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A graph-theoretic approach to the method of global Lyapunov functions
Author(s):
Hongbin
Guo;
Michael
Y.
Li;
Zhisheng
Shuai
Journal:
Proc. Amer. Math. Soc.
136
(2008),
2793-2802.
MSC (2000):
Primary 34D23, 92D30
Posted:
March 27, 2008
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Abstract:
A class of global Lyapunov functions is revisited and used to resolve a long-standing open problem on the uniqueness and global stability of the endemic equilibrium of a class of multi-group models in mathematical epidemiology. We show how the group structure of the models, as manifested in the derivatives of the Lyapunov function, can be completely described using graph theory.
References:
-
- 1.
- E. Beretta and V. Capasso, Global stability results for a multigroup SIR epidemic model, in: T. G. Hallam, L. J. Gross, and S. A. Levin (Eds.), Mathematical Ecology, World Scientific, Teaneck, NJ, 1988, pp. 317-342. MR 1040340 (91c:92074)
- 2.
- N. P. Bhatia and G. P. Szegö, Dynamical Systems: Stability Theory and Applications, Lecture Notes in Mathematics, Vol. 35, Springer, Berlin, 1967. MR 0219843 (36:2917)
- 3.
- H. I. Freedman and J. W. -H. So, Global stability and persistence of simple food chains, Math. Biosci. 76 (1985) 69-86. MR 809991 (87g:92040)
- 4.
- H. Guo and M. Y. Li, Global dynamics of a staged progression model for infectious diseases, Math. Biosci. Eng. 3 (2006) 513-525. MR 2217225 (2006m:92086)
- 5.
- H. W. Hethcote, An immunization model for a heterogeneous population, Theor. Popu. Biol. 14 (1978) 338-349. MR 520143 (80k:92009)
- 6.
- H. W. Hethcote and H. R. Thieme, Stability of the endemic equilibrium in epidemic models with subpopulations, Math. Biosci. 75 (1985) 205-227. MR 810991 (87c:92046)
- 7.
- W. Huang, K. L. Cooke, and C. Castillo-Chavez, Stability and bifurcation for a multiple-group model for the dynamics of HIV/AIDS transmission, SIAM J. Appl. Math. 52 (1992) 835-854. MR 1163809 (93d:92009)
- 8.
- D. E. Knuth, The Art of Computer Programming, Vol. 1, 3rd ed., Addison-Wesley, Reading, MA, 1997. MR 0286317 (44:3530)
- 9.
- A. Korobeinikov, A Lyapunov function for Leslie-Gower predator-prey models, Appl. Math. Lett. 14 (2001) 697-699. MR 1836072
- 10.
- A. Korobeinikov and P.K. Maini, A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence, Math. Biosci. Eng. 1 (2004) 57-60. MR 2130156 (2005k:92065)
- 11.
- A. Lajmanovich and J. A. York, A deterministic model for gonorrhea in a nonhomogeneous population, Math. Biosci. 28 (1976) 221-236. MR 0403726 (53:7537)
- 12.
- J. P. LaSalle, The Stability of Dynamical Systems, Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, PA, 1976. MR 0481301 (58:1426)
- 13.
- X. Lin and J. W.-H. So, Global stability of the endemic equilibrium and uniform persistence in epidemic models with subpopulations, J. Austral. Math. Soc. Ser. B 34 (1993) 282-295. MR 1197481 (93k:92010)
- 14.
- J. W. Moon, Counting Labelled Trees, Canadian Mathematical Congress, Montreal, 1970. MR 0274333 (43:98)
- 15.
- H. R. Thieme, Local stability in epidemic models for heterogeneous populations, in: V. Capasso, E. Grosso, and S.L. Paveri-Fontana (Eds.), Mathematics in Biology and Medicine, Lecture Notes in Biomathematics, Vol. 57, Springer, Berlin, 1985, pp. 185-189. MR 812889
- 16.
- H. R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton, NJ, 2003. MR 1993355 (2004m:92030)
- 17.
- P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci. 180 (2002) 29-48. MR 1950747 (2003m:92071)
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Additional Information:
Hongbin
Guo
Affiliation:
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, T6G 2G1, Canada
Address at time of publication:
Department of Mathematics and Statistics, York University, Toronto, Ontario, M3J 1P3, Canada
Email:
hguo@math.ualberta.ca
Michael
Y.
Li
Affiliation:
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, T6G 2G1, Canada
Email:
mli@math.ualberta.ca
Zhisheng
Shuai
Affiliation:
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, T6G 2G1, Canada
Email:
zshuai@math.ualberta.ca
DOI:
10.1090/S0002-9939-08-09341-6
PII:
S 0002-9939(08)09341-6
Keywords:
Lyapunov functions,
multi-group epidemic models,
global stability,
graph theory.
Received by editor(s):
November 8, 2006
Posted:
March 27, 2008
Communicated by:
Carmen C. Chicone
Copyright of article:
Copyright
2008,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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