A study of a seasonal influenza model in deterministic and stochastic environments

A study of a seasonal influenza model in deterministic and stochastic environments | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article A study of a seasonal influenza model in deterministic and stochastic environments Xiaoshan Zhang, Xinhong Zhang, Daqing Jiang This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4173109/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 02 Feb, 2025 Read the published version in Journal of Nonlinear Science → Version 1 posted 8 You are reading this latest preprint version Abstract Seasonal influenza occurs annually and is one of the most common infectious diseases in the world, posing a threat to public health security. Therefore, it is essential to study the dynamics of seasonal influenza to raise public awareness and implement scientific prevention measures. Huo et al. studied a deterministic seasonal influenza model in Gansu, China, in this paper, we extend their study starting from proving the local asymptotic stability of the endemic equilibrium. In addition, considering the non-negligible effects of environmental disturbance on the transmission of influenza viruses, we assume that the transmission rate in the deterministic model follows a log-normal Ornstein-Uhlenbeck process, therefore, we formulate the corresponding stochastic model. To analyze the dynamics of the stochastic model, we first verify the existence and uniqueness of the global positive solution. Next, by constructing suitable Lyapunov functions, we obtain sufficient conditions for the stationary distribution and the extinction of the disease. More precisely, we deduce that the seasonal influenza persist when R 0 s > 1 while it dies out when R 0 E < 1. Furthermore, we derive the exact expression of probability density function around the quasi-endemic equilibrium. Finally, we conduct numerical simulations to validate theoretical results and further explore the effects of environmental noise. Seasonal influenza Stationary distribution Density function Extinction Ornstein-Uhlenbeck process Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 02 Feb, 2025 Read the published version in Journal of Nonlinear Science → Version 1 posted Editorial decision: Revision requested 23 Aug, 2024 Reviews received at journal 24 Jun, 2024 Reviewers agreed at journal 25 Apr, 2024 Reviewers agreed at journal 05 Apr, 2024 Reviewers invited by journal 03 Apr, 2024 Submission checks completed at journal 28 Mar, 2024 Editor assigned by journal 28 Mar, 2024 First submitted to journal 26 Mar, 2024 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-4173109","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":285850862,"identity":"959a2f03-9618-4609-99b7-3b6399ff56eb","order_by":0,"name":"Xiaoshan Zhang","email":"","orcid":"","institution":"China University of Petroleum, East China","correspondingAuthor":false,"prefix":"","firstName":"Xiaoshan","middleName":"","lastName":"Zhang","suffix":""},{"id":285850863,"identity":"23f0546f-74da-4a33-8697-429171cb0cdc","order_by":1,"name":"Xinhong Zhang","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAxklEQVRIiWNgGAWjYDACCQaGA4wNQAZ7Y+PDD6Rp4TncbCxBrBYGsBaJ9DYBHmJ0yM/uMTxcuMMmTz7yYRtQv52cbgMBLYxzjiUcnnkmrdjwdmLbgwKGZGOzAwS0MEskHzjM23Y4cePsxHYDCYYDidsIaWGTSGyAaJl5sE2ChxgtPDBb5kswEqlFQiItAaglLXEDTyIwkA2I8Iv8jBzjz7xtNonz248/fPihwk6OoBY4MACrNCBWOdi6BlJUj4JRMApGwYgCAI8SRSnWvwAtAAAAAElFTkSuQmCC","orcid":"","institution":"China University of Petroleum, East China","correspondingAuthor":true,"prefix":"","firstName":"Xinhong","middleName":"","lastName":"Zhang","suffix":""},{"id":285850864,"identity":"7a319679-208e-497d-911c-daf4dcb84ee7","order_by":2,"name":"Daqing Jiang","email":"","orcid":"","institution":"China University of Petroleum, East China","correspondingAuthor":false,"prefix":"","firstName":"Daqing","middleName":"","lastName":"Jiang","suffix":""}],"badges":[],"createdAt":"2024-03-27 03:09:40","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-4173109/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-4173109/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s00332-025-10132-z","type":"published","date":"2025-02-02T15:57:52+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":75352031,"identity":"2ef90c93-2b23-45e1-b7d6-60e6d7687881","added_by":"auto","created_at":"2025-02-03 16:12:54","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":3785273,"visible":true,"origin":"","legend":"","description":"","filename":"JNS.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4173109/v1_covered_a8291375-c93b-4526-82d4-14b71167e00b.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"A study of a seasonal influenza model in deterministic and stochastic environments","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"journal-of-nonlinear-science","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"jnls","sideBox":"Learn more about [Journal of Nonlinear Science](http://link.springer.com/journal/332)","snPcode":"332","submissionUrl":"https://submission.nature.com/new-submission/332/3","title":"Journal of Nonlinear Science","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Seasonal influenza, Stationary distribution, Density function, Extinction, Ornstein-Uhlenbeck process","lastPublishedDoi":"10.21203/rs.3.rs-4173109/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-4173109/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eSeasonal influenza occurs annually and is one of the most common infectious diseases in the world, posing a threat to public health security. 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