How to Measure the Controllability of an Infectious Disease?

Quantifying how difficult it is to control an emerging infectious disease is crucial to public health decision-making, providing valuable evidence on if targeted interventions, e.g., quarantine and isolation, can contain spread or when population wide controls, e.g., lockdowns, are warranted. The disease reproduction number $R$ or growth rate $r$ are universally assumed to measure controllability because $R=1$ and $r=0$ define when infections stop growing and hence the state of critical stability. Outbreaks with larger $R$ or $r$ are therefore interpreted as less controllable and requiring more stringent interventions. We prove this common interpretation is impractical and incomplete. We identify a positive feedback loop among infections intrinsically underlying disease transmission and evaluate controllability from how interventions disrupt this loop. The epidemic gain and delay margins, which, respectively, define how much we can scale infections (this scaling is known as gain) or delay interventions on this loop before stability is lost, provide rigorous measures of controllability. Outbreaks with smaller margins necessitate more control effort. Using these margins, we quantify how presymptomatic spread, surveillance limitations, variant dynamics, and superspreading shape controllability and demonstrate that $R$ and $r$ measure controllability only when interventions do not alter timings between the infections and are implemented without delay. Our margins are easily computed, interpreted, and reflect complex relationships among interventions, their implementation, and epidemiological dynamics.