Numerical recipes for investigating endemic equilibria of age-structured SIR epidemics

Dimitri Breda; Stefano Maset; Rossana Vermiglio

doi:10.3934/dcds.2012.32.2675

Article Contents

2012, Volume 32, Issue 8: 2675-2699. Doi: 10.3934/dcds.2012.32.2675

This issue Previous Article Phase models and oscillators with time delayed coupling Next Article Dynamics of a delay differential equation with multiple state-dependent delays

Numerical recipes for investigating endemic equilibria of age-structured SIR epidemics

1.
Department of Mathematics and Computer Science, University of Udine, via delle Scienze 206, I33100 Udine
2.
Department of Mathematics and Computer Science, University of Trieste, via Valerio 12, I34127 Trieste

Received: April 30, 2011

Revised: June 30, 2011

Published: July 2012

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Abstract

The subject of this paper is the analysis of the equibria of a SIR type epidemic model, which is taken as a case study among the wide family of dynamical systems of infinite dimension. For this class of systems both the determination of the stationary solutions and the analysis of their local asymptotic stability are often unattainable theoretically, thus requiring the application of existing numerical tools and/or the development of new ones. Therefore, rather than devoting our attention to the SIR model's features, its biological and physical interpretation or its theoretical mathematical analysis, the main purpose here is to discuss how to study its equilibria numerically, especially as far as their stability is concerned. To this end, we briefly analyze the construction and solution of the system of nonlinear algebraic equations leading to the stationary solutions, and then concentrate on two numerical recipes for approximating the stability determining values known as the characteristic roots. An algorithm for the purpose is given in full detail. Two applications are presented and discussed in order to show the kind of results that can be obtained with these tools.

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Mathematics Subject Classification: Primary: 37M20, 65L07, 34L16; Secondary: 92D30.

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