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The final outcome and temporal solution of a carrier-borne epidemic model

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

Frank Ball  and
Damian Clancy
Frank Ball*
Affiliation:
University of Nottingham
Damian Clancy*
Affiliation:
University of Newcastle upon Tyne
*
∗Postal address: Department of Mathematics, University of Nottingham, University Park, Nottingham NG7 2RD, UK.
∗∗Postal address: Department of Mathematics and Statistics, University of Newcastle, Newcastle upon Tyne NE1 7RU, UK.
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Abstract

We consider a stochastic model for the spread of a carrier-borne epidemic amongst a closed homogeneously mixing population, in which a proportion 1 − π of infected susceptibles are directly removed and play no part in spreading the infection. The remaining proportion π become carriers, with an infectious period that follows an arbitrary but specified distribution. We give a construction of the epidemic process which directly exploits its probabilistic structure and use it to derive the exact joint distribution of the final size and severity of the carrier-borne epidemic, distinguishing between removed carriers and directly removed individuals. We express these results in terms of Gontcharoff polynomials. When the infectious period follows an exponential distribution, our model reduces to that of Downton (1968), for which we use our construction to derive an explicit expression for the time-dependent state probabilities.

Keywords

CARRIER-BORNE EPIDEMICSSIZE OF EPIDEMICAREA UNDER TRAJECTORY OF INFECTIVESGONTCHAROFF POLYNOMIALSTIME-DEPENDENT SOLUTION

MSC classification

Primary: 92D30: Epidemiology
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces
Type
Research Papers
Information
Journal of Applied Probability , Volume 32 , Issue 2 , June 1995 , pp. 304 - 315
Copyright
Copyright © Applied Probability Trust 1995 

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Footnotes

Work carried out in part while Damian Clancy was supported by an SERC research studentship at the University of Nottingham.

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