Vaccination and SARS-CoV-2 variants: how much containment is still needed? A quantitative assessment.

The paper specifies and calibrates a combined SIDARTHE‑V + data‑driven death model and reports quantitative findings (e.g., “at least 44%, 50%, 57% of lives saved” and death ranges 39k–196k, 35k–174k, 31k–146k; and that Close–Open saves no fewer than 23k lives vs Open–Close), all grounded in the stated assumptions, timeframe (Feb 2021–Jan 2022), R0 profiles, and Italian calibration; these are explicitly supported in the text and Methods (w(i) kernel, CFR0=0.0272, age‑prioritized vaccination, and Rt=R0 S(t)) . By contrast, the candidate solution’s core Lemma 1 (C_vacc(t) ≤ 1−V(t)) is false as stated: removing the highest‑risk V(t) quantile reduces the remaining-population mean CFR below the original mean, but not by a factor of (1−V). The rearrangement step conflates a bound on removed mass with a bound on the remaining average. The subsequent “CFR‑only” reductions (23%, 32%, 45%) rest on this faulty inequality. The ordering argument for Open–Close vs Close–Open treats S(t) as exogenous and thus fails to account for S(t)’s endogenous dependence on the entire R0(t) history. Other parts (incidence monotonicity under vaccination, Rt=R0 S herd‑threshold timings) align qualitatively with the paper but do not reproduce its calibrated numbers without the paper’s n(t) and initialization.