{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,3]],"date-time":"2026-06-03T18:28:12Z","timestamp":1780511292051,"version":"3.54.1"},"reference-count":24,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2022,3,11]],"date-time":"2022-03-11T00:00:00Z","timestamp":1646956800000},"content-version":"vor","delay-in-days":69,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"},{"start":{"date-parts":[[2022,1,1]],"date-time":"2022-01-01T00:00:00Z","timestamp":1640995200000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/doi.wiley.com\/10.1002\/tdm_license_1.1"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Complexity"],"published-print":{"date-parts":[[2022,1]]},"abstract":"<jats:p>\n                    Tuberculosis (TB) is caused by bacillus\n                    <jats:italic>Mycobacterium tuberculosis<\/jats:italic>\n                    (MTB). In this study, a mathematical model of tuberculosis (TB) is analyzed. The numerical behaviour of the considered model is analyzed including basic reproduction number and stability. We applied three numerical techniques to this model, i.e., nonstandard finite difference (NSFD) scheme, Runge\u2013Kutta method of order 4(RK\u20104), and forward Euler (FD) scheme. NSFD scheme preserves all the essential properties of the model. Acquired results corroborate that NSFD scheme converges for each step size. While the other two schemes failed to preserve some properties of the model such as positivity and convergence. A graphical comparison presented in this study confirms the numerical stability of the NSFD technique shown here is maintained over a large area.\n                  <\/jats:p>","DOI":"10.1155\/2022\/5147951","type":"journal-article","created":{"date-parts":[[2022,3,11]],"date-time":"2022-03-11T15:35:22Z","timestamp":1647012922000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Analysis of the Epidemic Biological Model of Tuberculosis (TB) via Numerical Schemes"],"prefix":"10.1155","volume":"2022","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7560-5317","authenticated-orcid":false,"given":"S.","family":"Kanwal","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"M.K.","family":"Siddiqui","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8763-6164","authenticated-orcid":false,"given":"E.","family":"Bonyah","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"K.","family":"Sarwar","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"T.S.","family":"Shaikh","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"N.","family":"Ahmed","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"311","published-online":{"date-parts":[[2022,3,11]]},"reference":[{"key":"e_1_2_11_1_2","unstructured":"WHO Global tuberculosis report 2019 https:\/\/www.who.int\/tb\/country\/data\/profiles\/en\/."},{"key":"e_1_2_11_2_2","unstructured":"Pakistan Bureau of Statistics Pakistan\u2019s 6th census: population of major cities census 2017 http:\/\/www.pbscensus.gov.pk\/."},{"key":"e_1_2_11_3_2","unstructured":"National Institute of Health National tuberculosis Pakistan 2021 http:\/\/www.ntp.gov.pk\/webdatabase.php."},{"key":"e_1_2_11_4_2","doi-asserted-by":"publisher","DOI":"10.2105\/ajph.52.6.1002"},{"key":"e_1_2_11_5_2","doi-asserted-by":"publisher","DOI":"10.1164\/arrd.1967.96.5.893"},{"key":"e_1_2_11_6_2","doi-asserted-by":"publisher","DOI":"10.1007\/s002850050069"},{"key":"e_1_2_11_7_2","volume-title":"Mathematical Models for the Disease Dynamics of Tuberculosis","author":"Castillo-Chavez C.","year":"1996"},{"key":"e_1_2_11_8_2","doi-asserted-by":"publisher","DOI":"10.3934\/mbe.2004.1.361"},{"key":"e_1_2_11_9_2","first-page":"365","article-title":"On the definition and the computation of the basic reproduction ratio Ro in models for infectious diseases in heterogeneous populations","volume":"28","author":"Heesterbeek J. 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Sci."},{"key":"e_1_2_11_16_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.heliyon.2018.e00631"},{"key":"e_1_2_11_17_2","doi-asserted-by":"publisher","DOI":"10.1515\/math-2019-0027"},{"key":"e_1_2_11_18_2","first-page":"1","article-title":"Spatio-temporal numerical modeling of auto-catalytic Brusselator model","volume":"64","author":"Ahmed N.","year":"2019","journal-title":"Romanian Journal of Physics"},{"key":"e_1_2_11_19_2","doi-asserted-by":"publisher","DOI":"10.1063\/1.5112831"},{"key":"e_1_2_11_20_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.jtbi.2018.01.026"},{"key":"e_1_2_11_21_2","doi-asserted-by":"publisher","DOI":"10.1002\/mma.5676"},{"key":"e_1_2_11_22_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.rinp.2021.104658"},{"key":"e_1_2_11_23_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.matcom.2019.03.012"},{"key":"e_1_2_11_24_2","unstructured":"TB country profiles TB country profiles 2018 https:\/\/www.who.int\/tb\/country\/data\/profiles\/en\/."}],"container-title":["Complexity"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1155\/2022\/5147951","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/full-xml\/10.1155\/2022\/5147951","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1155\/2022\/5147951","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,22]],"date-time":"2026-01-22T21:00:00Z","timestamp":1769115600000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1155\/2022\/5147951"}},"subtitle":[],"editor":[{"given":"C.","family":"Rajivganthi","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"editor"}]}],"short-title":[],"issued":{"date-parts":[[2022,1]]},"references-count":24,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2022,1]]}},"alternative-id":["10.1155\/2022\/5147951"],"URL":"https:\/\/doi.org\/10.1155\/2022\/5147951","archive":["Portico"],"relation":{},"ISSN":["1076-2787","1099-0526"],"issn-type":[{"value":"1076-2787","type":"print"},{"value":"1099-0526","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,1]]},"assertion":[{"value":"2022-01-08","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2022-02-16","order":2,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2022-03-11","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}],"article-number":"5147951"}}