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The SIS Great Circle Epidemic Model

Part of: Stochastic processes Local and nonlocal bifurcation theory

Published online by Cambridge University Press:  14 July 2016

Peter Neal
Peter Neal*
Affiliation:
University of Manchester
*
∗Postal address: School of Mathematics, University of Manchester, Alan Turing Building, Oxford Road, Manchester M13 9PL, UK. Email address: p.neal-2@manchester.ac.uk
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Abstract

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We consider a stochastic SIS model for the spread of an epidemic amongst a population of n individuals that are equally spaced upon the circumference of a circle. Whilst infectious, an individual, i say, makes both local and global infectious contacts at the points of homogeneous Poisson point processes. Global contacts are made uniformly at random with members of the entire population, whilst local contacts are made according to a contact distribution centred upon the infective. Individuals at the end of their infectious period return to the susceptible state and can be reinfected. The emphasis of the paper is on asymptotic results as the population size n → ∞. Therefore, a contact process with global infection is introduced representing the limiting behaviour as n → ∞ of the circle epidemics. A branching process approximation for the early stages of the epidemic is derived and the endemic equilibrium of a major outbreak is obtained. Furthermore, assuming exponential infectious periods, the probability of a major epidemic outbreak and the proportion of the population infectious in the endemic equilibrium are shown to satisfy the same equation which characterises the epidemic process.

Keywords

SIS epidemiccontact processgreat circle modelendemic equilibriumbranching process

MSC classification

Secondary: 92D30: Epidemiology 60G15: Gaussian processes 37G35: Attractors and their bifurcations
Type
Research Article
Information
Journal of Applied Probability , Volume 45 , Issue 2 , June 2008 , pp. 513 - 530
Copyright
Copyright © Applied Probability Trust 2008 

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