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From damage models to SIR epidemics and cascading failures

Part of:Stochastic processesApplicationsDistribution theory - Probability

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 theory60E05: Distributions62P05: Applications to actuarial sciences and financial mathematics

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 41 , Issue 1 , March 2009, pp. 247 - 269
Copyright
Copyright © Applied Probability Trust 2009 

References

Andersson, H. and Britton, T. (2000). Stochastic Epidemic Models and Their Statistical Analysis (Lecture Notes Statist. 151). Springer, New York.Google Scholar
Ball, F. (1983). A threshold theorem for the Reed–Frost chain-binomial epidemic. J. Appl. Prob. 20, 153–157.CrossRefGoogle Scholar
Ball, F. and Donnelly, P. (1995). Strong approximations for epidemic models. Stoch. Process. Appl. 55, 1–21.CrossRefGoogle Scholar
Ball, F. and Lyne, O. D. (2001). Stochastic multitype SIR epidemics among a population partitioned into households. Adv. Appl. Prob. 33, 99–123.CrossRefGoogle Scholar
Ball, F. and O'Neill, P. (1999). The distribution of general final state random variables for stochastic epidemic models. J. Appl. Prob. 36, 473–491.CrossRefGoogle Scholar
Ball, F., Mollison, D. and Scalia-Tomba, G. (1997). Epidemics with two levels of mixing. Ann. Appl. Prob. 7, 46–89.CrossRefGoogle Scholar
Charnet, R. and Gokhale, D. V. (2004). Statistical inference for damaged Poisson distribution. Commun. Statist. Simul. Comput. 33, 259–269.CrossRefGoogle Scholar
Consul, P. C. and Famoye, F. (2006). Lagrangian Probability Distributions. Birkhäuser, Boston, MA.Google Scholar
Daley, D. J. and Gani, J. (1999). Epidemic Modelling: An Introduction. Cambridge University Press.Google Scholar
Di Bucchcianico, A. (1997). Probabilistic and Analytical Aspects of the Umbral Calculus (CWI Tract 119), Amsterdam.Google Scholar
Dobson, I., Carreras, B. A. and Newman, D. E. (2005). A loading-dependent model of probabilistic cascading failure. Prob. Eng. Inf. Sci. 19, 15–32.CrossRefGoogle Scholar
Gathy, M. (2007). Distributions Lagrangiennes et de Pólya–Eggenberger généralisée. Working paper, Département de Mathématique, ULB, Bruxelles.Google Scholar
Gontcharoff, W. (1937). Détermination des Fonctions Entières par Interpolation. Hermann, Paris.Google Scholar
Johnson, N. L., Kotz, S. and Kemp, A. W. (1992). Univariate Discrete Distributions, 2nd edn. John Wiley, New York.Google Scholar
Lefèvre, C. (2006). On the outcome of a cascading failure model. Prob. Eng. Inf. Sci. 20, 413–427.Google Scholar
Lefèvre, C. (2007). First-crossing and ballot-type results for some nonstationary sequences. Adv. Appl. Prob. 39, 492–509.CrossRefGoogle Scholar
Lefèvre, C. and Picard, P. (1990). A nonstandard family of polynomials and the final size distribution of Reed–Frost epidemic processes. Adv. Appl. Prob. 22, 25–48.CrossRefGoogle Scholar
Lefèvre, C. and Picard, P. (2005). Nonstationarity and randomization in the Reed–Frost epidemic model. J. Appl. Prob. 42, 950–963.CrossRefGoogle Scholar
Lefèvre, C. and Utev, S. (1996). Comparing sums of exchangeable Bernoulli random variables. J. Appl. Prob. 33, 285–310.CrossRefGoogle Scholar
Lefèvre, C. and Utev, S. (1999). Branching approximation for the collective epidemic model. Methodology Comput. Appl. Prob. 1, 211–228.CrossRefGoogle Scholar
Lindley, D. V. and Singpurwalla, N. D. (2002). On exchangeable, causal and cascading failures. Statist. Sci. 2, 209–219.Google Scholar
Picard, P. and Lefèvre, C. (1990). A unified analysis of the final size and severity distribution in collective Reed–Frost epidemic processes. Adv. Appl. Prob. 22, 269–294.CrossRefGoogle Scholar
Picard, P. and Lefèvre, C. (1991). The dimension of Reed-Frost epidemic models with randomized susceptibility levels. Math. Biosci. 107, 225–233.CrossRefGoogle ScholarPubMed
Picard, P. and Lefèvre, C. (1996). First crossing of basic counting processes with lower non-linear boundaries: a unified approach through pseudopolynomials. I. Adv. Appl. Prob. 28, 853–876.CrossRefGoogle Scholar
Picard, P. and Lefèvre, C. (2003). On the first meeting or crossing of two independent trajectories for some counting processes. Stoch. Process. Appl. 104, 217–242.CrossRefGoogle Scholar
Rao, C. R. (1965). On discrete distributions arising out of methods of ascertainment. In Classical and Contagious Discrete Distributions, ed. Patil, G. P., Statistical Publishing Society, Calcutta, pp. 320–332.Google Scholar
Rao, C. R. and Rubin, H. (1964). On a characterization of the Poisson distribution. Sankhyā A 26, 295–298.Google Scholar
Rota, G.-C., Kahaner, D. and Odlyzko, A. (1973). On the foundations of combinatorial theory. VII. Finite operator calculus. J. Math. Anal. Appl. 42, 684–760.Google Scholar
Shanbhag, D. N. (1977). An extension of the Rao–Rubin characterization of the Poisson distribution. J. Appl. Prob. 14, 640–646.CrossRefGoogle Scholar
Takács, L. (1989). Ballots, queues and random graphs. J. Appl. Prob. 26, 103–112.CrossRefGoogle Scholar