Abstract
One of the simplest set of equations for the description of epidemics (the SEIR equations) has been much studied, and produces reasonable approximations to the dynamics of communicable disease. However, it has long been recognized that spatial and social structure are important if we are to understand the long-term persistence and detailed behaviour of disease. We will introduce three pair models which attempt to capture the underlying heterogeneous structure by studying the connections and correlations between individuals. Although modelling the correlations necessarily leads to more complex equations, this pair formulation naturally incorporates the local dynamical behaviour generating more realistic persistence. In common with other studies on childhood diseases we will focus our attention on measles, for which the case returns are particularly well documented and long running.
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Selected References
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- Aron J. L., Schwartz I. B. Seasonality and period-doubling bifurcations in an epidemic model. J Theor Biol. 1984 Oct 21;110(4):665–679. doi: 10.1016/s0022-5193(84)80150-2. [DOI] [PubMed] [Google Scholar]
- Black F. L. Measles endemicity in insular populations: critical community size and its evolutionary implication. J Theor Biol. 1966 Jul;11(2):207–211. doi: 10.1016/0022-5193(66)90161-5. [DOI] [PubMed] [Google Scholar]
- Bolker B. M., Grenfell B. T. Chaos and biological complexity in measles dynamics. Proc Biol Sci. 1993 Jan 22;251(1330):75–81. doi: 10.1098/rspb.1993.0011. [DOI] [PubMed] [Google Scholar]
- Bolker B. Chaos and complexity in measles models: a comparative numerical study. IMA J Math Appl Med Biol. 1993;10(2):83–95. doi: 10.1093/imammb/10.2.83. [DOI] [PubMed] [Google Scholar]
- Fine P. E., Clarkson J. A. Measles in England and Wales--I: An analysis of factors underlying seasonal patterns. Int J Epidemiol. 1982 Mar;11(1):5–14. doi: 10.1093/ije/11.1.5. [DOI] [PubMed] [Google Scholar]
- Johansen A. A simple model of recurrent epidemics. J Theor Biol. 1996 Jan 7;178(1):45–51. doi: 10.1006/jtbi.1996.0005. [DOI] [PubMed] [Google Scholar]
- Keeling M. J., Grenfell B. T. Disease extinction and community size: modeling the persistence of measles. Science. 1997 Jan 3;275(5296):65–67. doi: 10.1126/science.275.5296.65. [DOI] [PubMed] [Google Scholar]
- Olsen L. F., Schaffer W. M. Chaos versus noisy periodicity: alternative hypotheses for childhood epidemics. Science. 1990 Aug 3;249(4968):499–504. doi: 10.1126/science.2382131. [DOI] [PubMed] [Google Scholar]
- Rand D. A., Wilson H. B. Chaotic stochasticity: a ubiquitous source of unpredictability in epidemics. Proc Biol Sci. 1991 Nov 22;246(1316):179–184. doi: 10.1098/rspb.1991.0142. [DOI] [PubMed] [Google Scholar]
- Rhodes C. J., Anderson R. M. A scaling analysis of measles epidemics in a small population. Philos Trans R Soc Lond B Biol Sci. 1996 Dec 29;351(1348):1679–1688. doi: 10.1098/rstb.1996.0150. [DOI] [PubMed] [Google Scholar]
- Rhodes C. J., Anderson R. M. Power laws governing epidemics in isolated populations. Nature. 1996 Jun 13;381(6583):600–602. doi: 10.1038/381600a0. [DOI] [PubMed] [Google Scholar]
- Sato K., Matsuda H., Sasaki A. Pathogen invasion and host extinction in lattice structured populations. J Math Biol. 1994;32(3):251–268. doi: 10.1007/BF00163881. [DOI] [PubMed] [Google Scholar]
- Schenzle D. An age-structured model of pre- and post-vaccination measles transmission. IMA J Math Appl Med Biol. 1984;1(2):169–191. doi: 10.1093/imammb/1.2.169. [DOI] [PubMed] [Google Scholar]
- Schwartz I. B., Smith H. L. Infinite subharmonic bifurcation in an SEIR epidemic model. J Math Biol. 1983;18(3):233–253. doi: 10.1007/BF00276090. [DOI] [PubMed] [Google Scholar]