{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,26]],"date-time":"2026-03-26T15:22:42Z","timestamp":1774538562415,"version":"3.50.1"},"reference-count":34,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2025,2,20]],"date-time":"2025-02-20T00:00:00Z","timestamp":1740009600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Shanghai Agriculture Applied Technology Development Program, China","award":["T2022-03-01"],"award-info":[{"award-number":["T2022-03-01"]}]},{"name":"Shanghai Agriculture Applied Technology Development Program, China","award":["62366049"],"award-info":[{"award-number":["62366049"]}]},{"name":"Shanghai Agriculture Applied Technology Development Program, China","award":["62073217"],"award-info":[{"award-number":["62073217"]}]},{"name":"Shanghai Agriculture Applied Technology Development Program, China","award":["2021D01C113"],"award-info":[{"award-number":["2021D01C113"]}]},{"name":"Shanghai Agriculture Applied Technology Development Program, China","award":["24A110015"],"award-info":[{"award-number":["24A110015"]}]},{"name":"NSFC","award":["T2022-03-01"],"award-info":[{"award-number":["T2022-03-01"]}]},{"name":"NSFC","award":["62366049"],"award-info":[{"award-number":["62366049"]}]},{"name":"NSFC","award":["62073217"],"award-info":[{"award-number":["62073217"]}]},{"name":"NSFC","award":["2021D01C113"],"award-info":[{"award-number":["2021D01C113"]}]},{"name":"NSFC","award":["24A110015"],"award-info":[{"award-number":["24A110015"]}]},{"DOI":"10.13039\/100009110","name":"Natural Science Foundation of Xinjiang Uygur Autonomous Region","doi-asserted-by":"publisher","award":["T2022-03-01"],"award-info":[{"award-number":["T2022-03-01"]}],"id":[{"id":"10.13039\/100009110","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100009110","name":"Natural Science Foundation of Xinjiang Uygur Autonomous Region","doi-asserted-by":"publisher","award":["62366049"],"award-info":[{"award-number":["62366049"]}],"id":[{"id":"10.13039\/100009110","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100009110","name":"Natural Science Foundation of Xinjiang Uygur Autonomous Region","doi-asserted-by":"publisher","award":["62073217"],"award-info":[{"award-number":["62073217"]}],"id":[{"id":"10.13039\/100009110","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100009110","name":"Natural Science Foundation of Xinjiang Uygur Autonomous Region","doi-asserted-by":"publisher","award":["2021D01C113"],"award-info":[{"award-number":["2021D01C113"]}],"id":[{"id":"10.13039\/100009110","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100009110","name":"Natural Science Foundation of Xinjiang Uygur Autonomous Region","doi-asserted-by":"publisher","award":["24A110015"],"award-info":[{"award-number":["24A110015"]}],"id":[{"id":"10.13039\/100009110","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Key Research Projects of the University of Henan Province","award":["T2022-03-01"],"award-info":[{"award-number":["T2022-03-01"]}]},{"name":"Key Research Projects of the University of Henan Province","award":["62366049"],"award-info":[{"award-number":["62366049"]}]},{"name":"Key Research Projects of the University of Henan Province","award":["62073217"],"award-info":[{"award-number":["62073217"]}]},{"name":"Key Research Projects of the University of Henan Province","award":["2021D01C113"],"award-info":[{"award-number":["2021D01C113"]}]},{"name":"Key Research Projects of the University of Henan Province","award":["24A110015"],"award-info":[{"award-number":["24A110015"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>This paper proposes a multi-lingual S2IR rumor propagation model with white noise disturbances, aiming to study its dynamics and stochastic optimal control strategies. Firstly, a deterministic model is developed within a multi-lingual environment to identify rumor-free and rumor-spreading equilibria and calculate the basic reproduction number R0. Secondly, a stochastic model incorporating white noise perturbation is developed, and the uniqueness of its global positive solution is examined. Meanwhile, the asymptotic behaviors of the model\u2019s global solution near the steady states are discussed. Thirdly, the stochastic optimal control is designed to suppress the spread of rumors. Finally, the correctness and validity of the theoretical results are verified through numerical simulation.<\/jats:p>","DOI":"10.3390\/e27030217","type":"journal-article","created":{"date-parts":[[2025,2,20]],"date-time":"2025-02-20T11:03:37Z","timestamp":1740049417000},"page":"217","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Dynamic Analysis of the Multi-Lingual S2IR Rumor Propagation Model Under Stochastic Disturbances"],"prefix":"10.3390","volume":"27","author":[{"given":"Jinling","family":"Wang","sequence":"first","affiliation":[{"name":"Department of Automation, Shanghai Jiao Tong University, Shanghai 200240, China"},{"name":"School of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China"}]},{"given":"Jing","family":"Liao","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China"}]},{"given":"Jun-Guo","family":"Lu","sequence":"additional","affiliation":[{"name":"Department of Automation, Shanghai Jiao Tong University, Shanghai 200240, China"}]},{"given":"Jiarong","family":"Li","sequence":"additional","affiliation":[{"name":"Department of Automation, Shanghai Jiao Tong University, Shanghai 200240, China"}]},{"given":"Mei","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Zhoukou Normal University, Zhoukou 466001, China"}]}],"member":"1968","published-online":{"date-parts":[[2025,2,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1016\/j.physa.2018.04.082","article-title":"Rumor and authoritative information propagation model considering super spreading in complex social networks","volume":"506","author":"Zhang","year":"2018","journal-title":"Physica A"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"128096","DOI":"10.1016\/j.physa.2022.128096","article-title":"The impact of official rumor-refutation information on the dynamics of rumor spread","volume":"607","author":"Zhang","year":"2022","journal-title":"Physica A"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"109520","DOI":"10.1016\/j.paid.2019.109520","article-title":"Who falls for rumor? 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