In this project, fully interacting nonlinear multi-input control systems will be analyzed as optimal control problems. The motivation for this research comes from a systematic study of mathematical models for cancer treatment that combine various structures that form the tumor microenvironment. In modern oncology, a tumor is viewed as not just cancerous cells, but as a system of interacting components that in various ways aid and abet the tumor (e.g., the tumor vasculature), but also fight it (e.g., the immune system). Current treatments therefore are multi-targeted therapies that not only kill cancer cells, but also include antiangiogenic therapy, immunotherapy and many other options. The search for best ways of administering these therapeutic agents naturally leads to formulations as multi-input nonlinear optimal control problems. The aim of the research is to develop local syntheses of optimal controls for such systems, a difficult task even for single-input systems with only partial results known. These new results will be developed in connection with the study of systems that describe (i) cancer treatment within the complex context of the tumor microenvironment and (ii) optimal strategies for the control of the spread of diseases in epidemiology. In the latter field, mathematical models have been a relevant tool in analyzing the underlying dynamics, whereas in this project a much less explored optimal control approach to the problem will be pursued. While one aim is to develop general methods and procedures that have wide applicability, a second important aspect of the research is to provide qualitative and quantitative insights into the structure of optimal solutions for important real-life problems. Motivated by timely problems in biomedicine, like how to design metronomic chemotherapy protocols, challenging problems in optimal control theory will be considered whose solutions require developing new tools and methods. Because of its applied and interdisciplinary character, the project is expected to be of strong interest to students from mathematics and engineering and consequently it contains a substantial educational component. Efforts to attract women and minorities will be continued and expanded by reaching out to engineering students where participation of these groups is particularly low.
In modern oncology, a tumor is viewed as not just cancerous cells, but as a system of interacting components that in various ways aid and abet the tumor (e.g., the tumor vasculature), but also fight it (e.g., the immune system). Current treatments therefore are multi-targeted therapies that not only kill cancer cells, but also include antiangiogenic therapy, immunotherapy and many other options. It is very difficult and expensive to test complex multi-target protocols in medical trials. For this reason, the analysis of mathematical models becomes of intrinsic value. The medical community has become more and more aware that not only what drug is given, but also how it is administered, i.e., dosage, frequency and sequencing, can have a major impact on the outcome of the treatment. This led to a recently launched "Metronomics Global Health Initiative". The proposed research is motivated by the biomedical ideas of this initiative and the investigators believe that the tools of geometric optimal control are best suited to give mathematical answers to these questions and thus provide insights into how a metronomic protocol should be designed. Regarding a second topic, infectious diseases continue to be one of the most important health problems worldwide. In this project, the investigators seek theoretical results that can give practical guidelines how to pursue joint efforts of vaccination and treatment in an optimal way to maximize the effectiveness and minimize the social economic cost. Because of its applied and interdisciplinary character, the project is expected to be of strong interest to students from mathematics and engineering and consequently it contains a substantial educational component. Efforts to attract women and minorities will be continued and expanded by reaching out to engineering students where participation of these groups is particularly low.
