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Abstract

This study develops a fractional-order tumor-immune interaction model incorporating Caputo memory effects, delayed immune activation, and CTLA-4 checkpoint regulation. The model describes the coupled dynamics of tumor cells, CD4$^{+}$ T cells, IFN-$\gamma$, and CTLA-4, and extends classical integer-order tumor-immune models by accounting for hereditary immune responses and biologically motivated latency effects. Theoretical properties, including positivity, boundedness, equilibrium structure, and fractional-order stability, are examined to establish the biological and mathematical consistency of the model. The delayed fractional system is then investigated computationally by comparing several numerical methods, including finite difference discretization, Daubechies wavelet collocation, Euler wavelet collocation, and a predictor-corrector scheme. The simulations show consistent behavior across the numerical methods and indicate that both the fractional order and immune delay can substantially influence the timing and strength of tumor suppression. An exploratory optimal-control formulation for immune checkpoint inhibition is also introduced to illustrate how time-dependent therapeutic modulation may influence tumor burden and treatment effort. The main novelty of this work lies in integrating fractional memory, immune-response delay, and CTLA-4-mediated regulation within a unified tumor-immune modeling framework, together with an exploratory checkpoint-based treatment scenario. The numerical methods are used primarily to validate the consistency of the computed dynamics and to examine how memory and delay influence tumor suppression.

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