Immune Therapeutic Strategies Using Optimal Controls with $L^1$ and   $L^2$ Type Objectives
  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. 
Ouassim Bara, Seddik Djouadi, Judy Day and Suzanne Lenhart
math.OC