Where to cut to delay a pandemic with minimum disruption? Mathematical analysis based on the SIS model

The paper’s Theorem 3.1 asserts x(t) ≤ x̂(t) ≤ x̃(t) for SIS with identical initial conditions (equation (3.9)) . The first inequality x ≤ x̂ is correct and follows from the y = -log(1-x) transformation, a concave/convex tangent bound yielding the affine ŷ-dynamics (equation (3.7)) , and a monotone comparison for Metzler systems (as used around equations (3.1)–(3.7)) . However, the second inequality x̂ ≤ x̃ is false in general. A simple counterexample is A = 0 (no edges): the paper itself gives lim_{t→∞} x̂(t) = 1 - q e^{p} > 0 for β = 0 (Remark 3.1(iii)) , whereas the linearized solution is x̃(t) = p e^{-γ t} → 0, so x̂(t) > x̃(t) for large t. The proof of Lemma 3.2 relies on a derivative comparison dx̂/dt ≤ dx̃/dt (equations (3.18)–(3.19)) , but the step dx̂ = e^{-ŷ} dŷ and the sign of dŷ invalidate a uniform inequality over time; the argument also implicitly restricts to “above threshold” yet the lemma’s statement is unconditional . The candidate solution correctly proves x ≤ x̂ and x ≤ x̃ and gives the explicit counterexample showing no universal ordering between x̂ and x̃.