Fractional epidemics from quantum loops

TL;DR

This paper derives fractional epidemic models from quantum field theory principles, revealing how anomalous spreading behaviors naturally emerge from host-vacuum fluctuations, leading to new insights into super-spreading and epidemic dynamics.

Contribution

It introduces a first-principles derivation of fractional epidemic equations using non-equilibrium quantum field theory, connecting microscopic fluctuations to macroscopic anomalous spreading.

Findings

Fractional epidemic dynamics emerge from quantum field theory modeling.

Anomalous spreading involves Levy flights and temporal avalanches.

Effective reproductive number becomes a spectral dispersion relation.

Abstract

Classical compartmental models of epidemiology rely on well-mixed, local interaction approximations that fail to capture the heavy-tailed burst dynamics and long-range spatial correlations observed in real-world outbreaks. While fractional calculus is frequently employed to model these anomalous behaviors, fractional operators are introduced phenomenologically. In this work, we demonstrate that fractional space-time epidemic dynamics emerge naturally and rigorously from first principles using a non-equilibrium quantum field theory model. By mapping the stochastic contagion process to a gauge-mediated field theory via the Doi-Peliti formalism, we go beyond the static mean-field approximation to compute the full dynamical one-loop vacuum polarization. We prove that integrating out a dynamically fluctuating host vacuum generates anomalous momentum and frequency scaling. Transitioning back to coordinate space, this derives a coupled space-time fractional integro-differential equations, where the non-linear transmission vertex is governed by parabolic Riesz potentials and Riemann-Liouville time derivatives. We show that in the anomalous regime ($\alpha < 2$), local Debye screening is modified, facilitating L\'evy flight super-spreading and temporal avalanches. Consequently, the effective reproductive number ($R_{eff}$) ceases to be a scalar, transforming into a spectral dispersion relation bounded strictly by the ultraviolet spatial cutoff.