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Coupling of Two SIR Epidemic Models with Variable Susceptibilities and Infectivities

Part of: Limit theorems Graph theory

Published online by Cambridge University Press:  14 July 2016

Peter Neal
Peter Neal*
Affiliation:
University of Manchester
*
∗ Postal address: School of Mathematics, University of Manchester, Sackville Street, Manchester M60 1QD, UK. Email address: [email protected]
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Abstract

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The variable generalised stochastic epidemic model, which allows for variability in both the susceptibilities and infectivities of individuals, is analysed. A very different epidemic model which exhibits variable susceptibility and infectivity is the random-graph epidemic model. A suitable coupling of the two epidemic models is derived which enables us to show that, whilst the epidemics are very different in appearance, they have the same asymptotic final size distribution. The coupling provides a novel approach to studying random-graph epidemic models.

Keywords

SIR epidemicrandom graphbranching processcentral limit theoremcoupling

MSC classification

Primary: 92D30: Epidemiology
Secondary: 60F05: Central limit and other weak theorems 05C80: Random graphs
Type
Research Article
Information
Journal of Applied Probability , Volume 44 , Issue 1 , March 2007 , pp. 41 - 57
Copyright
Copyright © Applied Probability Trust 2007 

References

Andersson, H. (1998). Limit theorems for a random-graph epidemic model. Ann. Appl. Prob. 8, 1331–1349.CrossRefGoogle Scholar
Andersson, H. (1999). Epidemic models and social networks. Math. Scientist 24, 128–147.Google Scholar
Ball, F. (1985). Deterministic and stochastic epidemics with several kinds of susceptibles. Adv. Appl. Prob. 17, 1–22.Google Scholar
Ball, F. and Britton, T. (2005). An epidemic model with exposure-dependent severities. J. Appl. Prob. 42, 932–949.Google Scholar
Ball, F. and Clancy, D. (1993). The final size and severity of a generalised stochastic multitype epidemic model. Adv. Appl. Prob. 25, 721–736.CrossRefGoogle Scholar
Ball, F. and Clancy, D. (1995). The final outcome of an epidemic model with several different types of infective in a large population. J. Appl. Prob. 32, 579–590.CrossRefGoogle Scholar
Ball, F. and Lyne, O. (2001). Stochastic multitype SIR epidemics among a population partitioned into households. Adv. Appl. Prob. 33, 99–123.Google Scholar
Ball, F. and Neal, P. (2003). The great circle epidemic model. Stoch. Process. Appl. 107, 233–268.CrossRefGoogle Scholar
Ball, F., Mollison, D. and Scalia-Tomba, G. (1997). Epidemics with two levels of mixing. Ann. Appl. Prob. 7, 46–89.Google Scholar
Becker, N. and Starczak, D. (1998). The effect of random vaccine response on the vaccination coverage required to prevent epidemics. Math. Biosci. 154, 117–135.Google Scholar
Billingsley, P. (1968). Convergence of Probability Measures. John Wiley, New York.Google Scholar
Britton, T., Deijfen, M. and Martin-Löf, A. (2006). Generating simple random graphs with prescribed degree distribution. J. Statist. Phys. 124, 1377–1397.Google Scholar
Holst, L. (2001). Extreme value distributions for random coupon collector and birthday problems. Extremes 4, 129–145.CrossRefGoogle Scholar
Lefèvre, C. and Picard, P. (2005). Nonstationarity and randomization in the Reed–Frost epidemic model. J. Appl. Prob. 42, 950–963.Google Scholar
Lefèvre, C. and Utev, S. (1999). Branching approximation for the collective epidemic model. Methodology Comput. Appl. Prob. 1, 211–228.CrossRefGoogle Scholar
Neal, P. (2003). SIR epidemics on a Bernoulli random graph. J. Appl. Prob. 40, 779–782.Google Scholar
Neal, P. (2006). Multitype randomised Reed–Frost epidemics and epidemics upon random graphs. Ann. Appl. Prob. 16, 1166–1189.Google Scholar
Scalia-Tomba, G. (1985). Asymptotic final size distribution for some chain-binomial processes. Adv. Appl. Prob. 17, 477–495.Google Scholar
Scalia-Tomba, G. (1990). On the asymptotic final size distribution of epidemics in heterogeneous populations. In Stochastic Processes in Epidemic Theory (Lecture Notes Biomath. 86), Springer, New York, pp. 189–196.Google Scholar
Sellke, T. (1983). On the asymptotic distribution of the size of a stochastic epidemic. J. Appl. Prob. 20, 390–394.Google Scholar
Whittle, P. (1955). The outcome of a stochastic epidemic—a note on Bailey's paper. Biometrika 42, 116–122.Google Scholar