Estimation and optimal control of the multi-scale dynamics of the Covid-19

Estimation and optimal control of the multi-scale dynamics of the Covid-19 | 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 Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Estimation and optimal control of the multi-scale dynamics of the Covid-19 David Jaurès FOTSA MBOGNE, Stéphane Yanick TCHOUMI, Yannick KOUAKEP TCHAPTCHIE, and 5 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-298729/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 5 You are reading this latest preprint version Abstract This work aims at a better understanding and the optimal control of the spread of the new severe acute respiratory corona virus 2 (SARS-CoV-2). We first propose a multi-scale model giving insights on the virus population dynamics, the transmission process and the infection mechanism. We consider 10 compartments in the human population in order to take into accounts the effects of different specific mitigation policies. The population of viruses is also partitioned into 10 compartments corresponding respectively to each of the first nine human population compartments and the free viruses available in the environment. We show the global stability of the disease free equilibrium if a given threshold T_0 is less or equal to 1 and we provide how to compute the basic reproduction number R_0. A convergence index T_1 is also defined in order to estimate the speed at which the disease extincts and an upper bound to the time of extinction is given. The existence of the endemic equilibrium is conditional and its description is provided. We evaluate the sensitivity of R_0, T_0 and T_1 to control parameters such as the maximal human density allowed per unit of surface, the rate of disinfection both for people and environment, the mobility probability, the wearing mask probability or efficiency, and the human to human contact rate which results from the previous one. Except the maximal human density allowed per unit of surface, all those parameters have significant effects on the qualitative dynamics of the disease. The most significant is the probability of wearing mask followed by the probability of mobility and the disinfection rate. According to a functional cost taking into consideration economic impacts of SARS-CoV-2, we determine and discuss optimal fighting strategies. The study is applied to real available data from Cameroon and an estimation of model parameters is done. After several simulations, social distancing and the disinfection frequency appear as the main elements of the optimal control strategy. Applied Mathematics SARS-CoV-2 multi-scale modeling parameter estimation stability analysis time of extinction sensitivity analysis optimal control Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Full Text Cite Share Download PDF Status: Under Review Version 1 posted Reviewers invited by journal 10 Mar, 2021 Reviews received at journal 10 Mar, 2021 Editor invited by journal 05 Mar, 2021 First submitted to journal 03 Mar, 2021 Editor assigned by journal 03 Mar, 2021 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 In Review Editorial Policies 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-298729","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":16245914,"identity":"f6a653b9-676e-4b66-9bde-a386f6cd15fe","order_by":0,"name":"David Jaurès FOTSA MBOGNE","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA9UlEQVRIiWNgGAWjYDCCA0DEw8AGZn/4wMCQAGYQq4Vx5gyIFsYZhLQw8ECYjLN5iNHCd/yM4YE3DHzyuu3tD5tt2+zy+NkbGJsr8GiRPJNjcHAOA5vhtjNnDJtz25KLJXsOMDaewaPF4EBawmGgXxi33chhf5zbxpy44UYC+8MGfFrOPwNrsd92//nDZsu2epAWxka8Wm4kHwBpSdx2g8GwmbHtMGEtkjceHwD5JXnbmRzDxp5zxxNn9hxsxKuF73xi84c3DMdstx0//rDhR1l1Yj9780G8WsCA8d8xKAMSpQQ1gEANlP5DjOJRMApGwSgYaQAAzgtc7/tHx8QAAAAASUVORK5CYII=","orcid":"https://orcid.org/0000-0002-9090-0429","institution":"Université de Ngaoundéré, ENSAI","correspondingAuthor":true,"submittingAuthor":false,"prefix":"","firstName":"David","middleName":"Jaurès FOTSA","lastName":"MBOGNE","suffix":""},{"id":16245915,"identity":"a78ca809-96db-4fd1-b9d3-8e4f39831e97","order_by":1,"name":"Stéphane Yanick TCHOUMI","email":"","orcid":"","institution":"Universite de Ngaoundere","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Stéphane","middleName":"Yanick","lastName":"TCHOUMI","suffix":""},{"id":16245916,"identity":"131a9418-100c-4207-be06-5d252298bcd8","order_by":2,"name":"Yannick KOUAKEP TCHAPTCHIE","email":"","orcid":"","institution":"Universite de Ngaoundere","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Yannick","middleName":"KOUAKEP","lastName":"TCHAPTCHIE","suffix":""},{"id":16245917,"identity":"d6d8292c-1939-41da-a048-bc736e45e171","order_by":3,"name":"Vivient Corneille KAMLA","email":"","orcid":"","institution":"Universite de Ngaoundere","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Vivient","middleName":"Corneille","lastName":"KAMLA","suffix":""},{"id":16245918,"identity":"9291dfd5-85bd-49e8-bb8d-ef61f9aa19f2","order_by":4,"name":"Jean Claude KAMGANG","email":"","orcid":"","institution":"Universite de Ngaoundere","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Jean","middleName":"Claude","lastName":"KAMGANG","suffix":""},{"id":16245919,"identity":"bedde1e8-f085-440d-98d7-19207d1ad5ce","order_by":5,"name":"Duplex Elvis HOUPA DANGA","email":"","orcid":"","institution":"Universite de Ngaoundere","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Duplex","middleName":"Elvis HOUPA","lastName":"DANGA","suffix":""},{"id":16245920,"identity":"8edddc7b-3a16-4a29-ab6d-6eaac80c6182","order_by":6,"name":"Samuel BOWONG TSAKOU","email":"","orcid":"","institution":"Université de Douala: Universite de Douala","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Samuel","middleName":"BOWONG","lastName":"TSAKOU","suffix":""},{"id":16245921,"identity":"cfd5f2b8-ef6c-4d1e-85d7-934721635d45","order_by":7,"name":"David BEKOLLE","email":"","orcid":"","institution":"Universite de Ngaoundere","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"David","middleName":"","lastName":"BEKOLLE","suffix":""}],"badges":[],"createdAt":"2021-03-04 22:28:05","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-298729/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-298729/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":7771827,"identity":"4fdb2b48-f85c-46a9-a8b2-e830c418ccc4","added_by":"auto","created_at":"2021-04-07 22:33:46","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":40745,"visible":true,"origin":"","legend":"Flow chart of the Susceptible (S)- Exposed (E)- Infectious (I)- Quarantined (Q)- Hospitalized (H)-\nTreated (T)- Recovered (R)- Non Infectious Dead (D)- Infectious Dead (DI )- Buried (B) compartmental\nmodel with corresponding viruses subpopulations (Vk; k 2 fS;E; I; Q;H; T;R;D;DIg) and free viruses in\nthe environment (VF )","description":"","filename":"ScreenShot20210407at2.41.08PM.png","url":"https://assets-eu.researchsquare.com/files/rs-298729/v1/4fcf757f00b7d89a375b22ad.png"},{"id":7771826,"identity":"f2ad8eb1-a90c-4257-a1bc-ab26363f960c","added_by":"auto","created_at":"2021-04-07 22:33:46","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":51374,"visible":true,"origin":"","legend":"Asymptotic behavior of infectious dynamics depending on T0","description":"","filename":"ScreenShot20210407at2.41.25PM.png","url":"https://assets-eu.researchsquare.com/files/rs-298729/v1/d9aa8390145062cf240616b7.png"},{"id":7772046,"identity":"0910adc5-fa4d-4d6a-9d5a-06c5529dd3be","added_by":"auto","created_at":"2021-04-07 22:36:46","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":28268,"visible":true,"origin":"","legend":"Cumulative reported infectious from the 2nd of March to the 07th of June 2020.","description":"","filename":"ScreenShot20210407at2.41.33PM.png","url":"https://assets-eu.researchsquare.com/files/rs-298729/v1/0ab82ecffec150d0d817e5be.png"},{"id":7772513,"identity":"e72bcca2-bb6a-4d26-8ffc-4c7ea5852aab","added_by":"auto","created_at":"2021-04-07 22:39:46","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":40164,"visible":true,"origin":"","legend":"Effects of control parameters on R0, T0 and T1","description":"","filename":"ScreenShot20210407at2.41.41PM.png","url":"https://assets-eu.researchsquare.com/files/rs-298729/v1/2095bc38b667b9cfeba30f5c.png"},{"id":7771831,"identity":"3193ace8-efbd-40b7-a53c-95e92995c22f","added_by":"auto","created_at":"2021-04-07 22:33:46","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":57884,"visible":true,"origin":"","legend":"Optimal control dynamics of Covid-19 prevalence","description":"","filename":"ScreenShot20210407at2.41.51PM.png","url":"https://assets-eu.researchsquare.com/files/rs-298729/v1/d37d67af8a5bf9dba0f86156.png"},{"id":7772047,"identity":"116bb8ff-d794-4ada-8114-2558eb0dc624","added_by":"auto","created_at":"2021-04-07 22:36:46","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":29395,"visible":true,"origin":"","legend":"Controlled dynamics of Covid-19 prevalence","description":"","filename":"ScreenShot20210407at2.41.59PM.png","url":"https://assets-eu.researchsquare.com/files/rs-298729/v1/58f8a85565257c8b902ac5df.png"},{"id":13684372,"identity":"39a18cd4-bbf4-4b22-b300-d02a004b0d15","added_by":"auto","created_at":"2021-09-17 12:07:45","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":282152,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-298729/v1/d5d4e3e9-67cf-4078-b221-cf52b1773884.pdf"}],"financialInterests":"","formattedTitle":"Estimation and optimal control of the multi-scale dynamics of the Covid-19","fulltext":[{"header":"Full Text","content":"This preprint is available for \u003ca href='/article/rs-298729/latest.pdf' target='_blank'\u003edownload as a PDF\u003c/a\u003e."}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":true,"isPdf":false,"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":"SARS-CoV-2, multi-scale modeling, parameter estimation, stability analysis, time of extinction, sensitivity analysis, optimal control","lastPublishedDoi":"10.21203/rs.3.rs-298729/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-298729/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThis work aims at a better understanding and the optimal control of the spread of the new severe acute respiratory corona virus 2 (SARS-CoV-2). We first propose a multi-scale model giving insights on the virus population dynamics, the transmission process and the infection mechanism. We consider 10\u0026nbsp;compartments in the human population in order to take into accounts the effects of different specific mitigation policies. The population of viruses is also partitioned into 10 compartments corresponding respectively to each of the first nine human population compartments and the free viruses available in the environment. We show the global stability of the disease free equilibrium if a given threshold T_0 is less or equal to 1 and we provide how to compute the basic reproduction number R_0. A convergence index\u0026nbsp;T_1 is also defined in order to estimate the speed at which the disease extincts and an upper bound to the time of extinction is given. The existence of the endemic equilibrium is conditional and its description is provided. We evaluate the sensitivity of R_0, T_0 and\u0026nbsp;T_1 \u0026nbsp;to control parameters such as the maximal human density allowed per unit of surface, the rate of disinfection both for people and environment, the mobility probability, the wearing mask probability or efficiency, and the human to human contact rate which results from the previous one. Except the maximal human density allowed per unit of surface, all those parameters have significant effects on the qualitative dynamics of the disease. The most significant is the probability of wearing mask followed by the probability of mobility and the disinfection rate. According to a functional cost taking into consideration economic impacts of SARS-CoV-2, we determine and discuss optimal fighting strategies. The study is applied to real available data from Cameroon and an estimation of model parameters is done. After several simulations, social distancing and the disinfection frequency appear as the main elements of the optimal control strategy.\u003c/p\u003e","manuscriptTitle":"Estimation and optimal control of the multi-scale dynamics of the Covid-19","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2021-04-07 22:33:44","doi":"10.21203/rs.3.rs-298729/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"reviewersInvited","content":"","date":"2021-03-11T00:00:00+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2021-03-11T00:00:00+00:00","index":0,"fulltext":""},{"type":"editorInvited","content":"Journal of Nonlinear Science","date":"2021-03-06T00:00:00+00:00","index":"","fulltext":""},{"type":"submitted","content":"Journal of Nonlinear Science","date":"2021-03-04T03:11:23+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2021-03-04T00:00:00+00:00","index":"","fulltext":""}],"status":"published","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}}],"origin":"","ownerIdentity":"c9460f5f-0a1f-4d22-baa1-f8138ee5d89c","owner":[],"postedDate":"April 7th, 2021","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"under-review","subjectAreas":[{"id":3484471,"name":"Applied Mathematics"}],"tags":[],"updatedAt":"2021-05-29T14:26:40+00:00","versionOfRecord":[],"versionCreatedAt":"2021-04-07 22:33:44","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-298729","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-298729","identity":"rs-298729","version":["v1"]},"buildId":"_rKDHFSCF-2tw6eG1fWyD","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}