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Stochastic and deterministic analysis of SIS household epidemics

Part of: Stochastic processes Local and nonlocal bifurcation theory

Published online by Cambridge University Press:  08 September 2016

Peter Neal
Peter Neal*
Affiliation:
University of Manchester
*
∗ Postal address: School of Mathematics, University of Manchester, Sackville Street, Manchester M60 1QD, UK. Email address: p.neal-2@manchester.ac.uk
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Abstract

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We analyse SIS epidemics among populations partitioned into households. The analysis considers both the stochastic and deterministic models and, unlike in previous analyses, we consider general infectious period distributions. For the deterministic model, we prove the existence of an endemic equilibrium for the epidemic if and only if the threshold parameter, R*, is greater than 1. Furthermore, by utilising Markov chains we show that the total number of infectives converges to the endemic equilibrium as t → ∞. For the stochastic model, we prove a law of large numbers result for the convergence, to the deterministic limit, of the mean number of infectives per household. This is followed by the derivation of a Gaussian limit process for the fluctuations of the stochastic model.

Keywords

Stochastic modeldeterministic modelSIS epidemichousehold modelendemic equilibriumGaussian process

MSC classification

Primary: 92D30: Epidemiology
Secondary: 60G15: Gaussian processes 37G35: Attractors and their bifurcations
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 38 , Issue 4 , December 2006 , pp. 943 - 968
Copyright
Copyright © Applied Probability Trust 2006 

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