{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,5]],"date-time":"2026-01-05T14:45:19Z","timestamp":1767624319678,"version":"build-2065373602"},"reference-count":31,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2025,3,25]],"date-time":"2025-03-25T00:00:00Z","timestamp":1742860800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University (IMSIU)","award":["IMSIU-DDRSP2501"],"award-info":[{"award-number":["IMSIU-DDRSP2501"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper studies a model for competition between natural killer (NK) cells, cytotoxic T lymphocytes (CTLs) and tumor cells, and evaluates the outcomes in the absence and presence of chemotherapy treatment. The growth rate of the tumor is presumed to follow the classical logistic law. The model particularly emphasizes the rate-limiting recruitment of NK cells and CTL cells, which is activated by the presence of the tumor. It additionally includes the activation of CTL cells through debris produced by the lysis of tumor cells by NK cells, alongside the regulatory effect that NK cells have on CTL cells. Additionally, the model incorporates the reciprocal decreases in cell populations resulting from the interactions between tumor cells and immune cells, along with the impact of chemotherapy on all three types of cells. We analyze the stability of the equilibrium points. Utilizing parameter values that have been experimentally confirmed in the literature and applying some elementary principles of singularity theory, we investigate the bistability regimes anticipated by the model in the absence of chemotherapy, and evaluate the impact of model parameters on this behavior. This mathematical analysis serves to evaluate the effectiveness of chemotherapy treatment. We demonstrate that the interplay between the biological parameters in the model and those associated with chemotherapy can result in a range of treatment outcomes. The proposed mathematical analysis may serve as a valuable tool in directing the development of strategies for treatment interventions.<\/jats:p>","DOI":"10.3390\/sym17040492","type":"journal-article","created":{"date-parts":[[2025,3,25]],"date-time":"2025-03-25T12:18:52Z","timestamp":1742905132000},"page":"492","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Dynamics of a Symmetric Model of Competition Between Tumor and Immune Cells Under Chemotherapy"],"prefix":"10.3390","volume":"17","author":[{"given":"Abdelhamid","family":"Ajbar","sequence":"first","affiliation":[{"name":"Department of Chemical Engineering, College of Engineering, King Saud University, Riyadh 11421, Saudi Arabia"}]},{"given":"Rubayyi","family":"Alqahtani","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11623, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2025,3,25]]},"reference":[{"key":"ref_1","first-page":"229","article-title":"Global cancer statistics 2022: GLOBOCAN estimates of incidence and mortality worldwide for 36 cancers in 185 countries","volume":"74","author":"Bray","year":"2024","journal-title":"CA Cancer J. Clin."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"961","DOI":"10.3332\/ecancer.2019.961","article-title":"Innovative approaches for cancer treatment: Current perspectives and new challenges","volume":"13","author":"Pucci","year":"2019","journal-title":"Ecancermedicalscience"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"3722","DOI":"10.1007\/s11538-019-00640-x","article-title":"Mathematical Models of Cancer: When to predict novel therapies, and when not to","volume":"81","author":"Brady","year":"2019","journal-title":"Bull. Math. Biol."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1200\/CCI.19.00010","article-title":"Introduction to Mathematical Oncology","volume":"3","author":"Rockne","year":"2019","journal-title":"JCO Clin. Cancer Inform."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Apavaloaei, A., Hardy, M.P., Thibault, P., and Perreault, C. (2020). The origin and immune recognition of tumor-specific antigens. Cancers, 12.","DOI":"10.20944\/preprints202008.0649.v1"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"298","DOI":"10.1038\/s41568-021-00339-z","article-title":"Antigen presentation in cancer: Insights into tumour immunogenicity and immune evasion","volume":"21","author":"Jhunjhunwala","year":"2021","journal-title":"Nat. Rev. Cancer"},{"key":"ref_7","unstructured":"Rich, R.R., Fleisher, T.A., Shearer, W.T., Schroeder, H.W., Frew, A.J., and Weyand, C.M. (2019). Cytotoxic T Lymphocytes and Natural Killer Cells, in Clinical Immunology, Elsevier. [5th ed.]."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"671","DOI":"10.1038\/s41577-018-0061-z","article-title":"Natural killer cells and other innate lymphoid cells in cancer","volume":"18","author":"Chiossone","year":"2018","journal-title":"Nat. Rev. Immunol."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"614","DOI":"10.1038\/s41419-024-06976-0","article-title":"Natural Killer cells at the frontline in the fight against cancer","volume":"15","author":"Geindreau","year":"2024","journal-title":"Cell Death Dis."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2052","DOI":"10.1016\/j.cell.2024.03.037","article-title":"Principles and therapeutic applications of adaptive immunity","volume":"187","author":"Hongbo","year":"2024","journal-title":"Cell"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"646","DOI":"10.1142\/S0218202508002796","article-title":"On the foundations of cancer modelling: Selected topics, speculations, and perspectives","volume":"18","author":"Bellomo","year":"2008","journal-title":"Math. Models Methods Appl. Sci."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Mohammad Mirzaei, N., Tatarova, Z., Hao, W., Changizi, N., Asadpoure, A., and Zervantonakis, I.K. (2022). A PDE model of breast tumor progression in MMTV-PyMT Mice. J. Pers. Med., 12.","DOI":"10.3390\/jpm12050807"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Ghaffari Laleh, N., Loeffler, C.M.L., Grajek, J., Sta\u0148kov\u00e1, K., Pearson, A.T., and Muti, H.S. (2022). Classical mathematical models for prediction of response to chemotherapy and immunotherapy. PLoS Comput. Biol., 18.","DOI":"10.1371\/journal.pcbi.1009822"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"100534","DOI":"10.1016\/j.imu.2021.100534","article-title":"Prospect for application of mathematical models in combination cancer treatments","volume":"23","author":"Malinzi","year":"2023","journal-title":"Inform. Med. Unlocked"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Kuznetsov, V., Makalkin, I., Taylor, M., and Perelson, A. (1994). Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis. Bull. Math. Biol., 56.","DOI":"10.1016\/S0092-8240(05)80260-5"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Robertson-Tessi, M., Elkareh, A., and Goriely, A. (2012). A mathematical model of tumor-immune interactions. J. Theor. Biol., 294.","DOI":"10.1016\/j.jtbi.2011.10.027"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Eladdadi, A., Kim, P., and Mallet, D. (2014). Modeling tumor-immune dynamics. Mathematical Models of Tumor-Immune System Dynamics, Springer.","DOI":"10.1007\/978-1-4939-1793-8"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"L\u00f3pez, A.G., Seoane, J.M., and Sanju\u00e1n, M.A.F. (2014). A validated mathematical model of tumor growth including tumor-host interaction, cell-mediated immune response and chemotherapy. Bull. Math. Biol., 76.","DOI":"10.1007\/s11538-014-0037-5"},{"key":"ref_19","first-page":"7602","article-title":"Mathematical modelling for the role of CD4+ T cells in tumor-immune interactions","volume":"718","author":"Makhlouf","year":"2020","journal-title":"Comput. Math. Methods Med."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Song, G., Tian, T., and Zhang, X. (2020). A mathematical model of cell-mediated immune response to tumor. Math. Biosci. Eng., 18.","DOI":"10.3934\/mbe.2021020"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"7983","DOI":"10.1002\/mma.7706","article-title":"Modeling and analysis of nonlinear tumor-immune interaction under chemotherapy and radiotherapy","volume":"45","author":"Bashkirtseva","year":"2022","journal-title":"Math. Meth. Appl. Sci."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"116","DOI":"10.3934\/mmc.2023011","article-title":"Stability analysis of a targeted chemotherapy-cancer model","volume":"3","author":"Das","year":"2023","journal-title":"Math. Model Control."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1016\/j.apnum.2024.08.019","article-title":"Existence, uniqueness and Ulam-Hyers stability result for variable order fractional predator-prey system and it\u2019s numerical solution","volume":"207","author":"Kashif","year":"2025","journal-title":"Appl. Numer. Mat."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Feng, X., Liu, M., Jiang, Y., and Li, D. (2023). Dynamics and stability of a fractional-order tumor-immune interaction model with B-D functional response and immunotherapy. Fractal Fract., 7.","DOI":"10.3390\/fractalfract7020200"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Wiggins, S. (1990). Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer.","DOI":"10.1007\/978-1-4757-4067-7"},{"key":"ref_26","first-page":"69","article-title":"Singularities and Groups in Bifurcation Theory","volume":"2","author":"Golubitsky","year":"1988","journal-title":"Appl. Math. Sci."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Ajbar, A., and Alhumaizi, K. (2011). Dynamics of the Chemostat A Bifurcation Theory Approach, Chapman and Hall\/CRC.","DOI":"10.1201\/b11073"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1080\/13873950701742754","article-title":"New features of the software MatCont for bifurcation analysis of dynamical systems","volume":"14","author":"Dhooge","year":"2008","journal-title":"Math. Comput. Model. Dyn. Syst."},{"key":"ref_29","unstructured":"MATLAB (2018). Version 9.4.0 (R2018a), The MathWorks Inc."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"850","DOI":"10.1126\/science.1076514","article-title":"Cancer regression and autoimmunity in patients after clonal repopulation with antitumor lymphocytes","volume":"298","author":"Dudley","year":"2002","journal-title":"Science"},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Hirsch, M.W., Smale, S., and Devaney, R.L. (2012). Differential Equations, Dynamical Systems, and an Introduction to Chaos, Academic Press.","DOI":"10.1016\/B978-0-12-382010-5.00015-4"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/4\/492\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T16:59:58Z","timestamp":1760029198000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/4\/492"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,3,25]]},"references-count":31,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2025,4]]}},"alternative-id":["sym17040492"],"URL":"https:\/\/doi.org\/10.3390\/sym17040492","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2025,3,25]]}}}