Abstract: Based on Lyapunov  method and stochastic differential equation theory,  we proved the global existence and uniqueness of positive solution for a class of  stochastic SIQR epidemic model with nonmonotonic incidence rate driven by Lévy noise,  investigated the  asymptotic behavior of the solutions for  the  model near  the disease-free equilibrium point and the endemic equilibrium point of corresponding deterministic model, repectively, and analyzed  the influence of random noise on the dynamic behavior of the model.

Key words: Lévy noise, nonmonotonic incidence rate, stochastic SIQR epidemic model, existence and uniqueness, asymptotic property