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

Pre-exposure prophylaxis (PrEP) is an effective intervention for preventing HIV transmission, but high cost and uneven uptake raise challenges for resource allocation. While spillover effects, wherein PrEP use in one group reduces infections in others, are known to occur, they remain poorly quantified and rarely guide policy. We provide a comprehensive modeling study for PrEP spillover across risk groups, and develop analytic and numerical tools for its quantification. We first develop a compartmental model for HIV transmission that stratifies the population into interacting subgroups: heterosexual males (HETM), high- and low-risk heterosexual females (HETF-hi/HETF-lo) and men who have sex with men (MSM). The asymptotic stability of the disease-free equilibrium of the model is analyzed. Spillover is quantified by deriving an expression for the spillover-adjusted number needed to treat (NNT), a measure of the population-level impact of PrEP. Simulations show PrEP delivery to MSM yields substantial indirect benefits, particularly for HETF-lo, where spillover exceeds the direct effect by a factor of five. We show targeting HETF-hi outperforms direct PrEP delivery to HETM, emphasizing the importance of intra-group heterogeneity. To evaluate whether these results hold under more detailed assumptions, we embed our framework into the national HOPE model maintained by the Centers for Disease Control and Prevention (CDC) and conduct global sensitivity analysis using Sobol indices with Polynomial Chaos Expansion. This approach extends our analytical insights and quantifies how uncertainty in PrEP allocation propagates through complex dynamics. Further, this framework provides a numerical procedure for quantifying spillover where direct analysis is impractical. Our results show that spillover is a central driver of PrEP dynamics and that failing to account for it risks mis-allocating resources.

💡 Research Summary

The paper addresses a critical gap in HIV prevention modeling by quantifying spillover (indirect) effects of pre‑exposure prophylaxis (PrEP) across distinct risk groups. The authors first construct a tractable compartmental model that divides the population into four interacting sub‑populations: men who have sex with men (MSM), high‑risk heterosexual females (HETF‑hi), low‑risk heterosexual females (HETF‑lo), and heterosexual males (HETM). Each group is split into susceptible (S) and infected (I) compartments, and contact patterns are encoded through dimensionless mixing parameters (η, α, ξ). The model assumes 100 % PrEP efficacy for simplicity and incorporates recruitment, natural death, and disease‑induced mortality.

Mathematically, the authors prove positivity, boundedness, and invariance of the feasible region, and they analyze the disease‑free equilibrium (DFE) using the next‑generation matrix. The basic reproduction number (R_c) is expressed as the maximum of four terms, each containing a common MSM‑specific factor (f_{11}/(4K_1)). This demonstrates that MSM dynamics dominate overall transmission, motivating a focus on spillover from MSM to other groups. Local asymptotic stability of the DFE is shown for (R_c<1).

To capture spillover, the authors define the instantaneous incidence (\lambda_j(t,\theta)) and its annual cumulative counterpart (\lambda_{\text{year},j}). They derive a “spillover‑adjusted number needed to treat” (NNT) that multiplies the conventional NNT by a factor reflecting indirect infections averted in other groups. Numerical experiments calibrated to Georgia, USA, reveal that allocating PrEP to MSM yields indirect benefits to HETF‑lo that are five times larger than the direct effect, and that targeting HETF‑hi outperforms direct PrEP delivery to HETM.

Recognizing that real‑world policy models are far more complex, the authors embed their framework into the CDC’s national HOPE model, a high‑dimensional compartmental system. They employ polynomial chaos expansion (PCE) to propagate parameter uncertainty and compute Sobol sensitivity indices. The Sobol analysis confirms that MSM PrEP allocation contributes the largest first‑order effect and also dominates second‑order interactions, underscoring the centrality of spillover in shaping overall epidemic outcomes.

Overall, the study provides both an analytical tool (spillover‑aware NNT) and a numerical pipeline (PCE‑based Sobol analysis) for quantifying indirect benefits of PrEP. By demonstrating that neglecting spillover can lead to substantial misallocation of scarce resources, the work offers actionable insights for public‑health planners aiming to maximize HIV incidence reductions under budget constraints. Future extensions could incorporate partial adherence, cost‑effectiveness, and behavioral feedback mechanisms.