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.

Modelling and control of cholera on networks with a common water source

“…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).…”

“…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) …”

“…Specifically, if R 0 ≤ 1, then the disease-free equilibrium is globally asymptotically stable and cholera dies out from all patches; while if R 0 > 1, then there exists a unique endemic equilibrium that is globally asymptotically stable and cholera persists at an endemic level in all patches. The proofs of these global stability results utilize a new matrix-theoretic approach [27, Section 2] to the construction of Lyapunov functions (for the disease-free equilibrium) and a graph-theoretic approach (for the endemic equilibrium) recently developed in a series of papers [11,12,16,27]. For the second question, the star network structure allows the derivation of explicit formulas for target/type reproduction numbers [14,23,25] that can be used to measure the disease control and intervention strategies needed to eradicate cholera from all patches.…”

Modelling the impact of vaccination on infectious diseases dynamics

“…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…”

Global Stability for a Three-Species Food Chain Model in a Patchy Environment

“…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.…”