A Fractional-Order Tumor-Immune-Healthy Tissue Interaction Model: Mathematical Analysis and Numerical Investigation Using the Atangana-Baleanu Derivative

Gizachew Tirite Gellow

doi:doi:10.11648/j.sdcomput.20260102.12

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A Fractional-Order Tumor-Immune-Healthy Tissue Interaction Model: Mathematical Analysis and Numerical Investigation Using the Atangana-Baleanu Derivative

Gizachew Tirite Gellow^*

Published in Science Discovery Computers (Volume 1, Issue 2)

Received: 11 July 2026 Accepted: 22 August 2026 Published: 3 November 2026

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Abstract

Tumor progression is governed by nonlinear interactions among malignant cells, immune effector cells, and the surrounding healthy tissue. Such interactions are influenced by previous biological states, including accumulated immune stimulation, tissue damage, and cellular adaptation. Conventional integer-order models describe the instantaneous rate of change but do not explicitly represent this history dependence. In this work, a fractional-order tumor–immune–healthy tissue model is formulated using the Atangana–Baleanu Caputo (ABC) fractional derivative, whose nonsingular Mittag–Leffler kernel provides a nonlocal representation of biological memory. The model consists of three interacting populations: tumor cells, immune effector cells, and healthy tissue. Tumor proliferation is represented by logistic growth, immune-mediated tumor elimination and immune suppression are incorporated through nonlinear interaction terms, and healthy tissue regeneration is balanced against tumor-induced damage. The model is first analyzed mathematically. Under explicit regularity and parameter assumptions, local existence and uniqueness of solutions are established by a fixed-point argument. Positivity is obtained from the vector field on the boundary of the nonnegative orthant, and boundedness follows from comparison estimates. The fractional formulation is then converted into its equivalent Volterra integral equation and a history-weighted product-integration discretization with an implicit correction is implemented for numerical computation. Numerical illustrations are used to examine the effect of fractional order on tumor, immune, and healthy-tissue trajectories. The results indicate that decreasing the fractional order changes the transient response and introduces a stronger dependence on previous states, whereas the integer-order case is recovered at α=1. The study therefore provides a mathematically consistent framework for examining memory-dependent tumor dynamics and identifies directions for subsequent calibration with clinical or experimental data.

Published in	Science Discovery Computers (Volume 1, Issue 2)
DOI	10.11648/j.sdcomput.20260102.12
Page(s)	80-91
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2026. Published by Science Publishing Group

Keywords

Fractional Calculus, Atangana-Baleanu Caputo Derivative, Tumor-immune Interaction, Healthy Tissue, Mathematical Biology, Memory Effects, Fractional Differential Equations

References

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[15]	E. A. A. Ziada, Picard and Adomian solutions of nonlinear fractional differential equations system containing Atangana–Baleanu derivative, Journal of Engineering and Applied Science, 71, 31, 2024. https://doi.org/10.1186/s44147-024-00361-6
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Gellow, G. T. (2026). A Fractional-Order Tumor-Immune-Healthy Tissue Interaction Model: Mathematical Analysis and Numerical Investigation Using the Atangana-Baleanu Derivative. Science Discovery Computers, 1(2), 80-91. https://doi.org/10.11648/j.sdcomput.20260102.12

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ACS Style

Gellow, G. T. A Fractional-Order Tumor-Immune-Healthy Tissue Interaction Model: Mathematical Analysis and Numerical Investigation Using the Atangana-Baleanu Derivative. Sci. Discov. Comput. 2026, 1(2), 80-91. doi: 10.11648/j.sdcomput.20260102.12

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AMA Style

Gellow GT. A Fractional-Order Tumor-Immune-Healthy Tissue Interaction Model: Mathematical Analysis and Numerical Investigation Using the Atangana-Baleanu Derivative. Sci Discov Comput. 2026;1(2):80-91. doi: 10.11648/j.sdcomput.20260102.12

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@article{10.11648/j.sdcomput.20260102.12,
  author = {Gizachew Tirite Gellow},
  title = {A Fractional-Order Tumor-Immune-Healthy Tissue Interaction Model: Mathematical Analysis and Numerical Investigation Using the Atangana-Baleanu Derivative},
  journal = {Science Discovery Computers},
  volume = {1},
  number = {2},
  pages = {80-91},
  doi = {10.11648/j.sdcomput.20260102.12},
  url = {https://doi.org/10.11648/j.sdcomput.20260102.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sdcomput.20260102.12},
  abstract = {Tumor progression is governed by nonlinear interactions among malignant cells, immune effector cells, and the surrounding healthy tissue. Such interactions are influenced by previous biological states, including accumulated immune stimulation, tissue damage, and cellular adaptation. Conventional integer-order models describe the instantaneous rate of change but do not explicitly represent this history dependence. In this work, a fractional-order tumor–immune–healthy tissue model is formulated using the Atangana–Baleanu Caputo (ABC) fractional derivative, whose nonsingular Mittag–Leffler kernel provides a nonlocal representation of biological memory. The model consists of three interacting populations: tumor cells, immune effector cells, and healthy tissue. Tumor proliferation is represented by logistic growth, immune-mediated tumor elimination and immune suppression are incorporated through nonlinear interaction terms, and healthy tissue regeneration is balanced against tumor-induced damage. The model is first analyzed mathematically. Under explicit regularity and parameter assumptions, local existence and uniqueness of solutions are established by a fixed-point argument. Positivity is obtained from the vector field on the boundary of the nonnegative orthant, and boundedness follows from comparison estimates. The fractional formulation is then converted into its equivalent Volterra integral equation and a history-weighted product-integration discretization with an implicit correction is implemented for numerical computation. Numerical illustrations are used to examine the effect of fractional order on tumor, immune, and healthy-tissue trajectories. The results indicate that decreasing the fractional order changes the transient response and introduces a stronger dependence on previous states, whereas the integer-order case is recovered at α=1. The study therefore provides a mathematically consistent framework for examining memory-dependent tumor dynamics and identifies directions for subsequent calibration with clinical or experimental data.},
 year = {2026}
}

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TY  - JOUR
T1  - A Fractional-Order Tumor-Immune-Healthy Tissue Interaction Model: Mathematical Analysis and Numerical Investigation Using the Atangana-Baleanu Derivative
AU  - Gizachew Tirite Gellow
Y1  - 2026/11/03
PY  - 2026
N1  - https://doi.org/10.11648/j.sdcomput.20260102.12
DO  - 10.11648/j.sdcomput.20260102.12
T2  - Science Discovery Computers
JF  - Science Discovery Computers
JO  - Science Discovery Computers
SP  - 80
EP  - 91
PB  - Science Publishing Group
UR  - https://doi.org/10.11648/j.sdcomput.20260102.12
AB  - Tumor progression is governed by nonlinear interactions among malignant cells, immune effector cells, and the surrounding healthy tissue. Such interactions are influenced by previous biological states, including accumulated immune stimulation, tissue damage, and cellular adaptation. Conventional integer-order models describe the instantaneous rate of change but do not explicitly represent this history dependence. In this work, a fractional-order tumor–immune–healthy tissue model is formulated using the Atangana–Baleanu Caputo (ABC) fractional derivative, whose nonsingular Mittag–Leffler kernel provides a nonlocal representation of biological memory. The model consists of three interacting populations: tumor cells, immune effector cells, and healthy tissue. Tumor proliferation is represented by logistic growth, immune-mediated tumor elimination and immune suppression are incorporated through nonlinear interaction terms, and healthy tissue regeneration is balanced against tumor-induced damage. The model is first analyzed mathematically. Under explicit regularity and parameter assumptions, local existence and uniqueness of solutions are established by a fixed-point argument. Positivity is obtained from the vector field on the boundary of the nonnegative orthant, and boundedness follows from comparison estimates. The fractional formulation is then converted into its equivalent Volterra integral equation and a history-weighted product-integration discretization with an implicit correction is implemented for numerical computation. Numerical illustrations are used to examine the effect of fractional order on tumor, immune, and healthy-tissue trajectories. The results indicate that decreasing the fractional order changes the transient response and introduces a stronger dependence on previous states, whereas the integer-order case is recovered at α=1. The study therefore provides a mathematically consistent framework for examining memory-dependent tumor dynamics and identifies directions for subsequent calibration with clinical or experimental data.
VL  - 1
IS  - 2
ER  -

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Gizachew Tirite Gellow

Department of Applied Mathematics, Debre Tabor Universty, Debre Tabor, Ethiopia

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http://orcid.org/0009-0002-0756-0361

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APA Style

Gellow, G. T. (2026). A Fractional-Order Tumor-Immune-Healthy Tissue Interaction Model: Mathematical Analysis and Numerical Investigation Using the Atangana-Baleanu Derivative. Science Discovery Computers, 1(2), 80-91. https://doi.org/10.11648/j.sdcomput.20260102.12

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ACS Style

Gellow, G. T. A Fractional-Order Tumor-Immune-Healthy Tissue Interaction Model: Mathematical Analysis and Numerical Investigation Using the Atangana-Baleanu Derivative. Sci. Discov. Comput. 2026, 1(2), 80-91. doi: 10.11648/j.sdcomput.20260102.12

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AMA Style

Gellow GT. A Fractional-Order Tumor-Immune-Healthy Tissue Interaction Model: Mathematical Analysis and Numerical Investigation Using the Atangana-Baleanu Derivative. Sci Discov Comput. 2026;1(2):80-91. doi: 10.11648/j.sdcomput.20260102.12

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@article{10.11648/j.sdcomput.20260102.12,
  author = {Gizachew Tirite Gellow},
  title = {A Fractional-Order Tumor-Immune-Healthy Tissue Interaction Model: Mathematical Analysis and Numerical Investigation Using the Atangana-Baleanu Derivative},
  journal = {Science Discovery Computers},
  volume = {1},
  number = {2},
  pages = {80-91},
  doi = {10.11648/j.sdcomput.20260102.12},
  url = {https://doi.org/10.11648/j.sdcomput.20260102.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sdcomput.20260102.12},
  abstract = {Tumor progression is governed by nonlinear interactions among malignant cells, immune effector cells, and the surrounding healthy tissue. Such interactions are influenced by previous biological states, including accumulated immune stimulation, tissue damage, and cellular adaptation. Conventional integer-order models describe the instantaneous rate of change but do not explicitly represent this history dependence. In this work, a fractional-order tumor–immune–healthy tissue model is formulated using the Atangana–Baleanu Caputo (ABC) fractional derivative, whose nonsingular Mittag–Leffler kernel provides a nonlocal representation of biological memory. The model consists of three interacting populations: tumor cells, immune effector cells, and healthy tissue. Tumor proliferation is represented by logistic growth, immune-mediated tumor elimination and immune suppression are incorporated through nonlinear interaction terms, and healthy tissue regeneration is balanced against tumor-induced damage. The model is first analyzed mathematically. Under explicit regularity and parameter assumptions, local existence and uniqueness of solutions are established by a fixed-point argument. Positivity is obtained from the vector field on the boundary of the nonnegative orthant, and boundedness follows from comparison estimates. The fractional formulation is then converted into its equivalent Volterra integral equation and a history-weighted product-integration discretization with an implicit correction is implemented for numerical computation. Numerical illustrations are used to examine the effect of fractional order on tumor, immune, and healthy-tissue trajectories. The results indicate that decreasing the fractional order changes the transient response and introduces a stronger dependence on previous states, whereas the integer-order case is recovered at α=1. The study therefore provides a mathematically consistent framework for examining memory-dependent tumor dynamics and identifies directions for subsequent calibration with clinical or experimental data.},
 year = {2026}
}

Copy | Download

TY  - JOUR
T1  - A Fractional-Order Tumor-Immune-Healthy Tissue Interaction Model: Mathematical Analysis and Numerical Investigation Using the Atangana-Baleanu Derivative
AU  - Gizachew Tirite Gellow
Y1  - 2026/11/03
PY  - 2026
N1  - https://doi.org/10.11648/j.sdcomput.20260102.12
DO  - 10.11648/j.sdcomput.20260102.12
T2  - Science Discovery Computers
JF  - Science Discovery Computers
JO  - Science Discovery Computers
SP  - 80
EP  - 91
PB  - Science Publishing Group
UR  - https://doi.org/10.11648/j.sdcomput.20260102.12
AB  - Tumor progression is governed by nonlinear interactions among malignant cells, immune effector cells, and the surrounding healthy tissue. Such interactions are influenced by previous biological states, including accumulated immune stimulation, tissue damage, and cellular adaptation. Conventional integer-order models describe the instantaneous rate of change but do not explicitly represent this history dependence. In this work, a fractional-order tumor–immune–healthy tissue model is formulated using the Atangana–Baleanu Caputo (ABC) fractional derivative, whose nonsingular Mittag–Leffler kernel provides a nonlocal representation of biological memory. The model consists of three interacting populations: tumor cells, immune effector cells, and healthy tissue. Tumor proliferation is represented by logistic growth, immune-mediated tumor elimination and immune suppression are incorporated through nonlinear interaction terms, and healthy tissue regeneration is balanced against tumor-induced damage. The model is first analyzed mathematically. Under explicit regularity and parameter assumptions, local existence and uniqueness of solutions are established by a fixed-point argument. Positivity is obtained from the vector field on the boundary of the nonnegative orthant, and boundedness follows from comparison estimates. The fractional formulation is then converted into its equivalent Volterra integral equation and a history-weighted product-integration discretization with an implicit correction is implemented for numerical computation. Numerical illustrations are used to examine the effect of fractional order on tumor, immune, and healthy-tissue trajectories. The results indicate that decreasing the fractional order changes the transient response and introduces a stronger dependence on previous states, whereas the integer-order case is recovered at α=1. The study therefore provides a mathematically consistent framework for examining memory-dependent tumor dynamics and identifies directions for subsequent calibration with clinical or experimental data.
VL  - 1
IS  - 2
ER  -

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