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Couplings for locally branching epidemic processes

Part of: Special processes Markov processes

Published online by Cambridge University Press:  30 March 2016

A. D. Barbour
A. D. Barbour*
Affiliation:
Universität Zürich and National University of Singapore, Institut für Mathematik, Universität Zürich, Winterthurertrasse 190, CH-8057 Zürich. Email address: a.d.barbour@math.uzh.ch.
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Abstract

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The asymptotic behaviour of many locally branching epidemic models can, at least to first order, be deduced from the limit theory of two branching processes. The first is Whittle's (1955) branching approximation to the early stages of the epidemic, the phase in which approximately exponential growth takes place. The second is the susceptibility approximation; the backward branching process that approximates the history of the contacts that would lead to an individual becoming infected. The simplest coupling arguments for demonstrating the closeness of these branching process approximations do not keep the processes identical for quite long enough. Thus, arguments showing that the differences are unimportant are also needed. In this paper we show that, for some models, couplings can be constructed that are sufficiently accurate for this extra step to be dispensed with.

Keywords

Couplingepidemic processbranching process approximationdeterministic approximation

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 60J85: Applications of branching processes
Type
Part 3. Biological applications
Information
Journal of Applied Probability , Volume 51 , Issue A: Celebrating 50 Years of The Applied Probability Trust , December 2014 , pp. 43 - 56
Copyright
Copyright © Applied Probability Trust 2014 

References

Aldous, D. J. (2013). When knowing early matters: gossip, percolation and Nash equilibria. In Prokhorov and Contemporary Probability Theory (Proc. Math. Statist. 33), eds. Shiryaev, A. N., Varadhan, S. R. S. and Presman, E. L., Springer, Heidelberg, pp. 3–28.CrossRefGoogle Scholar
Bailey, N. T. J. (1953). The total size of a general stochastic epidemic. Biometrika 40, 177–185.CrossRefGoogle Scholar
Ball, F. (1983). The threshold behaviour of epidemic models. J. Appl. Prob. 20, 227–241.CrossRefGoogle Scholar
Barbour, A. D. (1975). The duration of the closed stochastic epidemic. Biometrika 62, 477–482.CrossRefGoogle Scholar
Barbour, A. D. (2010). Coupling a branching process to an infinite dimensional epidemic process. ESAIM Prob. Statist. 14, 53–64.CrossRefGoogle Scholar
Barbour, A. D., and Reinert, G. (2013a). Asymptotic behaviour of gossip processes and small-world networks. Adv. Appl. Prob. 45, 981–1010.CrossRefGoogle Scholar
Barbour, A. D., and Reinert, G. (2013b). Approximating the epidemic curve. Electron. J. Prob. 18, No. 54, 30 pp.CrossRefGoogle Scholar
Barbour, A. D., and Utev, S. (2004). Approximating the Reed-Frost epidemic process. Stoch. Process. Appl. 113, 173–197.CrossRefGoogle Scholar
Barbour, A. D., Holst, L., and Janson, S. (1992). Poisson approximation. Oxford University Press.CrossRefGoogle Scholar
Bartlett, M. S. (1949). Some evolutionary stochastic processes. J. R. Statist. Soc. B 11, 211–229.Google Scholar
Bhamidi, S., van der Hofstad, R., and Hooghiemstra, G. (2012). Universality for first passage percolation on sparse random graphs. Preprint. Available at http://arxiv.org/abs/1210.6839v1.Google Scholar
Chatterjee, S., and Durrett, R. (2011). Asymptotic behavior of Aldous' gossip process. Ann. Appl. Prob. 21, 2447–2482.CrossRefGoogle Scholar
Greenwood, M. (1931). On the statistical measure of infectiousness. J. Hygiene 31, 336–351.CrossRefGoogle ScholarPubMed
Kendall, D. G. (1956). Deterministic and stochastic epidemics in closed populations. In Proc. 3rd Berkeley Symp. Math. Statist. Prob., Vol. IV, University of California Press, Berkeley, pp. 149–165.Google Scholar
Kermack, W. O., and McKendrick, A. G. (1927). Contributions to the mathematical theory of epidemics. Proc. R. Soc. London A 115, 700–721.Google Scholar
McKendrick, A. G. (1926). Applications of mathematics to medical problems. Proc. Edinburgh Math. Soc. 44, 98–130.CrossRefGoogle Scholar
Sellke, T. (1983). On the asymptotic distribution of the size of a stochastic epidemic. J. Appl. Prob. 20, 390–394.CrossRefGoogle Scholar
Whittle, P. (1955). The outcome of a stochastic epidemic—a note on Bailey's paper. Biometrika 42, 116–122.Google Scholar