Analytical and numerical methods for spillover effects in prioritized PrEP for HIV prevention

The paper’s derivation of the NNT uses the same linearization route and chain-rule identities as the model, but it drops a necessary minus sign when moving from the Taylor expansion to the integral form (from its equation (2.15) to (2.17)) and states the final approximations without a sign convention, so the printed formulas yield a negative NNT whenever PrEP reduces infections. The paper also omits explicit conditions for neglecting the μ-weighted integral term when passing from its equation (2.19) to the short-horizon approximation (2.14). By contrast, the model keeps the sign correct, explains the sign convention, and supplies a sufficient bound for ignoring the μ-term. Aside from these issues, the core identities and steps—definition of incidence λj, sensitivities ∂Ij/∂Ek = γk_j/Sk, γk_j(0)=0—match the paper’s setup and are used consistently by the model. See the paper’s definitions and steps around (2.8), (2.10)–(2.12), (2.13)–(2.19), and Proposition 2.4 for details.