Analysis of a time-delayed HIV/AIDS epidemic model with education campaigns.

The paper’s Theorem 1.3 establishes: (i) I* is non-increasing in τ=q/η with explicit two-sided bounds and strictness conditions; (ii) if D0 ≥ (1/γ)∑j βj,uγj then I*→0 with an O(1/τ) rate when the inequality is strict; (iii) if D0 < (1/γ)∑j βj,uγj then I*→I*∞>0 solving D0=(μ/γ)∑j βj,uγj/(μ+βj,u I*∞). These statements and their proofs appear in the paper (statement and bounds in Theorem 1.3; proof via a scalar function G, its monotonicity in I, and an implicit-function argument for τ; limit analyses) . The candidate solution independently derives an equivalent scalar equation D0=Fτ(I*), proves Fτ is decreasing in I and non-increasing in τ, obtains the same bounds, and reproduces the same limiting cases and rate. Aside from a benign slip writing S0+S1+S2=N*−I* (should be N*−I*−R*), which does not affect the inequalities used, and a clarifying note that u drops out of the equilibrium algebra except via the u-dependent parameters βj,u, the candidate solution matches the paper’s result. The approaches differ in presentation (paper uses G and implicit differentiation; model uses Fτ and sign/monotonicity), but both are correct and consistent with Theorem 1.3 .