A diffusion model for mosquito control trials with spillover: application to calculations of power, sample size, and bias

Abstract Background Mosquito movement is ubiquitous and complicates the design of cluster randomized trials (CRTs) of vector control interventions because it can cause spillover between trial arms. There has been no analytical approach to quantify the impact of this spillover on trial power, required sample sizes, or bias. This precludes formal allowance for spillover in formulae for study power, and a precautionary principle is generally used in trial design, leading to CRTs with very large clusters and extensive buffer zones between arms. Methods The diffusion equation was solved to give a mathematical model for mosquito displacement in a cluster randomized trial. This provides an explicit function for calculating the bias in efficacy estimates due to mosquito movement. Substituting this into a conventional formula for the power gives adjusted sample size and power estimates and indicates how the exclusion of data from buffer zones affects bias and power. The method is generally applicable to trials with entomological outcomes, or to malaria trials with epidemiological outcomes, providing these can be related to biting densities of mosquitoes. It is illustrated with the baseline data from an intervention trial against Aedes aegypti in Côte d’Ivoire. Results Despite the small size of the Côte d’Ivoire trial, it has adequate power to detect true intervention efficacy of ≥0.4, even in the presence of considerable spillover. If the spillover extends over a wide distance, the power will be considerably increased by using buffer zones. Conclusions The analytical approach can be incorporated into power calculations for CRTs with geographical spillover, whenever the diffusion model is considered an acceptable approximation. The example suggests that the key to obtaining adequate power in the presence of spillover is the inclusion of a sufficiently large number of clusters, even if only a small amount of data can be obtained from each cluster. The method can be implemented in a publicly available R package.