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Second, we prove that the solutions of the model are uniformly bounded and discuss asymptotically stable of the rumor-free equilibrium point and the rumor-prevailing equilibrium point. Third, we propose an optimal control strategy for rumors to control the spread of rumors in the network. Finally, the above theories are verified by numerical simulation methods and the necessary conclusions are drawn.&lt;\/p&gt;&lt;\/abstract&gt;<\/jats:p>","DOI":"10.3934\/mbe.2022204","type":"journal-article","created":{"date-parts":[[2022,3,2]],"date-time":"2022-03-02T10:37:36Z","timestamp":1646217456000},"page":"4407-4428","source":"Crossref","is-referenced-by-count":12,"title":["Stability analysis and optimal control of a rumor propagation model based on two communication modes: friends and marketing account pushing"],"prefix":"10.3934","volume":"19","author":[{"given":"Ying","family":"Yu","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"name":"College of Information Science and Engineering, Yanshan University, Qinhuangdao, Hebei, China","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jiaomin","family":"Liu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jiadong","family":"Ren","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Cuiyi","family":"Xiao","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"name":"Liren College of Yanshan University, Qinhuangdao, Hebei, China","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"name":"Computer Virtual Technology and System Integration Laboratory of Hebei Province, China","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"name":"College of Mathematics and Information Technology, Hebei Normal University of Science and Technology, Qinhuangdao, Hebei, China","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"2321","reference":[{"key":"key-10.3934\/mbe.2022204-1","doi-asserted-by":"publisher","unstructured":"R. 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