Optimal control problem on cancer-obesity dynamics

This study introduces a novel fractional-order model to investigate the interplay between cancer and obesity and their treatment. Initially, we examine the solution’s existence and uniqueness for the proposed model. Additionally, we establish the boundedness of these solutions. Subsequently, we identify some potential equilibrium points of the cancer–obesity model and investigate their stability. To address the considered model, we propose fractional Euler’s and Adam’s methods. Theoretical and numerical analyses are conducted to assess the error estimates and performance of both methods with varying fractional-order derivatives. Moreover, we formulate an optimal control problem concerning cancer density and drug concentration. We delve into the existence of control and explore the first-order optimality conditions. We validate the analytical findings through numerical computations, demonstrating that administering drugs with control variables enhance immunity levels and reduce the burden of cancer.