Complex Dynamics Analysis of a Tumor–Immune Interaction Model Including Chemotherapy and the Allee Effect
- 1Department of Mathematics, Faculty of Art and Sciences, Çankaya University, 06815 Ankara, Türkiye
Abstract
In this study, we investigate the threshold-dependent dynamics of brain metastasis under chemotherapy using a nonlinear mathematical model that incorporates proliferating tumor cells, treatment-damaged cells, nonviable/necrotic cells, an associated immune response, and time-dependent treatment activity. The main objective is to determine how the Allee effect and treatment intensity jointly influence tumor persistence and elimination. The model is analyzed through equilibrium and local stability analyses, center manifold theory, and sensitivity and bifurcation analyses, complemented by numerical simulations under different treatment and initial tumor-burden scenarios. The theoretical analysis reveals that the Allee threshold plays a central role in determining the long-term tumor dynamics. In particular, increasing treatment intensity raises the smaller positive equilibrium, interpreted as a treatment-dependent effective Allee threshold, while decreasing the larger stable positive equilibrium, thereby narrowing the interval between the two positive equilibria. The two positive equilibria coalesce through a saddle-node bifurcation at a critical treatment level; for the baseline parameter set, this occurs at vc≈11.3349, beyond which no positive equilibrium exists. Sensitivity analysis identifies the parameters that most strongly influence the unstable and stable positive equilibria. Numerical simulations further reveal bistable behavior and demonstrate that, for the same treatment level, different initial tumor burdens can lead to either the tumor-free or a positive equilibrium. The numerically determined outcome boundary and phase-plane analysis further characterize this dependence on treatment intensity and initial tumor burden. Overall, the proposed framework provides a threshold-based characterization of the interaction between the Allee effect and time-dependent chemotherapy and demonstrates how treatment can influence both the equilibrium structure and the basin-dependent long-term dynamics of the system.
Related
- Fractional modeling of CD38–mediated multiple myeloma dynamics with immune interaction and therapy effects dynamics
- Kähler landscapes for complex neural network descents and guarantees including a search and destroy of the Calabi-Yau manifold
- Variational formulation for the dynamics of soft matter including inertia
- Hypocapnia in acute carbon monoxide poisoning: A complex interaction beyond metabolic compensation
- Ideal interaction dynamics of defects and background fields in condensed matter
BibTeX
@article{karaoglu2026complex,
title = {{Complex Dynamics Analysis of a Tumor–Immune Interaction Model Including Chemotherapy and the Allee Effect}},
author = {Esra Karaoğlu and Ceyda Nur Demirel},
journal = {Mathematics},
year = {2026},
month = oct,
doi = {10.3390/math14203696},
url = {https://www.mdpi.com/2227-7390/14/20/3696},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Complex Dynamics Analysis of a Tumor–Immune Interaction Model Including Chemotherapy and the Allee Effect
AU - Esra Karaoğlu
AU - Ceyda Nur Demirel
PY - 2026
DA - 2026/10/12
JO - Mathematics
DO - 10.3390/math14203696
UR - https://www.mdpi.com/2227-7390/14/20/3696
AB - In this study, we investigate the threshold-dependent dynamics of brain metastasis under chemotherapy using a nonlinear mathematical model that incorporates proliferating tumor cells, treatment-damaged cells, nonviable/necrotic cells, an associated immune response, and time-dependent treatment activity. The main objective is to determine how the Allee effect and treatment intensity jointly influence tumor persistence and elimination. The model is analyzed through equilibrium and local stability analyses, center manifold theory, and sensitivity and bifurcation analyses, complemented by numerical simulations under different treatment and initial tumor-burden scenarios. The theoretical analysis reveals that the Allee threshold plays a central role in determining the long-term tumor dynamics. In particular, increasing treatment intensity raises the smaller positive equilibrium, interpreted as a treatment-dependent effective Allee threshold, while decreasing the larger stable positive equilibrium, thereby narrowing the interval between the two positive equilibria. The two positive equilibria coalesce through a saddle-node bifurcation at a critical treatment level; for the baseline parameter set, this occurs at vc≈11.3349, beyond which no positive equilibrium exists. Sensitivity analysis identifies the parameters that most strongly influence the unstable and stable positive equilibria. Numerical simulations further reveal bistable behavior and demonstrate that, for the same treatment level, different initial tumor burdens can lead to either the tumor-free or a positive equilibrium. The numerically determined outcome boundary and phase-plane analysis further characterize this dependence on treatment intensity and initial tumor burden. Overall, the proposed framework provides a threshold-based characterization of the interaction between the Allee effect and time-dependent chemotherapy and demonstrates how treatment can influence both the equilibrium structure and the basin-dependent long-term dynamics of the system.
SN - 2227-7390
ER -