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

top
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

top

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

top
  1. R. M. Anderson, R. M. May. Infectious diseases of humans: dynamics and control, Oxford University Press, Oxford, UK, 1991.  
  2. J. Arino, P. van den Driessche. A multi-city epidemic model. Math. Popul. Stud., 10 (2003), 175-193.  Zbl1028.92021
  3. J. Arino, P. van den Driessche. The basic reproduction number in a multi-city compartmental epidemic model. LNCIS, 294 (2003), 135-142.  Zbl1057.92045
  4. F. Brauer. Some simple epidemic models. Math. Biosci. Engin., 3 (2006), 1-15.  Zbl1089.92042
  5. O. Diekmann, J. A. P. Heesterbeek. Mathematical epidemiology of infectious diseases: model building, analysis and interpretation. Wiley, 2000.  Zbl0997.92505
  6. J. K. Hale, S. M. Verduyn Lunel. Introduction to functional differential equations. Spring-Verlag, New York, 1993.  
  7. W. O. Kermack, A. G. McKendrick. A contribution to the mathematical theory of epidemics. Proc. Royal Soc. London, 115 (1927), 700-721.  Zbl53.0517.01
  8. 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.  Zbl1298.92100
  9. J. A. J. Metz, O. Diekmann. The dynamics of physiologically structured populations. Springer-Verlag, New York, 1986.  Zbl0614.92014
  10. K. Mischaikow, H. Smith, H. R. Thieme. Asymptotically autonomous semiflows: chain recurrence and Lyapunov functions. Trans. Amer. Math. Soc., 347 (1995), 1669-1685.  Zbl0829.34037
  11. J. D. Murray. Mathematical biology. 3rd ed., Springer-Verlag, New York, 2002.  Zbl1006.92001
  12. 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.  Zbl1088.92050
  13. 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.  
  14. W. Wang, X.-Q. Zhao. An epidemic model in a patchy environment. Math. Biosci., 190 (2004), 97-112.  Zbl1048.92030
  15. W. Wang, X.-Q. Zhao. An age-structured epidemic model in a patchy environment. SIAM J. Appl. Math., 65 (2005), 1597-1614.  Zbl1072.92045
  16. W. Wang, X.-Q. Zhao. An epidemic model with population dispersal and infection period. SIAM J. Appl. Math., 66 (2006), 1454-1472.  Zbl1094.92055

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.