{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,30]],"date-time":"2025-09-30T10:10:12Z","timestamp":1759227012445,"version":"3.44.0"},"reference-count":16,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2025,9,28]],"date-time":"2025-09-28T00:00:00Z","timestamp":1759017600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"the Fundamental Research Funds for the Central Universities","award":["25CX03003A"],"award-info":[{"award-number":["25CX03003A"]}]}],"content-domain":{"domain":["www.mdpi.com"],"crossmark-restriction":true},"short-container-title":["Axioms"],"abstract":"<jats:p>This study develops a stochastic SIQR epidemic model with mean-reverting Ornstein\u2013Uhlenbeck (OU) processes for both transmission rate \u03b2(t) and quarantine release rate k(t); this is distinct from existing non-white-noise stochastic epidemic models, most of which focus on single-parameter perturbation or only stability analysis. It synchronously embeds OU dynamics into two core epidemic parameters to capture asynchronous fluctuations between infection spread and control measures. It adopts a rare measure solution framework to derive rigorous infection extinction conditions, linking OU\u2019s ergodicity to long-term \u03b2+(t) averages. It obtains the explicit probability density function of the four-dimensional SIQR system, filling the gap of lacking quantifiable density dynamics in prior studies. Simulations validate that R0d&lt;1 ensures almost sure extinction, while R0e&gt;1 leads to stable stochastic persistence.<\/jats:p>","DOI":"10.3390\/axioms14100732","type":"journal-article","created":{"date-parts":[[2025,9,30]],"date-time":"2025-09-30T09:12:23Z","timestamp":1759223543000},"page":"732","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["The Density Function of the Stochastic SIQR Model with a Two-Parameters Mean-Reverting Process"],"prefix":"10.3390","volume":"14","author":[{"given":"Huina","family":"Zhang","sequence":"first","affiliation":[{"name":"College of Science, China University of Petroleum (East China), Qingdao 266580, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhiming","family":"Ni","sequence":"additional","affiliation":[{"name":"College of Science, China University of Petroleum (East China), Qingdao 266580, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Daqing","family":"Jiang","sequence":"additional","affiliation":[{"name":"College of Science, China University of Petroleum (East China), Qingdao 266580, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8308-6078","authenticated-orcid":false,"given":"Jianguo","family":"Sun","sequence":"additional","affiliation":[{"name":"College of Science, China University of Petroleum (East China), Qingdao 266580, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,9,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Anderson, R.M., and May, R.M. (1991). Infectious Diseases of Humans: Dynamics and Control, Oxford University Press.","DOI":"10.1093\/oso\/9780198545996.001.0001"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"539","DOI":"10.1007\/BF00164051","article-title":"Patterns in the effects of infectious diseases on population growth","volume":"29","author":"Diekmann","year":"1991","journal-title":"J. Math. Biol."},{"key":"ref_3","unstructured":"Gardiner, C.W. (1893). Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, Springer."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"6134","DOI":"10.3934\/era.2023312","article-title":"Dynamic analysis of a stochastic epidemic model incorporating the double epidemic hypothesis and Crowley-Martin incidence term","volume":"31","author":"Li","year":"2023","journal-title":"Electron. Res. 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