We　consider　the　long-term　properties　of　a　stochastic　SVIR　epidemic　model　with　saturation　incidence　rates　and　logistic　growth　in　this　paper.　We　firstly　derive　the　fitness　of　a　unique　global　positive　solution.　Then　we　construct　appropriate　Lyapunov　functions　and　obtain　condition　R0s＞1　for　existence　of　stationary　distribution,　and　conditions　for　persistence　in　the　mean.　Moreover,　conditions　including　R0e＜1　for　exponential　extinction　to　the　infected　individuals　are　figured　out.　Finally,　by　employing　Fokker-Planck　equation　and　stochastic　analysis,　we　derive　the　probability　density　function　around　the　quasi-endemic　equilibrium　point　when　critical　value　R0p＞1　is　valid.　Consequently,　some　examples　and　illustrative　simulations　are　carried　out　to　verify　the　main　theoretical　results.　©　2022　The　Franklin　Institute