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A graph-theoretic approach to the method of global Lyapunov functions. (English) Zbl 1155.34028

The authors study nonlinear \(n\)-group epidemic models of SEIR type. Under the assumption that the basic reproduction number \(R_0\) is bigger than one and that the transmission matrix \(B\) is irreducible the authors show that there exists a unique endemic equilibrium which is locally stable and globally attractive. For the proof, the authors use a Lyapunov function of the form \(V(x)=\sum_{k=1}^N a_k (x_k-\overline{x}_k \ln x_k)\), where \(x=(x_1,x_2,\ldots,x_N)^\top\in D\subseteq\mathbb{R}^N\), \(N\in\mathbb{N}\), is the state, \(\overline{x}\in D\) is the equilibrium and \(a_1,\ldots,a_N\in\mathbb{R}\) are some coefficients. To show that the function \(V(\cdot)\) fulfils \(\dot{V}(x)\leq 0\) for all \(x\in D\) some graph-theoretical arguments like special properties of unicyclic graphs are used.

MSC:

34C60 Qualitative investigation and simulation of ordinary differential equation models
34D23 Global stability of solutions to ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
92D30 Epidemiology
05C90 Applications of graph theory
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[1] 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. · Zbl 0684.92015
[2] 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. · Zbl 0155.42201
[3] H. I. Freedman and J. W.-H. So, Global stability and persistence of simple food chains, Math. Biosci. 76 (1985), no. 1, 69 – 86. · Zbl 0572.92025
[4] 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. · Zbl 1092.92040
[5] Herbert W. Hethcote, An immunization model for a heterogeneous population, Theoret. Population Biol. 14 (1978), no. 3, 338 – 349. · Zbl 0392.92009
[6] 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. · Zbl 0582.92024
[7] 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. · Zbl 0769.92023
[8] Donald E. Knuth, The art of computer programming. Vol. 1: Fundamental algorithms, Second printing, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont, 1969. · Zbl 0191.18001
[9] A. Korobeinikov, A Lyapunov function for Leslie-Gower predator-prey models, Appl. Math. Lett. 14 (2001), no. 6, 697 – 699. · Zbl 0999.92036
[10] 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. · Zbl 1062.92061
[11] Ana Lajmanovich and James A. Yorke, A deterministic model for gonorrhea in a nonhomogeneous population, Math. Biosci. 28 (1976), no. 3/4, 221 – 236. · Zbl 0344.92016
[12] J. P. LaSalle, The stability of dynamical systems, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1976. With an appendix: ”Limiting equations and stability of nonautonomous ordinary differential equations” by Z. Artstein; Regional Conference Series in Applied Mathematics. · Zbl 0364.93002
[13] 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. · Zbl 0778.92020
[14] J. W. Moon, Counting labelled trees, From lectures delivered to the Twelfth Biennial Seminar of the Canadian Mathematical Congress (Vancouver, vol. 1969, Canadian Mathematical Congress, Montreal, Que., 1970. · Zbl 0214.23204
[15] 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.
[16] Horst R. Thieme, Mathematics in population biology, Princeton Series in Theoretical and Computational Biology, Princeton University Press, Princeton, NJ, 2003. · Zbl 1054.92042
[17] 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. · Zbl 1015.92036
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