Unraveling the influence of the objective functional on epidemic optimal control: Insights from the SIR model

Abstract

In the application of optimal control theory to medical and biological problems, the dependence of the objective functional on the control variable is often subject to uncertainty. This study examines the effects of this dependency on the outcomes of optimal control problems in the context of disease control using the SIR model. We formulate two distinct optimal control problems: one for the control of disease spread through prophylactic vaccination, and another for the treatment of infected individuals. For each scenario, we propose four variations of the objective functional to capture the cost of control interventions, namely, quadratic state-independent, quadratic state-dependent, linear state-independent, and linear state-dependent. We also conduct numerical simulations to compare optimal control solutions across different weight parameters. While some qualitative characteristics of the control profiles are similar in certain scenarios, there are also notable differences suggesting that the choice of objective functional can substantially alter the resulting control profiles. Consequently, when there is uncertainty regarding the functional form of the objective and its relationship to the control parameter, it is recommended to evaluate multiple objectives and subsequently identify which solution is most suitable for practical implementation.