{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,22]],"date-time":"2026-09-22T00:54:35Z","timestamp":1790038475808,"version":"4.0.1"},"reference-count":35,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2025,6,3]],"date-time":"2025-06-03T00:00:00Z","timestamp":1748908800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0"},{"start":{"date-parts":[[2025,6,3]],"date-time":"2025-06-03T00:00:00Z","timestamp":1748908800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Sci Rep"],"DOI":"10.1038\/s41598-025-04416-3","type":"journal-article","created":{"date-parts":[[2025,6,3]],"date-time":"2025-06-03T10:56:16Z","timestamp":1748948176000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Mathematical analysis of modified blood glucose insulin model through fractal fractional operators"],"prefix":"10.1038","volume":"15","author":[{"given":"F.","family":"Gassem","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Abrar","family":"Zahir","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Arafa","family":"Dawood","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mohammed","family":"Almalahi","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Amjad","family":"Ali","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Khaled","family":"Aldwoah","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2025,6,3]]},"reference":[{"issue":"1","key":"4416_CR1","doi-asserted-by":"publisher","first-page":"25","DOI":"10.33545\/2616454X.2017.v1.i1a.7","volume":"1","author":"S Alam","year":"2017","unstructured":"Alam, S. et al. The history of diabetes: from olden days to discovering insulin. International Journal of Unani and Integrative Medicine 1(1), 25\u201328 (2017).","journal-title":"International Journal of Unani and Integrative Medicine"},{"issue":"4","key":"4416_CR2","first-page":"373","volume":"23","author":"AM Ahmed","year":"2002","unstructured":"Ahmed, A. M. et al. History of diabetes mellitus. Saudi Med. J. 23(4), 373\u2013378 (2002).","journal-title":"Saudi Med. J."},{"key":"4416_CR3","doi-asserted-by":"publisher","first-page":"136","DOI":"10.1007\/s002850050007","volume":"40","author":"A De Gaetano","year":"2000","unstructured":"De Gaetano, A. & Arino, O. Mathematical modelling of the intravenous glucose tolerance test. J. Math. Biol. 40, 136\u2013168 (2000).","journal-title":"J. Math. Biol."},{"issue":"4703","key":"4416_CR4","doi-asserted-by":"publisher","first-page":"373","DOI":"10.1136\/bmj.1.4703.373","volume":"1","author":"R Lawrence","year":"1951","unstructured":"Lawrence, R. Types of human diabetes. Br. Med. J. 1(4703), 373 (1951).","journal-title":"Br. Med. J."},{"issue":"10","key":"4416_CR5","doi-asserted-by":"publisher","first-page":"1747","DOI":"10.3390\/medicina59101747","volume":"59","author":"L-G Confederat","year":"2023","unstructured":"Confederat, L.-G., Condurache, M.-I., Alexa, R.-E. & Dragostin, O.-M. Particularities of urinary tract infections in diabetic patients: a concise review. Medicina 59(10), 1747 (2023).","journal-title":"Medicina"},{"key":"4416_CR6","first-page":"304","volume-title":"Excessive thirst, hunger, and urination in diabetes","author":"NS Gregory","year":"2023","unstructured":"Gregory, N. S. Excessive thirst, hunger, and urination in diabetes 304 (From Symptoms to Treatment, Introduction to Clinical Pharmacology, 2023)."},{"issue":"2","key":"4416_CR7","doi-asserted-by":"publisher","first-page":"221","DOI":"10.3390\/nu16020221","volume":"16","author":"IV Di Molfetta","year":"2024","unstructured":"Di Molfetta, I. V., Bordoni, L., Gabbianelli, R., Sagratini, G. & Alessandroni, L. Vitamin d and its role on the fatigue mitigation: A narrative review. Nutrients 16(2), 221 (2024).","journal-title":"Nutrients"},{"issue":"1","key":"4416_CR8","doi-asserted-by":"publisher","first-page":"414","DOI":"10.1186\/s12884-023-05657-8","volume":"23","author":"J-F Ke","year":"2023","unstructured":"Ke, J.-F., Liu, S., Ge, R.-L., Ma, L. & Li, M.-F. Associations of maternal pre-pregnancy bmi and gestational weight gain with the risks of adverse pregnancy outcomes in chinese women with gestational diabetes mellitus. BMC Pregnancy Childbirth. 23(1), 414 (2023).","journal-title":"BMC Pregnancy Childbirth."},{"key":"4416_CR9","first-page":"1","volume-title":"Relationship between disease and treatment factors in diabetes with vision difficulty: A cross-sectional, population-based analysis","author":"C Zajner","year":"2024","unstructured":"Zajner, C. et al. Relationship between disease and treatment factors in diabetes with vision difficulty: A cross-sectional, population-based analysis 1\u201312 (Lasers and Imaging Retina, Ophthalmic Surgery, 2024)."},{"issue":"12","key":"4416_CR10","doi-asserted-by":"publisher","first-page":"844","DOI":"10.3390\/fractalfract7120844","volume":"7","author":"K Aldwoah","year":"2023","unstructured":"Aldwoah, K., Almalahi, M. & Shah, K. Theoretical and numerical simulations on the hepatitis b virus model through a piecewise fractional order. Fractal Fract. 7(12), 844 (2023).","journal-title":"Fractal Fract."},{"issue":"10","key":"4416_CR11","doi-asserted-by":"publisher","first-page":"596","DOI":"10.3390\/fractalfract8100596","volume":"8","author":"T Alraqad","year":"2024","unstructured":"Alraqad, T. et al. Modeling ebola dynamics with a $$\\phi$$-piecewise hybrid fractional derivative approach. Fractal Fract. 8(10), 596 (2024).","journal-title":"Fractal Fract."},{"issue":"2","key":"4416_CR12","doi-asserted-by":"publisher","first-page":"4376","DOI":"10.3934\/math.2024216","volume":"9","author":"K Aldwoah","year":"2024","unstructured":"Aldwoah, K. et al. Mathematical analysis and numerical simulations of the piecewise dynamics model of malaria transmission: A case study in yemen. AIMS Math 9(2), 4376\u20134408 (2024).","journal-title":"AIMS Math"},{"key":"4416_CR13","doi-asserted-by":"publisher","DOI":"10.1016\/j.rinp.2021.104629","volume":"28","author":"OA Omar","year":"2021","unstructured":"Omar, O. A., Alnafisah, Y., Elbarkouky, R. A. & Ahmed, H. M. Covid-19 deterministic and stochastic modelling with optimized daily vaccinations in saudi arabia. Results Phys. 28, 104629 (2021).","journal-title":"Results Phys."},{"issue":"10","key":"4416_CR14","doi-asserted-by":"publisher","first-page":"1885","DOI":"10.1140\/epjs\/s11734-022-00460-6","volume":"231","author":"S Kumar","year":"2022","unstructured":"Kumar, S., Chauhan, R., Aly, A. A., Momani, S. & Hadid, S. A study on fractional hbv model through singular and non-singular derivatives. Eur. Phys. J. Spec. Top. 231(10), 1885\u20131904 (2022).","journal-title":"Eur. Phys. J. Spec. Top."},{"issue":"2","key":"4416_CR15","doi-asserted-by":"publisher","first-page":"939","DOI":"10.3390\/ijerph20020939","volume":"20","author":"M AlShurbaji","year":"2023","unstructured":"AlShurbaji, M., Kader, L. A., Hannan, H., Mortula, M. & Husseini, G. A. Comprehensive study of a diabetes mellitus mathematical model using numerical methods with stability and parametric analysis. Int. J. Environ. Res. Public Health 20(2), 939 (2023).","journal-title":"Int. J. Environ. Res. Public Health"},{"issue":"5","key":"4416_CR16","doi-asserted-by":"publisher","first-page":"4738","DOI":"10.3390\/ijms24054738","volume":"24","author":"I Azarova","year":"2023","unstructured":"Azarova, I., Polonikov, A. & Klyosova, E. Molecular genetics of abnormal redox homeostasis in type 2 diabetes mellitus. Int. J. Mol. Sci. 24(5), 4738 (2023).","journal-title":"Int. J. Mol. Sci."},{"key":"4416_CR17","doi-asserted-by":"crossref","unstructured":"M.\u00a0Yavuz, M.\u00a0ur Rahman, M.\u00a0Yildiz, H.\u00a0Joshi, Mathematical modeling of middle east respiratory syndrome coronavirus with bifurcation analysis. Contemporary Mathematics 3997\u20134012 (2024).","DOI":"10.37256\/cm.5320245004"},{"issue":"05","key":"4416_CR18","doi-asserted-by":"publisher","first-page":"2240183","DOI":"10.1142\/S0218348X22401831","volume":"30","author":"SS Alzaid","year":"2022","unstructured":"Alzaid, S. S., Chauhan, R., Kumar, S. & Alkahtani, B. S. T. A high order numerical scheme for fractal-fractional laser system with chaos study. Fractals 30(05), 2240183 (2022).","journal-title":"Fractals"},{"key":"4416_CR19","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1186\/s13661-020-01418-0","volume":"2020","author":"HM Ahmed","year":"2020","unstructured":"Ahmed, H. M., El-Borai, M. M., El Bab, A. O. & Ramadan, M. E. Approximate controllability of noninstantaneous impulsive hilfer fractional integrodifferential equations with fractional brownian motion. Boundary Value Problems 2020, 1\u201325 (2020).","journal-title":"Boundary Value Problems"},{"issue":"01","key":"4416_CR20","doi-asserted-by":"publisher","first-page":"2240032","DOI":"10.1142\/S0218348X22400321","volume":"30","author":"S Momani","year":"2022","unstructured":"Momani, S., Chauhan, R., Kumar, S. & Hadid, S. A theoretical study on fractional ebola hemorrhagic fever model. Fractals 30(01), 2240032 (2022).","journal-title":"Fractals"},{"key":"4416_CR21","volume-title":"Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications","author":"I Podlubny","year":"1998","unstructured":"Podlubny, I. Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications (Elsevier, 1998)."},{"key":"4416_CR22","doi-asserted-by":"publisher","first-page":"1565","DOI":"10.1007\/s11424-016-5198-x","volume":"29","author":"X Cao","year":"2016","unstructured":"Cao, X., Datta, A., Al Basir, F. & Roy, P. K. Fractional-order model of the disease psoriasis: a control based mathematical approach. J. Syst. Sci. Complex. 29, 1565\u20131584 (2016).","journal-title":"J. Syst. Sci. Complex."},{"key":"4416_CR23","doi-asserted-by":"publisher","DOI":"10.1016\/j.rinp.2021.103855","volume":"22","author":"S Kumar","year":"2021","unstructured":"Kumar, S., Chauhan, R., Abdel-Aty, A.-H. & Alharthi, M. R. A study on transmission dynamics of hiv\/aids model through fractional operators. Results Phys. 22, 103855 (2021).","journal-title":"Results Phys."},{"issue":"2","key":"4416_CR24","doi-asserted-by":"publisher","first-page":"145","DOI":"10.22436\/jmcs.028.02.03","volume":"28","author":"R Prasad","year":"2023","unstructured":"Prasad, R., Kumar, K. & Dohare, R. Caputo fractional order derivative model of zika virus transmission dynamics. J. Math. Comput. Sci 28(2), 145\u2013157 (2023).","journal-title":"J. Math. Comput. Sci"},{"issue":"6","key":"4416_CR25","doi-asserted-by":"publisher","first-page":"E667","DOI":"10.1152\/ajpendo.1979.236.6.E667","volume":"236","author":"RN Bergman","year":"1979","unstructured":"Bergman, R. N., Ider, Y. Z., Bowden, C. R. & Cobelli, C. Quantitative estimation of insulin sensitivity. Am. J. Physiol. Endocrinol. Metab. 236(6), E667 (1979).","journal-title":"Am. J. Physiol. Endocrinol. Metab."},{"key":"4416_CR26","unstructured":"Bergman, R. N., & Cobelli, C.\u00a0Minimal modeling, partition analysis, and the estimation of insulin sensitivity. In: Federation proceedings\u00a039 110\u2013115 (1980)."},{"issue":"1","key":"4416_CR27","doi-asserted-by":"publisher","first-page":"53","DOI":"10.1016\/S0025-5564(03)00044-0","volume":"184","author":"G Pillonetto","year":"2003","unstructured":"Pillonetto, G., Sparacino, G. & Cobelli, C. Numerical non-identifiability regions of the minimal model of glucose kinetics: superiority of bayesian estimation. Math. Biosci. 184(1), 53\u201367 (2003).","journal-title":"Math. Biosci."},{"issue":"2","key":"4416_CR28","doi-asserted-by":"publisher","first-page":"136","DOI":"10.1007\/s002850050007","volume":"40","author":"A De Gaetano","year":"2000","unstructured":"De Gaetano, A. & Arino, O. Mathematical modelling of the intravenous glucose tolerance test. J. Math. Biol. 40(2), 136\u2013168 (2000).","journal-title":"J. Math. Biol."},{"key":"4416_CR29","doi-asserted-by":"publisher","DOI":"10.1016\/j.rinp.2023.106892","volume":"52","author":"KI Ahmed","year":"2023","unstructured":"Ahmed, K. I., Adam, H. D., Youssif, M. & Saber, S. Different strategies for diabetes by mathematical modeling: Applications of fractal-fractional derivatives in the sense of atangana-baleanu. Results Phys. 52, 106892 (2023).","journal-title":"Results Phys."},{"key":"4416_CR30","doi-asserted-by":"crossref","unstructured":"Dubey,R. S., Baleanu, D., Mishra, M. N., & Goswami, P., Solution of modified bergman minimal blood glucose-insulin model using caputo-fabrizio fractional derivative. Comput. Model. Eng. Sci. 128 (3), 1247-1263 (2021).","DOI":"10.32604\/cmes.2021.015224"},{"key":"4416_CR31","unstructured":"France, J. & Thornley, J. H., Mathematical models in agriculture. Butterworths (1984)."},{"issue":"7","key":"4416_CR32","doi-asserted-by":"publisher","first-page":"1008","DOI":"10.3390\/e25071008","volume":"25","author":"A Deppman","year":"2023","unstructured":"Deppman, A., Meg\u00edas, E. & Pasechnik, R. Fractal derivatives, fractional derivatives and q-deformed calculus. Entropy 25(7), 1008 (2023).","journal-title":"Entropy"},{"key":"4416_CR33","doi-asserted-by":"publisher","first-page":"396","DOI":"10.1016\/j.chaos.2017.04.027","volume":"102","author":"A Atangana","year":"2017","unstructured":"Atangana, A. Fractal-fractional differentiation and integration: Connecting fractal calculus and fractional calculus to predict complex system. Chaos Solitons Fractals. 102, 396\u2013406 (2017).","journal-title":"Chaos Solitons Fractals."},{"issue":"1","key":"4416_CR34","first-page":"132","volume":"31","author":"P Kumar","year":"2024","unstructured":"Kumar, P., Kumar, A., Kumar, S. & Baleanu, D. A fractional order co-infection model between malaria and filariasis epidemic. Arab J. Basic. Appl. Sci. 31(1), 132\u2013153 (2024).","journal-title":"Arab J. Basic. Appl. Sci."},{"issue":"1","key":"4416_CR35","doi-asserted-by":"publisher","first-page":"9012","DOI":"10.1038\/s41598-023-35624-4","volume":"13","author":"A Selvam","year":"2023","unstructured":"Selvam, A. et al. Ulam-hyers stability of tuberculosis and covid-19 co-infection model under atangana-baleanu fractal-fractional operator. Sci. Rep. 13(1), 9012 (2023).","journal-title":"Sci. Rep."}],"container-title":["Scientific Reports"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.nature.com\/articles\/s41598-025-04416-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.nature.com\/articles\/s41598-025-04416-3","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.nature.com\/articles\/s41598-025-04416-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,3]],"date-time":"2025-06-03T10:56:21Z","timestamp":1748948181000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.nature.com\/articles\/s41598-025-04416-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,6,3]]},"references-count":35,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2025,12]]}},"alternative-id":["4416"],"URL":"https:\/\/doi.org\/10.1038\/s41598-025-04416-3","relation":{},"ISSN":["2045-2322"],"issn-type":[{"value":"2045-2322","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,6,3]]},"assertion":[{"value":"12 January 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 May 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 June 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Decaleration"}},{"value":"The authors declare no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}},{"value":"This research project does not involve any form of gender discrimination, religious biases, or caste-related considerations.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethical approval"}}],"article-number":"19452"}}