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This paper considers the classical SIR epidemic model driven by amultidimensional L\'evy jump process. We consecrate to develop a mathematicalmethod to obtain the asymptotic properties of the perturbed model. Our methoddiffers from previous approaches by the use of the comparison theorem, mutuallyexclusive possibilities lemma, and some new techniques of the stochasticdifferential systems. In this framework, we derive the threshold which candetermine the existence of a unique ergodic stationary distribution or theextinction of the epidemic. Numerical simulations about different perturbationsare realized to confirm the obtained theoretical results.

Developing new techniques for obtaining the threshold of a stochastic SIR epidemic model with 3-dimensional L\'evy process