SIR Epidemics in Interconnected Networks: threshold curve and phase transition

SIR Epidemics in Interconnected Networks: threshold curve and phase transition | 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 SIR Epidemics in Interconnected Networks: threshold curve and phase transition Saswata Das, Caterina Scoglio This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-3366972/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 28 Aug, 2024 Read the published version in Applied Network Science → Version 1 posted 7 You are reading this latest preprint version Abstract For simplicity of mathematical modelling of epidemic spreading, assumption is that hosts have identical rate of disease-causing contacts. However, in real world the scenario is different. The network-based framework allows us to capture the complex interdependencies and structural heterogeneity present in real-world systems. We examine two distinct scenarios involving the dynamics of Susceptible-Infected-Recovered (SIR) in interconnected networks. In the first part, we show how the epidemic threshold of a contact network changes as a result of being coupled with another network for a fixed infection strength. The model employed in this work considers both the contact networks and interconnections as generic. We have depicted the epidemic threshold curve for interconnected networks, considering the assumption that the infection could be initially present in either one or both of the networks. If the normalized infection strengths are above the threshold curve, the infection spreads, whereas if the normalized infection strengths are below the threshold curve, the disease does not spread. This is true for any level of interconnection. In the second part, we investigate the spillover phenomenon, where the disease in a novel host population network comes from a reservoir network. We have observed a clear phase transition when the number of links or the inter-network infection rate exceeds a certain threshold, keeping all other parameters constant. We observe two regimes for spillover: a major spillover region and a minor spillover region based on interpopulation links (fraction of links between two networks) and inter-network infection strength (infection rate between reservoir and host network). If the interpopulation links and inter-network infection strength are in the major spillover region, the spillover probability is high, while if the former parameters are in the minor spillover region, the spillover probability is low. When the number of infected individuals within a reservoir network is nearly equal, and the inter-network infection strength remains constant, the threshold number of links required to achieve the spillover threshold condition varies based on the network topology. Overall, this work contributes to the understanding of SIR dynamics in interconnected networks and sheds light on the behavior of epidemics in complex systems. Interconnected Networks Epidemic Threshold Spillover Phase Transition SIR Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 28 Aug, 2024 Read the published version in Applied Network Science → Version 1 posted Editorial decision: Revision requested 22 Apr, 2024 Reviews received at journal 12 Apr, 2024 Reviewers agreed at journal 10 Apr, 2024 Reviewers invited by journal 08 Mar, 2024 Submission checks completed at journal 25 Sep, 2023 Editor assigned by journal 25 Sep, 2023 First submitted to journal 18 Sep, 2023 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-3366972","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":235770287,"identity":"4de3d4c7-91fb-4897-9204-1eeaed043d19","order_by":0,"name":"Saswata Das","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA5ElEQVRIiWNgGAWjYHCDBAaGDwfALAOi1EuAtDDOIFkLMw8xWviljz/8+LPNro6/PfnxZ5szdYkN7M3bJPBpkezLMZbmbUuWkDjzzEw658bhxAaeY2V4tRic4WGQZtzGLMFwI8GMOefDgcQGiRwzvFrsz7A//vlzW72E/I30z58tPgAdJv8GvxYDHgYzCd5thyUMbuQYSDPcYAbawoNfi8QZHjNr3n/HJTeeeVMm2XPmsHEbT1qxBT4t/D3sj2/+OFPNL3c8ffOHH8fqZPvZD2+8gU8LJmAjTfkoGAWjYBSMAmwAANS7SsrpisV0AAAAAElFTkSuQmCC","orcid":"","institution":"Kansas State University","correspondingAuthor":true,"submittingAuthor":false,"prefix":"","firstName":"Saswata","middleName":"","lastName":"Das","suffix":""},{"id":235770290,"identity":"b59003e8-07b4-49a6-af91-a86ddb0a86df","order_by":1,"name":"Caterina Scoglio","email":"","orcid":"","institution":"Kansas State University","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Caterina","middleName":"","lastName":"Scoglio","suffix":""}],"badges":[],"createdAt":"2023-09-18 17:44:26","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-3366972/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-3366972/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s41109-024-00649-9","type":"published","date":"2024-08-28T15:57:12+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":63821249,"identity":"a9b65fce-aea3-4c3b-a53b-0cf288243017","added_by":"auto","created_at":"2024-09-02 16:12:56","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":601949,"visible":true,"origin":"","legend":"","description":"","filename":"FinalPaperSpringerCopy1.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3366972/v1_covered_5b52565e-313d-4644-b572-f6e62c1c17cb.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"SIR Epidemics in Interconnected Networks: threshold curve and phase transition","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":"applied-network-science","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"apns","sideBox":"Learn more about [Applied Network Science](http://appliednetsci.springeropen.com/)","snPcode":"41109","submissionUrl":"https://submission.nature.com/new-submission/41109/3","title":"Applied Network Science","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":true},"keywords":"Interconnected Networks, Epidemic Threshold, Spillover, Phase Transition, SIR","lastPublishedDoi":"10.21203/rs.3.rs-3366972/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-3366972/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eFor simplicity of mathematical modelling of epidemic spreading, assumption is that hosts have identical rate of disease-causing contacts. 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