Immune Therapeutic Strategies Using Optimal Controls with $L^1$ and $L^2$ Type Objectives

TL;DR

This paper formulates and analyzes optimal control strategies for immune response therapy using mathematical models and different objective functionals, providing insights into effective treatment protocols.

Contribution

It introduces a novel optimal control framework with $L^1$ and $L^2$ objectives for immune therapy, addressing singular controls and offering numerical solutions.

Findings

Optimal control strategies vary with $L^1$ and $L^2$ objectives.

Numerical simulations suggest effective drug treatment protocols.

Insights into immune response management and potential clinical applications.

Abstract

Therapeutic strategies to correct an excessive immune response to pathogenic infection is investigated as an optimal control problem. The control problem is formulated around a four dimensional mathematical model describing the inflammatory response to a pathogenic insult with two therapeutic control inputs which have either a direct pro- or anti-inflammatory effect in the given system. We use Pontryagin's maximum principle and discuss necessary optimality conditions. We consider both an $L^1$ type objective functional as well as an $L^2$ type objective. For the former, the presence of singular control will be addressed. For each case, numerical simulations using a nonlinear programming optimization solver to acquire different drug treatment strategies are presented and discussed. The results provide insight for possible treatment strategies and the methods could be a relevant tool for future practice to assist in better prediction of clinical outcomes and subsequently better treatment for patients.