2008
DOI: 10.1090/s0002-9939-08-09341-6
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A graph-theoretic approach to the method of global Lyapunov functions

Abstract: 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.

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Cited by 315 publications
(242 citation statements)
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“…Specifically, the matrix-theoretic approach using Perron eigenvectors in [27, Section 2] is applied to prove the global stability of the disease-free equilibrium (Theorem 4.1) while the graph-theoretic approach in [11,12,16,27] is used to establish the global stability of the endemic equilibrium (Theorem 4.2).…”
Section: Global Dynamicsmentioning
confidence: 99%
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“…Specifically, the matrix-theoretic approach using Perron eigenvectors in [27, Section 2] is applied to prove the global stability of the disease-free equilibrium (Theorem 4.1) while the graph-theoretic approach in [11,12,16,27] is used to establish the global stability of the endemic equilibrium (Theorem 4.2).…”
Section: Global Dynamicsmentioning
confidence: 99%
“…The weighed digraph (G, A) is depicted in Figure 2(a), and can be regarded as the disease transmission diagram (Figure 1) evaluated at the endemic equilibrium P * . Following the graph-theoretic approach in [27] (also see [11,12,16]), a Lyapunov functionD for Equation (1) …”
Section: J Imentioning
confidence: 99%
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“…To this end, we apply a Lyapunov function similar to those recently used by Guo et al [12], Korobeinikov and Maini [17] and McCluskey [22]. Such Lyapunov functions take advantages of the properties of the function…”
Section: Global Stability Of E *mentioning
confidence: 99%
“…Recently, a graph theoretic approach was proposed to construct Lyapunov functions for some general coupled systems of ordinary differential equations on networks, and the global stability was explored in [11,12]. We refer to [13,14] for recent applications.…”
Section: Introductionmentioning
confidence: 99%