{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,4]],"date-time":"2026-06-04T02:56:51Z","timestamp":1780541811960,"version":"3.54.1"},"reference-count":29,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2023,2,16]],"date-time":"2023-02-16T00:00:00Z","timestamp":1676505600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Basque Government","award":["IT1555-22"],"award-info":[{"award-number":["IT1555-22"]}]},{"name":"Basque Government","award":["KK-2022\/00090"],"award-info":[{"award-number":["KK-2022\/00090"]}]},{"name":"Basque Government","award":["PID2021-1235430B-C21"],"award-info":[{"award-number":["PID2021-1235430B-C21"]}]},{"name":"Basque Government","award":["PID2021-1235430B-C22"],"award-info":[{"award-number":["PID2021-1235430B-C22"]}]},{"name":"MCIN\/AEI","award":["IT1555-22"],"award-info":[{"award-number":["IT1555-22"]}]},{"name":"MCIN\/AEI","award":["KK-2022\/00090"],"award-info":[{"award-number":["KK-2022\/00090"]}]},{"name":"MCIN\/AEI","award":["PID2021-1235430B-C21"],"award-info":[{"award-number":["PID2021-1235430B-C21"]}]},{"name":"MCIN\/AEI","award":["PID2021-1235430B-C22"],"award-info":[{"award-number":["PID2021-1235430B-C22"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Cancer is dangerous and one of the major diseases affecting normal human life. In this paper, a fractional-order cancer model with stem cells and chemotherapy is analyzed to check the effects of infection in individuals. The model is investigated by the Sumudu transform and a very effective numerical method. The positivity of solutions with the ABC operator of the proposed technique is verified. Fixed point theory is used to derive the existence and uniqueness of the solutions for the fractional order cancer system. Our derived solutions analyze the actual behavior and effect of cancer disease in the human body using different fractional values. Modern mathematical control with the fractional operator has many applications including the complex and crucial study of systems with symmetry. Symmetry analysis is a powerful tool that enables the user to construct numerical solutions of a given fractional differential equation in a fairly systematic way. Such an analysis will provide a better understanding to control the of cancer disease in the human body.<\/jats:p>","DOI":"10.3390\/sym15020533","type":"journal-article","created":{"date-parts":[[2023,2,16]],"date-time":"2023-02-16T05:11:29Z","timestamp":1676524289000},"page":"533","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["Fractional Order Operator for Symmetric Analysis of Cancer Model on Stem Cells with Chemotherapy"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5181-4221","authenticated-orcid":false,"given":"Muhammad","family":"Azeem","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, The University of Lahore, Lahore 54590, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7616-0500","authenticated-orcid":false,"given":"Muhammad","family":"Farman","sequence":"additional","affiliation":[{"name":"Institute of Mathematics, Khwaja Fareed University of Engineering and Information Technology, Rahim Yar Khan 64200, Pakistan"},{"name":"Department of Computer Science and Mathematics, Lebanese American University, Beirut 5053, Lebanon"},{"name":"Mathematics Research Center, Department of Mathematics, Near East University, Near East Boulevard, Nicosia, Mersin 99138, Turkey"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9832-1424","authenticated-orcid":false,"given":"Ali","family":"Akg\u00fcl","sequence":"additional","affiliation":[{"name":"Department of Computer Science and Mathematics, Lebanese American University, Beirut 5053, Lebanon"},{"name":"Mathematics Research Center, Department of Mathematics, Near East University, Near East Boulevard, Nicosia, Mersin 99138, Turkey"},{"name":"Department of Mathematics, Art and Science Faculty, Siirt University, Siirt 56100, Turkey"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9320-9433","authenticated-orcid":false,"given":"Manuel","family":"De la Sen","sequence":"additional","affiliation":[{"name":"Department of Electricity and Electronics, Institute of Research and Development of Processes, Faculty of Science and Technology, University of the Basque Country, 48940 Leioa, Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,2,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1016\/j.critrevonc.2018.02.004","article-title":"Mathematical modeling of cancer metabolism","volume":"124","author":"Medina","year":"2018","journal-title":"Crit. Rev. Oncol.\/Hematol."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1683","DOI":"10.1142\/S0218202504003799","article-title":"Mathematical topics on the modeling of multicellular systems in competition between tumor and immune cells","volume":"14","author":"Bellomo","year":"2004","journal-title":"Math. Models Methods Appl. Sci."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"153","DOI":"10.5055\/jom.2009.0016","article-title":"Medicinal use of cannabis in the United States: Historical perspectives, current trends, and future directions","volume":"5","author":"Aggarwal","year":"2009","journal-title":"J. Opioid. Manag."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1835","DOI":"10.1016\/j.camwa.2009.08.028","article-title":"On the global existence of solutions to a class of fractional differential equations","volume":"59","author":"Baleanu","year":"2010","journal-title":"Comp. Math. Appl."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"527","DOI":"10.1016\/j.amc.2005.11.072","article-title":"Approximations of fractional integrals and Caputo fractional derivatives","volume":"178","author":"Odibat","year":"2006","journal-title":"Appl. Math. Comput."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Baleanu, D., Guvenc, Z.B., and Machado, J. (2010). New Trends in Nanotechnology and Fractional Calculus Applications, Springer.","DOI":"10.1007\/978-90-481-3293-5"},{"key":"ref_7","unstructured":"Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006). Theory and Applications of Fractional Differential Equations, Elsevier."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"203875","DOI":"10.1155\/2013\/203875","article-title":"The analytical solutions of some fractional ordinary differential equations by Sumudu transform method","volume":"2013","author":"Bulut","year":"2013","journal-title":"Abstr. Appl. Anal."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"71","DOI":"10.21833\/ijaas.2018.02.012","article-title":"Dynamical transmission and effect of smoking in society","volume":"5","author":"Ahmad","year":"2018","journal-title":"Int. J. Adv. Appl. Sci."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"123","DOI":"10.21833\/ijaas.2018.01.016","article-title":"Dynamical behavior of SIR epidemic model with non-integer time fractional derivatives: A mathematical analysis","volume":"5","author":"Ahmad","year":"2018","journal-title":"Int. J. Adv. Appl. Sci."},{"key":"ref_11","first-page":"20201405","article-title":"Key questions for modelling COVID-19 exit strategies","volume":"287","author":"Thompson","year":"2020","journal-title":"Proc. R. Soc."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"3068","DOI":"10.1016\/j.camwa.2012.02.002","article-title":"A numeric analytic method for approximating a giving up smoking model containing fractional derivatives","volume":"64","author":"Erturk","year":"2012","journal-title":"Comput. Math. Appl."},{"key":"ref_13","first-page":"13","article-title":"Numerical Solution of Fractional Order Epidemic Model of a Vector Born Disease by Laplace Adomian Decomposition Method","volume":"49","author":"Haq","year":"2017","journal-title":"Punjab Univ. J. Math."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1432521","DOI":"10.1080\/23311835.2018.1432521","article-title":"Mathematical analysis of HIV\/AIDS infection model with Caputo-Fabrizio fractional derivative","volume":"5","author":"Bushnaq","year":"2018","journal-title":"Cogent Math. Stat."},{"key":"ref_15","first-page":"8792","article-title":"Properties of a new fractional derivative without singular kernel","volume":"1","author":"Losada","year":"2015","journal-title":"Prog. Fract. Differ. Appl."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"2639","DOI":"10.1016\/j.aej.2020.04.027","article-title":"A control of glucose level in insulin therapies for the development of artificial pancreas byAtangana Baleanu fractional derivative","volume":"59","author":"Farman","year":"2020","journal-title":"Alex. Eng. J."},{"key":"ref_17","first-page":"753","article-title":"Dynamical Transmission of Coronavirus model with Analysis and Simulation","volume":"127","author":"Farman","year":"2021","journal-title":"Comput. Model. Eng. Sci."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"3551","DOI":"10.1016\/j.aej.2021.09.006","article-title":"Effect of Vaccination to Control COVID-19 with Fractal-Fractional Operator","volume":"61","author":"Amin","year":"2022","journal-title":"Alex. Eng. J."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"2062","DOI":"10.1016\/j.aej.2021.07.040","article-title":"Modeling and Analysis of Fractional Order Ebola Virus Model with Mittag-Lefler Kernel","volume":"61","author":"Farman","year":"2022","journal-title":"Alex. Eng. J."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"104178","DOI":"10.1016\/j.rinp.2021.104178","article-title":"Analysis of fractal-fractional model of tumor-immune interaction","volume":"25","author":"Ahmad","year":"2021","journal-title":"Results Phys."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"9334","DOI":"10.1002\/mma.7360","article-title":"Modeling of Fractional Order COVID-19 Epidemic Model with Quarantine and Social Distancing","volume":"44","author":"Farman","year":"2021","journal-title":"Math. Method Appl. Sci."},{"key":"ref_22","first-page":"1946","article-title":"Modeling and numerical investigation of fractional-order bovine babesiosis disease","volume":"37","author":"Aqeel","year":"2022","journal-title":"Numer Methods Partial Differ Equ."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Naeem, M., Zidan, A.M., Nonlaopon, K., Syam, M.I., Al-Zhour, Z., and Shah, R. (2021). A New Analysis of Fractional-Order Equal-Width Equations via Novel Techniques. Symmetry, 13.","DOI":"10.3390\/sym13050886"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Theswan, S., Ntouyas, S.K., Ahmad, B., and Tariboon, J. (2022). Existence Results for Nonlinear Coupled Hilfer Fractional Differential Equations with Nonlocal Riemann Liouville and Hadamard-Type Iterated Integral Boundary Conditions. Symmetry, 14.","DOI":"10.3390\/sym14091948"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Iqbal, Z., Mac\u00edas-D\u00edaz, J.E., Ahmed, N., Javaid, A., Rafiq, M., and Raza, A. (2022). Analytical and Numerical Boundedness of a Model with Memory Effects for the Spreading of Infectious Diseases. Symmetry, 14.","DOI":"10.3390\/sym14122540"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Alyobi, S., Shah, R., Khan, A., Shah, N.A., and Nonlaopon, K. (2022). Fractional Analysis of Nonlinear Boussinesq Equation under Atangana Baleanu Caputo Operator. Symmetry, 14.","DOI":"10.3390\/sym14112417"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"38","DOI":"10.1016\/j.cjph.2020.02.003","article-title":"A fractional order optimal 4D chaotic financial model with Mittag-Leffler law","volume":"65","author":"Atangana","year":"2020","journal-title":"Chin. J. Phys."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1953","DOI":"10.1016\/j.aej.2019.12.025","article-title":"Cancer treatment by stem cells and chemotherapy as a mathematical model with numerical simulations","volume":"59","author":"Alqudah","year":"2020","journal-title":"Alex. Eng. J."},{"key":"ref_29","unstructured":"Alqudah, M.A. (2019, January 12\u201316). Mathematical model of stem cells therapy for the treatment of cancer. Proceedings of the International Conference on Computational Methods in Applied Sciences (IC2MAS19), Istanbul, Turkey."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/2\/533\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T18:38:03Z","timestamp":1760121483000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/2\/533"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,2,16]]},"references-count":29,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2023,2]]}},"alternative-id":["sym15020533"],"URL":"https:\/\/doi.org\/10.3390\/sym15020533","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,2,16]]}}}