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A graph-theoretic method for the basic reproduction number in continuous time epidemiological models

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Abstract

In epidemiological models of infectious diseases the basic reproduction number \({\mathcal{R}_0}\) is used as a threshold parameter to determine the threshold between disease extinction and outbreak. A graph-theoretic form of Gaussian elimination using digraph reduction is derived and an algorithm given for calculating the basic reproduction number in continuous time epidemiological models. Examples illustrate how this method can be applied to compartmental models of infectious diseases modelled by a system of ordinary differential equations. We also show with these examples how lower bounds for \({\mathcal{R}_0}\) can be obtained from the digraphs in the reduction process.

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Author notes

  1. Tomás de-Camino-Beck

    Present address: Department of Entomology, Penn State University, ASI Bldg, State College, PA, 16802, USA

Authors and Affiliations

  1. Department of Mathematical and Statistical Sciences, Centre for Mathematical Biology, University of Alberta, Edmonton, AB, T6G 2G1, Canada

    Tomás de-Camino-Beck & Mark A. Lewis

  2. Department of Mathematics and Statistics, University of Victoria, Victoria, BC, V8W 3R4, Canada

    P. van den Driessche

Authors
  1. Tomás de-Camino-Beck
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  2. Mark A. Lewis
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  3. P. van den Driessche
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Correspondence to Tomás de-Camino-Beck.

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de-Camino-Beck, T., Lewis, M.A. & van den Driessche, P. A graph-theoretic method for the basic reproduction number in continuous time epidemiological models. J. Math. Biol. 59, 503–516 (2009). https://doi.org/10.1007/s00285-008-0240-9

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  • DOI: https://doi.org/10.1007/s00285-008-0240-9

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