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

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.

💡 Research Summary

The authors introduce a unifying concept called “epidemic momentum,” defined as the prevalence weighted by each infected individual’s remaining potential to transmit infection. Mathematically, momentum Y(t) can be expressed either as a convolution of the incidence curve ι(t) with the reduced reproduction number Rα (Y(t)=∫₀^∞ι(t−α)Rα/R₀ dα) or equivalently as Y(t)=ι(t)−∫₀^∞ι(t−α)g(α)dα, where g(α) is the generation‑interval distribution. This definition works for the most general renewal‑equation framework and collapses to the ordinary prevalence for the classic SIR and SEIR models, which assume constant infectiousness over an exponentially distributed infectious period.

By pairing the susceptible fraction X(t) with epidemic momentum Y(t), the authors derive a simple differential relationship dY/dX = –1 + (1/R₀)/X that is independent of the specific functional form of the force of infection F(t). Integrating yields a universal phase‑plane trajectory:  Y(X) = Y₀ + (X₀ – X) – (1/R₀) ln(X₀/X). All epidemic models, regardless of complexity, trace out the same family of curves in the (X, Y) plane; the only model‑specific effect is a re‑parameterisation of time. The authors formalise this by defining a conserved quantity  C(X,Y) = Y + (1/R₀) V(X R₀) with V(u)=u−1−ln u, which remains constant along any trajectory and equals the peak momentum Ŷ. This constitutes a previously unrecognised conservation law for infectious‑disease dynamics, analogous to a Hamiltonian or charge‑momentum invariant in physics.

The conserved quantity enables direct calculation of two epidemiologically crucial quantities without external assumptions: (i) the level of prior immunity, z₋ = 1 – X₋, where X₋ is the left‑hand intersection of the trajectory with the X‑axis; and (ii) the final epidemic size, z₊ = X₋ – X₊, where X₊ is the right‑hand intersection. Thus, given only the basic reproduction number R₀, one can infer both the pre‑epidemic immune fraction and the total proportion infected, resolving the identifiability problem that traditionally forces model fitting to conflate these two effects.

Furthermore, the authors show that the exponential growth and decay rates of incidence, force of infection, and momentum are identical and satisfy  1/R₀ X₋ = L