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 $Γ$-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.

💡 Research Summary

The manuscript develops a rigorous optimal‑control framework for epidemic mitigation when the underlying disease dynamics follow a SEIR compartmental model with a realistic operational delay in vaccine deployment. The state variables (susceptible s, exposed e, infected i, recovered r) evolve according to standard flow equations, but the transmission rate is reduced by a non‑pharmaceutical intervention (NPI) control h(t) and the susceptible pool can be vaccinated at rate u(t). The vaccination control is constrained to be zero before a prescribed delay τ_delay, reflecting production or distribution lead times, while both controls are bounded by maximal feasible levels (u_max, h_max).

The authors first establish well‑posedness of the controlled system. By invoking Carathéodory’s existence theorem and exploiting the positivity of the effective transmission rate (β − h(t) > 0), they prove existence, uniqueness, and global boundedness of solutions in the simplex s+e+i+r = 1. A series of lemmas shows that s(t) stays strictly positive, i(t) remains positive for any finite horizon, and both e(t) and i(t) converge to zero as t → ∞. Barbalat’s lemma is used to turn integrability of i(t) into asymptotic extinction, guaranteeing that the epidemic eventually dies out while the susceptible fraction converges to a finite limit s_∞.

A central contribution is the treatment of a hard state constraint i(t) ≤ I_max, which models an absolute ICU capacity limit. Directly imposing such a constraint leads to numerical difficulties, so the authors introduce a Moreau–Yosida regularization. The original constrained objective J is augmented with a quadratic penalty (1/2ε)∫