Healthcare-Strain-Aware Epidemic Control Using State-Constrained Optimization

Abstract

A state-constrained optimal-control problem is formulated to prevent healthcare-system overload dynamically while balancing intervention costs. The formulation leads to an eight-compartment model (S, U, D, T, H, R, E, V), which includes a separate compartment for hospitalisation and a sigmoidal mortality rate \(\bar{\mu}_h(Q)\) that depends on the available capacity of the healthcare system. The objective is to minimise the combined disease burden and intervention costs while imposing the hard constraint H (t) \(\le\) Q. Using Pontryagin’s Maximum Principle extended to state-constrained problems, the adjoint equations and the characterisation of the optimal time-dependent controls for transmission reduction, testing/detection, clinical-treatment efficiency, and vaccination are obtained. Numerical simulations under different capacity levels and cost assumptions show that the optimal policies involve early scaling-up of non-pharmaceutical interventions and testing/detection to keep hospitalisations low, followed by a transition towards vaccination and efficient use of clinical resources. The inclusion of the state constraint leads to 15–30% higher implementation costs but avoids mortality peaks associated with violation of the healthcare-capacity constraint.