Optimal Control of an Epidemic with Intervention Design

This paper investigates the optimal control of an epidemic governed by a SEIR model with an operational delay in vaccination. We address the mathematical challenge of imposing hard healthcare capacity constraints (e.g., ICU limits) over an infinite time horizon. To rigorously bridge the gap between theoretical constraints and numerical tractability, we employ a variational framework based on Moreau--Yosida regularization and establish the connection between finite- and infinite-horizon solutions via $\Gamma$-convergence. The necessary conditions for optimality are derived using the Pontryagin Maximum Principle, allowing for the characterization of boundary-maintenance arcs where the optimal strategy maintains the infection level precisely at the capacity boundary. Numerical simulations illustrate these theoretical findings, quantifying the shadow prices of infection and costs associated with intervention delays.