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(PDF) Session 04 : Modelling and Math Biology Oscillations in epidemic models : the role of infection and recovery times
<!DOCTYPE html> <html > <head> <meta charset="utf-8"> <meta rel="search" type="application/opensearchdescription+xml" href="/open_search.xml" title="Academia.edu"> <meta content="width=device-width, initial-scale=1" name="viewport"> <meta name="google-site-verification" content="bKJMBZA7E43xhDOopFZkssMMkBRjvYERV-NaN4R6mrs"> <meta name="csrf-param" content="authenticity_token" /> <meta name="csrf-token" content="_9wi17D4iUbAK3Ab3YCWkFecwpjkItNxaUohDbazPqP_fFSW9zsvuAA24TaWK27jwIk2vSSoUFsRzalFyPWQbw" /> <meta name="citation_title" content="Session 04 : Modelling and Math Biology Oscillations in epidemic models : the role of infection and recovery times" /> <meta name="citation_publication_date" content="2012/01/01" /> <meta name="citation_author" content="Guillermo Abramson" /> <meta name="twitter:card" content="summary" /> <meta name="twitter:url" content="https://www.academia.edu/99639249/Session_04_Modelling_and_Math_Biology_Oscillations_in_epidemic_models_the_role_of_infection_and_recovery_times" /> <meta name="twitter:title" content="Session 04 : Modelling and Math Biology Oscillations in epidemic models : the role of infection and recovery times" /> <meta name="twitter:description" content="Traditional epidemic models consider that individual processes occur at constant rates. That is, an infected individual has a constant probability per unit time of recovering from infection after contagion. This assumption certainly fails for almost" /> <meta name="twitter:image" content="http://a.academia-assets.com/images/twitter-card.jpeg" /> <meta property="fb:app_id" content="2369844204" /> <meta property="og:type" content="article" /> <meta property="og:url" content="https://www.academia.edu/99639249/Session_04_Modelling_and_Math_Biology_Oscillations_in_epidemic_models_the_role_of_infection_and_recovery_times" /> <meta property="og:title" content="Session 04 : Modelling and Math Biology Oscillations in epidemic models : the role of infection and recovery times" /> <meta property="og:image" content="http://a.academia-assets.com/images/open-graph-icons/fb-paper.gif" /> <meta property="og:description" content="Traditional epidemic models consider that individual processes occur at constant rates. That is, an infected individual has a constant probability per unit time of recovering from infection after contagion. This assumption certainly fails for almost" /> <meta property="article:author" content="https://uncu.academia.edu/GAbramson" /> <meta name="description" content="Traditional epidemic models consider that individual processes occur at constant rates. That is, an infected individual has a constant probability per unit time of recovering from infection after contagion. This assumption certainly fails for almost" /> <title>(PDF) Session 04 : Modelling and Math Biology Oscillations in epidemic models : the role of infection and recovery times</title> <link rel="canonical" href="https://www.academia.edu/99639249/Session_04_Modelling_and_Math_Biology_Oscillations_in_epidemic_models_the_role_of_infection_and_recovery_times" /> <script async src="https://www.googletagmanager.com/gtag/js?id=G-5VKX33P2DS"></script> <script> window.dataLayer = window.dataLayer || []; function gtag(){dataLayer.push(arguments);} gtag('js', new Date()); gtag('config', 'G-5VKX33P2DS', { cookie_domain: 'academia.edu', send_page_view: false, }); gtag('event', 'page_view', { 'controller': "single_work", 'action': "show", 'controller_action': 'single_work#show', 'logged_in': 'false', 'edge': 'unknown', // Send nil if there is no A/B test bucket, in case some records get logged // with missing data - that way we can distinguish between the two cases. // ab_test_bucket should be of the form <ab_test_name>:<bucket> 'ab_test_bucket': null, }) </script> <script> var $controller_name = 'single_work'; var $action_name = "show"; var $rails_env = 'production'; var $app_rev = '0d8900916299500186ee259308143dbb44917529'; var $domain = 'academia.edu'; var $app_host = "academia.edu"; var $asset_host = "academia-assets.com"; var $start_time = new Date().getTime(); var $recaptcha_key = "6LdxlRMTAAAAADnu_zyLhLg0YF9uACwz78shpjJB"; var $recaptcha_invisible_key = "6Lf3KHUUAAAAACggoMpmGJdQDtiyrjVlvGJ6BbAj"; var $disableClientRecordHit = false; </script> <script> window.require = { config: function() { return function() {} } } </script> <script> window.Aedu = window.Aedu || {}; window.Aedu.hit_data = null; window.Aedu.serverRenderTime = new Date(1740589690000); window.Aedu.timeDifference = new Date().getTime() - 1740589690000; </script> <script type="application/ld+json">{"@context":"https://schema.org","@type":"ScholarlyArticle","abstract":"Traditional epidemic models consider that individual processes occur at constant rates. That is, an infected individual has a constant probability per unit time of recovering from infection after contagion. This assumption certainly fails for almost all infectious diseases, in which the infection time usually follows a probability distribution more or less spread around a mean value. We show a general treatment for an SIRS model in which both the infected and the immune phases admit such a description. The general behavior of the system shows transitions between endemic and oscillating situations that could be relevant in many real scenarios. The interaction with the other main source of oscillations, seasonality, will also be discussed.","author":[{"@context":"https://schema.org","@type":"Person","name":"Guillermo Abramson","url":"https://uncu.academia.edu/GAbramson"}],"contributor":[],"dateCreated":"2023-04-04","dateModified":"2024-11-28","datePublished":"2012-01-01","headline":"Session 04 : Modelling and Math Biology Oscillations in epidemic models : the role of infection and recovery 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That is, an infected individual has a constant probability per unit time of recovering from infection after contagion. This assumption certainly fails for almost all infectious diseases, in which the infection time usually follows a probability distribution more or less spread around a mean value. We show a general treatment for an SIRS model in which both the infected and the immune phases admit such a description. The general behavior of the system shows transitions between endemic and oscillating situations that could be relevant in many real scenarios. The interaction with the other main source of oscillations, seasonality, will also be discussed.","ai_title_tag":"Modeling Epidemic Oscillations with Infection Recovery Dynamics","publication_date":"2012,,"},"document_type":"paper","pre_hit_view_count_baseline":null,"quality":"high","language":"en","title":"Session 04 : Modelling and Math Biology Oscillations in epidemic models : the role of infection and recovery times","broadcastable":true,"draft":null,"has_indexable_attachment":true,"indexable":true}}["work"]; window.loswp.workCoauthors = [176215349]; window.loswp.locale = "en"; window.loswp.countryCode = "SG"; window.loswp.cwvAbTestBucket = ""; window.loswp.designVariant = "ds_vanilla"; window.loswp.fullPageMobileSutdModalVariant = "full_page_mobile_sutd_modal"; window.loswp.useOptimizedScribd4genScript = false; window.loginModal = {}; window.loginModal.appleClientId = 'edu.academia.applesignon'; window.userInChina = "false";</script><script defer="" src="https://accounts.google.com/gsi/client"></script><div class="ds-loswp-container"><div class="ds-work-card--grid-container"><div class="ds-work-card--container js-loswp-work-card"><div class="ds-work-card--cover"><div class="ds-work-cover--wrapper"><div class="ds-work-cover--container"><button class="ds-work-cover--clickable js-swp-download-button" data-signup-modal="{"location":"swp-splash-paper-cover","attachmentId":100675305,"attachmentType":"pdf"}"><img alt="First page of “Session 04 : Modelling and Math Biology Oscillations in epidemic models : the role of infection and recovery times”" class="ds-work-cover--cover-thumbnail" src="https://0.academia-photos.com/attachment_thumbnails/100675305/mini_magick20230404-1-7kwns5.png?1680610590" /><img alt="PDF Icon" class="ds-work-cover--file-icon" src="//a.academia-assets.com/images/single_work_splash/adobe_icon.svg" /><div class="ds-work-cover--hover-container"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span><p>Download Free PDF</p></div><div class="ds-work-cover--ribbon-container">Download Free PDF</div><div class="ds-work-cover--ribbon-triangle"></div></button></div></div></div><div class="ds-work-card--work-information"><h1 class="ds-work-card--work-title">Session 04 : Modelling and Math Biology Oscillations in epidemic models : the role of infection and recovery times</h1><div class="ds-work-card--work-authors ds-work-card--detail"><a class="ds-work-card--author js-wsj-grid-card-author ds2-5-body-md ds2-5-body-link" data-author-id="176215349" href="https://uncu.academia.edu/GAbramson"><img alt="Profile image of Guillermo Abramson" class="ds-work-card--author-avatar" src="//a.academia-assets.com/images/s65_no_pic.png" />Guillermo Abramson</a></div><div class="ds-work-card--detail"><p class="ds-work-card--detail ds2-5-body-sm">2012</p><div class="ds-work-card--work-metadata"><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">visibility</span><p class="ds2-5-body-sm" id="work-metadata-view-count">…</p></div><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">description</span><p class="ds2-5-body-sm">5 pages</p></div><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">link</span><p class="ds2-5-body-sm">1 file</p></div></div><script>(async () => { const workId = 99639249; const worksViewsPath = "/v0/works/views?subdomain_param=api&work_ids%5B%5D=99639249"; const getWorkViews = async (workId) => { const response = await fetch(worksViewsPath); if (!response.ok) { throw new Error('Failed to load work views'); } const data = await response.json(); return data.views[workId]; }; // Get the view count for the work - we send this immediately rather than waiting for // the DOM to load, so it can be available as soon as possible (but without holding up // the backend or other resource requests, because it's a bit expensive and not critical). const viewCount = await getWorkViews(workId); const updateViewCount = (viewCount) => { try { const viewCountNumber = parseInt(viewCount, 10); if (viewCountNumber === 0) { // Remove the whole views element if there are zero views. document.getElementById('work-metadata-view-count')?.parentNode?.remove(); return; } const commaizedViewCount = viewCountNumber.toLocaleString(); const viewCountBody = document.getElementById('work-metadata-view-count'); if (!viewCountBody) { throw new Error('Failed to find work views element'); } viewCountBody.textContent = `${commaizedViewCount} views`; } catch (error) { // Remove the whole views element if there was some issue parsing. document.getElementById('work-metadata-view-count')?.parentNode?.remove(); throw new Error(`Failed to parse view count: ${viewCount}`, error); } }; // If the DOM is still loading, wait for it to be ready before updating the view count. if (document.readyState === "loading") { document.addEventListener('DOMContentLoaded', () => { updateViewCount(viewCount); }); // Otherwise, just update it immediately. } else { updateViewCount(viewCount); } })();</script></div><p class="ds-work-card--work-abstract ds-work-card--detail ds2-5-body-md">Traditional epidemic models consider that individual processes occur at constant rates. That is, an infected individual has a constant probability per unit time of recovering from infection after contagion. This assumption certainly fails for almost all infectious diseases, in which the infection time usually follows a probability distribution more or less spread around a mean value. We show a general treatment for an SIRS model in which both the infected and the immune phases admit such a description. The general behavior of the system shows transitions between endemic and oscillating situations that could be relevant in many real scenarios. The interaction with the other main source of oscillations, seasonality, will also be discussed.</p><div class="ds-work-card--button-container"><button class="ds2-5-button js-swp-download-button" data-signup-modal="{"location":"continue-reading-button--work-card","attachmentId":100675305,"attachmentType":"pdf","workUrl":"https://www.academia.edu/99639249/Session_04_Modelling_and_Math_Biology_Oscillations_in_epidemic_models_the_role_of_infection_and_recovery_times"}">See full PDF</button><button class="ds2-5-button ds2-5-button--secondary js-swp-download-button" data-signup-modal="{"location":"download-pdf-button--work-card","attachmentId":100675305,"attachmentType":"pdf","workUrl":"https://www.academia.edu/99639249/Session_04_Modelling_and_Math_Biology_Oscillations_in_epidemic_models_the_role_of_infection_and_recovery_times"}"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span>Download PDF</button></div><div class="ds-signup-banner-trigger-container"><div class="ds-signup-banner-trigger ds-signup-banner-trigger-control"></div></div><div class="ds-signup-banner ds-signup-banner-control"><div id="ds-signup-banner-close-button"><button class="ds2-5-button ds2-5-button--secondary ds2-5-button--inverse"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">close</span></button></div><div class="ds-signup-banner-ctas" data-impression-entity-id="99639249" data-impression-entity-type="2" data-impression-source="signup-banner"><img src="//a.academia-assets.com/images/academia-logo-capital-white.svg" /><h4 class="ds2-5-heading-serif-sm">Sign up for access to the world's latest research</h4><button class="ds2-5-button ds2-5-button--inverse ds2-5-button--full-width js-swp-download-button" data-signup-modal="{"location":"signup-banner"}">Sign up for free<span class="material-symbols-outlined" style="font-size: 20px" translate="no">arrow_forward</span></button></div><div class="ds-signup-banner-divider"></div><div class="ds-signup-banner-reasons"><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Get notified about relevant papers</span></div><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Save papers to use in your research</span></div><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Join the discussion with peers</span></div><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Track your impact</span></div></div></div><script>(() => { // Set up signup banner show/hide behavior: // 1. 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That is, an infected individual has a constant probability per unit time of recovering from infection after contagion. This assumption certainly fails for almost all infectious diseases, in which the infection time usually follows a probability distribution more or less spread around a mean value. We show a general treatment for an SIRS model in which both the infected and the immune phases admit such a description. The general behavior of the system shows transitions between endemic and oscillating situations that could be relevant in many real scenarios. The interaction with the other main source of oscillations, seasonality, is also discussed.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Epidemic oscillations: Interaction between delays and seasonality","attachmentId":71301399,"attachmentType":"pdf","work_url":"https://www.academia.edu/55426202/Epidemic_oscillations_Interaction_between_delays_and_seasonality","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/55426202/Epidemic_oscillations_Interaction_between_delays_and_seasonality"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="1" data-entity-id="116551384" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/116551384/A_Model_for_Reinfections_and_the_Transition_of_Epidemics">A Model for Reinfections and the Transition of Epidemics</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="282724578" href="https://independent.academia.edu/YannisYortsos">Yannis Yortsos</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Viruses</p><p class="ds-related-work--abstract ds2-5-body-sm">Reinfections of infected individuals during a viral epidemic contribute to the continuation of the infection for longer periods of time. In an epidemic, contagion starts with an infection wave, initially growing exponentially fast until it reaches a maximum number of infections, following which it wanes towards an equilibrium state of zero infections, assuming that no new variants have emerged. If reinfections are allowed, multiple such infection waves might occur, and the asymptotic equilibrium state is one in which infection rates are not negligible. This paper analyzes such situations by expanding the traditional SIR model to include two new dimensionless parameters, ε and θ, characterizing, respectively, the kinetics of reinfection and a delay time, after which reinfection commences. We find that depending on these parameter values, three different asymptotic regimes develop. For relatively small θ, two of the regimes are asymptotically stable steady states, approached either mo...</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"A Model for Reinfections and the Transition of Epidemics","attachmentId":112649191,"attachmentType":"pdf","work_url":"https://www.academia.edu/116551384/A_Model_for_Reinfections_and_the_Transition_of_Epidemics","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/116551384/A_Model_for_Reinfections_and_the_Transition_of_Epidemics"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="2" data-entity-id="87356287" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/87356287/Realistic_Distributions_of_Infectious_Periods_in_Epidemic_Models_Changing_Patterns_of_Persistence_and_Dynamics">Realistic Distributions of Infectious Periods in Epidemic Models: Changing Patterns of Persistence and Dynamics</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="25528693" href="https://ncsu.academia.edu/AlunLloyd">Alun Lloyd</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Theoretical Population Biology, 2001</p><p class="ds-related-work--abstract ds2-5-body-sm">Most mathematical models used to study the epidemiology of childhood viral diseases, such as measles, describe the period of infectiousness by an exponential distribution. The effects of including more realistic descriptions of the infectious period within SIR (susceptible infec-tious recovered) models are studied. Less dispersed distributions are seen to have two important epidemiological consequences. First, less stable behaviour is seen within the model: incidence patterns become more complex. Second, disease persistence is diminished: in models with a finite population, the minimum population size needed to allow disease persistence increases. The assumption made concerning the infectious period distribution is of a kind routinely made in the formulation of mathematical models in population biology. Since it has a major effect on the central issues of population persistence and dynamics, the results of this study have broad implications for mathematical modellers of a wide range of biological systems.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Realistic Distributions of Infectious Periods in Epidemic Models: Changing Patterns of Persistence and Dynamics","attachmentId":91588375,"attachmentType":"pdf","work_url":"https://www.academia.edu/87356287/Realistic_Distributions_of_Infectious_Periods_in_Epidemic_Models_Changing_Patterns_of_Persistence_and_Dynamics","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/87356287/Realistic_Distributions_of_Infectious_Periods_in_Epidemic_Models_Changing_Patterns_of_Persistence_and_Dynamics"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="3" data-entity-id="37714721" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/37714721/Oscillations_in_a_patchy_environment_disease_model">Oscillations in a patchy environment disease model</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="51631063" href="https://independent.academia.edu/PDriessche">P. van Den Driessche</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Mathematical Biosciences, 2008</p><p class="ds-related-work--abstract ds2-5-body-sm">For a single patch SIRS model with a period of immunity of fixed length, recruitment-death demographics, disease related deaths and mass action incidence, the basic reproduction number R 0 is identified. It is shown that the disease-free equilibrium is globally asymptotically stable if R 0 < 1. For R 0 > 1, local stability of the endemic equilibrium and Hopf bifurcation analysis about this equilibrium are carried out. Moreover, a practical numerical approach to locate the bifurcation values for a characteristic equation with delay-dependent coefficients is provided. For a two patch SIRS model with travel, it is shown that there are several threshold quantities determining its dynamic behavior and that travel can reduce oscillations in both patches; travel may enhance oscillations in both patches; or travel can switch oscillations from one patch to another.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Oscillations in a patchy environment disease model","attachmentId":57707169,"attachmentType":"pdf","work_url":"https://www.academia.edu/37714721/Oscillations_in_a_patchy_environment_disease_model","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/37714721/Oscillations_in_a_patchy_environment_disease_model"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="4" data-entity-id="84361965" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/84361965/Seasonal_dynamics_in_an_SIR_epidemic_system">Seasonal dynamics in an SIR epidemic system</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="199896151" href="https://independent.academia.edu/nadirsari">nadir sari</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Mathematical Biology, 2013</p><p class="ds-related-work--abstract ds2-5-body-sm">We consider a seasonally forced SIR epidemic model where periodicity occurs in the contact rate. This periodical forcing represents successions of school terms and holidays. The epidemic dynamics are described by a switched system. Numerical studies in such a model have shown the existence of periodic solutions. First, we analytically prove the existence of an invariant domain D containing all periodic (harmonic and subharmonic) orbits. Then, using different scales in time and variables, we rewrite the SIR model as a slow-fast dynamical system and we establish the existence of a macroscopic attractor domain K , included in D, for the switched dynamics. The existence of a unique harmonic solution is also proved for any value of the magnitude of the seasonal forcing term which can be interpreted as an annual infection. Subharmonic solutions can be seen as epidemic outbreaks. Our theoretical results allow us to exhibit quantitative characteristics about epidemics, such as the maximal period between major outbreaks and maximal prevalence.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Seasonal dynamics in an SIR epidemic system","attachmentId":89411977,"attachmentType":"pdf","work_url":"https://www.academia.edu/84361965/Seasonal_dynamics_in_an_SIR_epidemic_system","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/84361965/Seasonal_dynamics_in_an_SIR_epidemic_system"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="5" data-entity-id="111747515" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/111747515/Modeling_of_Asymptotically_Periodic_Outbreaks_a_long_term_SIRW2_description_of_COVID_19">Modeling of Asymptotically Periodic Outbreaks: a long-term SIRW2 description of COVID-19?</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="135038711" href="https://independent.academia.edu/AlessandroVeneziani">Alessandro Veneziani</a></div><p class="ds-related-work--metadata ds2-5-body-xs">arXiv (Cornell University), 2022</p><p class="ds-related-work--abstract ds2-5-body-sm">As the outbreak of COVID-19 enters its third year, we have now enough data to analyse the behavior of the pandemic with mathematical models over a long period of time. The pandemic alternates periods of high and low infections, in a way that sheds a light on the nature of mathematical model that can be used for reliable predictions. The main hypothesis of the model presented here is that the oscillatory behavior is a structural feature of the outbreak, even without postulating a time-dependence of the coefficients. As such, it should be reflected by the presence of limit cycles as asymptotic solutions. This stems from the introduction of (i) a non-linear waning immunity based on the concept of immunity booster (already used for other pathologies); (ii) a fine description of the compartments with a discrimination between individuals infected/vaccinated for the first time, and individuals already infected/vaccinated, undergoing to new infections/doses. We provide a proof-of-concept that our novel model is capable of reproducing long-term oscillatory behavior of many infectious diseases, and, in particular, the periodic nature of the waves of infection. Periodic solutions are inherent to the model, and achieved without changing parameter values in time. This may represent an important step in the long-term modeling of COVID-19 and similar diseases, as the natural, unforced behavior of the solution shows the qualitative characteristics observed during the COVID-19 pandemic.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Modeling of Asymptotically Periodic Outbreaks: a long-term SIRW2 description of COVID-19?","attachmentId":109192098,"attachmentType":"pdf","work_url":"https://www.academia.edu/111747515/Modeling_of_Asymptotically_Periodic_Outbreaks_a_long_term_SIRW2_description_of_COVID_19","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/111747515/Modeling_of_Asymptotically_Periodic_Outbreaks_a_long_term_SIRW2_description_of_COVID_19"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="6" data-entity-id="99022266" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/99022266/An_Epidemic_Model_Structured_by_Host_Immunity">An Epidemic Model Structured by Host Immunity</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="32528038" href="https://independent.academia.edu/MaiaMartcheva">Maia Martcheva</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Biological Systems, 2006</p><p class="ds-related-work--abstract ds2-5-body-sm">In this paper we introduce a physiologically structured SIR epidemic model where the individuals are distributed according to their immune status. An individual immune status is assumed to increase during the infectious period and remain unchanged after the recovery. Recovered individuals can become reinfected at a rate which is a decreasing function of their immune status. We find that the possibility of reinfection of recovered individuals results in subthreshold endemic equilibria. The differential immunity of the infectious individuals leads to multiple nontrivial equilibria in the superthreshold case. We present an example that has exactly three nontrivial equilibria. We also analyze the local stability of equilibria.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"An Epidemic Model Structured by Host Immunity","attachmentId":100218778,"attachmentType":"pdf","work_url":"https://www.academia.edu/99022266/An_Epidemic_Model_Structured_by_Host_Immunity","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/99022266/An_Epidemic_Model_Structured_by_Host_Immunity"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="7" data-entity-id="55426196" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/55426196/Mathematical_modeling_of_the_spread_of_infectious_diseases">Mathematical modeling of the spread of infectious diseases</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="176215349" href="https://uncu.academia.edu/GAbramson">Guillermo Abramson</a></div><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Mathematical modeling of the spread of infectious diseases","attachmentId":71301388,"attachmentType":"pdf","work_url":"https://www.academia.edu/55426196/Mathematical_modeling_of_the_spread_of_infectious_diseases","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/55426196/Mathematical_modeling_of_the_spread_of_infectious_diseases"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="8" data-entity-id="73992897" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/73992897/Destabilization_of_epidemic_models_with_the_inclusion_of_realistic_distributions_of_infectious_periods">Destabilization of epidemic models with the inclusion of realistic distributions of infectious periods</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="25528693" href="https://ncsu.academia.edu/AlunLloyd">Alun Lloyd</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2001</p><p class="ds-related-work--abstract ds2-5-body-sm">Most mathematical models used to understand the dynamical patterns seen in the incidence of childhood viral diseases, such as measles, employ a simple, but epidemiologically unrealistic, description of the infection and recovery process. The inclusion of more realistic descriptions of the recovery process is shown to cause a signi¢cant destabilization of the model. When there is seasonal variation in disease transmission this destabilization leads to the appearance of complex dynamical patterns with much lower levels of seasonality than previously predicted. More generally, this study illustrates how detailed dynamical properties of a model may depend in an important way on the assumptions made in the formulation of the model.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Destabilization of epidemic models with the inclusion of realistic distributions of infectious periods","attachmentId":83818637,"attachmentType":"pdf","work_url":"https://www.academia.edu/73992897/Destabilization_of_epidemic_models_with_the_inclusion_of_realistic_distributions_of_infectious_periods","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/73992897/Destabilization_of_epidemic_models_with_the_inclusion_of_realistic_distributions_of_infectious_periods"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="9" data-entity-id="12076904" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/12076904/Qualitative_Analysis_of_a_Discrete_SIR_Epidemic_Model">Qualitative Analysis of a Discrete SIR Epidemic Model</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="3387364" href="https://ijceronline.academia.edu/IJCEROnline">Independent IJCEROnline</a></div><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Qualitative Analysis of a Discrete SIR Epidemic Model","attachmentId":37398568,"attachmentType":"pdf","work_url":"https://www.academia.edu/12076904/Qualitative_Analysis_of_a_Discrete_SIR_Epidemic_Model","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/12076904/Qualitative_Analysis_of_a_Discrete_SIR_Epidemic_Model"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div></div></div><div class="ds-sticky-ctas--wrapper js-loswp-sticky-ctas hidden"><div class="ds-sticky-ctas--grid-container"><div class="ds-sticky-ctas--container"><button class="ds2-5-button js-swp-download-button" data-signup-modal="{"location":"continue-reading-button--sticky-ctas","attachmentId":100675305,"attachmentType":"pdf","workUrl":null}">See full PDF</button><button class="ds2-5-button ds2-5-button--secondary js-swp-download-button" data-signup-modal="{"location":"download-pdf-button--sticky-ctas","attachmentId":100675305,"attachmentType":"pdf","workUrl":null}"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span>Download PDF</button></div></div></div><div class="ds-below-fold--grid-container"><div class="ds-work--container js-loswp-embedded-document"><div class="attachment_preview" data-attachment="Attachment_100675305" style="display: none"><div class="js-scribd-document-container"><div class="scribd--document-loading js-scribd-document-loader" style="display: block;"><img alt="Loading..." src="//a.academia-assets.com/images/loaders/paper-load.gif" /><p>Loading Preview</p></div></div><div style="text-align: center;"><div class="scribd--no-preview-alert js-preview-unavailable"><p>Sorry, preview is currently unavailable. 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