The Stability of a Tumor–Macrophages Model with Caputo Fractional Operator

This study proposes a fractional-order model in the Caputo sense to describe the interaction between tumor and immune macrophages by assuming that the pro-tumor macrophages induce a Holling type-II response to the tumor. Then, the basic properties of the solutions to the model are studied. Local stability analysis is conducted at each of the equilibria in the model, and a numerical study is performed with varying activation rates of type-II or pro-tumor macrophages and the order of the fractional operator. The numerical findings suggest that type-I or anti-tumor macrophages can stabilize the system if the activation rate of type-II or pro-tumor macrophages is low. Still, for a higher value of the activation rate for type-II or pro-tumor macrophages, the proliferation of tumor cells is uncontrollable and the system becomes unstable. Furthermore, the stability of the system decreases as the order of the fractional operator increases.