{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T20:07:58Z","timestamp":1775851678782,"version":"3.50.1"},"reference-count":48,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2022,3,27]],"date-time":"2022-03-27T00:00:00Z","timestamp":1648339200000},"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 China","doi-asserted-by":"publisher","award":["U1703262, 62163035\uff0c12026235"],"award-info":[{"award-number":["U1703262, 62163035\uff0c12026235"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Tianshan Innovation Team Program","award":["2020D14017"],"award-info":[{"award-number":["2020D14017"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>This paper is devoted to investigating the impact of the recurrence of rumors and individual behaviors and control strategies related to rumor spreading in online social networks. To do this, a novel susceptible-hesitating-infected-latent-recovered (SHILR) rumor propagation model in heterogeneous networks is presented. Firstly, based on the relevant mean-field equations of the model, the threshold value is examined to demonstrate the existence and stability of rumor-free\/spreading equilibrium with the help of the algebraic method. Secondly, the global stabilities of the equilibria are analyzed through applying Lyapunov stability theory and LaSalle\u2019s invariance principle. Next, the optimal control is proposed by taking advantage of Pontryagin\u2019s maximum principle for reducing the number of infected individuals with minimum cost. Moreover, some numerical examples are carried out to test the theoretical results. Finally, combined with practice, a model application is presented.<\/jats:p>","DOI":"10.3390\/e24040464","type":"journal-article","created":{"date-parts":[[2022,3,27]],"date-time":"2022-03-27T21:27:51Z","timestamp":1648416471000},"page":"464","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":26,"title":["Dynamic Analysis and Optimal Control of Rumor Spreading Model with Recurrence and Individual Behaviors in Heterogeneous Networks"],"prefix":"10.3390","volume":"24","author":[{"given":"Xinru","family":"Tong","sequence":"first","affiliation":[{"name":"College of Mathematics and System Science, Xinjiang University, Urumqi 830046, China"}]},{"given":"Haijun","family":"Jiang","sequence":"additional","affiliation":[{"name":"College of Mathematics and System Science, Xinjiang University, Urumqi 830046, China"}]},{"given":"Xiangyong","family":"Chen","sequence":"additional","affiliation":[{"name":"School of Automation and Electrical Engineering, Linyi University, Linyi 276007, China"}]},{"given":"Shuzhen","family":"Yu","sequence":"additional","affiliation":[{"name":"College of Mathematics and System Science, Xinjiang University, Urumqi 830046, China"}]},{"given":"Jiarong","family":"Li","sequence":"additional","affiliation":[{"name":"College of Mathematics and System Science, Xinjiang University, Urumqi 830046, China"}]}],"member":"1968","published-online":{"date-parts":[[2022,3,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1086\/220916","article-title":"Rumor and public opinion","volume":"57","author":"Peterson","year":"1951","journal-title":"Am. 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