Optimising malaria and dengue case management through optimal diagnostic test allocation under budget constraints

In 2023 Burkina Faso faced hyperendemic Plasmodium falciparum malaria alongside the largest dengue epidemic recorded in West Africa. Both present as undifferentiated fever, and rapid diagnostic tests (RDTs) are the practical means of distinguishing them at the point of care. We formulate a 15-dimensional ordinary differential equation model coupling malaria and dengue transmission through two vector populations (Anopheles and Aedes), in which the two screening rates are control variables competing for a single pointwise-constrained diagnostic budget. We establish well-posedness, derive the basic and invasion reproduction numbers by the next-generation matrix method, and characterise the optimal controls via the Pontryagin Maximum Principle with the Karush-Kuhn-Tucker conditions, so that the allocation depends explicitly on the shadow price of the budget. A forward-backward sweep solves the boundary value problem, cross-validated against a sequential quadratic Hamiltonian scheme. Calibration to 104 weeks of Kadiogo Province surveillance attains coefficients of determination on the log(Y+1) scale of 0.82 (malaria) and 0.72 (dengue), with R₀^m=1.14 and R₀^d=1.57. The optimal dynamic allocation outperforms every heuristic tested, including two that are themselves time-varying: it reduces disease burden by 24.6% relative to standard malaria-first practice at the baseline budget and by over 32% at double the budget, and outperforms malaria-first in every one of 200 parameter draws under  ± 30% uncertainty. The gain comes from the two epidemics peaking months apart, which in an epidemic year follows from malaria opening the season near its endemic level while dengue starts from a small seed.