Long-run bifurcation analysis aims to describe the asymptotic behavior of a dynamical system. One of the main objectives of mathematical epidemiology is to determine the acute threshold between an infection’s persistence and its elimination. In this study, we use a more comprehensive SVIR epidemic model with large jumps to tackle this and related challenging problems in epidemiology. The huge discontinuities arising from the complexity of the problem are modelled by four independent, tempered, $\alpha$-stable quadratic Lévy processes. A new analytical method is used and for the proposed stochastic model, the critical value ${\mathfrak{R}}_0^{🟉}$ is calculated. For strictly positive value of ${\mathfrak{R}}_0^{🟉}$, the stationary and ergodic properties of the perturbed model are verified (continuation scenario). However, for a strictly negative value of ${\mathfrak{R}}_0^{🟉}$, the model predicts that the infection will vanish exponentially (disappearance scenario). The current study incorporates a large number of earlier works and provides a novel analytical method that can successfully handle numerous stochastic models. This innovative approach can successfully handle a variety of stochastic models in a wide range of applications. For the tempered $\alpha$-stable processes, the Rosinski (2007) algorithm with a specific Lévy measure is implemented as a numerical application. It is concluded that both noise intensities and parameter $\alpha$ have a great influence on the dynamical transition of the model as well as on the shape of its associated probability density function.