Modeling the dynamics of mixed immunotherapy and chemotherapy for the treatment of immunogenic tumor

We investigate a mathematical model that delineates the nonlinear dynamics of tumor-immune interplay by considering the roles of immunotherapy and chemotherapy. The proposed model explores a system of coupled nonlinear ordinary differential equations (ODEs), involving tumor cells, cytotoxic T-lymphocytes (CD8+T cells), macrophages, dendritic cells, regulatory T-cells (Tregs), IL-10, TGF-beta\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta$$\end{document}, IL-12, IFN-gamma\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma$$\end{document} and the concentration of chemotherapeutic drug. We use optimal control theory to understand the dynamics under what conditions the immune system can eradicate tumor cells. The control problem is solved with an objective functional that minimizes the tumor cell population and maximizes the immune components. The basic properties of optimal control theory are established through the boundedness of solutions for each state variable. Our optimal control theory is characterized by coupling the state variables with costates. Additionally, our study investigates the uniqueness property of the optimal control problem within a small time window. Subsequently, we explored the methods employed to estimate the system parameters. Finally, we demonstrate numerically that the optimal control strategy minimizes the burden of tumor cells and maximizes immune cell populations under different scenarios. Moreover, we provide corresponding biological implications.