Bang-Bang Optimal Control of Vaccination in Metapopulation Epidemics With Linear Cost Structures

In this paper, we study optimal vaccination in a heterogeneous metapopulation epidemic model with state and mixed control-state constraints and a linear cost on the control, which better reflects real-world expenses compared to quadratic penalties. Within a general SIR / SEIR K-patch framework, we formulate the resulting control problem on a finite horizon, prove the existence of optimal control, and derive the necessary conditions via Pontryagin’s maximum principle extended to handle both shipment and per-region capacity limits. We use Pontryagin’s result to rule out the existence of singular arcs and obtain a complete characterization of the optimal policy as bang-bang, with at most one switching time per region per week. Numerical experiments on networks of 3 to 8 cities confirm that the predicted switching structure arises from the linear cost assumption, and we compare these findings with optimal quadratic-cost policies.