Stability analysis and optimal control of tumour-immune interaction problem using fractional order derivative

In this study, we propose a tumour-immune interaction model using Caputo–Fabrizio fractional order derivative. The conditions for the well-posedness of the solution are examined. The stability of the endemic equilibrium point is derived and its stability is proved using Routh–Hurwitz criteria. The solution is approximated using a shifted Legendre polynomial at Gauss–Legendre collocation points, which is compared with the numerical results of the Adams–Bashforth scheme in the interval [0,1]. We have also proposed a fractional optimal control problem and proved the necessary optimality conditions. The optimal system is solved using the forward–backward sweep method (FBSM) with the Adams–Bashforth predictor–corrector numerical method. We have demonstrated that the antigenicity of tumours plays a crucial role in activating immune cells, suggesting that enhancing tumour antigenicity could improve immunotherapeutic outcomes. The effects of fractional-order derivatives and the proliferation rate of the Michaelis–Menten term are observed. Moreover, the impact of other model parameters on the system is highlighted through numerical results. Finally, the reduction in tumour cells and the increase of active immune cells are demonstrated in the presence of optimal control.