Assessing Prevention for Positives: Cost-Utility Assessment of Behavioral Interventions for Reducing HIV Transmission

Abstract

Typical studies of HIV behavioral interventions measure relative risk reduction for HIV transmission. Here, we also consider the health benefits of such interventions on secondary transmission. In addition, a sensitivity analysis explores the potential additional benefits that may accrue if partners of those in the intervention group also adopt the risk reducing behavior. To do this, we developed an ordinary differential equation (ODE) model to analyze the cost and utility (measured in quality-adjusted life years, or QALYs) of a published behavioral HIV intervention that aims to reduce the risk of transmission from HIV-infected persons to their sexual partners. The ODE model maps measurements of behavioral risk reduction parameters, estimated from sampling, into costs and QALYs. Monte Carlo sampling was used to perform a probabilistic sensitivity analysis to quantify uncertainty in costs and QALYs due to parameter estimation error for the behavioral HIV intervention. The results suggest that the behavioral intervention is most likely to be cost-saving or, at least, cost-effective. The analysis highlights the step of converting uncertainty about estimates of mean values of parameters that are commonly reported in the literature to uncertainty about the costs and health benefits of an intervention. It also shows the potential importance of considering secondary transmission of HIV and the partial adoption of behavior change by partners of the individuals who undergo the intervention.

Appendix

The ODE for S(t) in (8.1) and the discounted cost and QALY equations in (8.4) and (8.5) drive the analysis. The term S(t) is readily solvable in closed form when the time-varying parameters are assumed to be piecewise constant on a sequence of intervals in (8.1). For the behavioral interventions that we modeled, we assumed that these parameters were indeed piecewise constant on intervals. In particular, the functional form of (8.1) for the studies that we analyzed is

$$ \mathrm{ d}S(t)/\mathrm{ d}t={a_i}+{b_i}S(t),\ \mathrm{ for}\ t\in [{\tau_i},\;{\tau_{i+1 }}) $$

where τ ₀ = 0, τ ₁ is the time of initiation of the behavioral intervention, τ ₂ − τ ₁ is the duration of the intervention, τ ₃ is the time through which infections are counted for the purpose of the endpoint of the study, and τ ₄ = ∞ allows S(t) to be defined for all t ≥ 0. We assumed that τ _i  ≤ τ _{i + 1} for i = 0, 1, 2, 3, meaning that a _i  = λ _S (t) and b _i = −[μ _S (t) + γ(t) + β(t) + α _S (t)] were constant for t in [τ _i , τ _i+1). The constants may differ depending on the intervention for each i.

The solution is straightforward by a quick change of variables for each interval [τ _i , τ _i+1). If we set T(t) = a _i  + b _i S(t), we obtain dT(t)/dt = b _i T(t), which has solution T(t) = c _i exp[b _i t] for some constant c _i . Solving for S(t) we get

$$ S(t)={c_i}\exp [{b_i}t]/{b_i}-{a_i}/{b_i},\ \mathrm{ for}\ t\in [{\tau_i},\;{\tau_{i+1 }}). $$

The value of c _i is determined by the preceding equation, which implies c _i  = b _i s _i  + a _i , and the initial condition S(τ ₀) = s ₀. We evaluate this first for i = 0 to obtain

$$ S(t)={s_0}\exp [{b_0}t]/{b_0}-{a_0}(\exp [{b_0}t]-1)/{b_0},\ \mathrm{ for}\ t\in [{\tau_0},\;{\tau_1}). $$

This determines s _i+1 = S(τ _i+1) = s _i exp[b _i (τ _i+1 − τ _i )] − a _i /b _i (exp[b _i (τ _i+1 − τ _i )] − 1), which we sequentially evaluate for i = 0, 1, 2, 3. Thus, a closed-form expression for S(t) is found by iterating over each time interval, with the parameters of the intervention set to constants in each interval. A similar analysis can be used obtain an iterated closed-form solution for the number of individuals infected up to a given time t.

This analysis was used to debug the spreadsheet code, which implements a forward Euler finite difference approximation to the ODE. The spreadsheet code additionally can be modified to have more flexibility than allowed by the piecewise linear assumption for the parameters.