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An Sir Epidemic Model on a Population with Random Network and Household Structure, and Several Types of Individuals

Part of: Markov processes Graph theory

Published online by Cambridge University Press:  04 January 2016

Frank Ball  and
David Sirl
Frank Ball*
Affiliation:
University of Nottingham
David Sirl*
Affiliation:
University of Nottingham
*
∗ Postal address: School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK. Email address: frank.ball@nottingham.ac.uk
∗∗ Current address: School of Mathematics, Loughborough University, Loughborough, Leicestershire LE11 3TU, UK. Email address: d.sirl@lboro.ac.uk
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Abstract

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We consider a stochastic SIR (susceptible → infective → removed) epidemic model with several types of individuals. Infectious individuals can make infectious contacts on two levels, within their own ‘household’ and with their neighbours in a random graph representing additional social contacts. This random graph is an extension of the well-known configuration model to allow for several types of individuals. We give a strong approximation theorem which leads to a threshold theorem for the epidemic model and a method for calculating the probability of a major outbreak given few initial infectives. A multitype analogue of a theorem of Ball, Sirl and Trapman (2009) heuristically motivates a method for calculating the expected size of such a major outbreak. We also consider vaccination and give some short numerical illustrations of our results.

Keywords

Couplingfinal outcomehouseholdslocal and global contactsmultitype branching processmultitype epidemic processmultitype random graphthreshold theorem

MSC classification

Primary: 92D30: Epidemiology
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 05C80: Random graphs
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 44 , Issue 1 , March 2012 , pp. 63 - 86
Copyright
© Applied Probability Trust 

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