On the relation of the COVID-19 reproduction number to the explosive timescales: the case of Italy

What the paper proves analytically for SIRD is sound but one-sided: Eq. (6) derives the two nonzero eigenvalues λ1,2 = (X ± √(X^2 − 4Y))/2 with X = βS − βI − (γ+µ) and Y = β I (γ+µ), and Eq. (7) shows Re λ1,2 > 0 iff X > 0; from S − I < 1 this indeed yields the forward implication “existence of an explosive timescale ⇒ Rt > 1” (their Eq. (8)) . The paper then asserts “their absence implies Rt < 1,” but that reverse direction is not established by Eq. (8) and is false in general: from β(S − I) ≤ γ+µ one only gets β/(γ+µ) ≤ 1/(S − I) ≥ 1, which does not force Rt < 1. The paper’s broader claim that the gap between the two explosive timescales shrinks as Rt approaches 1 is supported numerically (Figs. 3–5 and surrounding text) but not derived analytically in SIRD . By contrast, the model’s solution reproduces the Jacobian/eigenvalue structure and correctly proves: (i) if Rt ≤ 1 then X ≤ 0 for all t, hence no explosive timescale anywhere (the correct contrapositive of the paper’s analytical result), and (ii) near the disease-free regime (I → 0, S → 1), the timescale gap obeys |τexp,f − τexp,s| ≍ |Rt − 1|/((γ+µ) Rt I), so coalescence along sequences with I → 0 forces Rt → 1. This provides the missing analytic underpinning (with explicit early-epidemic assumptions) that the paper only observes empirically; it also correctly notes that away from the DFE, coalescence (Δ = 0) can occur without Rt ≈ 1. The only caveat in the model’s derivation is that the closed-form gap formula |1/λ+ − 1/λ−| = √Δ/Y presumes real positive eigenvalues (Δ ≥ 0); this holds in the regime used (near DFE and Rt > 1 with small I), but should be stated explicitly. The NGM expressions used for Rt, including Rt = β/(γ+µ) for SIRD computed at the DFE with S(0)=1, match the paper’s appendix derivations .