{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,31]],"date-time":"2026-07-31T20:03:00Z","timestamp":1785528180903,"version":"3.56.0"},"reference-count":42,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2023,4,6]],"date-time":"2023-04-06T00:00:00Z","timestamp":1680739200000},"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>In this paper, a hybrid variable-order mathematical model for multi-vaccination COVID-19 is analyzed. The hybrid variable-order derivative is defined as a linear combination of the variable-order integral of Riemann\u2013Liouville and the variable-order Caputo derivative. A symmetry parameter \u03c3 is presented in order to be consistent with the physical model problem. The existence, uniqueness, boundedness and positivity of the proposed model are given. Moreover, the stability of the proposed model is discussed. The theta finite difference method with the discretization of the hybrid variable-order operator is developed for solving numerically the model problem. This method can be explicit or fully implicit with a large stability region depending on values of the factor \u0398. The convergence and stability analysis of the proposed method are proved. Moreover, the fourth order generalized Runge\u2013Kutta method is also used to study the proposed model. Comparative studies and numerical examples are presented. We found that the proposed model is also more general than the model in the previous study; the results obtained by the proposed method are more stable than previous research in this area.<\/jats:p>","DOI":"10.3390\/sym15040869","type":"journal-article","created":{"date-parts":[[2023,4,6]],"date-time":"2023-04-06T02:29:58Z","timestamp":1680748198000},"page":"869","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Numerical Simulation for a Hybrid Variable-Order Multi-Vaccination COVID-19 Mathematical Model"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7428-5799","authenticated-orcid":false,"given":"Nasser","family":"Sweilam","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0351-9679","authenticated-orcid":false,"given":"Seham","family":"Al-Mekhlafi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Education, Sana\u2019a University, Sana\u2019a 1247, Yemen"},{"name":"Department of Engineering Mathematics and Physics, Future University in Egypt, New Cairo 11835, Egypt"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Reem","family":"Salama","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef 62521, Egypt"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Tagreed","family":"Assiri","sequence":"additional","affiliation":[{"name":"Department of mathematics, Faculty of Science, Umm Al-Qura University, Makkah 21961, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,4,6]]},"reference":[{"key":"ref_1","unstructured":"United States Food and Drug Administration (2021, June 17). 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