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.


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
Full Text: DOI


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