{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,13]],"date-time":"2026-03-13T06:43:27Z","timestamp":1773384207972,"version":"3.50.1"},"reference-count":35,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2023,5,16]],"date-time":"2023-05-16T00:00:00Z","timestamp":1684195200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of People\u2019s Republic of China","doi-asserted-by":"publisher","award":["62163035"],"award-info":[{"award-number":["62163035"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of People\u2019s Republic of China","doi-asserted-by":"publisher","award":["U1703262"],"award-info":[{"award-number":["U1703262"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of People\u2019s Republic of China","doi-asserted-by":"publisher","award":["ZYYD2022A05"],"award-info":[{"award-number":["ZYYD2022A05"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of People\u2019s Republic of China","doi-asserted-by":"publisher","award":["XJDX1401"],"award-info":[{"award-number":["XJDX1401"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Special Project for Local Science and Technology Development Guided by the Central Government","award":["62163035"],"award-info":[{"award-number":["62163035"]}]},{"name":"Special Project for Local Science and Technology Development Guided by the Central Government","award":["U1703262"],"award-info":[{"award-number":["U1703262"]}]},{"name":"Special Project for Local Science and Technology Development Guided by the Central Government","award":["ZYYD2022A05"],"award-info":[{"award-number":["ZYYD2022A05"]}]},{"name":"Special Project for Local Science and Technology Development Guided by the Central Government","award":["XJDX1401"],"award-info":[{"award-number":["XJDX1401"]}]},{"name":"Xinjiang Key Laboratory of Applied Mathematics","award":["62163035"],"award-info":[{"award-number":["62163035"]}]},{"name":"Xinjiang Key Laboratory of Applied Mathematics","award":["U1703262"],"award-info":[{"award-number":["U1703262"]}]},{"name":"Xinjiang Key Laboratory of Applied Mathematics","award":["ZYYD2022A05"],"award-info":[{"award-number":["ZYYD2022A05"]}]},{"name":"Xinjiang Key Laboratory of Applied Mathematics","award":["XJDX1401"],"award-info":[{"award-number":["XJDX1401"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>With the development of the Internet, it is more convenient for people to obtain information, which also facilitates the spread of rumors. It is imperative to study the mechanisms of rumor transmission to control the spread of rumors. The process of rumor propagation is often affected by the interaction of multiple nodes. To reflect higher-order interactions in rumor-spreading, hypergraph theories are introduced in a Hyper-ILSR (Hyper-Ignorant\u2013Lurker\u2013Spreader\u2013Recover) rumor-spreading model with saturation incidence rate in this study. Firstly, the definition of hypergraph and hyperdegree is introduced to explain the construction of the model. Secondly, the existence of the threshold and equilibrium of the Hyper-ILSR model is revealed by discussing the model, which is used to judge the final state of rumor propagation. Next, the stability of equilibrium is studied by Lyapunov functions. Moreover, optimal control is put forward to suppress rumor propagation. Finally, the differences between the Hyper-ILSR model and the general ILSR model are shown in numerical simulations.<\/jats:p>","DOI":"10.3390\/e25050805","type":"journal-article","created":{"date-parts":[[2023,5,17]],"date-time":"2023-05-17T02:04:05Z","timestamp":1684289045000},"page":"805","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Dynamical Analysis of Hyper-ILSR Rumor Propagation Model with Saturation Incidence Rate"],"prefix":"10.3390","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1219-5680","authenticated-orcid":false,"given":"Xuehui","family":"Mei","sequence":"first","affiliation":[{"name":"College of Mathematics and System Science, Xinjiang University, Urumqi 830046, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ziyu","family":"Zhang","sequence":"additional","affiliation":[{"name":"College of Mathematics and System Science, Xinjiang University, Urumqi 830046, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0310-6842","authenticated-orcid":false,"given":"Haijun","family":"Jiang","sequence":"additional","affiliation":[{"name":"College of Mathematics and System Science, Xinjiang University, Urumqi 830046, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,5,16]]},"reference":[{"key":"ref_1","unstructured":"Liubov, B. 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