Relaxation Dynamics of SIR-Flocks with Random Epidemic States

The PDF studies the SIR‑flock model (eq. (1.6)) and proves Theorem 4.1: (1) for each i, Wi(t) → (S∞i, 0, R∞i), and (2) under aij = (||xi − xj|| + L)^{−γ} and min_i b_i > κ1 (N−1)/L^γ, the average infection satisfies (1/N)∑i Ii(t) ≤ e^{−λ t} with λ = min_i b_i − κ1 (N−1)/L^γ > 0. The model and conditions are stated explicitly in the paper and theorem statement (including the exponential decay claim) . The paper’s proof shows monotonicity (Si nonincreasing, Ri nondecreasing) and concludes Ii(t) → 0 from Ṙi = b_i Ii, then uses an integrating‑factor inequality for ∑i e^{b_i t} Ii and the bound aij ≤ L^{−γ} . The candidate solution is correct and gives a clean alternative: (i) Ii ∈ L^1 because R_i converges; (ii) Ii′ is uniformly bounded using aij ≤ L^{−γ}, implying uniform continuity; hence Ii(t) → 0 by the standard “nonnegative, uniformly continuous, integrable ⇒ limit zero” fact; and (iii) for the average Ī, it derives directly Ī′ ≤ −λ Ī and applies Grönwall to get Ī(t) ≤ e^{−λ t}. This directly matches the theorem’s stated decay and slightly streamlines that step relative to the integrating‑factor route. Overall, the paper’s result is correct, and the model’s argument is also correct, with a different but compatible proof. For completeness, the paper also presents a relaxed condition ensuring exponential decay of ∑i Ii via a matrix comparison argument (Corollary 4.1) .