In this paper, the dynamical behavior of a delayed SIQR stochastic epidemic model with Lévy noise is presented and studied. First, we prove the existence and uniqueness of positive solution. Then, we establish the threshold $R_0^l$ as a sufficient condition for the extinction and persistence in mean of the disease. Finally, some numerical simulations are presented to support our theoretical results and we infer that the white and Lévy noises affect the transmission dynamics of the system.