## Global stability of multi-group SEIR epidemic models with distributed delays and nonlinear transmission.(English)Zbl 1254.92085

Summary: The dynamics of multi-group SEIR epidemic models with distributed and infinite delay and nonlinear transmission are investigated. We derive the basic reproduction number $$R_{0}$$ and establish that the global dynamics are completely determined by the values of $$R_{0}$$: if $$R_{0}\leq 1$$, then the disease-free equilibrium is globally asymptotically stable; if $$R_{0}>1$$, then there exists a unique endemic equilibrium which is globally asymptotically stable. Our results contain those for single-group SEIR models with distributed and infinite delays. In the proof of global stability of the endemic equilibrium, we exploit a graph-theoretical approach to the method of Lyapunov functionals. The biological significance of the results is also discussed.

### MSC:

 92D30 Epidemiology 37N25 Dynamical systems in biology
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### References:

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