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 "Dragon King" events observed in real-world outbreaks. We propose a unified framework that bridges microscopic agent-based simulations 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, demonstrating that the macroscopic "Social Friction" ($\beta$) is a direct emergent property of microscopic "Collision Shocks". This framework provides a useful descriptive tool for public health planning, emphasizing the need to manage extreme volatility rather than deterministic averages.