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

Published online by Cambridge University Press:  26 March 2009

J. Li  and
X. Zou
J. Li
Affiliation:
Department of Applied Mathematics University of Western Ontario London, Ontario, Canada N6A 5B7
X. Zou*
Affiliation:
Department of Applied Mathematics University of Western Ontario London, Ontario, Canada N6A 5B7
*
[email protected]
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Abstract

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

Keywords

infectious diseaseSIR modellatent periodpatchnon-localinfectiondispersionmultiple outbreaks
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 4 , Issue 2: Delay equations in biology , 2009 , pp. 92 - 118
Copyright
© EDP Sciences, 2009

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