A profound understanding of influenza transmission mechanisms is crucial for developing effective control strategies. This study first establishes a deterministic SAIV influenza model incorporating vaccination and asymptomatic carriers. The basic reproduction number $R_0$ is derived using the next-generation matrix method, followed by an analysis of equilibrium existence and stability. The existence of a transcritical bifurcation in the deterministic model is rigorously demonstrated through bifurcation theory. Subsequently, a stochastic SAIV model incorporating both white noise and Lévy jumps is formulated based on the deterministic framework. By employing Lyapunov function theory, the existence, uniqueness, and non-negative boundedness of the global positive solution are proven, ensuring biological plausibility. Furthermore, threshold conditions for disease extinction and persistence are analytically derived, providing a theoretical delineation of influenza fade-out or endemic transmission. Finally, model parameters are estimated using empirical influenza data, validating the accuracy of the deterministic model and the reliability of stochastic dynamical predictions. The results indicate that limited medical resources facilitate sustained transmission and reduce eradication probability, while environmental stochasticity further emphasizes the critical role of optimized resource allocation in influenza containment.