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Zbl 1155.34028
Guo, Hongbin; Li, Michael Y.; Shuai, Zhisheng
A graph-theoretic approach to the method of global Lyapunov functions. (English)
[J] Proc. Am. Math. Soc. 136, No. 8, 2793-2802 (2008). ISSN 0002-9939; ISSN 1088-6826/e

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.
[Stephan Trenn (Ilmenau)]

MSC 2000:: *34C60 Applications of qualitative theory of ODE
34D23 Global stability
34D20 Lyapunov stability of ODE
92D30 Epidemiology
05C90 Appl. of graph theory

Keywords: Lyapunov functions; multi-group epidemic models; global stability

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