{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:43:17Z","timestamp":1760150597842,"version":"build-2065373602"},"reference-count":39,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2023,11,27]],"date-time":"2023-11-27T00:00:00Z","timestamp":1701043200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["62163035","ZYYD2022A05","2022TSYCLJ0004","2022D01B111","XJDX1401"],"award-info":[{"award-number":["62163035","ZYYD2022A05","2022TSYCLJ0004","2022D01B111","XJDX1401"]}]},{"name":"Central Government","award":["62163035","ZYYD2022A05","2022TSYCLJ0004","2022D01B111","XJDX1401"],"award-info":[{"award-number":["62163035","ZYYD2022A05","2022TSYCLJ0004","2022D01B111","XJDX1401"]}]},{"name":"Tianshan Talent Program","award":["62163035","ZYYD2022A05","2022TSYCLJ0004","2022D01B111","XJDX1401"],"award-info":[{"award-number":["62163035","ZYYD2022A05","2022TSYCLJ0004","2022D01B111","XJDX1401"]}]},{"name":"Natural Science Foundation of Xinjiang Uygur Autonomous Region","award":["62163035","ZYYD2022A05","2022TSYCLJ0004","2022D01B111","XJDX1401"],"award-info":[{"award-number":["62163035","ZYYD2022A05","2022TSYCLJ0004","2022D01B111","XJDX1401"]}]},{"name":"Xinjiang Key Laboratory of Applied Mathematics","award":["62163035","ZYYD2022A05","2022TSYCLJ0004","2022D01B111","XJDX1401"],"award-info":[{"award-number":["62163035","ZYYD2022A05","2022TSYCLJ0004","2022D01B111","XJDX1401"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The presence of information asymmetry can hinder the public\u2019s ability to make well-informed decisions, resulting in unwarranted suspicion and the widespread dissemination of rumors. Therefore, it is crucial to provide individuals with consistent and dependable scientific education. Regular popular science education is considered a periodic impulsive intervention to mitigate the impact of information asymmetry and promote a more informed and discerning public. Drawing on these findings, this paper proposes a susceptible-hesitant-infected-refuting-recovered (SHIDR) rumor-spreading model to explain the spread of rumors. The model incorporates elements such as time delay, nonlinear incidence, and refuting individuals. Firstly, by applying the comparison theorem of an impulsive differential equation, we calculate two thresholds for rumor propagation. Additionally, we analyze the conditions of global attractiveness of the rumor-free periodic solution. Furthermore, we consider the condition for the rumor\u2019s permanence. Finally, numerical simulations are conducted to validate the accuracy of our findings. The results suggest that increasing the proportion of impulsive vaccination, reducing the impulsive period, or prolonging the delay time can effectively suppress rumors.<\/jats:p>","DOI":"10.3390\/e25121590","type":"journal-article","created":{"date-parts":[[2023,11,27]],"date-time":"2023-11-27T11:54:48Z","timestamp":1701086088000},"page":"1590","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Stability Analysis of a Delayed Rumor Propagation Model with Nonlinear Incidence Incorporating Impulsive Vaccination"],"prefix":"10.3390","volume":"25","author":[{"given":"Yuqian","family":"Zhou","sequence":"first","affiliation":[{"name":"College of Mathematics and System Sciences, Xinjiang University, Urumqi 830017, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0310-6842","authenticated-orcid":false,"given":"Haijun","family":"Jiang","sequence":"additional","affiliation":[{"name":"College of Mathematics and System Sciences, Xinjiang University, Urumqi 830017, China"},{"name":"School of Mathematics and Statistics, Yili Normal University, Yining 835000, China"}]},{"given":"Xupeng","family":"Luo","sequence":"additional","affiliation":[{"name":"School of Mathematics and Physics, Guangxi Minzu University, Nanning 530006, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9954-5435","authenticated-orcid":false,"given":"Shuzhen","family":"Yu","sequence":"additional","affiliation":[{"name":"School of Mathematics Science, Xinjiang Normal University, Urumqi 830017, China"}]}],"member":"1968","published-online":{"date-parts":[[2023,11,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"385","DOI":"10.1080\/00952990.2020.1753759","article-title":"What We Can Learn from COVID-19 Outbreak in Iran about the Importance of Alcohol Use Education","volume":"46","author":"Shalbafan","year":"2020","journal-title":"Am. 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