The SIR epidemic model from a PDE point of view

Fábio Augusto da Costa Carvalho Chalub; Max O. Souza

doi:10.1016/j.mcm.2010.05.036

The SIR epidemic model from a PDE point of view

Fábio Augusto da Costa Carvalho Chalub, Max O. Souza

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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 language	English
Pages (from-to)	1568-1574
Journal	Mathematical And Computer Modelling
Volume	53
Issue number	7-8
DOIs	https://doi.org/10.1016/j.mcm.2010.05.036
Publication status	Published - Apr 2011

Keywords

Conservation laws
Epidemiology
Markov chains
Thermodynamical limit

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@article{d6427a7960e34b3a878a76ce5d3de90a,

title = "The SIR epidemic model from a PDE point of view",

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.",

keywords = "laws, endemicity, Markov, chains, Conservation, limit, Thermodynamical, Epidemiology, mathematical-theory, Conservation laws , Epidemiology , Markov chains , Thermodynamical limit",

author = "Chalub, {F{\'a}bio Augusto da Costa Carvalho} and Souza, {Max O.}",

note = "Sem PDF conforme Despacho. FACCC is partially supported by FCT/Portugal, grants PTDC/MAT/68615/2006 and PTDC/MAT/66426/2006, PTDC/FIS/7093/2006 and Financiamento Base 2009 ISFL-1-297. MOS is partially supported by CNPQ grant 309616/2009-3 and FAPERJ grant 110.174/2009. Both FACC and MOS are partially supported by the bilateral agreement Brazil-Portugal (CAPES-FCT).",

year = "2011",

month = apr,

doi = "10.1016/j.mcm.2010.05.036",

language = "English",

volume = "53",

pages = "1568--1574",

journal = "Mathematical And Computer Modelling",

issn = "0895-7177",

publisher = "Elsevier Science B.V., Amsterdam.",

number = "7-8",

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T1 - The SIR epidemic model from a PDE point of view

AU - Chalub, Fábio Augusto da Costa Carvalho

AU - Souza, Max O.

N1 - Sem PDF conforme Despacho. FACCC is partially supported by FCT/Portugal, grants PTDC/MAT/68615/2006 and PTDC/MAT/66426/2006, PTDC/FIS/7093/2006 and Financiamento Base 2009 ISFL-1-297. MOS is partially supported by CNPQ grant 309616/2009-3 and FAPERJ grant 110.174/2009. Both FACC and MOS are partially supported by the bilateral agreement Brazil-Portugal (CAPES-FCT).

PY - 2011/4

Y1 - 2011/4

N2 - 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.

AB - 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.

KW - laws

KW - endemicity

KW - Markov

KW - chains

KW - Conservation

KW - limit

KW - Thermodynamical

KW - Epidemiology

KW - mathematical-theory

KW - Conservation laws

KW - Epidemiology

KW - Markov chains

KW - Thermodynamical limit

U2 - 10.1016/j.mcm.2010.05.036

DO - 10.1016/j.mcm.2010.05.036

M3 - Article

VL - 53

SP - 1568

EP - 1574

JO - Mathematical And Computer Modelling

JF - Mathematical And Computer Modelling

SN - 0895-7177

IS - 7-8

ER -