{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,15]],"date-time":"2026-05-15T01:55:16Z","timestamp":1778810116294,"version":"3.51.4"},"reference-count":32,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2024,9,6]],"date-time":"2024-09-06T00:00:00Z","timestamp":1725580800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Two novel crossover models for breast cancer that incorporate \u03a8-Caputo fractal variable-order fractional derivatives, fractal fractional-order derivatives, and variable-order fractional stochastic derivatives driven by variable-order fractional Brownian motion and the crossover model for breast cancer that incorporates Atangana\u2013Baleanu Caputo fractal variable-order fractional derivatives, fractal fractional-order derivatives, and variable-order fractional stochastic derivatives driven by variable-order fractional Brownian motion are presented here, where we used a simple nonstandard kernel function \u03a8(t) in the first model and a non-singular kernel in the second model. Moreover, we evaluated our models using actual statistics from Saudi Arabia. To ensure consistency with the physical model problem, the symmetry parameter \u03b6 is introduced. We can obtain the fractal variable-order fractional Caputo and Caputo\u2013Katugampola derivatives as special cases from the proposed \u03a8-Caputo derivative. The crossover dynamics models define three alternative models: fractal variable-order fractional model, fractal fractional-order model, and variable-order fractional stochastic model over three-time intervals. The stability of the proposed model is analyzed. The \u03a8-nonstandard finite-difference method is designed to solve fractal variable-order fractional and fractal fractional models, and the Toufik\u2013Atangana method is used to solve the second crossover model with the non-singular kernel. Also, the nonstandard modified Euler\u2013Maruyama method is used to study the variable-order fractional stochastic model. Numerous numerical tests and comparisons with real data were conducted to validate the methods\u2019 efficacy and support the theoretical conclusions.<\/jats:p>","DOI":"10.3390\/sym16091172","type":"journal-article","created":{"date-parts":[[2024,9,6]],"date-time":"2024-09-06T07:42:06Z","timestamp":1725608526000},"page":"1172","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Comparative Study of Crossover Mathematical Model of Breast Cancer Based on \u03a8-Caputo Derivative and Mittag-Leffler Laws: Numerical Treatments"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7428-5799","authenticated-orcid":false,"given":"Nasser H.","family":"Sweilam","sequence":"first","affiliation":[{"name":"Mathematics Department, Faculty of Science, Cairo University, Giza 12613, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0351-9679","authenticated-orcid":false,"given":"Seham M.","family":"Al-Mekhlafi","sequence":"additional","affiliation":[{"name":"Mathematics Department, Faculty of Education, Sana\u2019a University, Sana\u2019a 1247, Yemen"},{"name":"Jadara University Research Center, Jadara University, Irbid 21110, Jordan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Waleed S.","family":"Abdel Kareem","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Suez University, Suez 43511, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ghader","family":"Alqurishi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Suez University, Suez 43511, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,9,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"505","DOI":"10.1001\/jamaoncol.2015.0735","article-title":"The global burden of cancer 2013","volume":"1","author":"Fitzmaurice","year":"2015","journal-title":"JAMA Oncol."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1495","DOI":"10.1158\/1055-9965.EPI-15-0535","article-title":"International variation in female breast cancer incidence and mortality rate international variation in female breast cancer rates","volume":"24","author":"DeSantis","year":"2015","journal-title":"Cancer Epidemiol. 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