Tumor–immune dynamics: stability, optimal control, and global sensitivity analysis

Nacima Moussouni; Louiza Dehbi; Fares Yazid; Hicham Saber; Abdelkader Moumen; Khaled Aldwoah; Mohamed Bouye; Nacima Moussouni; Louiza Dehbi; Fares Yazid; Hicham Saber; Abdelkader Moumen; Khaled Aldwoah; Mohamed Bouye

doi:10.3934/math.20261219

AIMS Mathematics

2026, Volume 11, Issue 9: 30782-30814. doi: 10.3934/math.20261219

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Theory article

Tumor–immune dynamics: stability, optimal control, and global sensitivity analysis

1.
Faculty of Medicine, Amar Telidji University of Laghouat, 03000 Laghouat, Algeria
2.
L2CSP Laboratory, Mouloud Mammeri University of Tizi-Ouzou, Hasnaoua Ⅱ Campus, P.O. Box 17 RP, 15000, Tizi-Ouzou, Algeria
3.
Laboratory of Pure and Applied Mathematics, Amar Telidji University of Laghouat, P.O. Box 37 G, 03000 Laghouat, Algeria
4.
Department of Mathematics, College of Science, University of Ha'il, 55473 Ha'il, Saudi Arabia
5.
Department of Mathematics, Faculty of Science, Islamic University of Madinah, Madinah 42351, Saudi Arabia
6.
Department of Mathematics, Faculty of Sciences, King Khalid University, P.O. Box 9004, Abha, Saudi Arabia

Received: 27 February 2026 Revised: 31 August 2026 Accepted: 08 September 2026 Published: 21 September 2026
MSC : 92C50, 93C10

Oncology is one of the most important fields in modern medicine. In this work, we aimed to model tumor–immune interactions using a system of ordinary differential equations. Our objective of this modeling was to help control tumor cell growth (with a carrying capacity that is a function of time) in the human body while determining an optimal treatment strategy. To achieve this goal, we first introduced our model and studied its main characteristics. We then investigated the existence and uniqueness of solutions. Afterward, we analyzed the stability of the system. Next, we addressed the optimal control problem theoretically using Pontryagin's Maximum Principle and, numerically using direct and indirect methods, implemented it in MATLAB. We also performed numerical simulations corresponding to different treatment scenarios. Finally, we studied the sensitivity of the model parameters by means of Partial Rank Correlation Coefficients (PRCC) to identify the most influential ones.
- tumor–immune cells,
- Pontryagin's Maximum Principle,
- partial rank correlation coefficient (PRCC),
- Rosenzweig–MacArthur model
Citation: Nacima Moussouni, Louiza Dehbi, Fares Yazid, Hicham Saber, Abdelkader Moumen, Khaled Aldwoah, Mohamed Bouye. Tumor–immune dynamics: stability, optimal control, and global sensitivity analysis[J]. AIMS Mathematics, 2026, 11(9): 30782-30814. doi: 10.3934/math.20261219

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Abstract

Oncology is one of the most important fields in modern medicine. In this work, we aimed to model tumor–immune interactions using a system of ordinary differential equations. Our objective of this modeling was to help control tumor cell growth (with a carrying capacity that is a function of time) in the human body while determining an optimal treatment strategy. To achieve this goal, we first introduced our model and studied its main characteristics. We then investigated the existence and uniqueness of solutions. Afterward, we analyzed the stability of the system. Next, we addressed the optimal control problem theoretically using Pontryagin's Maximum Principle and, numerically using direct and indirect methods, implemented it in MATLAB. We also performed numerical simulations corresponding to different treatment scenarios. Finally, we studied the sensitivity of the model parameters by means of Partial Rank Correlation Coefficients (PRCC) to identify the most influential ones.

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