1 Centre de recherche du CHU Sainte-Justine, Département de Mathématiques et de Statistique, Université de Montréal, Montreal, Canada.
2 Department of Mathematics and Statistics, Concordia University, Montreal, Canada.

We propose a unified stochastic SIR model driven by Lévy noise. The model is structural enough to allow for time-dependency, nonlinearity, discontinuity, demography, and environmental disturbances. We present concise results on the existence and uniqueness of positive global solutions and investigate the extinction and persistence of the novel model. Examples and simulations are provided to illustrate the main results.