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From damage models to SIR epidemics and cascading failures

Part of: Distribution theory - Probability Stochastic processes Applications

Published online by Cambridge University Press:  01 July 2016

Maude Gathy  and
Claude Lefèvre
Maude Gathy*
Affiliation:
Université Libre de Bruxelles
Claude Lefèvre*
Affiliation:
Université Libre de Bruxelles
*
∗ Postal address: Université Libre de Bruxelles, Département de Mathématique, Campus de la Plaine CP 210, B-1050 Bruxelles, Belgium.
∗ Postal address: Université Libre de Bruxelles, Département de Mathématique, Campus de la Plaine CP 210, B-1050 Bruxelles, Belgium.
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Abstract

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This paper is concerned with a nonstationary Markovian chain of cascading damage that constitutes an iterated version of a classical damage model. The main problem under study is to determine the exact distribution of the total outcome of this process when the cascade of damages finally stops. Two different applications are discussed, namely the final size for a wide class of SIR (susceptible → infective → removed) epidemic models and the total number of failures for a system of components in reliability. The starting point of our analysis is the recent work of Lefèvre (2007) on a first-crossing problem for the cumulated partial sums of independent parametric distributions, possibly nonstationary but stable by convolution. A key mathematical tool is provided by a nonstandard family of remarkable polynomials, called the generalised Abel–Gontcharoff polynomials. Somewhat surprisingly, the approach followed will allow us to relax some model assumptions usually made in epidemic theory and reliability. To close, approximation by a branching process is also investigated to a certain extent.

Keywords

First-crossing problemstability by convolutiondamage modelgeneralised Abel–Gontcharoff polynomialepidemic processcascading failurebinomialMarkov–Pólya and hypergeometric distributionstotal outcome distributionbranching approximation

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory 60E05: Distributions 62P05: Applications to actuarial sciences and financial mathematics
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 41 , Issue 1 , March 2009 , pp. 247 - 269
Copyright
Copyright © Applied Probability Trust 2009 

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