Delay distributions limit the identifiability of rapid variations in epidemics

Public-health surveillance systems rely on downstream indicators to infer latent infection incidence, but delays and observation noise provide only an indirect and temporally distorted view of the underlying epidemic process. Reconstructing upstream epidemic trajectories from these observations is therefore an ill-posed inverse problem, in which different reconstruction assumptions may produce different trajectories that remain consistent with the observed data. Here, we develop a general spectral framework that quantifies the statistical distinguishability of candidate upstream trajectories under delayed and noisy observations. We show that epidemiological delay distributions impose a frequency-dependent limit on the information that surveillance observations retain about upstream epidemic dynamics. This limitation propagates to epidemiological inference, making some quantities substantially more sensitive to reconstruction assumptions than others and rendering distinct event-impact profiles difficult to distinguish from downstream observations.