Susceptibility Sets and the Final Outcome of Collective Reed–Frost Epidemics
- 21 Downloads
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.
KeywordsTotal size Severity Susceptibility set Symmetric sampling procedure Gontcharoff polynomial General final state random variables
Mathematic Subject Classification (2010)92D30 60K99 05C80
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.
- Barbour AD, Mollison D (1990) Epidemics and random graphs. In: Gabriel J-P, Lefèvre C, Picard P (eds) Stochastic processes in epidemic theory. lecture notes in biomathematics, vol 86. Springer, Heidelberg, pp 86–89Google Scholar
- Lefèvre C (1990) Stochastic epidemic models for SIR infectious diseases: a brief survey of the recent general theory. In: Gabriel J-P, Lefèvre C, Picard P (eds) Stochastic processes in epidemic theory. Lecture notes in biomathematics, vol 86. Springer, Heidelberg, pp 1–12Google Scholar
- Lefèvre C, Picard P (1995) Collective epidemic processes: a general modelling approach to the final outcome of SIR infectious diseases. In: Mollison D (ed) Epidemic models: their structure and relation to data. Cambridge University Press, pp 53–70Google Scholar
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.