Fractional order delay differential model of a tumor-immune system with vaccine efficacy: Stability, bifurcation and control

The aim of this paper is to present a mathematical model of tumor vaccine efficacy in the context of immunotherapy treatment. The model is governed by fractional-order delay differential equations with a control variable. Transforming growth factor beta (TGF-β) inhibition alone gives limited clinical advantages, but when combined with a tumor vaccine, it can dramatically increase the immune system's anti-tumor response. The mathematical model takes into account tumor growth dynamics, TGF-β concentrations, cytotoxic effector cells, and regulatory T-cells. The non-negativity of the solutions to such a model has been examined. The steady-state stability and Hopf bifurcation of tumor time delays τ are investigated. An optimal fractional-order control problem is derived and analyzed in the presence of immunotherapy treatments. Using Adams–Bashforth–Moulton predictor–corrector algorithms, we illustrate numerical examples that confirm the analytical findings. The optimal control treatment method reduces the load of tumor cells while increasing the effector cells.