From damage models to SIR epidemics and cascading failures

Abstract

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.

Classification of Asymptotic Behavior in a Stochastic SIR Model

“…Since epidemic models were first introduced by Kermack and McKendrick in [15,16], the study on mathematical models has been flourished. Much attention has been devoted to analyzing, predicting the spread, and designing controls of infectious diseases in host populations; see [1,2,4,6,8,18,19,15,16,26,27] and the references therein. One of classic epidemic models is the SIR (Susceptible-Infected-Removed) model that is suitable for modeling some diseases with permanent immunity such as rubella, whooping cough, measles, smallpox, etc.…”