We　investigate　the　global　asymptotic　stabilities　of　disease-free　equilibrium　and　endemic　equilibrium　of　the　deterministic　susceptible-exposed-infected-recovered　epidemic　model　(short　for　SEIR　model).　The　basic　reproduction　number　R-0,　depends　on　constant　contact　rate　beta　and　natural　death　rate　d　and　other　parameters　as　well,　indicates　the　critical　value　of　stability,　and　completely　determines　the　dynamical　behavior　of　the　deterministic　model.　After　taking　the　perturbations　of　the　environments　into　account,　the　corresponding　stochastic　SEIR　model　with　generalized　nonlinear　incidence　is　discussed　in　existence　and　uniqueness,　the　extinction　in　the　mean,　and　the　existence　of　the　unique　stationary　distribution　as　well.　As　a　consequence,　we　carry　out　several　numerical　simulations　to　support　the　main　theoretical　results　of　this　paper.　(C)　2019　International　Association　for　Mathematics　and　Computers　in　Simulation　(IMACS).　Published　by　Elsevier　B.Y.　All　rights　reserved.