•Classify the asymptotic behavior SIR models with noise perturbing to linear terms.•Construct a threshold value based on coefficients.•The negativity of threshold implies the extinction of the model at exponential rate.•If it is positive, the solution converges to unique stationary measure in total variation.•We can use this technique to classify models with noise perturbing to non linear terms. In this paper, we classify the asymptotic behavior for a class of stochastic SIR epidemic models represented by stochastic differential systems where the Brownian motions and Lévy jumps perturb to the linear terms of each equation. We construct a threshold value between permanence and extinction and develop the ergodicity of the underlying system. It is shown that the transition probabilities converge in total variation norm to the invariant measure. Our results can be considered as a significant contribution in studying the long term behavior of stochastic differential models because there are no restrictions imposed to the parameters of models. Techniques used in proving results of this paper are new and suitable to deal with other stochastic models in biology where the noises may perturb to nonlinear terms of equations or with delay equations.