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Fractional epidemics from quantum loops

Jose Jesus Bernal-Alvarado, David Delepine

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 ($α< 2$), local Debye screening is modified, facilitating Lévy 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.

Fractional epidemics from quantum loops

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 (), local Debye screening is modified, facilitating Lévy flight super-spreading and temporal avalanches. Consequently, the effective reproductive number () ceases to be a scalar, transforming into a spectral dispersion relation bounded strictly by the ultraviolet spatial cutoff.

Paper Structure

This paper contains 16 sections, 39 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Dynamical Vacuum Polarization (Beyond the Static Limit)
  3. Emergence of Fractional Scaling and the Dressed Propagator
  4. The Fractional Integro-Differential SIR Equations
  5. Relation to standard definition for fractional derivatives
  6. The Temporal Limit: Recovering the Riemann-Liouville Derivative
  7. The Spatial Limit: Recovering the Riesz Fractional Laplacian
  8. Computation of the Effective Reproductive Number ($R_{eff}(k,\omega)$)
  9. Static limit ($\omega \to 0$): Condition: $D_{red} k^2 \gg |\omega|$
  10. The High-Frequency Limit ($|\omega| \gg D_{red} k^2$)
  11. Epidemiological Consequences
  12. The Fallacy of Local Herd Immunity
  13. Microscopic Constraints on Macroscopic Spread
  14. Warning Signs of Avalanches
  15. Mobility effects
  16. ...and 1 more sections

Figures (4)

  • Figure 1: One-loop Feynman diagram for the dynamical vacuum polarization $\Pi(k,\omega)$ (pathogen self-energy). The external pathogen field $\varphi$, carrying momentum $k$ and frequency $\omega$, fluctuates into a virtual pair composed of a Susceptible host $S$ and an Infected host $I$. The loop represents the integral over all possible internal momenta $q$ and frequencies $\Omega$ of the fluctuating host vacuum. The left vertex corresponds to the transmission absorption event ($\beta_0$), while the right vertex corresponds to the subsequent emission ($g$).
  • Figure 2: Tree-level Feynman diagram representing the fundamental transmission vertex. A susceptible host field $S$ with momentum $p$ absorbs a quantum of the pathogen gauge field $\varphi$ carrying momentum $k$, transitioning into the infected state $I$ with conserved momentum $p+k$. The bare coupling constant for this local interaction is the transmission rate $\beta_0$.
  • Figure 3: Tree-level Feynman diagram of the pathogen emission vertex. The infected host population $I$ acts as the dynamical source current for the mediating gauge field, shedding the pathogen $\varphi$ into the environment. The strength of this shedding mechanism is governed by the bare emission coupling $g$.
  • Figure 4: Tree-level Feynman diagram of the pathogen emission vertex. The infected host population $I$ acts as the dynamical source current for the mediating gauge field, shedding the pathogen $\varphi$ into the environment. The strength of this shedding mechanism is governed by the bare emission coupling $g$.