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A deterministic epidemic model taking account of repeated contacts between the same individuals

Published online by Cambridge University Press:  14 July 2016

O. Diekmann ,
M. C. M. de Jong  and
J. A. J. Metz
O. Diekmann*
Affiliation:
CWI, EEW
M. C. M. de Jong*
Affiliation:
ID-DLO
J. A. J. Metz*
Affiliation:
EEW, International Institute for Applied Systems Analysis
*
∗Postal address: Mathematical Institue, Utrecht University, P.O. Box 80.010, 3508TA Utrecht, The Netherlands. E-mail address: [email protected]
∗∗Postal address: ID-DLO, PO Box 65, 8200 AB Lelystad.
∗∗∗Postal address: EEW, section Theoretical Biology, University of Leiden, PO Box 9516, 2300 RA Leiden. The Netherlands.
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Abstract

We introduce a certain population contact structure and derive, in three different ways, the final size equation for a quite general superimposed epidemic process. The contact structure is characterized by the following two properties: (i) each individual contacts exactly k other individuals; (ii) these k acquaintances are a random sample of the (infinite) population.

Keywords

Contact structurebasic reproduction ratiofinal size

MSC classification

Primary: 92D30: Epidemiology
Type
Research Papers
Information
Journal of Applied Probability , Volume 35 , Issue 2 , June 1998 , pp. 448 - 462
Copyright
Copyright © Applied Probability Trust 1998 

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References

Andersson, H. (1997). Epidemics in a population with social structures. Math. Biosc. 140, 79–84.CrossRefGoogle Scholar
Barbour, A., and Mollison, D. (1990). Epidemics and random graphs. In Stochastic Processes in Epidemic Theory, Lect. Notes in Biomath, eds. Gabriel, J.-P., Lefèvre, C. and Picard, P. 86, 86–89.CrossRefGoogle Scholar
Ball, F., Mollison, D., and Scalia-Tomba, G. (1997). Epidemics with two levels of mixing. Ann. Appl. Prob. 7, 46–89.CrossRefGoogle Scholar
Becker, N.G., and Dietz, K. (1995). The effect of the household distribution on transmission and control of highly infectious diseases. Math. Biosc. 127, 207–219.CrossRefGoogle ScholarPubMed
Bender, E.A., and Canfield, E.R. (1978). The asymptotic number of labelled graphs with given degree sequences. J. Combinat. Theory 24, 296–307.CrossRefGoogle Scholar
Blanchard, Ph., Bolz, G.F. and Krüger, T. (1990). Modelling AIDS-epidemics or any venereal disease on random graphs. In Stochastic Processes in Epidemic Theory, Lect. Notes in Biomath, eds. Gabriel, J.-P., Lefèvre, C. and Picard, P. 86, 104–117.CrossRefGoogle Scholar
Bollobás, B. (1985). Random Graphs. Academic Press, London.Google Scholar
De Jong, M.C. M., Diekmann, O., and Metz, J.A. J. In preparation.Google Scholar
Diekmann, O., Dietz, K., and Heesterbeek, J.A. P. (1991). The basic reproduction ratio for sexually transmitted diseases, part 1: theoretical considerations. Math. Biosc. 107, 325–339.CrossRefGoogle Scholar
Dietz, K., and Hadeler, K.P. (1988). Epidemiological models for sexually transmitted diseases. J. Math. Biol. 26, 1–25.CrossRefGoogle ScholarPubMed
Durrett, R. (1995). Spatial epidemic models. In Epidemic Models: Their Structure and Relation to Data, ed. Mollison, D. Cambridge University Press. pp. 187–201.Google Scholar
Kermack, W.O., and McKendrick, A.G. (1927). Contributions to the mathematical theory of epidemics, Part 1. In Proc. R. Soc. Edinb. A 115, 700–721. (Reprinted in (1991). Bull. Math. Biol. 53, 33–55.)Google Scholar
Ludwig, D. (1975). Final size distributions for epidemics. Math. Biosc. 23, 33–46.CrossRefGoogle Scholar
Martin-Löf, A. (1986). Symmetric sampling procedures, general epidemic processes and their threshold limit theorems. J. Appl. Prob. 23, 265–282.CrossRefGoogle Scholar
Metz, J.A. J. (1978). The epidemic in a closed population with all susceptibles equally vulnerable; some results for large susceptible populations and small initial infections. Acta Biotheoretica 27, 75–123.CrossRefGoogle Scholar
Molloy, M., and Reed, B. (1995). A critical point for random graphs with a given degree sequence. Random Struct. Alg. 6, 161–179.CrossRefGoogle Scholar
Rand, D.A. (1997). Correlation equations and pair approximations for spatial ecologies. In Theoretical Ecology 2, ed. MacGlade, J. Blackwell.Google Scholar
Van Druten, J.A. M., de Boo, Th., Reintjes, A.G. M., Jager, J.C., Heisterkamp, S.H., Coutinho, R.A., Bos, J.M., and Ruitenberg, E.J. (1988). Reconstruction and prediction of spread of HIV infection in populations of homosexual men. In Statistical Analysis and Mathematical Modelling of AIDS, eds. Jager, J.C. and Ruitenberg, E.J. Oxford University Press, pp. 52–76.Google Scholar