{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,24]],"date-time":"2026-09-24T19:18:39Z","timestamp":1790277519234,"version":"4.1.0"},"reference-count":31,"publisher":"Public Library of Science (PLoS)","issue":"4","license":[{"start":{"date-parts":[[2025,4,1]],"date-time":"2025-04-01T00:00:00Z","timestamp":1743465600000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["www.plosone.org"],"crossmark-restriction":false},"short-container-title":["PLoS ONE"],"abstract":"<jats:sec>\n<jats:title>Objectives<\/jats:title>\n<jats:p>The current research investigations present the numerical solutions of the anthrax disease system in animals by designing a machine learning stochastic procedure. The mathematical anthrax disease system in animals is classified into susceptible, infected, recovered and vaccinated.<\/jats:p>\n<\/jats:sec>\n<jats:sec>\n<jats:title>Method<\/jats:title>\n<jats:p>A Runge-Kutta solver is applied to collect the dataset, which decreases the mean square error by dividing into training as 78%, testing 12% and verification 10%. The proposed stochastic computing technique is performed through the logistic sigmoid activation function, and a single hidden layer construction, twenty-seven numbers of neurons, and optimization through the Bayesian regularization for the mathematical anthrax disease system in animals.<\/jats:p>\n<\/jats:sec>\n<jats:sec>\n<jats:title>Finding<\/jats:title>\n<jats:p>The designed procedure\u2019s correctness is authenticated through the results overlapping and reducible absolute error, which are calculated around 10<jats:sup>-05<\/jats:sup> to 10<jats:sup>-08<\/jats:sup> for each case of the model. The best training performances are performed as 10<jats:sup>-10<\/jats:sup> to 10<jats:sup>-12<\/jats:sup> of the model. Moreover, the statistical performances in terms of regression coefficient, error histogram, and state transition values enhance the reliability of the proposed stochastic machine learning approach.<\/jats:p>\n<\/jats:sec>\n<jats:sec>\n<jats:title>Novelty<\/jats:title>\n<jats:p>The designed scheme is not applied before to get the numerical results of the anthrax disease system in animals.<\/jats:p>\n<\/jats:sec>","DOI":"10.1371\/journal.pone.0320327","type":"journal-article","created":{"date-parts":[[2025,4,1]],"date-time":"2025-04-01T21:35:48Z","timestamp":1743543348000},"page":"e0320327","update-policy":"https:\/\/doi.org\/10.1371\/journal.pone.corrections_policy","source":"Crossref","is-referenced-by-count":8,"title":["A machine learning computational approach for the mathematical anthrax disease system in animals"],"prefix":"10.1371","volume":"20","author":[{"given":"Zulqurnain","family":"Sabir","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4831-1249","authenticated-orcid":true,"given":"Eman","family":"Simbawa","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"340","published-online":{"date-parts":[[2025,4,1]]},"reference":[{"issue":"2","key":"pone.0320327.ref001","doi-asserted-by":"crossref","first-page":"323","DOI":"10.5455\/javar.2022.i599","article-title":"Review of anthrax: a disease of farm animals","volume":"9","author":"ME Alam","year":"2022","journal-title":"J Adv Vet Anim Res"},{"issue":"5","key":"pone.0320327.ref002","doi-asserted-by":"crossref","first-page":"370","DOI":"10.3390\/pathogens9050370","article-title":"Current status and trends in prophylaxis and management of anthrax disease","volume":"9","author":"V Savransky","year":"2020","journal-title":"Pathogens"},{"key":"pone.0320327.ref003","first-page":"1","article-title":"A study on the AH1N1\/09 influenza transmission model with the fractional Caputo\u2013Fabrizio derivative","author":"S Rezapour","year":"2020","journal-title":"Adv Difference Equat"},{"key":"pone.0320327.ref004","doi-asserted-by":"crossref","first-page":"389","DOI":"10.11113\/mjfas.v17n4.2062","article-title":"Mathematical modeling of cancer treatments with fractional derivatives: an overview","volume":"17","author":"MF Farayola","year":"2021","journal-title":"Malays J Fundam Appl Sci"},{"key":"pone.0320327.ref005","first-page":"1","article-title":"Existence of solutions for tripled system of fractional differential equations involving cyclic permutation boundary conditions","author":"MM Matar","year":"2020","journal-title":"Boundary Value Problems"},{"key":"pone.0320327.ref006","doi-asserted-by":"crossref","first-page":"283","DOI":"10.1016\/j.aej.2024.03.059","article-title":"A review of fractional order epidemic models for life sciences problems: Past, present and future","volume":"95","author":"KS Nisar","year":"2024","journal-title":"Alexandria Eng J"},{"issue":"1","key":"pone.0320327.ref007","doi-asserted-by":"crossref","first-page":"481","DOI":"10.1186\/s13662-020-02937-x","article-title":"A mathematical analysis of a system of Caputo\u2013Fabrizio fractional differential equations for the anthrax disease model in animals","author":"S Rezapour","year":"2020","journal-title":"Advances in Difference Equations"},{"key":"pone.0320327.ref008","unstructured":"Chavez CC, Feng Z, Huang W. 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