Epidemic Conditions with Temporary Link Deactivation on a Network SIR Disease Model

The paper derives R0 = β⟨k⟩/γ from İ = β[SI] − γI using [SI](0) ≈ ⟨k⟩I0 as I0 → 0, then uses the pairwise ODE and the homogeneous-degree closure [ABC] ≈ ((⟨k⟩−1)/⟨k⟩)([AB][BC]/B) to obtain lim I0→0 I0^{-1} [ṠI](0) and hence p∗1 = β(⟨k⟩/2 − 3/2) − γ; since S̈ = −β[ṠI], p∗1 is the concavity threshold for S. Differentiating İ gives p∗2 = p∗1 − γ + γ^2/(β⟨k⟩), the concavity threshold for I. The paper also states the positivity conditions: p∗1 > 0 iff ⟨k⟩ > 3 and R0 > 2⟨k⟩/(⟨k⟩ − 3), and p∗2 > 0 iff p∗1 > 0 and γ < p∗1 R0/(R0 − 1) (with R0 > 1). All of these match the candidate solution step-by-step, including the initialization with all non-[SS]/[SI] (and hatted) compartments zero at t = 0 and the first-order neglect of O(I0^2) terms. The ODEs and closure used by the paper are exactly those used by the candidate, and the resulting formulas and inequalities coincide. Citations: paper’s ODEs and closure (eqs. (10)–(11), (4) ), R0 derivation (eqs. (12)–(13) ), p∗1 computation (eqs. (14)–(17) ), p∗2 computation and positivity conditions (eqs. (18)–(19) and discussion ), and the initialization used in the I0 → 0 analysis (Figure 7 setup ).