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Global stability of the endemic equilibrium of multigroup SIR epidemic models. (English) Zbl 1148.34039
This paper provides an analysis of the global dynamics of a class of multigroup SIR epidemic models, with varying group sizes, in terms of the so-called basic reproduction number ${R}_{0}$. The incidence susceptible-infectious between groups is formulated as bilinear, giving rise to a constant nonnegative contact matrix model, which is assumed to be irreducible. Some well-known results from the graph theory applied to the study of irreducible nonnegative matrices, allow to construct a suitable Lyapunov function. As a consequence, the following result is established: 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 in the interior of the feasible region.

##### MSC:
 34D23 Global stability of ODE 92D30 Epidemiology