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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. [ABSTRACT FROM AUTHOR]

• A mathematical model of the interactions between Th1 and Th2 cells with Treg cell inhibition and stochastic effects is developed. • Sufficient conditions for different asymptotic phases of Th1 and Th2 responses under the Treg cell regulation are obtained. • Numerical simulations are performed to identify the basins of attraction of different steady states and illustrate different dynamical scenarios. • The effect of stochastic effect under the Treg cell regulation is discussed. • The switching probability and the mean residence time are applied to study the effect of noise. T cells differentiate into Th1 or Th2 cells upon maturation to influence different patterns of the immune response. Th1 and Th2 cells regulate each other and their responses are inhibited by Treg cells. With noisy external stimulation, Th1/Th2 cell differentiation can be dynamically balanced. The underlying mechanisms of Th cell differentiation under Treg cell inhibition and the extrinsic noise effects are not yet completely understood. In this paper, a mathematical model of the interactions between Th1 and Th2 cells with Treg cell inhibition and stochastic effects is developed to study the preference of outcomes and the noise-induced hopping among different states. First, we provide the conditions for different asymptotic phases of Th1 and Th2 responses under Treg cell regulation. Numerical simulations are applied to calculate the switching probability and the mean residence time to study how the noise affects the attractiveness of different states. Our results support that due to the stronger inhibitory effect of Treg cells on Th1 cell development, the high-Th2-low-Th1 state is more attractive under small noise effects. Additionally, we show that the attractiveness of the states is affected mainly by the extrinsic noise in Th2 cell signaling. [ABSTRACT FROM AUTHOR]