietz and J.A.P. Heesterbeek. Bernoulli was ahead of modern epidemiology. Nature, 408:513–514, 2000

    Article  Google Scholar 

  48. K. Dietz and J.A.P. Heesterbeek. Daniel Bernoulli’s epidemiological model revisited. Math. Biosci., 180:1–21, 2002

    Article  MATH  MathSciNet  Google Scholar 

  49. W. Farr. Progress of epidemics. Second report of the Registrar General of England, pages 91–8, 1840

    Google Scholar 

  50. W.H. Hamer. Epidemic disease in England. Lancet, 1:733–739, 1906

    Google Scholar 

  51. J. Brownlee. Statistical studies in immunity: the theory of an epidemic. Proc. R. Soc. Edn., 26:484–521, 1906

    Google Scholar 

  52. R. Ross. Report on the Prevention of Malaria in Mauritius. London, 1908

    Google Scholar 

  53. R. Ross. The Prevention of Malaria (2nd ed.). Murray, London, 1911

    Google Scholar 

  54. R. Ross and H.P. Hudson. An application of the theories of probably to the study of a priori pathometry, III. Proc. R. Soc. A, 93:225–240, 1917

    Article  Google Scholar 

  55. W.O. Kermack and A.G. McKendrick. Contributions to the mathematical theory of epidemics I. Proc. R. Soc. Lond., 115:700–721, 1927

    Article  Google Scholar 

  56. W.O. Kermack and A.G. McKendrick. Contributions to the mathematical theory of epidemics II. Proc. R. Soc. Lond., 138:55–83, 1932

    Article  MATH  Google Scholar 

  57. W.O. Kermack and A.G. McKendrick. Contributions to the mathematical theory of epidemics III. Proc. R. Soc. Lond., 141:94–112, 1933

    Article  MATH  Google Scholar 

  58. A. Hirsch. Handbook of Geographical and Historical Pathology, Volume 1, translated by Charles Creighton, New Sydenham Soc., London, 1883

    Google Scholar 

  59. W.O. Kermack and A.G. McKendrick. Contributions to the mathematical theory of epidemics I–III. Bull. Math. Biol., 53:33–118, 1991

    Google Scholar 

  60. M.S. Bartlett. The critical community size for measles in the United States. J. R. Statist. Soc., 123:37–44, 1960

    Google Scholar 

  61. M.S. Bartlett. Stochastic Population Models in Ecology and Epidemiology. Methuen, London, 1960

    MATH  Google Scholar 

  62. S.F. Dowell. Seasonal variation in host susceptibility and cycles of certain infectious diseases. Emerg. Infect. Dis., 7:369–374, 2001

    Google Scholar 

  63. P.E.M. Fine and J.A. Clarkson. Measles in England and Wales–I: An analysis of factors underlying seasonal patterns. Int. J. Epidemiol., 11:5–14, 1982

    Article  Google Scholar 

  64. B. Finkenstadt and B.T. Grenfell. Time series modelling of childhood infectious diseases: A dynamical systems approach. J. R. Stat. Soc. C, 49:187–205, 2000

    Article  MathSciNet  Google Scholar 

  65. N.M. Ferguson, D.J. Nokes, and R.M. Anderson. Dynamical complexity in age-structured models of the transmission of measles virus. Math. Biosci., 138:101–130, 1996

    Article  MATH  Google Scholar 

  66. E.B. Wilson and J. Worcester. The law of mass action in epidemiology. Proc. Natl. Acad. Sci., 31:24–34, 1945.

    Article  MathSciNet  Google Scholar 

  67. E.B. Wilson and J. Worcester. The law of mass action in epidemiology II. Proc. Natl. Acad. Sci., 31:109–116, 1945

    Article  MathSciNet  Google Scholar 

  68. H.W. Hethcote, H.W. Stech, and P. van den Driessche. Stability analysis for models of diseases without immunity. J. Math. Biol., 13:185–198, 1981

    MATH  MathSciNet  Google Scholar 

  69. W. Liu, S.A. Levin, and Y. Iwasa. Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models. J. Math. Biol., 23:187–204, 1986

    Article  MATH  MathSciNet  Google Scholar 

  70. W. Liu, H.W. Hethcote, and S.A. Levin. Dynamical behavior of epidemiological models with nonlinear incidence rates. J. Math. Biol., 25:359–380, 1987

    Article  MATH  MathSciNet  Google Scholar 

  71. P. Rohani, M.J. Keeling, and B.T. Grenfell. The interplay between determinism and stochasticity in childhood diseases. Am. Nat., 159:469–481, 2002

    Article  Google Scholar 

  72. M.J. Keeling, P. Rohani, and B.T. Grenfell. Seasonally forced disease dynamics explored as switching between attractors. Physica D, 148:317–335, 2002

    Article  Google Scholar 

  73. R.M. May. Simple mathematical models with very complicated dynamics. Nature, 261:459–467, 1976

    Article  Google Scholar 

  74. W.M. Schaffer and M. Kot. Nearly one dimensional dynamics in an epidemic. J. Theor. Biol., 112:403–427, 1985

    Article  MathSciNet  Google Scholar 

  75. L.F. Olsen and W.M. Schaffer. Chaos versus noisy periodicity: Alternative hypothesis for childhood epidemics. Science, 249:499–504, 1990

    Article  Google Scholar 

  76. M. Kot, D.J. Graser, G.L. Truty, W.M. Schaffer, and L.F. Olsen. Changing criteria for imposing order. Ecol. Model., 43:75–110, 1988

    Article  Google Scholar 

  77. W.M. Schaffer. Order and chaos in ecological systems. Ecology, 66:93–106, 1985

    Article  Google Scholar 

  78. L.F. Olsen, G.L. Truty, and W.M. Schaffer. Oscillations and chaos in epidemics: A nonlinear dyamics study of six childhood diseases in Copenhagen, Denmark. Theor. Pop. Biol., 33:344–370, 1988

    Article  MATH  MathSciNet  Google Scholar 

  79. D.A. Rand and H.B. Wilson. Chaotic stochasticity: a ubiquitous source of unpredictability in epidemics. Proc. R. Soc. Lond. B, 246:179–184, 1991

    Article  Google Scholar 

  80. G. Sugihara, B.T. Grenfell, and R.M. May. Distinguishing error from chaos in ecological time series. Philos. Trans. R. Soc. Lond. B, 330:235–251, 1990

    Article  Google Scholar 

  81. S. Ellner, A.R. Gallant, and J. Theiler. Detecting nonlinearity and chaos in epidemic data. In D. Mollison, editor, Epidemic models: their structure and relation to data, pages 229–247. Cambridge University Press, Cambridge, 1995

    Google Scholar 

  82. B.T. Grenfell, A. Kleczkowski, S.P. Ellner, and B.M. Bolker. Measles as a case study in nonlinear forecasting and chaos. Philos. Trans. R. Soc. Lond. A, 348:515–530, 1994.

    Article  MATH  Google Scholar 

  83. W.M. Schaffer. Can nonlinear dynamics elucidate mechanisms in ecology and epidemiology? IMA J. Math. Appl. Med. Biol., 2:221–252, 1985

    Article  MATH  MathSciNet  Google Scholar 

  84. C. Torrence and G.P. Compo. A practical guide to wavelet analysis. Bull. Am. Meteor. Soc., 79:61–78, 1998

    Article  Google Scholar 

  85. B.T. Grenfell, O.N. Bjornstad, and J. Kappey. Travelling waves and spatial hierarchies in measles epidemics. Nature, 414:716–723, 2001

    Article  Google Scholar 

  86. P. Rohani, D.J.D. Earn, and B.T. Grenfell. Opposite patterns of synchrony in sympatric disease metapopulations. Science, 286:968–971, 1999

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Guelph, Guelph, ON, N1G 2W1, Canada

    Chris T. Bauch

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  1. Chris T. Bauch
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Editors and Affiliations

  1. Department of Mathematics, University of British Columbia, Vancouver, B.C. V6T 1Z2, Canada

    Fred Brauer

  2. Department of Mathematics and Statistics, University of Victoria, 3060 STN CSC, Victoria, B.C. V8W 3R4, Canada

    Pauline van den Driessche

  3. Center for Disease Modeling Department of Mathematics and Statistics, York University, Toronto, Ontario, M3J 1P3, Canada

    Jianhong Wu

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Bauch, C.T. (2008). The Role of Mathematical Models in Explaining Recurrent Outbreaks of Infectious Childhood Diseases. In: Brauer, F., van den Driessche, P., Wu, J. (eds) Mathematical Epidemiology. Lecture Notes in Mathematics, vol 1945. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-78911-6_11

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