Fractional-order pro-tumor and anti-tumor macrophages model: Dynamical analysis and optimal control

The development of new techniques for anti-cancer therapy has become a very active area of research in recent years. One of the promising directions of anti-cancer therapy aims at using the genetically engineered macrophages as killers of tumor cells. The present work introduces a new fractional-order mathematical model to simulate the nonlinear interaction between genetically engineered macrophages and tumor cells. The proposed model incorporates the pro-tumor impacts and anti-tumor impacts which are triggered by the macrophages. The existence and uniqueness of positive solutions of the model are proved. Then, the stability conditions for the equilibrium points are obtained. The bifurcation diagrams explore the impacts of crucial parameters on the dynamical behaviors of the model. Finally, the optimal control scheme is employed to efficiently eliminate tumor cells and inhibit their prevalence in the body. Using Adms-Bashforth-Moulton method, numerical experiments are carried out to validate theoretical analysis.