In this paper, a stochastic epidemic model with logistic growth and saturated treatment is formulated to probe the effect of white noise on population. We show that the proposed autonomous stochastic model possess a unique and non-negative solution. We obtain the sufficient conditions for extinction of the infectious disease and persistent in the mean of the stochastic autonomous epidemic model with probability one. That is, if ℛ0˜<1, under some parametric conditions then the infection goes to extinction with probability one and if ℛ0˜>1, under some parametric conditions then the infection persist with probability one. By using Has’minskii’s theory of periodic solutions, we show that the stochastic non-autonomous epidemic model has at least one nontrivial positive θ-periodic solution. Finally, the theoretical results are illustrated by numerical simulations which obtains some additional interesting phenomena. •A stochastic epidemic model with logistic growth and saturated treatment is formulated.•The sufficient conditions for the extinction and persistence are derived.•The existence of periodic solutions of the stochastic non-autonomous system is obtained.•Irregularity of stochastic variation and range of fluctuation depends on the strength of white noise.•Numerical investigations conforms the dynamics of stochastic model.