{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,11]],"date-time":"2026-06-11T19:05:03Z","timestamp":1781204703761,"version":"3.54.1"},"reference-count":23,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2023,1,19]],"date-time":"2023-01-19T00:00:00Z","timestamp":1674086400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["11801249"],"award-info":[{"award-number":["11801249"]}]},{"name":"National Natural Science Foundation of China","award":["ZR2020MF040"],"award-info":[{"award-number":["ZR2020MF040"]}]},{"name":"National Natural Science Foundation of China","award":["319312101-01"],"award-info":[{"award-number":["319312101-01"]}]},{"name":"Nature Science Foundation of Shandong Province China","award":["11801249"],"award-info":[{"award-number":["11801249"]}]},{"name":"Nature Science Foundation of Shandong Province China","award":["ZR2020MF040"],"award-info":[{"award-number":["ZR2020MF040"]}]},{"name":"Nature Science Foundation of Shandong Province China","award":["319312101-01"],"award-info":[{"award-number":["319312101-01"]}]},{"name":"Liaocheng University","award":["11801249"],"award-info":[{"award-number":["11801249"]}]},{"name":"Liaocheng University","award":["ZR2020MF040"],"award-info":[{"award-number":["ZR2020MF040"]}]},{"name":"Liaocheng University","award":["319312101-01"],"award-info":[{"award-number":["319312101-01"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Human society always wants a safe environment from pollution and infectious diseases, such as COVID-19, etc. To control COVID-19, we have started the big effort for the discovery of a vaccination of COVID-19. Several biological problems have the aspects of symmetry, and this theory has many applications in explaining the dynamics of biological models. In this research article, we developed the stochastic COVID-19 mathematical model, along with the inclusion of a vaccination term, and studied the dynamics of the disease through the theory of symmetric dynamics and ergodic stationary distribution. The basic reproduction number is evaluated using the equilibrium points of the proposed model. For well-posedness, we also test the given problem for the existence and uniqueness of a non-negative solution. The necessary conditions for eradicating the disease are also analyzed along with the stationary distribution of the proposed model. For the verification of the obtained result, simulations of the model are performed.<\/jats:p>","DOI":"10.3390\/sym15020285","type":"journal-article","created":{"date-parts":[[2023,1,19]],"date-time":"2023-01-19T06:22:10Z","timestamp":1674109330000},"page":"285","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Extinction and Ergodic Stationary Distribution of COVID-19 Epidemic Model with Vaccination Effects"],"prefix":"10.3390","volume":"15","author":[{"given":"Humera","family":"Batool","sequence":"first","affiliation":[{"name":"School of Mathematics, Liaocheng University, Liaocheng 252000, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Weiyu","family":"Li","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Suzhou University of Science and Technology, Suzhou 215000, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0657-0103","authenticated-orcid":false,"given":"Zhonggui","family":"Sun","sequence":"additional","affiliation":[{"name":"School of Mathematics, Liaocheng University, Liaocheng 252000, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,1,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"497","DOI":"10.1016\/S0140-6736(20)30183-5","article-title":"Clinical features of patients infected with 2019 novel coronavirus in Wuhan, China","volume":"395","author":"Huang","year":"2020","journal-title":"Lancet"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Ding, Y., Jiao, J., Zhang, Q., Zhang, Y., and Ren, X. 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