{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,23]],"date-time":"2026-01-23T16:10:18Z","timestamp":1769184618815,"version":"3.49.0"},"reference-count":42,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2022,5,23]],"date-time":"2022-05-23T00:00:00Z","timestamp":1653264000000},"content-version":"vor","delay-in-days":142,"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>In this paper, we consider a stochastic epidemic model under regime\u2010switching. We show that our model is well\u2010posed by proving the existence of solution, positivity, and uniqueness. Furthermore, we investigate sufficient conditions for the existence of ergodic stationary distribution to the model, which shows that the hepatitis B epidemic will persist. In addition, we establish sufficient conditions for the extinction of the hepatitis B epidemic. We perform numerical simulations to confirm and illustrate the theoretical results.<\/jats:p>","DOI":"10.1155\/2022\/2347414","type":"journal-article","created":{"date-parts":[[2022,5,23]],"date-time":"2022-05-23T17:50:31Z","timestamp":1653328231000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Stochastic Hybrid Hepatitis B Epidemic Model with Markovian Switching"],"prefix":"10.1155","volume":"2022","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6566-8181","authenticated-orcid":false,"given":"A.","family":"El Koufi","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2409-6513","authenticated-orcid":false,"given":"N.","family":"Seshagiri Rao","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2022,5,23]]},"reference":[{"key":"e_1_2_9_1_2","doi-asserted-by":"publisher","DOI":"10.1007\/s40471-020-00243-4"},{"key":"e_1_2_9_2_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.chaos.2020.110174"},{"key":"e_1_2_9_3_2","doi-asserted-by":"publisher","DOI":"10.1007\/s10489-020-01770-9"},{"key":"e_1_2_9_4_2","doi-asserted-by":"publisher","DOI":"10.1038\/s41598-020-62715-3"},{"key":"e_1_2_9_5_2","doi-asserted-by":"publisher","DOI":"10.1016\/s0092-8240(73)80057-6"},{"key":"e_1_2_9_6_2","doi-asserted-by":"publisher","DOI":"10.1007\/s40995-020-00960-x"},{"key":"e_1_2_9_7_2","doi-asserted-by":"publisher","DOI":"10.1016\/s0895-7177(97)00165-9"},{"key":"e_1_2_9_8_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.rinp.2021.104472"},{"key":"e_1_2_9_9_2","doi-asserted-by":"publisher","DOI":"10.1155\/2019\/9275051"},{"key":"e_1_2_9_10_2","doi-asserted-by":"publisher","DOI":"10.1155\/2022\/8093696"},{"key":"e_1_2_9_11_2","doi-asserted-by":"publisher","DOI":"10.1080\/07362994.2016.1244644"},{"key":"e_1_2_9_12_2","doi-asserted-by":"publisher","DOI":"10.1007\/s40065-014-0119-0"},{"key":"e_1_2_9_13_2","doi-asserted-by":"publisher","DOI":"10.1080\/17513758.2020.1833993"},{"key":"e_1_2_9_14_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.chaos.2018.01.036"},{"key":"e_1_2_9_15_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.aml.2019.106017"},{"key":"e_1_2_9_16_2","doi-asserted-by":"publisher","DOI":"10.1186\/s13662-021-03335-7"},{"key":"e_1_2_9_17_2","doi-asserted-by":"publisher","DOI":"10.1063\/5.0013066"},{"key":"e_1_2_9_18_2","unstructured":"https:\/\/www.who.int\/news-room\/fact-sheets\/detail\/hepatitis-b."},{"key":"e_1_2_9_19_2","doi-asserted-by":"publisher","DOI":"10.1080\/17513758.2019.1600750"},{"key":"e_1_2_9_20_2","doi-asserted-by":"publisher","DOI":"10.1137\/10081856x"},{"key":"e_1_2_9_21_2","doi-asserted-by":"publisher","DOI":"10.1016\/s0304-4149(01)00126-0"},{"key":"e_1_2_9_22_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.nonrwa.2013.10.010"},{"key":"e_1_2_9_23_2","doi-asserted-by":"publisher","DOI":"10.1155\/2020\/5898456"},{"key":"e_1_2_9_24_2","doi-asserted-by":"publisher","DOI":"10.1080\/07362994.2021.1981382"},{"key":"e_1_2_9_25_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2018.11.005"},{"key":"e_1_2_9_26_2","doi-asserted-by":"publisher","DOI":"10.3390\/sym12030331"},{"key":"e_1_2_9_27_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.apm.2013.07.037"},{"key":"e_1_2_9_28_2","doi-asserted-by":"publisher","DOI":"10.1137\/060649343"},{"key":"e_1_2_9_29_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.cnsns.2017.11.028"},{"key":"e_1_2_9_30_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2014.07.012"},{"key":"e_1_2_9_31_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2006.12.032"},{"key":"e_1_2_9_32_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.jtbi.2010.08.030"},{"key":"e_1_2_9_33_2","doi-asserted-by":"publisher","DOI":"10.1186\/s13662-021-03238-7"},{"key":"e_1_2_9_34_2","doi-asserted-by":"publisher","DOI":"10.3934\/dcds.2019249"},{"key":"e_1_2_9_35_2","doi-asserted-by":"publisher","DOI":"10.1007\/s12190-020-01453-1"},{"key":"e_1_2_9_36_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.matcom.2021.08.003"},{"key":"e_1_2_9_37_2","doi-asserted-by":"publisher","DOI":"10.1051\/mmnp\/2021047"},{"key":"e_1_2_9_38_2","doi-asserted-by":"publisher","DOI":"10.1142\/p473"},{"key":"e_1_2_9_39_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.nahs.2016.01.002"},{"key":"e_1_2_9_40_2","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(77)90073-8"},{"key":"e_1_2_9_41_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.spa.2006.12.001"},{"key":"e_1_2_9_42_2","doi-asserted-by":"publisher","DOI":"10.1137\/s0036144500378302"}],"container-title":["Complexity"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1155\/2022\/2347414","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/full-xml\/10.1155\/2022\/2347414","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1155\/2022\/2347414","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,22]],"date-time":"2026-01-22T21:04:02Z","timestamp":1769115842000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1155\/2022\/2347414"}},"subtitle":[],"editor":[{"given":"Tongqian","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"role":"editor","vocabulary":"crossref"}]}],"short-title":[],"issued":{"date-parts":[[2022,1]]},"references-count":42,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2022,1]]}},"alternative-id":["10.1155\/2022\/2347414"],"URL":"https:\/\/doi.org\/10.1155\/2022\/2347414","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":"2021-11-01","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2022-04-26","order":2,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2022-05-23","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}],"article-number":"2347414"}}