Optimal Control of Epidemiological SEIR models with L1-Objectives and Control-State Constraints

Abstract

Optimal control is an important tool to determine vaccination policies for infectious diseases. SEIR compartment models have been developed to describe the spread of a disease transmitted horizontally. Most of the literature on SEIR models deals with cost functions that are quadratic with respect to the control variable, the rate of vaccination. In this paper, we consider L-1-type objectives that are linear with respect to the control variable. Various control, mixed control state and pure state constraints are imposed. For each type of constraint, we discuss the necessary optimality conditions of the Maximum Principle and compute optimal control strategies that satisfy the necessary optimality conditions with high accuracy. Since the control variable appears linearly in the Hamiltonian, the optimal control is a concatenation of bang-bang arcs, singular arcs or boundary arcs. For bang-bang controls, we are able to check second-order sufficient conditions.