  This paper considers the classical SIR epidemic model driven by a
multidimensional L\'evy jump process. We consecrate to develop a mathematical
method to obtain the asymptotic properties of the perturbed model. Our method
differs from previous approaches by the use of the comparison theorem, mutually
exclusive possibilities lemma, and some new techniques of the stochastic
differential systems. In this framework, we derive the threshold which can
determine the existence of a unique ergodic stationary distribution or the
extinction of the epidemic. Numerical simulations about different perturbations
are realized to confirm the obtained theoretical results.
