{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,1]],"date-time":"2026-04-01T16:07:46Z","timestamp":1775059666375,"version":"3.50.1"},"reference-count":34,"publisher":"MDPI AG","issue":"18","license":[{"start":{"date-parts":[[2021,9,14]],"date-time":"2021-09-14T00:00:00Z","timestamp":1631577600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100006242","name":"Universiti Malaysia Sabah","doi-asserted-by":"publisher","award":["SDK0058"],"award-info":[{"award-number":["SDK0058"]}],"id":[{"id":"10.13039\/501100006242","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Applied Sciences"],"abstract":"<jats:p>The aim of this work is to present the numerical results of the influenza disease nonlinear system using the feed forward artificial neural networks (ANNs) along with the optimization of the combination of global and local search schemes. The genetic algorithm (GA) and active-set method (ASM), i.e., GA-ASM, are implemented as global and local search schemes. The mathematical nonlinear influenza disease system is dependent of four classes, susceptible S(u), infected I(u), recovered R(u) and cross-immune individuals C(u). For the solutions of these classes based on influenza disease system, the design of an objective function is presented using these differential system equations and its corresponding initial conditions. The optimization of this objective function is using the hybrid computing combination of GA-ASM for solving all classes of the influenza disease nonlinear system. The obtained numerical results will be compared by the Adams numerical results to check the authenticity of the designed ANN-GA-ASM. In addition, the designed approach through statistical based operators shows the consistency and stability for solving the influenza disease nonlinear system.<\/jats:p>","DOI":"10.3390\/app11188549","type":"journal-article","created":{"date-parts":[[2021,9,14]],"date-time":"2021-09-14T23:34:11Z","timestamp":1631662451000},"page":"8549","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Soft Computing Paradigms to Find the Numerical Solutions of a Nonlinear Influenza Disease Model"],"prefix":"10.3390","volume":"11","author":[{"given":"Zulqurnain","family":"Sabir","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, Hazara University, Mansehra 21300, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1793-5905","authenticated-orcid":false,"given":"Ag Asri","family":"Ag Ibrahim","sequence":"additional","affiliation":[{"name":"Faculty of Computing and Informatics, Universiti Malaysia Sabah, Jalan UMS, Kota Kinabalu 88400, Sabah, Malaysia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9953-822X","authenticated-orcid":false,"given":"Muhammad Asif Zahoor","family":"Raja","sequence":"additional","affiliation":[{"name":"Future Technology Research Center, National Yunlin University of Science and Technology, 123 University Road, Section 3, Douliou 64002, Yunlin, Taiwan"}]},{"given":"Kashif","family":"Nisar","sequence":"additional","affiliation":[{"name":"Faculty of Computing and Informatics, Universiti Malaysia Sabah, Jalan UMS, Kota Kinabalu 88400, Sabah, Malaysia"}]},{"given":"Muhammad","family":"Umar","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Hazara University, Mansehra 21300, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8657-3800","authenticated-orcid":false,"given":"Joel J. P. C.","family":"Rodrigues","sequence":"additional","affiliation":[{"name":"Post-Graduation Program in Electrical Engineering, Federal University of Piau\u00ed (UFPI), Teresina 64049-550, Brazil"},{"name":"Covilh\u00e3 Delegation, Instituto de Telecomunica\u00e7\u00f5es, 6201-001 Covilh\u00e3, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7008-1366","authenticated-orcid":false,"given":"Samy R.","family":"Mahmoud","sequence":"additional","affiliation":[{"name":"GRC Department, Faculty of Applied Studies, King Abdulaziz University, Jeddah 21589, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2021,9,14]]},"reference":[{"key":"ref_1","unstructured":"World Health Organization (WHO) (2021, May 16). Influenza Overview. 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