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Threshold behaviour and final outcome of an epidemic on a random network with household structure

Part of: Markov processes Special processes Graph theory

Published online by Cambridge University Press:  01 July 2016

Frank Ball ,
David Sirl  and
Pieter Trapman
Frank Ball*
Affiliation:
The University of Nottingham
David Sirl*
Affiliation:
The University of Nottingham
Pieter Trapman*
Affiliation:
University Medical Center Utrecht and Vrije Universiteit Amsterdam
*
∗ Postal address: School of Mathematical Sciences, The University of Nottingham, University Park, Nottingham, NG7 2RD, UK.
∗ Postal address: School of Mathematical Sciences, The University of Nottingham, University Park, Nottingham, NG7 2RD, UK.
∗∗∗∗ Postal address: Stochastics Section, Faculty of Sciences, Vrije Universiteit Amsterdam, De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands. Email address: [email protected]
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Abstract

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In this paper we consider a stochastic SIR (susceptible→infective→removed) epidemic model in which individuals may make infectious contacts in two ways, both within ‘households’ (which for ease of exposition are assumed to have equal size) and along the edges of a random graph describing additional social contacts. Heuristically motivated branching process approximations are described, which lead to a threshold parameter for the model and methods for calculating the probability of a major outbreak, given few initial infectives, and the expected proportion of the population who are ultimately infected by such a major outbreak. These approximate results are shown to be exact as the number of households tends to infinity by proving associated limit theorems. Moreover, simulation studies indicate that these asymptotic results provide good approximations for modestly sized finite populations. The extension to unequal-sized households is discussed briefly.

Keywords

Branching processcouplingepidemic processfinal outcomehouseholdslocal and global contactsrandom graphsusceptibility setthreshold theorem

MSC classification

Primary: 92D30: Epidemiology 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 05C80: Random graphs 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 41 , Issue 3 , September 2009 , pp. 765 - 796
Copyright
Copyright © Applied Probability Trust 2009 

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