Fleeing lockdown and its impact on the size of epidemic outbreaks in the source and target regions - a COVID-19 lesson

Fleeing lockdown and its impact on the size of epidemic outbreaks in the source and target regions - a COVID-19 lesson | 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 Fleeing lockdown and its impact on the size of epidemic outbreaks in the source and target regions - a COVID-19 lesson Maria Vittoria Barbarossa, Norbert Bogya, Attila Dénes, Gergely Röst, and 2 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-82993/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 The COVID-19 pandemic forced authorities worldwide to implement moderate to severe restrictions in order to slow down or suppress the spread of the disease. It has been observed in several countries that a significant number of people fled a city or a region just before strict lockdown measures were implemented. This behavior carries the risk of seeding a large number of infections all at once in regions with otherwise small number of cases. In this work, we investigate the effect of fleeing on the size of an epidemic outbreak in the region under lockdown, and also in the region of destination. We propose a mathematical model that is suitable to describe the spread of an infectious disease over multiple geographic regions. Our approach is flexible to characterize the transmission of different viruses. As an example, we consider the COVID-19 outbreak in Italy. Projection of different scenarios shows that (i) timely and stricter intervention could have significantly lowered the number of cumulative cases in Italy, and (ii) fleeing at the time of lockdown possibly played a minor role in the spread of the disease in the country. Mathematical and Theoretical Biology Epidemiology Infectious Diseases disease dynamics COVID-19 final size lockdown two patches control measures Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Full Text 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. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. 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-82993","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":2709933,"identity":"10fe13f3-96b5-4d7d-9759-325a343a53d5","order_by":0,"name":"Maria Vittoria Barbarossa","email":"data:image/png;base64,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","orcid":"https://orcid.org/0000-0001-8788-1709","institution":"Frankfurt Institute for Advanced Studies","correspondingAuthor":true,"submittingAuthor":false,"prefix":"","firstName":"Maria","middleName":"Vittoria","lastName":"Barbarossa","suffix":""},{"id":2709934,"identity":"99d06ecd-4d11-4435-b5dd-61c2f9aae9c4","order_by":1,"name":"Norbert Bogya","email":"","orcid":"","institution":"Bolyai Institute, University of Szeged","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Norbert","middleName":"","lastName":"Bogya","suffix":""},{"id":2709935,"identity":"564d9f6d-afe5-4e97-8906-0e472a413c20","order_by":2,"name":"Attila Dénes","email":"","orcid":"https://orcid.org/0000-0003-1827-7932","institution":"Bolyai Institute, University of Szeged","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Attila","middleName":"","lastName":"Dénes","suffix":""},{"id":2709936,"identity":"b7220e2a-b8b8-4970-a44a-e72775c67f78","order_by":3,"name":"Gergely Röst","email":"","orcid":"https://orcid.org/0000-0001-9476-3284","institution":"Bolyai Institute, University of Szeged","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Gergely","middleName":"","lastName":"Röst","suffix":""},{"id":2709937,"identity":"0839aa62-7cb5-4d12-b866-1dae9f25fdb0","order_by":4,"name":"Hridya Vinod Varma","email":"","orcid":"","institution":"Interdisciplinary Center for Scientific Computing, Heidelberg University","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Hridya","middleName":"Vinod","lastName":"Varma","suffix":""},{"id":2709938,"identity":"5d74d66a-021a-40ce-af22-34eebf03ecd4","order_by":5,"name":"Zsolt Vizi","email":"","orcid":"","institution":"Bolyai Institute, University of Szeged","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Zsolt","middleName":"","lastName":"Vizi","suffix":""}],"badges":[],"createdAt":"2020-09-24 11:41:18","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":false,"conflictsOfInterestStatement":true,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false,"coiExplicitlySet":false},"doi":"10.21203/rs.3.rs-82993/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-82993/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":2584068,"identity":"7d333ac9-0940-4246-84ff-2220107a26f0","added_by":"auto","created_at":"2020-09-24 18:30:11","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":44430,"visible":true,"origin":"","legend":"Model structure for the transmission dynamics of an infectious disease, on the example of\nCOVID-19. Solid arrows indicate transition from one compartment to another, dashed arrows indicate virus\ntransmission due to contact with infectives. Upon infection, susceptible (S) individuals enter a\nlatent/presymptomatic phase (E). After symptoms onset, infections may be detected (I) or remain undetected (U).\nDetected cases might become severe and require hospitalization (H). Infected individuals who recovered from a\ndetected (R) or an undetected (RU) infection, as well as patients who died (D) upon infections, are removed from\nthe chain of transmission.","description":"","filename":"F1.png","url":"https://assets-eu.researchsquare.com/files/rs-82993/v1/F1.png"},{"id":2584069,"identity":"8448bedd-e6ae-473c-b32f-7294c9b3f590","added_by":"auto","created_at":"2020-09-24 18:30:11","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":85391,"visible":true,"origin":"","legend":"Detected infected cases and deaths depending on (a) the lockdown time T and (b) the\nfleeing fraction of exposed/undetected individuals at the time of the lockdown. An initial outbreak\nstarts off with 20 cases and two deaths in region A (population 25 million), where lockdown is established at time T,\nimmediately followed by lockdown-induced migration of a fraction of the unconstrained population from region A to\nthe disease-free region B (population 50 million). (a) The lockdown time T varies from one to twelve weeks after the\ninitial reporting, while the fleeing fraction is fixed to 1% of the unconstrained population in region A; (b) 0.01% to\n50% of the unconstrained population moves from region A to region B at the fixed time T = 21 days after the initial\nreporting. At the time of lockdown, control measures are applied in both regions in order to restrict contacts by\n60% and reduce transmission.","description":"","filename":"F2.png","url":"https://assets-eu.researchsquare.com/files/rs-82993/v1/F2.png"},{"id":2584070,"identity":"b3b79125-c2a1-489a-8329-cd0edb6111e7","added_by":"auto","created_at":"2020-09-24 18:30:11","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":249594,"visible":true,"origin":"","legend":"Sensitivity of cases in region B depending on the lockdown time T, the fraction of\npopulation leaving region A, the strength of intervention measures in reducing contacts, and the\npopulation size of region B. An initial outbreak starts off with 20 detected cases and two deaths in region A\n(population 25 million) which is locked at time T (horizontal axes indicate days after the initial reporting) and\nlockdown-induced migration (vertical axes) occurs. At the time of lockdown, control measures are applied to restrict\ncontacts by 5% (first row), 30% (second row) or 80% (third row), and maintained for two months. Color codes\nrepresent cases/deaths projected at the end of the two months following model (1): (a) the number of detected cases\n(left column) and deaths (right column) in region A, (b) detected case numbers in region B depending on the size of\nthe population in region B with respect to that of region A (columns).","description":"","filename":"F3.png","url":"https://assets-eu.researchsquare.com/files/rs-82993/v1/F3.png"},{"id":2584071,"identity":"071ee5c2-b72c-4797-aadc-f50bbb2e2f66","added_by":"auto","created_at":"2020-09-24 18:30:11","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":261651,"visible":true,"origin":"","legend":"How long should control measures be in place? An initial outbreak starts off with 20 cases and\ntwo deaths in region A (population 25 million) which is then locked at time T (horizontal axes indicate days after the\ninitial reporting) and lockdown-induced migration (vertical axes) occurs. In both regions, at the time of lockdown,\ncontrol measures are applied to restrict contacts by 5% (first row), 30% (second row) or 60% (third row), and\nmaintained for 7 days after the peak in the daily incidence in the respective region is reached. Color codes represent\nthe duration in days of the control period in region A (first column) and region B (second to fourth columns), also\ndepending on the population size in region B (being half, twice or four times the population in region A).","description":"","filename":"F4.png","url":"https://assets-eu.researchsquare.com/files/rs-82993/v1/F4.png"},{"id":2584072,"identity":"f8c736c7-5ae6-4e31-b7c9-54c9481879d6","added_by":"auto","created_at":"2020-09-24 18:30:12","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":103106,"visible":true,"origin":"","legend":"Cumulative detected cases in region B three months after lockdown of region A\ndepending on the fraction φ of population leaving region A, the intervention time TB in region B,\nthe effectiveness of intervention measures in reducing contacts, and the population size of region B.\nRegion A is isolated 21 days after the beginning of the outbreak and migration towards region B occurs. For each\npanel, the vertical axis denotes the variation in the fleeing fraction from A (\u001eφ, between 0.1% and 0.5). On the\nhorizontal axis, the reaction time TB indicates how many days passed for control measures in region B to be applied\nsince isolation of region A. Control measures are applied in region B to restrict contacts by 5% (first row), 30%\n(second row) or 50% (third row), and maintained until the end of the simulations. Cumulative detected cases are\nprojected also depending on the size of population in region B with respect to that in region A (columns). Note the\ndifferent scaling of the color legend in the panels.","description":"","filename":"F5.png","url":"https://assets-eu.researchsquare.com/files/rs-82993/v1/F5.png"},{"id":2584073,"identity":"a1132a53-c9e0-44d5-bac6-2312199eb9ec","added_by":"auto","created_at":"2020-09-24 18:30:12","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":68166,"visible":true,"origin":"","legend":"Model fit for the early dynamics of the COVID-19 outbreak in Italy. Reported data (dots)\nand model results (crosses) for daily incidence (left panel), deaths (middle panel) and hospitalized cases (right panel)\nin region A (Lombardy, Emilia-Romagna, Marche, Piedmont, Veneto; red) and region B (rest of the country; blue).\nParameter values are estimated as indicated in three different time intervals (separated by vertical lines in the\nfigure): pre-lockdown (February 24 – March 8), first lockdown measures (March 9 – 21), and extended lockdown\nmeasures (March 22 – May 4), cf. Supplementary Material.","description":"","filename":"F6.png","url":"https://assets-eu.researchsquare.com/files/rs-82993/v1/F6.png"},{"id":2584074,"identity":"6894df93-0430-4094-a6a3-1b37db4b0665","added_by":"auto","created_at":"2020-09-24 18:30:13","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":30656,"visible":true,"origin":"","legend":"Scenario comparison for the cumulative cases of COVID-19 in Italy as of May 4, 2020. Cumulative\ndetected cases in region A and region B are simulated for different scenarios: (baseline) the setting as of fit in\nFigure 6; For all other scenarios severe restrictions (parameter set as of March 22) are in place from the lockdown\ntime: (SC2) lockdown on March 9 (a) with and (b) without fleeing population at lockdown; (SC3) lockdown\nanticipated to March 2 (a) with and (b) without fleeing population at lockdown.","description":"","filename":"F7.png","url":"https://assets-eu.researchsquare.com/files/rs-82993/v1/F7.png"},{"id":2584075,"identity":"ed99b0df-446a-468c-8fb1-83ff9274738d","added_by":"auto","created_at":"2020-09-24 18:30:13","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":26470,"visible":true,"origin":"","legend":"Partial rank correlation coefficients (PRCCs) of seven main model parameters for (a)\nreproduction number, (b) final size (S1) and (c) the number of hospitalized cases at peak in region\nA. Parameters with PRCC larger than zero are positively correlated with the quantity of interest, whereas\nparameters with negative PRCC are negatively correlated. Parameters were varied within the ranges given in the\nSupplementary Material.","description":"","filename":"F8.png","url":"https://assets-eu.researchsquare.com/files/rs-82993/v1/F8.png"},{"id":13535164,"identity":"15c69f83-24fb-468b-b503-1a305f1d96ca","added_by":"auto","created_at":"2021-09-17 01:27:29","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":2283641,"visible":true,"origin":"","legend":"","description":"","filename":"PreprintRS.pdf","url":"https://assets-eu.researchsquare.com/files/rs-82993/v1_covered.pdf"},{"id":2584077,"identity":"e089a3fc-04e6-41eb-b4ba-da12743e1f61","added_by":"auto","created_at":"2020-09-24 18:30:15","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":2257499,"visible":true,"origin":"","legend":"","description":"","filename":"PreprintRS.pdf","url":"https://assets-eu.researchsquare.com/files/rs-82993/v1_stamped.pdf"},{"id":2584076,"identity":"159aba09-817a-43e8-9cb4-4c114ed31ef1","added_by":"auto","created_at":"2020-09-24 18:30:13","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":2159403,"visible":true,"origin":"","legend":"","description":"","filename":"PreprintRS.pdf","url":"https://assets-eu.researchsquare.com/files/rs-82993/v1/PreprintRS.pdf"}],"financialInterests":"","formattedTitle":"\u003cp\u003eFleeing lockdown and its impact on the size of epidemic outbreaks in the source and target regions - a COVID-19 lesson\u003c/p\u003e","fulltext":[{"header":"Full Text","content":"\u003cp\u003eThis preprint is available for \u003ca href='/article/rs-82993/latest.pdf' target='_blank'\u003edownload as a PDF\u003c/a\u003e.\u003c/p\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"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":"disease dynamics, COVID-19, final size, lockdown, two patches, control measures","lastPublishedDoi":"10.21203/rs.3.rs-82993/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-82993/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThe COVID-19 pandemic forced authorities worldwide to implement moderate to severe restrictions in order to slow down or suppress the spread of the disease. It has been observed in several countries that a significant number of people fled a city or a region just before strict lockdown measures were implemented. This behavior carries the risk of seeding a large number of infections all at once in regions with otherwise small number of cases. In this work, we investigate the effect of fleeing on the size of an epidemic outbreak in the region under lockdown, and also in the region of destination.\u0026nbsp;We propose a mathematical model that is suitable to describe the spread of an infectious disease over multiple geographic regions. Our approach is flexible to characterize the transmission of different viruses. As an example, we consider the COVID-19 outbreak in Italy. Projection of different scenarios shows that (i) timely and stricter intervention could have significantly lowered the number of cumulative cases in Italy, and (ii) fleeing at the time of lockdown possibly played a minor role in the spread of the disease in the country.\u003c/p\u003e","manuscriptTitle":"Fleeing lockdown and its impact on the size of epidemic outbreaks in the source and target regions - a COVID-19 lesson","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2020-09-24 18:30:10","doi":"10.21203/rs.3.rs-82993/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":"8b9c26bb-9558-4d5d-bcb2-bd7028113442","owner":[],"postedDate":"September 24th, 2020","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":605874,"name":"Mathematical and Theoretical Biology"},{"id":605875,"name":"Epidemiology"},{"id":605876,"name":"Infectious Diseases"}],"tags":[],"updatedAt":"2020-09-24T18:30:10+00:00","versionOfRecord":[],"versionCreatedAt":"2020-09-24 18:30:10","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-82993","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-82993","identity":"rs-82993","version":["v1"]},"buildId":"cBFmMYwuxLRRLfASyISRj","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}