Feedback design to measure the effect of therapies in controlling cancer using the fractional approach

In this paper, we developed a cancer dynamical system that incorporates the interaction of tumor cells, immune systems, and drug reaction systems to investigate the impact and therapeutic implications of a fractional order Caputo derivative's with memory effects. The solutions of the proposed system are shown to be bounded and positive. The existence and uniqueness of the solutions of the suggested model are examined using a few fixed-point theorems. The global stability of the system is examined through the use of Lyapunov's first derivative functions. For various fractional values, solutions to the fractional order system are obtained with the help of a fractional operator with a power law kernel. The kernel also checks for unique solutions and verification of the scheme through mathematical analysis using novel approaches. Next, a simulation of the derived iterative technique is made for various fractional orders against the real data at different fractional order values. This fractional order model can be used to investigate the dynamics of tumor cells, the interactions between tumor cells and immune cells, and the effects of medications on the disease. The proposed system's controllability and observability are further addressed by using various therapies as inputs and normal cells as output. A linear control system with a closed-loop design, in which the drug is the input and treated cells are the output, is used to investigate the influence of cancer treatments on different patients.