Fig. 1

Illustration of approach. Beginning with malaria-endemic countries in sub-Saharan Africa, the study first (A) identified all administrative units with an international border (red) and (B) the administrative unit centroids of border areas (red) and non-border areas (orange). From there, mixing matrices were created for each cluster of 8 administrative units. Panels (C–F) show an example cluster with the international border indicated with a bolded line. Grey polygons indicate border administrative units, yellow polygons indicate non-border units. For each cluster the following were identified: (C) a seed point (red), the seed point’s three nearest neighbours within the same country (orange), and four nearest neighbours outside of the country (blue). Panel (D) shows the travel time raster developed by the Malaria Atlas Project [22] overlaid on the example cluster area. Each pixel (1 km x 1 km) represents the travel time to cross the pixel in minutes according to the fastest mode of transportation, with an additional one-hour time penalty added at the border. Panel (E) shows the 0.1 × 0.1 degree grid surface used in partnership with travel times and trip durations to calculate the mixing matrix. Dots represent the population sizes taken from the centroids of the grid and were obtained from WorldPop [23]. Finally, panel (F) displays the normalized mixing probabilities between units using the gravity model based on population sizes and travel times, aggregated from the grid to the administrative unit level