Document Type

Article

Source of Publication

Mathematical Modelling and Numerical Simulation with Applications

Publication Date

6-30-2026

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-γ, 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.

ISSN

2791-8564

Publisher

Necmettin Erbakan University Press

Volume

6

Issue

2

First Page

435

Last Page

486

Disciplines

Computer Sciences | Medicine and Health Sciences

Keywords

Caputo fractional derivative, fractional differential equations, immunotherapy modeling, memory effects, time-delay systems, Tumor-immune interactions, wavelet collocation methods

Scopus ID

105044800104

Creative Commons License

Creative Commons Attribution 4.0 International License
This work is licensed under a Creative Commons Attribution 4.0 International License.

Indexed in Scopus

yes

Open Access

yes

Open Access Type

Hybrid: This publication is openly available in a subscription-based journal/series

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