An equilibrium of an autonomous differential system is said to be globally stable with respect to an open set

$D$ if it is asymptotically stable and its basin of attraction contains

$D$. The global stability problem asks to find additional conditions under which the only equilibrium in a simply connected domain is globally stable provided that it is locally stable. The paper gives such conditions using higher dimensional generalizations of the criteria of Bendixson and Dulac of planar systems and a local version of the

${C}^{1}$ closing lemma of Pugh. An epidemiological model is studied as an application.