Susceptibility Sets and the Final Outcome of Collective Reed–Frost Epidemics

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Abstract

This paper is concerned with exact results for the final outcome of stochastic SIR (susceptible \(\rightarrow \) infective \(\rightarrow \) recovered) epidemics among a closed, finite and homogeneously mixing population. The factorial moments of the number of initial susceptibles who ultimately avoid infection by such an epidemic are shown to be intimately related to the concept of a susceptibility set. This connection leads to simple, probabilistically illuminating proofs of exact results concerning the total size and severity of collective Reed–Frost epidemic processes, in terms of Gontcharoff polynomials, first obtained in a series of papers by Claude Lefèvre and Philippe Picard. The proofs extend easily to include general final state random variables defined on SIR epidemics, and also to multitype epidemics.

Keywords

Total size Severity Susceptibility set Symmetric sampling procedure Gontcharoff polynomial General final state random variables 

Mathematic Subject Classification (2010)

92D30 60K99 05C80 

Notes

Acknowledgements

I am grateful to Karen Guy for helpful discussions. This work was supported by the Engineering and Physical Sciences Research Council [grant number GR/L56282], providing partial support.

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© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity of NottinghamNottinghamUK

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