In　this　paper,　the　dynamics　of　a　stochastic　susceptible–infected–removed　model　in　a　population　with　varying　size　is　investigated.　We　firstly　show　that　the　stochastic　epidemic　model　has　a　unique　global　positive　solution　with　any　positive　initial　value.　Then　we　verify　that　random　perturbations　lead　to　extinction　when　some　conditions　are　being　valid.　Moreover,　we　prove　that　the　solution　of　the　stochastic　epidemic　model　is　persistent　in　the　mean　by　building　up　a　suitable　Lyapunov　function　and　using　generalized　Itô＇s　formula.　Further,　the　stochastic　epidemic　model　admits　a　stationary　distribution　around　the　endemic　equilibrium　when　parameters　satisfy　some　sufficient　conditions.　To　end　this　contribution　and　to　check　the　validity　of　the　main　results,　numerical　simulations　are　separately　carried　out　to　illustrate　these　results.　©　2017　Elsevier　B.V.