Abstract Within the context of SEIR models, we consider a lockdown that is both imposed and lifted at an early stage of an epidemic. We show that, in these models, although such a lockdown may delay deaths, it eventually does not avert a significant number of fatalities. Therefore, in these models, the efficacy of a lockdown cannot be gauged by simply comparing figures for the deaths at the end of the lockdown with the projected figure for deaths by the same date without the lockdown. We provide a simple but robust heuristic argument to explain why this conclusion should generalize to more elaborate compartmental models. We qualitatively discuss some conditions, beyond the scope of simple models, under which a lockdown might increase or decrease the epidemic's final toll in the real world. We conclude that SEIR models and their generalizations provide no reliable quantitative evidence that the Indian lockdown has averted any deaths from the COVID-19 pandemic.