Probability of HIV Transmission During Acute Infection in Rakai, Uganda

Abstract

Accurate estimates of the probability of HIV transmission during various stages of infection are needed to inform epidemiological models. Very limited information is available about the probability of transmission during acute HIV infection. We conducted a secondary analysis of published data from the Rakai, Uganda seroconversion study. Mathematical and computer-based models were used to quantify the per-act and per-partnership transmission probabilities during acute and chronic HIV infection, and to estimate how many of the transmission events reported in the Rakai study were due to acute-phase HIV transmission. The average per-act transmission probability during acute infection equaled 0.03604 vs. 0.00084 for chronic HIV infection. Overall, HIV was transmitted during acute infection in 46.5% of 23 “incident index partner couples.” Acute-phase transmission accounted for 89.1% of all transmission events in the first 20 months of follow-up. These results highlight the substantial risk of transmission during acute HIV infection.

Appendix

As in the main text, let α(d) denote the per-act transmission probability on day d of the acute-phase of infection. The overall risk of HIV transmission from an acutely infected person to his or her sex partner is then $\gamma = 1 - (1 - \alpha (d_{1} )){\text{ }}(1 - \alpha (d_{2} )) \ldots (1 - \alpha (d_{n} )),$ where n is the total number of unprotected sex acts during the acute-phase of infection and α(d _k ) denotes the per-act transmission probability associated with the kth sex act, which occurs on day d _k . This per-partnership transmission risk value can be approximated by $\gamma _{{\text{A}}} = 1 - (1 - \alpha _{{\text{A}}} )^{n} ,$ where $\alpha _{{\text{A}}} = (\alpha (d_{1} ) + \alpha (d_{2} ) + \cdots + \alpha (d_{n} ))/n$ is the corresponding average per-act transmission probability.

Claim The average per-act transmission probability-based approximation, γ_A, never overestimates the true per-partnership transmission risk, γ (i.e., γ_A ≤ γ).

Proof First we will establish the following lemma: if x ₁, x ₂,..., x _n are positive real numbers, then $x_{1} *x_{2} * \cdots *x_{n} \le ((x_{1} + x_{2} + \cdots + x_{n} )/n)^{n}$. The proof is by induction on n. The claim is trivial for n = 1. Assume the claim holds for some n ≥ 1. Then $x_{1} *x_{2} * \cdots *x_{n} *x_{{n + 1}} \le x_{{n + 1}} *((x_{1} + x_{2} + \cdots + x_{n} )/n)^{n}$. To prove that $x_{1} *x_{2} * \cdots *x_{n} *x_{{n + 1}} \le ((x_{1} + x_{2} + \cdots + x_{n} + x_{{n + 1}} )/(n + 1))^{{n + 1}}$ it suffices to show that $x_{{n + 1}} *((x_{1} + x_{2} + \cdots + x_{n} )/n)^{n} \le ((x_{1} + x_{2} + \cdots + x_{n} + x_{{n + 1}} )/(n + 1))^{{n + 1}}$, or equivalently, that $f(y) = (x + y)^{{n + 1}} - ((n + 1)^{{n + 1}} /n^{n} )x^{n} y \ge 0$ (here, $x = x_{1} + x_{2} + \cdots + x_{n}$ and y = x _n+1). It is easily established through differential calculus that the minimum value of f(y) equals 0 (this occurs when y = x/n) and therefore f(y) ≥ 0 whenever x, y ≥ 0.

To show that γ_A ≤ γ, let x _k  = 1 − α(d _k ) for k = 1, 2,..., n and notice that $\alpha _{{\text{A}}} = (\alpha (d_{1} ) + \alpha (d_{2} ) + \cdots + \alpha (d_{n} ))/n = 1 - (x_{1} + x_{2} + \cdots + x_{n} )/n$. Then, γ_A ≤ γ, if and only if ($1 - \alpha (d_{1} ))*(1 - \alpha (d_{2} ))* \cdots *(1 - \alpha (d_{n} )) \le (1 - \alpha _{{\text{A}}} )^{n}$, or equivalently, $x_{1} *x_{2} * \cdots *x_{n} \le ((x_{1} + x_{2} + \cdots + x_{n} )/n)^{n}$, which is true by the lemma.