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

Mathematical Modelling of Natural Phenomena (2009)

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

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topLi, 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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- W. Wang, X.-Q. Zhao. An epidemic model with population dispersal and infection period. SIAM J. Appl. Math., 66 (2006), 1454-1472. Zbl1094.92055

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