Impact of a cost functional on the optimal control and the cost-effectiveness: control of a spreading infection as a case study

TL;DR

This paper investigates how the choice of the objective functional influences optimal control strategies and cost-effectiveness in epidemic models, finding that control costs and effectiveness are minimally affected by the functional's form.

Contribution

It demonstrates that variations in the control cost functional have limited impact on optimal controls and outcomes in epidemic models, highlighting the importance of cost dependencies on state variables.

Findings

Control form and effectiveness are minimally affected by the control cost functional.

Cost dependencies on state variables can be very significant.

Cost-effectiveness analysis supports qualitative comparisons of different functionals.

Abstract

In applications of the optimal control theory to problems in medicine and biology, the dependency of the objective functional on the control itself is often a matter of controversy. In this paper, we explore the impact of the dependency using reasonably simple \emph{SIR} and \emph{SEIRS} epidemic models. To qualitatively compare the outcomes for different objective functionals, we apply the cost-effectiveness analysis. Our result shows that, at least for the comparatively inexpensive controls, the variation of the power at the controls in a biologically feasible range does not significantly affect the forms of the optimal controls and the corresponding optimal state solutions. Moreover, the costs and effectiveness are affected even less. At the same time, the dependency of the cost on the state variables can be very significant.