The　dynamical　properties　of　a　stochastic　susceptible-infected　epidemic　model　with　Logistic　growth　are　investigated　in　this　paper.　We　show　that　the　stochastic　model　admits　a　nonnegative　solution　by　using　the　Lyapunov　function　method.　We　then　obtain　that　the　infected　individuals　are　persistent　under　some　simple　conditions.　As　a　consequence,　a　simple　sufficient　condition　that　guarantees　the　extinction　of　the　infected　individuals　is　presented　with　a　couple　of　illustrative　examples.