From Stochastic Shocks to Macroscopic Tails: The Moyal Distribution as a Unified Framework for Epidemic Dynamics

Traditional epidemiological models often fail to characterize the extreme volatility and heavy-tailed \new{Dragon King events (statistically and mechanistically exceptional outliers in transmission that deviate from standard scale-free distributions)}% observed in real-world outbreaks. We propose a unified model that bridges microscopic stochastic individual-based SIR with macroscopic wave decomposition using the Moyal probability density function. By treating viral transmission as a stochastic collision process, we derive a Moyal-Poisson mixture that describes secondary case distributions. Our model successfully recovers the extreme superspreading events in SARS, MERS, and COVID-19 data that standard Negative Binomial models systematically miss. Furthermore, we apply spectral decomposition to pandemic waves in Germany, \new{illustrating} that the macroscopic Social Friction ($β_w$) is a direct emergent property of microscopic Collision Shocks. This framework provides a useful \new{retrospective} descriptive tool for public health planning, emphasizing the need to manage extreme volatility rather than deterministic averages.