A graph-theoretic approach to the method of global Lyapunov functions

Guo, Hongbin; Li, Michael; Shuai, Zhisheng

doi:10.1090/S0002-9939-08-09341-6

A graph-theoretic approach to the method of global Lyapunov functions
HTML articles powered by AMS MathViewer

by Hongbin Guo, Michael Y. Li and Zhisheng Shuai PDF

Proc. Amer. Math. Soc. 136 (2008), 2793-2802 Request permission

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

E. Beretta and V. Capasso, Global stability results for a multigroup SIR epidemic model, Mathematical ecology (Trieste, 1986) World Sci. Publ., Teaneck, NJ, 1988, pp. 317–342. MR 1040340
N. P. Bhatia and G. P. Szegő, Dynamical systems: Stability theory and applications, Lecture Notes in Mathematics, No. 35, Springer-Verlag, Berlin-New York, 1967. MR 0219843
H. I. Freedman and J. W.-H. So, Global stability and persistence of simple food chains, Math. Biosci. 76 (1985), no. 1, 69–86. MR 809991, DOI 10.1016/0025-5564(85)90047-1
Hongbin Guo and Michael Y. Li, Global dynamics of a staged progression model for infectious diseases, Math. Biosci. Eng. 3 (2006), no. 3, 513–525. MR 2217225, DOI 10.3934/mbe.2006.3.513
Herbert W. Hethcote, An immunization model for a heterogeneous population, Theoret. Population Biol. 14 (1978), no. 3, 338–349. MR 520143, DOI 10.1016/0040-5809(78)90011-4
Herbert W. Hethcote and Horst R. Thieme, Stability of the endemic equilibrium in epidemic models with subpopulations, Math. Biosci. 75 (1985), no. 2, 205–227. MR 810991, DOI 10.1016/0025-5564(85)90038-0
Wen Zhang Huang, Kenneth L. Cooke, and Carlos Castillo-Chavez, Stability and bifurcation for a multiple-group model for the dynamics of HIV/AIDS transmission, SIAM J. Appl. Math. 52 (1992), no. 3, 835–854. MR 1163809, DOI 10.1137/0152047
Donald E. Knuth, The art of computer programming. Vol. 1: Fundamental algorithms, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. Second printing. MR 0286317
A. Korobeinikov, A Lyapunov function for Leslie-Gower predator-prey models, Appl. Math. Lett. 14 (2001), no. 6, 697–699. MR 1836072, DOI 10.1016/S0893-9659(01)80029-X
Andrei Korobeinikov and Philip K. Maini, A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence, Math. Biosci. Eng. 1 (2004), no. 1, 57–60. MR 2130156, DOI 10.3934/mbe.2004.1.57
Ana Lajmanovich and James A. Yorke, A deterministic model for gonorrhea in a nonhomogeneous population, Math. Biosci. 28 (1976), no. 3-4, 221–236. MR 403726, DOI 10.1016/0025-5564(76)90125-5
J. P. LaSalle, The stability of dynamical systems, Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1976. With an appendix: “Limiting equations and stability of nonautonomous ordinary differential equations” by Z. Artstein. MR 0481301
Xiao Dong Lin and Joseph 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), no. 3, 282–295. MR 1197481, DOI 10.1017/S0334270000008900
J. W. Moon, Counting labelled trees, Canadian Mathematical Monographs, No. 1, Canadian Mathematical Congress, Montreal, Que., 1970. From lectures delivered to the Twelfth Biennial Seminar of the Canadian Mathematical Congress (Vancouver, 1969). MR 0274333
Horst R. Thieme, Local stability in epidemic models for heterogeneous populations, Mathematics in biology and medicine (Bari, 1983) Lecture Notes in Biomath., vol. 57, Springer, Berlin, 1985, pp. 185–189. MR 812889, DOI 10.1007/978-3-642-93287-8_{2}6
Horst R. Thieme, Mathematics in population biology, Princeton Series in Theoretical and Computational Biology, Princeton University Press, Princeton, NJ, 2003. MR 1993355
P. van den Driessche and James Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci. 180 (2002), 29–48. John A. Jacquez memorial volume. MR 1950747, DOI 10.1016/S0025-5564(02)00108-6