The Interplay between Determinism and Stochasticity in Childhood Diseases

An important issue in the history of ecology has been the study of the relative importance of deterministic forces and processes noise in shaping the dynamics of ecological populations. We address this question by exploring the temporal dynamics of two childhood infections, measles and whooping cough, in England and Wales. We demonstrate that epidemics of whooping cough are strongly influenced by stochasticity; fully deterministic approaches cannot achieve even a qualitative fit to the observed data. In contrast, measles dynamics are extremely well explained by a deterministic model. These differences are shown to be caused by their contrasting responses to dynamical noise due to different infectious periods.

[1]  R. Pearl,et al.  On the Rate of Growth of the Population of the United States since 1790 and Its Mathematical Representation. , 1920, Proceedings of the National Academy of Sciences of the United States of America.

[2]  A. J. Lotka,et al.  Elements of Physical Biology. , 1925, Nature.

[3]  A. J. Lotka.  Elements of Physical Biology. , 1925, Nature.

[4]  W. O. Kermack,et al.  A contribution to the mathematical theory of epidemics , 1927 .

[5]  A. Nicholson,et al.  The Balance of Animal Populations.—Part I. , 1935 .

[6]  A. Nicholson,et al.  The Self-Adjustment of Populations to Change , 1957 .

[7]  M. Bartlett.  Measles Periodicity and Community Size , 1957 .

[8]  R M May,et al.  Biological Populations with Nonoverlapping Generations: Stable Points, Stable Cycles, and Chaos , 1974, Science.

[9]  K. Dietz,et al.  The Incidence of Infectious Diseases under the Influence of Seasonal Fluctuations , 1976 .

[10]  R. May,et al.  Regulation and Stability of Host-Parasite Population Interactions: I. Regulatory Processes , 1978 .

[11]  M. Hassell.  The dynamics of arthropod predator-prey systems. , 1979, Monographs in population biology.

[12]  P. Fine,et al.  THE RECURRENCE OF WHOOPING COUGH: POSSIBLE IMPLICATIONS FOR ASSESSMENT OF VACCINE EFFICACY , 1982, The Lancet.

[13]  I B Schwartz,et al.  Infinite subharmonic bifurcation in an SEIR epidemic model , 1983, Journal of mathematical biology.

[14]  H. Hethcote,et al.  Measles and rubella in the United States. , 1983, American journal of epidemiology.

[15]  D. Schenzle.  An age-structured model of pre- and post-vaccination measles transmission. , 1984, IMA journal of mathematics applied in medicine and biology.

[16]  M. Kot,et al.  Nearly one dimensional dynamics in an epidemic. , 1985, Journal of theoretical biology.

[17]  P. Fine,et al.  The efficiency of measles and pertussis notification in England and Wales. , 1985, International journal of epidemiology.

[18]  Robert M. May,et al.  Population Biology of Microparasitic Infections , 1986 .

[19]  M. Yamaguti,et al.  Chaos and Fractals , 1987 .

[20]  M. Kliegman.  Nelson Essentials of Pediatrics , 1989 .

[21]  B T Grenfell,et al.  Pertussis in England and Wales: an investigation of transmission dynamics and control by mass vaccination , 1989, Proceedings of the Royal Society of London. B. Biological Sciences.

[22]  L. Olsen,et al.  Chaos versus noisy periodicity: alternative hypotheses for childhood epidemics. , 1990, Science.

[23]  G Sugihara,et al.  Distinguishing error from chaos in ecological time series. , 1990, Philosophical transactions of the Royal Society of London. Series B, Biological sciences.

[24]  H. B. Wilson,et al.  Chaotic stochasticity: a ubiquitous source of unpredictability in epidemics , 1991, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[25]  E. Miller,et al.  The epidemiology of pertussis in England and Wales. , 1992, Communicable disease report. CDR review.

[26]  Bryan T. Grenfell,et al.  Chance and Chaos in Measles Dynamics , 1992 .

[27]  Robert M. May,et al.  Dynamics of metapopulations : habitat destruction and competitive coexistence , 1992 .

[28]  A. J. Hall.  Infectious diseases of humans: R. M. Anderson & R. M. May. Oxford etc.: Oxford University Press, 1991. viii + 757 pp. Price £50. ISBN 0-19-854599-1 , 1992 .

[29]  B. Bolker,et al.  Chaos and biological complexity in measles dynamics , 1993, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[30]  Ralf Engbert,et al.  Chance and chaos in population biology—Models of recurrent epidemics and food chain dynamics , 1994 .

[31]  D. Tilman.  Competition and Biodiversity in Spatially Structured Habitats , 1994 .

[32]  D J Nokes,et al.  Dynamical complexity in age-structured models of the transmission of the measles virus: epidemiological implications at high levels of vaccine uptake. , 1996, Mathematical biosciences.

[33]  B. Grenfell,et al.  (Meta)population dynamics of infectious diseases. , 1997, Trends in ecology & evolution.

[34]  Paul Glendinning,et al.  Stability, instability and chaos , by Paul Glendinning. Pp. 402. £45. 1994. ISBN 0 521 41553 5 (hardback); £17.95 ISBN 0 521 42566 2 (paperback) (Cambridge). , 1997, The Mathematical Gazette.

[35]  A. Hastings,et al.  Stochastic Dynamics and Deterministic Skeletons: Population Behavior of Dungeness Crab , 1997 .

[36]  T. Clutton‐Brock,et al.  Noise and determinism in synchronized sheep dynamics , 1998, Nature.

[37]  Costantino,et al.  Moving toward an unstable equilibrium: saddle nodes in population systems , 1998 .

[38]  A. R. Gallant,et al.  Noise and Nonlinearity in Measles Epidemics: Combining Mechanistic and Statistical Approaches to Population Modeling , 1998, The American Naturalist.

[39]  Nils Chr. Stenseth,et al.  Population dynamics of the Indian meal moth:Demographic stochasticity and delayed regulatory mechanisms. , 1998 .

[40]  D. Earn,et al.  Population dynamic interference among childhood diseases , 1998, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[41]  B Grenfell,et al.  Stochastic dynamics and a power law for measles variability. , 1999, Philosophical transactions of the Royal Society of London. Series B, Biological sciences.

[42]  William W. Murdoch,et al.  Large-amplitude cycles of Daphnia and its algal prey in enriched environments , 1999, Nature.

[43]  C. Zimmer.  Life After Chaos , 1999, Science.

[44]  R. Marino.  Nelson Essentials of Pediatrics , 1999 .

[45]  D. Earn,et al.  Opposite patterns of synchrony in sympatric disease metapopulations. , 1999, Science.

[46]  D. Earn,et al.  A simple model for complex dynamical transitions in epidemics. , 2000, Science.

[47]  Bärbel Finkenstädt,et al.  Time series modelling of childhood diseases: a dynamical systems approach , 2000 .

[48]  Pejman Rohani,et al.  Impact of immunisation on pertussis transmission in England and Wales , 2000, The Lancet.

[49]  M. Begon,et al.  Invasion sequence affects predator–prey dynamics in a multi-species interaction , 2000, Nature.

[50]  D. Earn,et al.  Pertussis transmission in England and Wales , 2000, The Lancet.

[51]  Pejman Rohani,et al.  Seasonnally forced disease dynamics explored as switching between attractors , 2001 .

[52]  B T Grenfell,et al.  Age, sex, density, winter weather, and population crashes in Soay sheep. , 2001, Science.