Epidemic "momentum" and a conservation law for infectious disease dynamics

TL;DR

This paper introduces the concept of epidemic momentum, a conserved quantity in infectious disease dynamics, which helps explain the similarity of epidemic curves and separates transmissibility from population immunity.

Contribution

It proposes a unifying framework based on epidemic momentum that reveals a conservation law, enabling better inference of transmissibility and immunity from epidemic data.

Findings

Epidemic trajectories follow contours of a conserved quantity.

Epidemic momentum disentangles transmissibility from immunity.

Reveals the true final size of outbreaks.

Abstract

Infectious disease outbreaks have precipitated a profusion of mathematical models. Epidemic curves predicted by these models are typically qualitatively similar, despite distinct model assumptions, but there is no theoretical explanation for this similarity in terms of any recognised common structure. In addition, fits of epidemic models to time series conflate pathogen transmissibility with pre-existing population immunity, so only a single composite parameter can be inferred. Here, we introduce a unifying concept of "epidemic momentum" -- prevalence weighted by potential to infect -- which is more informative than prevalence, yet analytically tractable. Epidemic momentum reveals a common underlying geometry in which outbreak trajectories always follow contours of a conserved quantity. This previously unrecognised conservation law constrains how epidemics can unfold, enabling us to disentangle transmissibility from prior immunity and to infer each separately from the same time series. We illustrate the significance of these insights with a novel reappraisal of the transmissibility of influenza during the 1918 pandemic. Beyond resolving an apparent identifiability problem, epidemic momentum also exposes the true final size of an outbreak and a universal phase-plane description that links generic renewal models to the classical SIR system. A broader concept of "population momentum" has the potential to illuminate seemingly intractable nonlinear dynamical processes in many other areas of science.