New developments in the application of optimal control theory to therapeutic protocols

Optimal control theory is one of the most important tools in the development of new therapeutic protocols for treating infections. In this work, we present an algorithm able to deal with high-dimensional problems with bounded controls. The optimal solution is obtained by minimizing a positive-definite treatment cost function. Our method, based on Pontryagin's Minimum Principle and the coordinate cyclic descent method, allows solving problems of varied nature. In this paper, and by way of example, therapeutic enhancement of the immune response to invasion by pathogenic attack is addressed as an optimal control problem. The generic mathematical model used describes the evolution of the disease by means of four non-linear, ordinary differential equations. The model is characterized by the concentration of pathogens, plasma cells, antibodies and a numerical value that indicates the relative characteristic of an organ damaged by disease. From a system theory point of view, drugs can be interpreted as control inputs. Therapies based on separate application of the agents are presented in previous studies. We shall present the more general problem in this paper, considering combined therapies and bounded controls. Finally, we present several numerical simulations.