Evaluating the potential of HIV self-testing to reduce HIV incidence in EHE districts: a modeling study

The paper proves the pointwise and uniform bounds on the susceptible fraction’s derivative, |Σ̇(t)| ≤ (Λ/n(t) + λ_a + μ_s − μ_e)|1−Σ(t)| and |Σ̇(t)| ≤ Λ/n_min + λ_a + μ_s − μ_e, for the five-compartment system (A1), and derives the corollary bounds (A7)-(A8) via the mean value/Taylor theorem, all in Supplement A (see the model definition (A1), the stated bounds (A5)-(A6), and the corollary (A7)-(A8) ). The candidate solution derives the same inequalities by explicitly computing Σ̇ through the quotient rule, introducing aggregate transmission and mortality fractions, then bounding them with λ_a^⋆ ≥ max{λ_a,λ_u,λ_s,λ_d} and μ_s ≥ μ_i ≥ μ_e to control terms. This proof is algebraically clean and yields the same results (and the same corollary via the integral mean value theorem). The only substantive difference is that the model explicitly states the mild assumption μ_i ≥ μ_e to justify a nonnegative deviation term, whereas the paper compresses the bounding step; otherwise, the arguments agree on all key steps and conclusions (compare Theorem statements and proofs in Supplement A and the corollary discussion ).