Africa, where only one-third of the population resides within 2 km of functioning routes, carries over 85% of the world’s infant HIV burden. Drones hold great potential for transporting samples crucial to the early infant diagnosis of HIV in hard-hit regions. We consider a resource-limited environment where efficient sample transfer between scattered clinics and laboratories is indispensable. We first develop a mixed-integer non-linear program to model this complex supply chain and optimize the clinics-labs-transportation mode assignment. Despite highly non-linear objectives and constraints, we derive a tractable linear approximation. Using country-level data from Mozambique, we find that the bottleneck is caused by a confluence of prolonged transportation time, long waiting at the clinics until a transportation opportunity becomes available, and batching delay at the labs. A drone network relieves the bottleneck in three ways: (i) Drones significantly reduce sample transportation time; (ii) By eliminating waiting for transportation opportunities, drones remarkably decrease clinic delays while increasing dispatch frequencies; (iii) Increased dispatch frequency smooths clinic-lab sample flow which decreases the variation in random time between creation of two sample batches at labs. This, in turn, decreases the total lab delay. Our analysis also demonstrates that effectiveness-first designs compromise fairness less than equity-first designs do effectiveness. We finally find that optimizing fairness concentrates drones in villages rather than cities. Conversely, optimizing effectiveness allocates drones mainly to urban areas: i.e., in cities, drones impact many infants less while in villages, they impact fewer infants more. Therefore, the network structure should be tailored to the policy-maker’s needs.