Generalization of the Kermack-McKendrick SIR Model to a Patchy Environment for a Disease with Latency

J. Li; X. Zou

Mathematical Modelling of Natural Phenomena (2009)

  • Volume: 4, Issue: 2, page 92-118
  • ISSN: 0973-5348

Abstract

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In this paper, with the assumptions that an infectious disease has a fixed latent period in a population and the latent individuals of the population may disperse, we reformulate an SIR model for the population living in two patches (cities, towns, or countries etc.), which is a generalization of the classic Kermack-McKendrick SIR model. The model is given by a system of delay differential equations with a fixed delay accounting for the latency and non-local terms caused by the mobility of the individuals during the latent period. We analytically show that the model preserves some properties that the classic Kermack-McKendrick SIR model possesses: the disease always dies out, leaving a certain portion of the susceptible population untouched (called final sizes). Although we can not determine the two final sizes, we are able to show that the ratio of the final sizes in the two patches is totally determined by the ratio of the dispersion rates of the susceptible individuals between the two patches. We also explore numerically the patterns by which the disease dies out, and find that the new model may have very rich patterns for the disease to die out. In particular, it allows multiple outbreaks of the disease before it goes to extinction, strongly contrasting to the classic Kermack-McKendrick SIR model.

How to cite

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Li, J., and Zou, X.. "Generalization of the Kermack-McKendrick SIR Model to a Patchy Environment for a Disease with Latency." Mathematical Modelling of Natural Phenomena 4.2 (2009): 92-118. <http://eudml.org/doc/222218>.

@article{Li2009,
abstract = { In this paper, with the assumptions that an infectious disease has a fixed latent period in a population and the latent individuals of the population may disperse, we reformulate an SIR model for the population living in two patches (cities, towns, or countries etc.), which is a generalization of the classic Kermack-McKendrick SIR model. The model is given by a system of delay differential equations with a fixed delay accounting for the latency and non-local terms caused by the mobility of the individuals during the latent period. We analytically show that the model preserves some properties that the classic Kermack-McKendrick SIR model possesses: the disease always dies out, leaving a certain portion of the susceptible population untouched (called final sizes). Although we can not determine the two final sizes, we are able to show that the ratio of the final sizes in the two patches is totally determined by the ratio of the dispersion rates of the susceptible individuals between the two patches. We also explore numerically the patterns by which the disease dies out, and find that the new model may have very rich patterns for the disease to die out. In particular, it allows multiple outbreaks of the disease before it goes to extinction, strongly contrasting to the classic Kermack-McKendrick SIR model. },
author = {Li, J., Zou, X.},
journal = {Mathematical Modelling of Natural Phenomena},
keywords = {infectious disease; SIR model; latent period; patch; non-local infection; dispersion; multiple outbreaks; non-local infection},
language = {eng},
month = {3},
number = {2},
pages = {92-118},
publisher = {EDP Sciences},
title = {Generalization of the Kermack-McKendrick SIR Model to a Patchy Environment for a Disease with Latency},
url = {http://eudml.org/doc/222218},
volume = {4},
year = {2009},
}

TY - JOUR
AU - Li, J.
AU - Zou, X.
TI - Generalization of the Kermack-McKendrick SIR Model to a Patchy Environment for a Disease with Latency
JO - Mathematical Modelling of Natural Phenomena
DA - 2009/3//
PB - EDP Sciences
VL - 4
IS - 2
SP - 92
EP - 118
AB - In this paper, with the assumptions that an infectious disease has a fixed latent period in a population and the latent individuals of the population may disperse, we reformulate an SIR model for the population living in two patches (cities, towns, or countries etc.), which is a generalization of the classic Kermack-McKendrick SIR model. The model is given by a system of delay differential equations with a fixed delay accounting for the latency and non-local terms caused by the mobility of the individuals during the latent period. We analytically show that the model preserves some properties that the classic Kermack-McKendrick SIR model possesses: the disease always dies out, leaving a certain portion of the susceptible population untouched (called final sizes). Although we can not determine the two final sizes, we are able to show that the ratio of the final sizes in the two patches is totally determined by the ratio of the dispersion rates of the susceptible individuals between the two patches. We also explore numerically the patterns by which the disease dies out, and find that the new model may have very rich patterns for the disease to die out. In particular, it allows multiple outbreaks of the disease before it goes to extinction, strongly contrasting to the classic Kermack-McKendrick SIR model.
LA - eng
KW - infectious disease; SIR model; latent period; patch; non-local infection; dispersion; multiple outbreaks; non-local infection
UR - http://eudml.org/doc/222218
ER -

References

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