{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,20]],"date-time":"2026-05-20T01:47:06Z","timestamp":1779241626059,"version":"3.51.4"},"reference-count":41,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2023,2,9]],"date-time":"2023-02-09T00:00:00Z","timestamp":1675900800000},"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\/C22"],"award-info":[{"award-number":["PID2021-1235430B-C21\/C22"]}]},{"name":"MCIN\/AEI 269.10.13039\/501100011033","award":["IT1555-22"],"award-info":[{"award-number":["IT1555-22"]}]},{"name":"MCIN\/AEI 269.10.13039\/501100011033","award":["KK-2022\/00090"],"award-info":[{"award-number":["KK-2022\/00090"]}]},{"name":"MCIN\/AEI 269.10.13039\/501100011033","award":["PID2021-1235430B-C21\/C22"],"award-info":[{"award-number":["PID2021-1235430B-C21\/C22"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Despite the existence of a secure and reliable immunization, measles, also known as rubeola, continues to be a leading cause of fatalities globally, especially in underdeveloped nations. For investigation and observation of the dynamical transmission of the disease with the influence of vaccination, we proposed a novel fractional order measles model with a constant proportional (CP) Caputo operator. We analysed the proposed model\u2019s positivity, boundedness, well-posedness, and biological viability. Reproductive and strength numbers were also verified to examine how the illness dynamically behaves in society. For local and global stability analysis, we introduced the Lyapunov function with first and second derivatives. In order to evaluate the fractional integral operator, we used different techniques to invert the PC and CPC operators. We also used our suggested model\u2019s fractional differential equations to derive the eigenfunctions of the CPC operator. There is a detailed discussion of additional analysis on the CPC and Hilfer generalised proportional operators. Employing the Laplace with the Adomian decomposition technique, we simulated a system of fractional differential equations numerically. Finally, numerical results and simulations were derived with the proposed measles model. The intricate and vital study of systems with symmetry is one of the many applications of contemporary fractional mathematical control. A strong tool that makes it possible to create numerical answers to a given fractional differential equation methodically is symmetry analysis. It is discovered that the proposed fractional order model provides a more realistic way of understanding the dynamics of a measles epidemic.<\/jats:p>","DOI":"10.3390\/sym15020468","type":"journal-article","created":{"date-parts":[[2023,2,10]],"date-time":"2023-02-10T03:52:49Z","timestamp":1676001169000},"page":"468","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":57,"title":["Modelling and Analysis of a Measles Epidemic Model with the Constant Proportional Caputo Operator"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7616-0500","authenticated-orcid":false,"given":"Muhammad","family":"Farman","sequence":"first","affiliation":[{"name":"Institute of Mathematics, Khawaja 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, 99138 Mersin, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Aamir","family":"Shehzad","sequence":"additional","affiliation":[{"name":"Institute of Mathematics, Khawaja Fareed University of Engineering and Information Technology, Rahim Yar Khan 64200, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"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, 99138 Mersin, Turkey"},{"name":"Department of Mathematics, Art and Science Faculty, Siirt University, 56100 Siirt, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0286-7244","authenticated-orcid":false,"given":"Dumitru","family":"Baleanu","sequence":"additional","affiliation":[{"name":"Department of Computer Science and Mathematics, Lebanese American University, Beirut 5053, Lebanon"},{"name":"Department of Mathematics, Cankaya University, 06790 Ankara, Turkey"},{"name":"Insitute of Space Sciences, 077125 Magurele, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9320-9433","authenticated-orcid":false,"given":"Manuel De la","family":"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":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,2,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Griffin, D.E. 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