Cancer model and its possible control-A Z-type control approach.

This paper investigates the dynamics of a three-dimensional nonlinear cancer model involving interactions among cancer cells, normal cells, and immune cells. By performing a linear stability analysis of the equilibria and investigating the Hopf bifurcation in relation to the immune cell growth rate, we reveal the possibility of chaotic behavior when radiation is absent. However, with the appropriate implementation of radiotherapy, the cancer model demonstrates stable solutions, transitioning from chaotic oscillations through period-halving bifurcation. Additionally, we propose and examine an indirect Z-control mechanism within the cancer model. Our findings indicate that using the indirect Z-controller on the immune population successfully manages chaos and adjusts the cancer cell density to a desired level. Through extensive investigation, we demonstrate the robustness of the Z-controller in managing oscillations and provide insights into determining the minimum number of immune cells needed to achieve a predetermined cancer cell density. This study underscores the importance of control mechanisms in mitigating cancer progression and highlights the potential of Z-control for therapeutic intervention strategies.