Time to sustained control of the effective reproduction number as a failure-time estimand: a county-level analysis of the first SARS-CoV-2 wave in the United States.

Subnational analyses of emerging pathogen transmission typically model the reproduction number (Rt) as a continuous outcome and estimate average associations with its level. Operationally, however, response planning depends on a different quantity: which jurisdictions bring transmission below the epidemic threshold (Rt<1), and how quickly. We formulate time to control as a failure-time estimand and show that doing so exposes a structural feature that linear regression does not accommodate: a jurisdiction cannot enter the risk set until enough cases accumulate to estimate Rt. Cases accumulate faster where transmission is faster, so entry time depends on the outcome process itself and on factors that drive early spread. We account for this outcome-dependent observability using a mixed-effects Cox proportional hazards model with calendar-time left truncation and state-level Gaussian frailties, applied to 2,956 US counties observed between January 22 and July 15, 2020. The primary event was defined as the first three-day sustained reduction of mean Rt below one. Control was rare (171 counties, 5.8%). Each standard-deviation increase in log population density was associated with a lower rate of control (hazard ratio 0.54; 95% confidence interval 0.44 - 0.67) and higher policy stringency with a faster rate (HR 1.50; CI 1.09 - 2.05). Evaluating jurisdictions by their rate of reaching an operational target - rather than by average transmission level - provides a clearer, decision-relevant metric for prospective allocation decisions and public health interventions. Keywords: SARS-CoV-2, COVID-19, reproduction number, survival analysis, Cox model, county-level, United States, policy stringency, population density.