Inference for Within- and Between-Partnership Transmission Rates for HIV Infection

HIV transmission within serodiscordant couples remains a significant public health challenge, particularly in sub-Saharan Africa. Estimating the rate of such infection, alongside the rates of introduction of infection from outside the partnership, is a special case of the more general epidemiological challenge of inferring intensities of within- and between-group intensities of transmission. This study presents a stochastic susceptible-infected (SI) pair model for estimating key epidemiological parameters governing HIV transmission within and between couples, which we further extend to account for gender-specific differences in infection dynamics. Using a likelihood-based inference approach, we estimate transmission parameters and associated uncertainty from observed data. These values can be used to inform infection prevention strategies for HIV, and the methodology proposed can be generalised to other epidemiological settings.

💡 Research Summary

This paper addresses the challenge of quantifying HIV transmission within stable heterosexual couples by separating infection into two distinct processes: internal transmission (τ), which occurs from an infected partner to the susceptible partner within the same couple, and external transmission (λ), which represents infection acquired from the broader community. The authors develop a stochastic susceptible‑infected (SI) pair model that tracks the three possible pair states—both susceptible (SS), one infected and one susceptible (SI), and both infected (II). The dynamics are described by a simple system of ordinary differential equations (ODEs): dP_SS/dt = –2λP_SS, dP_SI/dt = 2λP_SS – (λ + τ)P_SI, and dP_II/dt = (λ + τ)P_SI. Because the system is linear, closed‑form solutions are available, allowing the expected numbers of each pair type at any time t to be expressed explicitly in terms of λ and τ.

To capture gender‑specific effects, the model is extended to four states (SS, IₘS𝒻, SₘI𝒻, IₘI𝒻) with separate external infection rates for males (λₘ) and females (λ𝒻) and distinct internal transmission rates from male to female (τₘ→𝒻) and from female to male (τ𝒻→ₘ). This richer model still admits analytical solutions, preserving the tractability of the basic framework while allowing investigation of asymmetries in transmission.

Empirical data come from a retrospective cohort of 1,802 stable couples in Mwanza, Tanzania, followed for two years (Hugonnet et al., 2002). At baseline there were 1,742 SS couples, 43 SI couples (22 male‑infected, 21 female‑infected), and 17 II couples. After two years the counts changed to 1,721 SS, 58 SI (33 male‑infected, 25 female‑infected), and 23 II. These observations provide the necessary information to estimate λ and τ.

The authors link model predictions to observed counts via a multinomial likelihood. Because observations are available only at two time points (t = 0 and t = T = 2 years), the log‑likelihood simplifies considerably. The external rate λ can be solved analytically as λ̂ = (1/2T) log(N₀^SS/N_T^SS), yielding λ̂ ≈ 0.003 yr⁻¹ (≈0.3 % per year). The internal rate τ is obtained by re‑parameterising τ = (2ϕ + 1)λ, solving for ϕ using the change in SI couples, and then back‑calculating τ. For the gender‑specific model, λ is split into λₘ = 2λq and λ𝒻 = 2λ(1 – q), while τₘ→𝒻 and τ𝒻→ₘ are expressed as 2λ(q + θₘ) and 2λ(1 – q + θ𝒻) respectively. Closed‑form expressions for q, θₘ, and θ𝒻 are derived from the observed gender‑specific SI transitions.

Maximum‑likelihood estimates (MLE) give λ̂ ≈ 0.003 yr⁻¹ and τ̂ ≈ 0.02 yr⁻¹, indicating that internal transmission is roughly six to seven times stronger than external acquisition in this setting. The gender‑specific analysis suggests that male‑to‑female transmission (τₘ→𝒻) is about 1.5 times higher than female‑to‑male transmission (τ𝒻→ₘ), implying that interventions targeting infected men could have a disproportionate impact on reducing couple‑level incidence.

Uncertainty is quantified by evaluating the Hessian matrix of the negative log‑likelihood at the MLE, inverting it to obtain the covariance matrix, and constructing approximate 95 % confidence intervals. Both λ and τ are statistically significant, though the confidence interval for τ is relatively wide, reflecting the limited temporal resolution (only two observation points) and the simplifying assumption that partners remain in the same pair for the entire study period.

The paper’s contributions are threefold: (1) it provides a parsimonious yet analytically tractable pair‑based SI model that directly maps to observable pair‑state transitions; (2) it demonstrates how closed‑form solutions combined with a multinomial likelihood enable straightforward maximum‑likelihood estimation of within‑ and between‑partner transmission rates; (3) it extends the framework to incorporate gender‑specific transmission pathways, yielding insights relevant for targeted HIV prevention strategies such as male‑focused testing, treatment, and pre‑exposure prophylaxis.

Limitations include the omission of partnership duration variability, the effects of antiretroviral therapy (ART) or pre‑exposure prophylaxis (PrEP) during follow‑up, and the assumption of no re‑infection within a pair after the first transmission event. Moreover, the data consist of only two time points, which restricts the ability to assess model fit and to detect time‑varying transmission dynamics. Future work should incorporate longitudinal observations, explicit partnership formation and dissolution processes, and treatment effects to refine estimates and enhance the model’s applicability to other sexually transmitted infections and epidemiological contexts.