https://doi.org/10.1140/epjp/s13360-026-07747-w
Regular Article
Existence results and stability analysis for a fractional spatiotemporal brain tumor and immune system interaction
1
Department of Mathematics, Laboratory of Applied Mathematics and Scientific Computing, Sultan Moulay Slimane University, BP 523, 23000, Beni Mellal, Morocco
2
Department of Mathematics, Sonari College, 785690, Sonari, Assam, India
3
Mathematics Research Center, Near East University, Mersin-10, Nicosia, Türkiye
4
Research Center of Applied Mathematics, Khazar University, Baku, Azerbaijan
5
LERSEM Laboratory, ENCGJ University Chouaib Doukkali, El Jadida, Morocco
6
Department of Mathematics, Mathematics Research Center, Near East University, Mersin-10, Nicosia, Türkiye
7
Faculty of Medicine, Department of Biostatistics and Medical Informatics, Karadeniz Technical University, Trabzon, Türkiye
8
International Centre for Interdisciplinary Research in Sciences, The University of Lahore, Lahore, Pakistan
9
Symbiosis Institute of Technology, Hyderabad Campus, Symbiosis International (Deemed University), Pune, India
a
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Received:
30
October
2025
Accepted:
25
April
2026
Published online:
26
May
2026
Abstract
In this paper, we present an innovative model that provides a comprehensive framework for understanding the complex interactions between glioma cells and immune cells in the progression of brain cancer. This study stands out by addressing the limitations of traditional diffusion models, which often fall short in capturing the irregular and non-local movements of cells within the tumor microenvironment. To overcome these challenges, we employ fractional time derivatives and the fractional Laplacian operator, offering a more accurate depiction of the anomalous diffusion processes crucial to cancer dynamics. Our research begins with a rigorous mathematical analysis, establishing the global existence of a unique mild solution, thereby providing a solid theoretical foundation for the model. A key contribution of our study is the introduction of the “invasion threshold,” a critical parameter inspired by the next-generation operator from epidemiological modeling. This threshold serves as a powerful tool for determining the existence and stability of equilibrium points, offering valuable insights into the conditions that either promote or inhibit tumor growth.
Key words: 2020 AMS Subject Classification / 37N25 / 35R11
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© The Author(s), under exclusive licence to Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2026
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
