{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,13]],"date-time":"2026-01-13T08:38:24Z","timestamp":1768293504865,"version":"3.49.0"},"reference-count":52,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2021,4,7]],"date-time":"2021-04-07T00:00:00Z","timestamp":1617753600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100006261","name":"Taif University","doi-asserted-by":"publisher","award":["TURSP-2020\/77"],"award-info":[{"award-number":["TURSP-2020\/77"]}],"id":[{"id":"10.13039\/501100006261","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The current study aims to design an integrated numerical computing-based scheme by applying the Levenberg\u2013Marquardt backpropagation (LMB) neural network to solve the nonlinear susceptible (S), infected (I) and recovered (R) (SIR) system of differential equations, representing the spreading of infection along with its treatment. The solutions of both the categories of spreading infection and its treatment are presented by taking six different cases of SIR models using the designed LMB neural network. A reference dataset of the designed LMB neural network is established with the Adam numerical scheme for each case of the spreading infection and its treatment. The approximate outcomes of the SIR system based on the spreading infection and its treatment are presented in the training, authentication and testing procedures to adapt the neural network by reducing the mean square error (MSE) function using the LMB. Studies based on the proportional performance and inquiries based on correlation, error histograms, regression and MSE results establish the efficiency, correctness and effectiveness of the proposed LMB neural network scheme.<\/jats:p>","DOI":"10.3390\/sym13040618","type":"journal-article","created":{"date-parts":[[2021,4,7]],"date-time":"2021-04-07T21:49:06Z","timestamp":1617832146000},"page":"618","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":19,"title":["Computational Intelligent Paradigms to Solve the Nonlinear SIR System for Spreading Infection and Treatment Using Levenberg\u2013Marquardt Backpropagation"],"prefix":"10.3390","volume":"13","author":[{"given":"Muhammad","family":"Umar","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, Hazara University, Mansehra 21120, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zulqurnain","family":"Sabir","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Hazara University, Mansehra 21120, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9953-822X","authenticated-orcid":false,"given":"Muhammad Asif","family":"Zahoor Raja","sequence":"additional","affiliation":[{"name":"Future Technology Research Center, National Yunlin University of Science and Technology, Douliou, Yunlin 64002, Taiwan"},{"name":"Department of Electrical and Computer Engineering, Attock Campus, COMSATS University Islamabad, Attock 43600, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4274-4927","authenticated-orcid":false,"given":"Manoj","family":"Gupta","sequence":"additional","affiliation":[{"name":"Department of Electronics and Communication Engineering, JECRC University Jaipur, Jaipur 303905, Rajasthan, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2601-2803","authenticated-orcid":false,"given":"Dac-Nhuong","family":"Le","sequence":"additional","affiliation":[{"name":"Institute of Research and Development, Duy Tan University, Danang 550000, Vietnam"},{"name":"Faculty of Information Technology, Duy Tan University, Danang 550000, Vietnam"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ayman A.","family":"Aly","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, College of Engineering, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4228-2355","authenticated-orcid":false,"given":"Yolanda","family":"Guerrero-S\u00e1nchez","sequence":"additional","affiliation":[{"name":"Department of Anathomy and Pscicobiology, Faculty of Medicine, University of Murcia, 30100-Murcia, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,4,7]]},"reference":[{"key":"ref_1","unstructured":"Diekmann, O., and Heesterbeek, J.A. (2000). Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation, John Wiley & Sons."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Brauer, F., and Castillo-Chavez, C. (2001). Mathematical Models in Population Biology and Epidemiology, Springer. 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