In　this　paper　we　investigate　the　persistence　and　extinction　of　a　stochastic　epidemic　model　with　a　varying　population　environment　in　the　long-term　behavior.　Our　model　consists　of　two　stochastic　differential　equations:　one　for　the　susceptible　individuals　in　which　the　transmission　rate　is　disturbed　by　white　noise,　one　for　the　exposed　individuals　in　which　the　same　perturbation　occurs,　and　one　ordinary　differential　equation　in　which　describes　the　infective　individuals　in　a　varying　population　environment.　We　derive　sufficient　conditions　for　the　extinction　and　persistence　of　the　epidemic　model　depending　on　the　constant　contact　rate.　Moreover,　we　carry　out　several　numerical　simulations　to　illustrate　the　main　results　of　this　contribution.　©　2016　Elsevier　B.V.