The SIR epidemic model from a PDE point of view

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Abstract

We present a derivation of the classical Susceptible-Infected-Removed (SIR) and Susceptible-Infected-Removed-Susceptible (SIRS) models through a mean-field approximation from a discrete version of SIR(S). We then obtain a hyperbolic forward Kolmogorov equation, and show that its projected characteristics recover the standard SIR(S) model. Moreover, for the SIRS model, we show that the long time limit of the SIRS model will be a Dirac measure supported on the corresponding isolated equilibria. For the SIR model, we show that the long time limit is a Radon measure supported in a segment of nonisolated equilibria. (C) 2010 Elsevier Ltd. All rights reserved.
Original languageEnglish
Pages (from-to)1568-1574
JournalMathematical And Computer Modelling
Volume53
Issue number7-8
DOIs
Publication statusPublished - Apr 2011

Keywords

  • Conservation laws
  • Epidemiology
  • Markov chains
  • Thermodynamical limit

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