State-dependent non-identifiability of the reproduction number under adaptive behavior: an empirical characterization from COVID-19 mobility

The basic reproduction number R0 confounds pathogen biology with adaptive human contact behavior. Earlier epidemiological--economic theory predicted a forward-looking behavioral contact response but could not test it in the absence of appropriate behavioral data. Using directly measured mobility as an observable proxy for contact, we (i) estimate the behavioral response function directly from data; (ii) show that the biology/behavior decomposition and hence the behavioral correction to R0 is not identified from an epidemic trajectory, the apparent constant-contact R0 being one endpoint of an observational-equivalence class that fits the factual curve identically yet diverges under counterfactual; and (iii) characterize that divergence ("what R0 deletes") as state-dependent, unimodal in counterfactual severity and vanishing when behavior saturates. We then show that, across US jurisdictions, the correction is empirically bounded because risk-responsiveness and behavioral non-saturation are confounded (r=-0.57, n=51): where behavior could compensate, it was already maximal, and where it was not maximal it did not respond. What R0 deletes is thus real and structurally characterizable yet empirically modest here, for reasons the framework itself supplies.