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

TL;DR

This work introduces a Moyal-distribution-based framework to unify microscopic transmission heterogeneity and macroscopic epidemic waves. By treating infectiousness as a per-individual collision shock drawn from a Moyal distribution and aggregating these into a CTMC, the authors derive a Moyal-Poisson mixture that captures the heavy tails and Dragon King superspreading events—features often missed by Negative Binomial models. The macroscopic epidemic curve is linked to a Moyal PDF through the Survivor Mean, enabling a continuous, analytically tractable description of multi-wave dynamics via spectral decomposition; this is demonstrated in Germany (2020–2023) with nine variant-driven waves. The approach introduces the Social Friction width $\beta_w$ as a physically interpretable parameter guiding public health strategies toward mitigating extreme volatility in outbreaks.

Abstract

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" ($β$) 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.

Figures (5)

• Figure 1: Stochastic simulations of an SIR model with Moyal-distributed infectiousnessgillespie1977exactgillespie1976general The plot shows the number of infected individuals over time for multiple simulation runs. The transmission rate $\beta$ for each infected individual is drawn from a Moyal distribution, with the mean $\beta$ matched to 0.4. The multiple red curves represent different realizations of the stochastic model, illustrating the variability in epidemic trajectories. The peaked curves show successful epidemics that reach a high number of infected individuals, while the flat line near the bottom represents simulations where the epidemic failed to establish and quickly died out. The x-axis represents time in days, and the y-axis represents the number of infected individuals.
• Figure 2: Divergence of Epidemic Peaks and the "Statistical Gap" (N=10,000, $\sigma$=4.0). This figure illustrates the critical discrepancy between deterministic mean-field predictions and realized stochastic risk in a high-overdispersion regime. The Standard ODE SIR (blue dashed line), calibrated to the global theoretical mean of the Moyal distribution, predicts a peak intensity of 9,054 cases. In contrast, the All Mean (green solid line) appears significantly diluted (Peak $I \approx 5,237$) due to the high frequency of early stochastic extinctions (gray background trajectories) inherent in Moyal-driven dynamics.The Survivor Mean (red solid line), representing the average of realized outbreaks, demonstrates a peak of 7,934 cases. While this specific parameterization results in an overshoot of approximately -1,120 cases relative to the deterministic model, the visualization confirms that successful outbreaks follow an accelerated, "super-exponential" growth phase. The divergence highlights the "Statistical Gap": deterministic models represent a mathematical average that fails to characterize the actual velocity and volatility experienced during a sustained superspreading event.
• Figure 3: The Statistical Gap. The Standard ODE (blue dashed) assumes a smooth average.The Survivor Mean of the Stochastic Moyal model (red solid) shows an accelerated peak and an asymmetric decline, driven by the "momentum" of early superspreading shocks.
• Figure 4: Empirical Validation of Offspring Distributions via Stochastic Moyal-Poisson Mixture. This multi-panel analysis compares the descriptive power of the proposed Moyal Estocástico model (red solid line) against the standard Negative Binomial framework (blue dashed line) across three landmark infectious disease outbreaks: SARS (2003), MERS (2015), and COVID-19 (HK).
• Figure 5: Spectral Decomposition. The Moyal model successfully decomposes the 3-year pandemic into 9 distinct variant-driven waves.