Fractional-Order Framework for Gambling Dynamics with Delay and Control Strategies

Fractional-Order Framework for Gambling Dynamics with Delay and Control Strategies | 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 Fractional-Order Framework for Gambling Dynamics with Delay and Control Strategies Poonam Yadav, Komal Bansal, Trilok Mathur, Shivi Agarwal This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9521975/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract This work develops a fractional-order mathematical model to investigate the dynamics of gambling behavior within a population, explicitly accounting for memory effects. Individuals are categorized into non-gamblers, occasional gamblers, addicted gamblers, and recovered individuals, with transitions driven by behavioral interactions and recovery mechanisms. The model utilizes the Caputo fractional derivative to represent nonlocal and hereditary features inherent in human decision-making. Analytical results establish existence, uniqueness, and boundedness of the solutions, and the basic reproduction number $\mathcal{R}_0$ is determined via the next-generation matrix approach. Stability analysis indicates that the gambler-free equilibrium is locally asymptotically stable when $\mathcal{R}_0 1$. A fractional-order delay extension is formulated to incorporate relapse behavior, and conditions for Hopf bifurcation are derived. Numerical simulations confirm the theoretical findings, highlighting the influence of fractional-order memory and time-delay effects on the persistence of gambling addiction. Additionally, the convergence behavior of the proposed numerical scheme for varying fractional orders $\eta$ is examined. This study provides a rigorous quantitative framework for designing behavioral intervention strategies and underscores the significance of fractional dynamics in modeling addiction processes. Caputo derivative Gambling dynamics Convergence Analysis Delayed model Hopf bifurcation Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted 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-9521975","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":639865883,"identity":"da2b9a1f-72e2-47a4-947a-af4dd410cca9","order_by":0,"name":"Poonam Yadav","email":"","orcid":"","institution":"Birla Institute of Technology and Science, Pilani","correspondingAuthor":false,"prefix":"","firstName":"Poonam","middleName":"","lastName":"Yadav","suffix":""},{"id":639865884,"identity":"e5436f01-ac21-4627-ab12-cb95d9d7b99d","order_by":1,"name":"Komal Bansal","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAx0lEQVRIiWNgGAWjYBACCRDxoIKBBybAg1stspaEMww8MKVEaklsI0YpDEi2dyd+SJx3WMae/ewBhh81DDLmhLRI85zdLJG47TAPD09eAmPPMQYeywYCWuQkcjcAtaQB/ZJjwMDbwMBjcICwls0/EucAtfC/MWD8S4wWaYncbRKJDTY8PBI5BsxE2SLZc3abRcIxoJYb7xIOyxyTIKxF4njv5hsfaiTs2ftzDz58U2NjT1ALEuBhOACNJxK0jIJRMApGwSjACgAUOTfkKzIa1AAAAABJRU5ErkJggg==","orcid":"","institution":"VIT-AP University","correspondingAuthor":true,"prefix":"","firstName":"Komal","middleName":"","lastName":"Bansal","suffix":""},{"id":639865885,"identity":"f29adff6-1405-40d3-8f78-5a711583da64","order_by":2,"name":"Trilok Mathur","email":"","orcid":"","institution":"Birla Institute of Technology and Science, Pilani","correspondingAuthor":false,"prefix":"","firstName":"Trilok","middleName":"","lastName":"Mathur","suffix":""},{"id":639865886,"identity":"7f67ed07-f205-4b35-acfc-c018429ceb60","order_by":3,"name":"Shivi Agarwal","email":"","orcid":"","institution":"Birla Institute of Technology and Science, Pilani","correspondingAuthor":false,"prefix":"","firstName":"Shivi","middleName":"","lastName":"Agarwal","suffix":""}],"badges":[],"createdAt":"2026-04-25 04:08:14","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-9521975/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9521975/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":109297794,"identity":"8ec7c9fb-c45d-4026-8bf8-d6d6d9def5cd","added_by":"auto","created_at":"2026-05-15 09:05:48","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":2211274,"visible":true,"origin":"","legend":"","description":"","filename":"gambling41.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9521975/v1_covered_2e4895ce-c46d-41fb-9978-d6077e1e522f.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Fractional-Order Framework for Gambling Dynamics with Delay and Control Strategies","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Caputo derivative, Gambling dynamics, Convergence Analysis, Delayed model, Hopf bifurcation","lastPublishedDoi":"10.21203/rs.3.rs-9521975/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9521975/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"This work develops a fractional-order mathematical model to investigate the dynamics of gambling behavior within a population, explicitly accounting for memory effects. Individuals are categorized into non-gamblers, occasional gamblers, addicted gamblers, and recovered individuals, with transitions driven by behavioral interactions and recovery mechanisms. The model utilizes the Caputo fractional derivative to represent nonlocal and hereditary features inherent in human decision-making. Analytical results establish existence, uniqueness, and boundedness of the solutions, and the basic reproduction number $\\mathcal{R}_0$ is determined via the next-generation matrix approach. Stability analysis indicates that the gambler-free equilibrium is locally asymptotically stable when $\\mathcal{R}_0 \u003c 1$, whereas the gambler-persistence equilibrium remains stable for $\\mathcal{R}_0 \u003e 1$. A fractional-order delay extension is formulated to incorporate relapse behavior, and conditions for Hopf bifurcation are derived. Numerical simulations confirm the theoretical findings, highlighting the influence of fractional-order memory and time-delay effects on the persistence of gambling addiction. Additionally, the convergence behavior of the proposed numerical scheme for varying fractional orders $\\eta$ is examined. This study provides a rigorous quantitative framework for designing behavioral intervention strategies and underscores the significance of fractional dynamics in modeling addiction processes.","manuscriptTitle":"Fractional-Order Framework for Gambling Dynamics with Delay and Control Strategies","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-05-14 19:02:19","doi":"10.21203/rs.3.rs-9521975/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"441a4b2e-7535-42f2-bc16-3a42a9417754","owner":[],"postedDate":"May 14th, 2026","published":true,"recentEditorialEvents":[{"type":"reviewerAgreed","content":"257976291733336006291162127553121021203","date":"2026-05-13T16:12:20+00:00","index":26,"fulltext":""},{"type":"reviewerAgreed","content":"125909616719182622972978014473109876756","date":"2026-05-06T13:08:19+00:00","index":24,"fulltext":""},{"type":"reviewerAgreed","content":"298922837177443222401484374005260147490","date":"2026-05-06T12:24:39+00:00","index":23,"fulltext":""},{"type":"reviewerAgreed","content":"194419972254315013408420445553376373598","date":"2026-05-06T10:11:44+00:00","index":22,"fulltext":""},{"type":"reviewerAgreed","content":"216319167960014539793187468695102233360","date":"2026-05-06T05:16:12+00:00","index":21,"fulltext":""},{"type":"reviewersInvited","content":"15","date":"2026-05-06T05:11:34+00:00","index":"","fulltext":""}],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2026-05-14T19:02:19+00:00","versionOfRecord":[],"versionCreatedAt":"2026-05-14 19:02:19","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-9521975","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-9521975","identity":"rs-9521975","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}