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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 ${R}_{0}\le 1$, 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