Gómez-Corral, A and López‐García, M (2018) Perturbation analysis in finite LD‐QBD processes and applications to epidemic models. Numerical Linear Algebra with Applications, 25 (5). ISSN 1070-5325
Abstract
In this paper, our interest is in the perturbation analysis of level‐dependent quasi‐birth‐and‐death (LD‐QBD) processes, which constitute a wide class of structured Markov chains. An LD‐QBD process has the special feature that its space of states can be structured by levels (groups of states), so that a tridiagonal‐by‐blocks structure is obtained for its infinitesimal generator. For these processes, a number of algorithmic procedures exist in the literature in order to compute several performance measures while exploiting the underlying matrix structure; among others, these measures are related to first‐passage times to a certain level L(0) and hitting probabilities at this level, the maximum level visited by the process before reaching states of level L(0), and the stationary distribution. For the case of a finite number of states, our aim here is to develop analogous algorithms to the ones analyzing these measures, for their perturbation analysis. This approach uses matrix calculus and exploits the specific structure of the infinitesimal generator, which allows us to obtain additional information during the perturbation analysis of the LD‐QBD process by dealing with specific matrices carrying probabilistic insights of the dynamics of the process. We illustrate the approach by means of applying multitype versions of the susceptible‐infective (SI) and susceptible‐infective‐susceptible (SIS) epidemic models to the spread of antibiotic‐sensitive and antibiotic‐resistant bacterial strains in a hospital ward.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2018 The Authors. Numerical Linear Algebra With Applications Published by John Wiley & Sons Ltd. This is an open access article under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits unrestricted use, distribution and reproduction in any medium, provided the original work is properly cited. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/ |
Keywords: | epidemic model; hitting time; matrix calculus; perturbation analysis; QBD process |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Applied Mathematics (Leeds) |
Funding Information: | Funder Grant number MRC MR/N014855/1 |
Depositing User: | Symplectic Publications |
Date Deposited: | 10 Jan 2018 12:22 |
Last Modified: | 28 Sep 2018 14:54 |
Status: | Published |
Publisher: | Wiley |
Identification Number: | 10.1002/nla.2160 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:126035 |
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