We develop a rigorous mathematical framework for measles transmission that integrates the impact of Lévy noise to account for environmental disturbances. First, the model is formulated, and the uniqueness and existence of a positive solution are investigated. A stochastic threshold is derived to ensure sufficient conditions for the persistence and extinction of the disease. It is demonstrated that the solution curves of the system fluctuate in the neighborhood of the disease-free state of the underlying deterministic model when ${\mathbb{R}}_D<1$. We establish sufficient conditions for both extinction and persistence of the disease in the proposed stochastic system. The obtained results demonstrate the ability of the model to characterize the long-term dynamics of measles transmission. Artificial neural networks (ANNs) are employed to approximate the dynamics of the system with Lévy noise. The ANN model, trained on time-series data from stochastic simulations, accurately predicts transitions among the susceptible, exposed, infected, first-dose vaccinated, second-dose vaccinated, and recovered populations. The model's ability to capture stochastic dynamics driven by Lévy noise highlights the potential of ANNs in approximating complex epidemic models. The results validate the theoretical findings and demonstrate that stochastic environmental perturbations can significantly influence the transmission of epidemic diseases. As noise levels increase, a population's ability to sustain the spread of illness is limited, underscoring the critical role of the environment in shaping epidemic dynamics.