Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-27T09:44:23.478Z Has data issue: false hasContentIssue false
  • English
  • Français

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
*
xzou@uwo.ca
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

R. M. Anderson, R. M. May. Infectious diseases of humans: dynamics and control, Oxford University Press, Oxford, UK, 1991.
J. Arino, P. van den Driessche. A multi-city epidemic model. Math. Popul. Stud., 10 (2003), 175-193.
J. Arino, P. van den Driessche. The basic reproduction number in a multi-city compartmental epidemic model. LNCIS, 294 (2003), 135-142.
F. Brauer. Some simple epidemic models. Math. Biosci. Engin., 3 (2006), 1-15.
O. Diekmann, J. A. P. Heesterbeek. Mathematical epidemiology of infectious diseases: model building, analysis and interpretation. Wiley, 2000.
J. K. Hale, S. M. Verduyn Lunel. Introduction to functional differential equations. Spring-Verlag, New York, 1993.
W. O. Kermack, A. G. McKendrick. A contribution to the mathematical theory of epidemics. Proc. Royal Soc. London, 115 (1927), 700-721.
Y.-H. Hsieh, P. van den Driessche, L. Wang. Impact of travel between patches for spatial spread of disease. Bull. Math. Biol., 69 (2007), 1355-1375.
J. A. J. Metz, O. Diekmann. The dynamics of physiologically structured populations. Springer-Verlag, New York, 1986.
K. Mischaikow, H. Smith, H. R. Thieme. Asymptotically autonomous semiflows: chain recurrence and Lyapunov functions. Trans. Amer. Math. Soc., 347 (1995), 1669-1685.
J. D. Murray. Mathematical biology. 3rd ed., Springer-Verlag, New York, 2002.
M. Salmani, P. van den Driessche. A model for disease transmission in a patchy environment. Disc. Cont. Dynam. Syst. Ser. B, 6 (2006), 185-202.
H. R. Thieme, C. Castillo-Chavez. Asymptotically autonomous epidemic models, in Mathematical Population Dynamics: Analysis of Heterogeneity, Vol. 1: Theory of Epidemics (O. Arino, D. Axelrod, M. Kimmel, M. Langlais eds.), pp. 33-50, Wuerz, 1995.
W. Wang, X.-Q. Zhao. An epidemic model in a patchy environment. Math. Biosci., 190 (2004), 97-112.
W. Wang, X.-Q. Zhao. An age-structured epidemic model in a patchy environment. SIAM J. Appl. Math., 65 (2005), 1597-1614.
W. Wang, X.-Q. Zhao. An epidemic model with population dispersal and infection period. SIAM J. Appl. Math., 66 (2006), 1454-1472.