Asymptotic behavior of the final number of susceptible individuals in generalized epidemic processes
type: Item
instance of: scholarly article
title: Asymptotic behavior of the final number of susceptible individuals in generalized epidemic processes
zbMATH Open document ID: 0897.92032
DOI: 10.1007/BF02104666
author: S. A. Utev
author: Claude Lefèvre
published in: Siberian Mathematical Journal
publication date: 1998-10-06T00:00:00Z
review text: Many epidemic models describe the spread of infectious diseases by the so-called Susceptible-Infective-Removed scheme. In the literature, the standard model used to describe an epidemic is the so-called generalized epidemic process defined as follows: Consider a population of \(n\) susceptible individuals and let an epidemic start by introducing \(m\) infected individuals in the system. We suppose that each pair of individuals contact at the points of a homogeneous Poisson process of rate \(\beta\) and that the contacts are independent in different pairs. Every susceptible, if contacted with an infective, is assumed immediately infectious. Each infective \(i\), initial or subsequent, remains infectious for a time of random duration \(D_i\). The random times \(D_i\) are assumed independent and distributed identically with some random variable \(D\). After the period \(D_i\) the infective \(i\) dies or immunizes and becomes inessential for the further infection process; it is regarded as removed from the population. It is clear that with probability 1 the infection process of the epidemic terminates at some finite time when there are no infectives in the population. In the present article, we are interested in the final number of the susceptibles that survive the disease. Henceforth, we suppose that the initial number of susceptibles increases. In other words, we consider a sequence of generalized epidemic models with parameters \((n,m_n,\beta_n,D)\), \(n\to \infty \). Observe that the random variable \(D\) is assumed independent of \(n\), which is rather natural in practice. Let \(S_n (\infty)\), with law \({\mathcal L} [S_n (\infty)]\), be the final number of susceptibles. In our recent article Ann. Probab. 23, No. 3, 1139-1162 (1995; Zbl 0833.60023), we studied the Poisson approximation of the distribution \({\mathcal L} [S_n (\infty)]\) as \(n\to \infty\). Our purpose here is to supplement that study by examining the weak convergence of \(S_n (\infty)\) to an arbi…
Mathematics Subject Classification ID: 92D30
Mathematics Subject Classification ID: 60F05
zbMATH DE Number: 1023927
zbMATH Keywords: homogeneous Poisson process
zbMATH Keywords: generalized epidemic models
zbMATH Keywords: Poisson approximation
MaRDI profile type: Publication
cites work: Q4101645
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full work available at URL: https://doi.org/10.1007/bf02104666
OpenAlex ID: W2004781843
description: scientific article; zbMATH DE number 1023927