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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)= k=1 N a k (x k -x ¯ k lnx k ), where x=(x 1 ,x 2 ,...,x N ) D N , N, is the state, x ¯D is the equilibrium and a 1 ,...,a N are some coefficients. To show that the function V(·) fulfils V ˙(x)0 for all xD some graph-theoretical arguments like special properties of unicyclic graphs are used.

MSC:
34C60Qualitative investigation and simulation of models (ODE)
34D23Global stability of ODE
34D20Stability of ODE
92D30Epidemiology
05C90Applications of graph theory