{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,1]],"date-time":"2026-02-01T05:07:19Z","timestamp":1769922439605,"version":"3.49.0"},"reference-count":39,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2024,5,31]],"date-time":"2024-05-31T00:00:00Z","timestamp":1717113600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12371488"],"award-info":[{"award-number":["12371488"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["20K03755"],"award-info":[{"award-number":["20K03755"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Japan Society for the Promotion of Science","award":["12371488"],"award-info":[{"award-number":["12371488"]}]},{"name":"Japan Society for the Promotion of Science","award":["20K03755"],"award-info":[{"award-number":["20K03755"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Tumor\u2013immune system interactions are very complicated, being highly nonlinear and not well understood. A large number of tumors can potentially weaken the immune system through various mechanisms such as secreting cytokines that suppress the immune response. In this paper, we propose a tumor\u2013immune system interaction model with a nonmonotonic immune response function and adoptive cellular immunotherapy (ACI). The model has a tumor-free equilibrium and at most three tumor-presence equilibria (low, moderate and high ones). The stability of all equilibria is studied by analyzing their characteristic equations. The consideration of nonmonotonic immune response results in a series of bifurcations such as the saddle-node bifurcation, transcritical bifurcation, Hopf bifurcation and Bogdanov\u2013Takens bifurcation. In addition, numerical simulation results show the coexistence of periodic orbits and homoclinic orbits. Interestingly, along with various bifurcations, we also found two bistable scenarios: the coexistence of a stable tumor-free as well as a high-tumor-presence equilibrium and the coexistence of a stable-low as well as a high-tumor-presence equilibrium, which can show symmetric and antisymmetric properties in a range of model parameters and initial cell concentrations. The new findings indicate that under ACI, patients can possibly reach either a stable tumor-free state or a low-tumor-presence state in the presence of nonmonotonic immune response once the immune system is activated.<\/jats:p>","DOI":"10.3390\/sym16060676","type":"journal-article","created":{"date-parts":[[2024,6,3]],"date-time":"2024-06-03T10:06:01Z","timestamp":1717409161000},"page":"676","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Modeling the Nonmonotonic Immune Response in a Tumor\u2013Immune System Interaction"],"prefix":"10.3390","volume":"16","author":[{"given":"Yu","family":"Liu","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China"},{"name":"School of Information Technology, Guangxi Police College, Nanning 530028, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yuhang","family":"Ma","sequence":"additional","affiliation":[{"name":"School of Public Health, Nanjing Medical University, Nanjing 211166, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Cuihong","family":"Yang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4372-2487","authenticated-orcid":false,"given":"Zhihang","family":"Peng","sequence":"additional","affiliation":[{"name":"School of Public Health, Nanjing Medical University, Nanjing 211166, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0004-3234-1210","authenticated-orcid":false,"given":"Yasuhiro","family":"Takeuchi","sequence":"additional","affiliation":[{"name":"College of Science and Engineering, Aoyama Gakuin University, Kanagawa 252-5258, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4327-5904","authenticated-orcid":false,"given":"Malay","family":"Banerjee","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur 208016, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yueping","family":"Dong","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China"},{"name":"Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan 430079, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,5,31]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1111\/nyas.12105","article-title":"Cancer Immunoediting: Antigens, mechanisms and implications to cancer immunotherapy","volume":"1284","author":"Vesely","year":"2013","journal-title":"Ann. N. Y. Acad. Sci."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1199173","DOI":"10.3389\/fimmu.2023.1199173","article-title":"T cells, NK cells, and tumor-associated macrophages in cancer immunotherapy and the current state of the art of drug delivery systems","volume":"14","author":"Yang","year":"2023","journal-title":"Front. Immunol."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"687","DOI":"10.1158\/1078-0432.CCR-14-1860","article-title":"Immune escape mechanisms as a guide for cancer immunotherapy","volume":"21","author":"Beatty","year":"2015","journal-title":"Clin. Cancer Res."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"868695","DOI":"10.3389\/fphar.2022.868695","article-title":"The evasion mechanisms of cancer immunity and drug intervention in the tumor microenvironment","volume":"13","author":"Kim","year":"2022","journal-title":"Front. Pharmacol."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Tan, S., Li, D., and Zhu, Z. (2020). T cells, Cancer immunotherapy: Pros, cons and beyond. Biomed. Pharmacother., 124.","DOI":"10.1016\/j.biopha.2020.109821"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Farkona, S., Diamandis, E.P., and Blasutig, I.M. (2016). Cancer immunotherapy: The beginning of the end of cancer?. BMC Med., 14.","DOI":"10.1186\/s12916-016-0623-5"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"651","DOI":"10.1038\/s41577-020-0306-5","article-title":"A guide to cancer immunotherapy: From T cell basic science to clinical practice","volume":"20","author":"Waldman","year":"2020","journal-title":"Nat. Rev. Immunol."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"2","DOI":"10.1007\/s11538-010-9526-3","article-title":"Interactions between the immune system and cancer: A brief review of non-spatial mathematical models","volume":"73","author":"Eftimie","year":"2011","journal-title":"Bull. Math. Biol."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1080\/10273660108833067","article-title":"A mathematical tumor model with immune resistance and drug therapy: An optimal control approach","volume":"3","author":"Radunskaya","year":"2001","journal-title":"Computat. Math. Methods Med."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"7950","DOI":"10.1158\/0008-5472.CAN-05-0564","article-title":"A validated mathematical model of cell-mediated immune response to tumor growth","volume":"65","author":"Radunskaya","year":"2005","journal-title":"Cancer Res."},{"key":"ref_11","first-page":"55","article-title":"Mathematical modeling on helper T cells in a tumor immune system","volume":"19","author":"Dong","year":"2014","journal-title":"Discret. Contin. Dyn. Syst. Ser. B"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1485","DOI":"10.1007\/s11538-012-9722-4","article-title":"A mathematical model of the enhancement of tumor vaccine efficacy by immunotherapy","volume":"74","author":"Wilson","year":"2012","journal-title":"Bull. Math. Biol."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1016\/j.jtbi.2015.10.034","article-title":"The re-polarisation of M2 and M1 macrophages and its role on cancer outcomes","volume":"390","author":"Eftimie","year":"2016","journal-title":"J. Theor. Biol."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"758","DOI":"10.1016\/j.apm.2020.06.042","article-title":"Mathematical modeling and bifurcation analysis of pro- and anti-tumor macrophages","volume":"88","author":"Shu","year":"2020","journal-title":"Appl. Math. Model."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"614","DOI":"10.1016\/j.mcm.2007.02.032","article-title":"Metamodeling tumor-immune systeminteraction, tumor evasion and immunotherapy","volume":"47","year":"2008","journal-title":"Math. Comput. Model."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1016\/j.jtbi.2019.03.002","article-title":"Mathematical modeling of tumor-immune cell interactions","volume":"469","author":"Mahlbacher","year":"2019","journal-title":"J. Theor. Biol."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"2405","DOI":"10.1137\/20M1378065","article-title":"Dynamics of a diffusive nutrient-phytoplankton-zooplankton model with spatio-temporal delay","volume":"81","author":"Tao","year":"2021","journal-title":"SIAM J. Appl. Math."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"441","DOI":"10.1007\/s10255-022-1088-2","article-title":"A delayed HIV infection model with the homeostatic proliferation of CD4 +T cells","volume":"38","author":"Xu","year":"2022","journal-title":"Acta Math. Appl. Sin. Engl. Ser."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1007\/s00332-023-09907-z","article-title":"Nilpotent singularities and periodic perturbation of a GI\u03b2 model: A pathway to glucose disorder","volume":"33","author":"Tao","year":"2023","journal-title":"J. Nonlinear Sci."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"295","DOI":"10.1007\/BF02460644","article-title":"Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis","volume":"56","author":"Kuznetsov","year":"1994","journal-title":"Bull. Math. Biol."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1007\/s002850050127","article-title":"Modeling immunotherapy of the tumor-immune interaction","volume":"37","author":"Kirschner","year":"1998","journal-title":"J. Math. Biol."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"S36","DOI":"10.30707\/LiB5.2Dritschel","article-title":"A mathematical model of cytotoxic and helper T cell interactions in a tumour microenvironment","volume":"5","author":"Dritschel","year":"2018","journal-title":"Lett. Biomath."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1059","DOI":"10.1007\/s11538-017-0263-8","article-title":"Ordinary differential equation models for adoptive immunotherapy","volume":"80","author":"Talkington","year":"2018","journal-title":"Bull. Math. Biol."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Song, G., Liang, G., Tian, T., and Zhang, X. (2022). Mathematical modeling and analysis of tumor chemotherapy. Symmetry, 14.","DOI":"10.3390\/sym14040704"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"100796","DOI":"10.1016\/j.neo.2022.100796","article-title":"Mathematical modeling of radiotherapy and its impact on tumor interactions with the immune system","volume":"28","author":"Bekker","year":"2022","journal-title":"Neoplasia"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"785","DOI":"10.1038\/s43588-022-00377-z","article-title":"Mathematical modeling of cancer immunotherapy for personalized clinical translation","volume":"2","author":"Butner","year":"2022","journal-title":"Nat. Comput. Sci."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Tang, S., Li, S., Wang, X., Xiao, Y., and Checke, R.A. (2023). Hormetic and synergistic effects of cancer treatments revealed by modelling combinations of radio-or chemotherapy with immunotherapy. BMC Cancer, 23.","DOI":"10.1186\/s12885-023-11542-6"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1892","DOI":"10.1137\/22M1533979","article-title":"Bifurcation analysis in a tumor-immune system interaction model with dendritic cell therapy and immune response delay","volume":"83","author":"Zhang","year":"2023","journal-title":"SIAM J. Appl. Math."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"24","DOI":"10.1038\/s41368-021-00131-7","article-title":"Tumor microenvironment and immune evasion in head and neck squamous cell carcinoma","volume":"13","author":"Elmusrati","year":"2021","journal-title":"Int. J. Oral Sci."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"S185","DOI":"10.1016\/j.semcancer.2015.03.004","article-title":"Immune evasion in cancer: Mechanistic basis and therapeutic strategies","volume":"35","author":"Vinay","year":"2015","journal-title":"Semin. Cancer Biol."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1267","DOI":"10.1101\/gad.314617.118","article-title":"Roles of the immune system in cancer: From tumor initiation to metastatic progression","volume":"32","author":"Gonzalez","year":"2018","journal-title":"Genes. Dev."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"4931","DOI":"10.1158\/0008-5472.CAN-15-3567","article-title":"Mathematical modeling of cancer immunotherapy and its synergy with radiotherapy","volume":"76","author":"Serre","year":"2016","journal-title":"Cancer Res."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"334","DOI":"10.1080\/17513758.2016.1231347","article-title":"Dual role of delay effects in a tumour-immune system","volume":"11","author":"Yu","year":"2017","journal-title":"J. Biol. Dyn."},{"key":"ref_34","unstructured":"Perko, L. (2013). Differential Equations and Dynamical Systems, Springer. [3rd ed.]."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"786","DOI":"10.1006\/jmaa.1996.0468","article-title":"Bifurcation analysis of a chemostat model with a distributed delay","volume":"204","author":"Ruan","year":"1996","journal-title":"J. Math. Anal. Appl."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Marsden, J.E., and McKracken, M. (1976). The Hopf Bifurcation and Its Applications, Springer. [1st ed.].","DOI":"10.1007\/978-1-4612-6374-6"},{"key":"ref_37","unstructured":"Wiggins, S. (2003). Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer. [2nd ed.]."},{"key":"ref_38","first-page":"100534","article-title":"Prospect for application of mathematical models in combination cancer treatments","volume":"23","author":"Malinzi","year":"2021","journal-title":"Int. J. Med. Inform."},{"key":"ref_39","first-page":"541","article-title":"Nonlinear dynamics in tumor-immune system interaction models with delays","volume":"26","author":"Ruan","year":"2021","journal-title":"Discrete Contin. Dyn. Syst. Ser. B"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/6\/676\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:51:40Z","timestamp":1760107900000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/6\/676"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,5,31]]},"references-count":39,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2024,6]]}},"alternative-id":["sym16060676"],"URL":"https:\/\/doi.org\/10.3390\/sym16060676","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,5,31]]}}}