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(PDF) Emergence of oscillations in a simple epidemic model with demographic data
<!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="QPuccmBE5BqfLgZBBcMt3huBzly-fQin_OvBPc1Vr9LsgG3ikv58ffIbYppjQsqn8BQ_RIsbxQ01E52HKfzyJA" /> <meta name="citation_title" content="Emergence of oscillations in a simple epidemic model with demographic data" /> <meta name="citation_journal_title" content="Royal Society Open Science" /> <meta name="citation_author" content="Raj Saha" /> <meta name="twitter:card" content="summary" /> <meta name="twitter:url" content="https://www.academia.edu/52080924/Emergence_of_oscillations_in_a_simple_epidemic_model_with_demographic_data" /> <meta name="twitter:title" content="Emergence of oscillations in a simple epidemic model with demographic data" /> <meta name="twitter:description" content="A simple susceptible–infectious–removed epidemic model for smallpox, with birth and death rates based on historical data, produces oscillatory dynamics with remarkably accurate periodicity. Stochastic population data cause oscillations to be" /> <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/52080924/Emergence_of_oscillations_in_a_simple_epidemic_model_with_demographic_data" /> <meta property="og:title" content="Emergence of oscillations in a simple epidemic model with demographic data" /> <meta property="og:image" content="http://a.academia-assets.com/images/open-graph-icons/fb-paper.gif" /> <meta property="og:description" content="A simple susceptible–infectious–removed epidemic model for smallpox, with birth and death rates based on historical data, produces oscillatory dynamics with remarkably accurate periodicity. Stochastic population data cause oscillations to be" /> <meta property="article:author" content="https://independent.academia.edu/RajSaha22" /> <meta name="description" content="A simple susceptible–infectious–removed epidemic model for smallpox, with birth and death rates based on historical data, produces oscillatory dynamics with remarkably accurate periodicity. Stochastic population data cause oscillations to be" /> <title>(PDF) Emergence of oscillations in a simple epidemic model with demographic data</title> <link rel="canonical" href="https://www.academia.edu/52080924/Emergence_of_oscillations_in_a_simple_epidemic_model_with_demographic_data" /> <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 = '2b2763d861eba4fa7a8ab85b59b94c082c9d888c'; 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(1740604835000); window.Aedu.timeDifference = new Date().getTime() - 1740604835000; </script> <script type="application/ld+json">{"@context":"https://schema.org","@type":"ScholarlyArticle","abstract":"A simple susceptible–infectious–removed epidemic model for smallpox, with birth and death rates based on historical data, produces oscillatory dynamics with remarkably accurate periodicity. Stochastic population data cause oscillations to be sustained rather than damped, and data analysis regarding the oscillations provides insights into the same set of population data. Notably, oscillations arise naturally from the model, instead of from a periodic forcing term or other exogenous mechanism that guarantees oscillation: the model has no such mechanism. These emergent natural oscillations display appropriate periodicity for smallpox, even when the model is applied to different locations and populations. The model and datasets, in turn, offer new observations about disease dynamics and solution trajectories. These results call for renewed attention to relatively simple models, in combination with datasets from real outbreaks.","author":[{"@context":"https://schema.org","@type":"Person","name":"Raj Saha","url":"https://independent.academia.edu/RajSaha22"}],"contributor":[],"dateCreated":"2021-09-12","headline":"Emergence of oscillations in a simple epidemic model with demographic data","image":"https://attachments.academia-assets.com/69510193/thumbnails/1.jpg","inLanguage":"en","keywords":[],"publication":"Royal Society Open Science","publisher":{"@context":"https://schema.org","@type":"Organization","name":"The Royal Society"},"sourceOrganization":[{"@context":"https://schema.org","@type":"EducationalOrganization","name":null}],"thumbnailUrl":"https://attachments.academia-assets.com/69510193/thumbnails/1.jpg","url":"https://www.academia.edu/52080924/Emergence_of_oscillations_in_a_simple_epidemic_model_with_demographic_data"}</script><style type="text/css">@media(max-width: 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window.loswp.work = {"work":{"id":52080924,"created_at":"2021-09-12T23:34:13.917-07:00","from_world_paper_id":172750174,"updated_at":"2024-11-29T00:54:23.482-08:00","_data":{"abstract":"A simple susceptible–infectious–removed epidemic model for smallpox, with birth and death rates based on historical data, produces oscillatory dynamics with remarkably accurate periodicity. Stochastic population data cause oscillations to be sustained rather than damped, and data analysis regarding the oscillations provides insights into the same set of population data. Notably, oscillations arise naturally from the model, instead of from a periodic forcing term or other exogenous mechanism that guarantees oscillation: the model has no such mechanism. These emergent natural oscillations display appropriate periodicity for smallpox, even when the model is applied to different locations and populations. The model and datasets, in turn, offer new observations about disease dynamics and solution trajectories. These results call for renewed attention to relatively simple models, in combination with datasets from real outbreaks.","publisher":"The Royal Society","ai_title_tag":"Natural Oscillations in an Epidemic Model of Smallpox","publication_name":"Royal Society Open Science"},"document_type":"paper","pre_hit_view_count_baseline":null,"quality":"high","language":"en","title":"Emergence of oscillations in a simple epidemic model with demographic data","broadcastable":false,"draft":null,"has_indexable_attachment":true,"indexable":true}}["work"]; window.loswp.workCoauthors = [144642595]; window.loswp.locale = "en"; window.loswp.countryCode = "SG"; window.loswp.cwvAbTestBucket = ""; window.loswp.designVariant = "ds_vanilla"; window.loswp.fullPageMobileSutdModalVariant = "control"; 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":69510193,"attachmentType":"pdf"}"><img alt="First page of “Emergence of oscillations in a simple epidemic model with demographic data”" class="ds-work-cover--cover-thumbnail" src="https://0.academia-photos.com/attachment_thumbnails/69510193/mini_magick20210912-29241-orx8zv.png?1631514903" /><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">Emergence of oscillations in a simple epidemic model with demographic data</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="144642595" href="https://independent.academia.edu/RajSaha22"><img alt="Profile image of Raj Saha" class="ds-work-card--author-avatar" src="//a.academia-assets.com/images/s65_no_pic.png" />Raj Saha</a></div><div class="ds-work-card--detail"><p class="ds-work-card--detail ds2-5-body-sm">Royal Society Open Science</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">13 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 = 52080924; 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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">A simple susceptible–infectious–removed epidemic model for smallpox, with birth and death rates based on historical data, produces oscillatory dynamics with remarkably accurate periodicity. Stochastic population data cause oscillations to be sustained rather than damped, and data analysis regarding the oscillations provides insights into the same set of population data. Notably, oscillations arise naturally from the model, instead of from a periodic forcing term or other exogenous mechanism that guarantees oscillation: the model has no such mechanism. These emergent natural oscillations display appropriate periodicity for smallpox, even when the model is applied to different locations and populations. The model and datasets, in turn, offer new observations about disease dynamics and solution trajectories. These results call for renewed attention to relatively simple models, in combination with datasets from real outbreaks.</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":69510193,"attachmentType":"pdf","workUrl":"https://www.academia.edu/52080924/Emergence_of_oscillations_in_a_simple_epidemic_model_with_demographic_data"}">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":69510193,"attachmentType":"pdf","workUrl":"https://www.academia.edu/52080924/Emergence_of_oscillations_in_a_simple_epidemic_model_with_demographic_data"}"><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="52080924" 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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The model incorporates the spread of awareness among individuals and, moreover, a small inflow of imported cases. These cases prevent stochastic extinctions when we simulate the epidemics and, hence, they allow to check whether the average dynamics for the fraction of infected individuals are accurately predicted by the ODE model. Stochastic simulations confirm the existence of sustained oscillations for different types of random networks, with a sharp transition from a nonoscillatory asymptotic regime to a periodic one as the alerting rate of susceptible individuals increases from very small values. This abrupt transition to periodic epidemics of high amplitude is quite accurately predicted by the Hopf-bifurcation curve computed from the ODE model using the alerting rate and the infection transmission rate for aware individuals as tuning parameters.</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":"Robustness of behaviorally induced oscillations in epidemic models under a low rate of imported cases","attachmentId":89620499,"attachmentType":"pdf","work_url":"https://www.academia.edu/84683452/Robustness_of_behaviorally_induced_oscillations_in_epidemic_models_under_a_low_rate_of_imported_cases","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/84683452/Robustness_of_behaviorally_induced_oscillations_in_epidemic_models_under_a_low_rate_of_imported_cases"><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="23791325" 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/23791325/TURNING_POINTS_AND_RELAXATION_OSCILLATION_CYCLES_IN_SIMPLE_EPIDEMIC_MODELS">TURNING POINTS AND RELAXATION OSCILLATION CYCLES IN SIMPLE EPIDEMIC MODELS</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="46058495" href="https://independent.academia.edu/ChunhuaShan">Chunhua Shan</a><span>, </span><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="339387" href="https://ualberta.academia.edu/MichaelLi">Michael Y Li</a></div><p class="ds-related-work--abstract ds2-5-body-sm">We study the interplay between effects of disease burden on the host population and the effects of population growth on the disease incidence, in an epidemic model of SIR type with demography and disease-caused death. We revisit the classical problem of periodicity in incidences of certain autonomous diseases. Under the assumption that the host population has a small intrinsic growth rate, using singular perturbation techniques and the phenomenon of the delay of stability loss due to turning points, we prove that large-amplitude relaxation oscillation cycles exist for an open set of model parameters. Simulations are provided to support our theoretical results. Our results offer new insight into the classical periodicity problem in epidemiology. Our approach relies on analysis far away from the endemic equilibrium and contrasts sharply with the method of Hopf bifurcations.</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":"TURNING POINTS AND RELAXATION OSCILLATION CYCLES IN SIMPLE EPIDEMIC MODELS","attachmentId":44200402,"attachmentType":"pdf","work_url":"https://www.academia.edu/23791325/TURNING_POINTS_AND_RELAXATION_OSCILLATION_CYCLES_IN_SIMPLE_EPIDEMIC_MODELS","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/23791325/TURNING_POINTS_AND_RELAXATION_OSCILLATION_CYCLES_IN_SIMPLE_EPIDEMIC_MODELS"><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="99639249" 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/99639249/Session_04_Modelling_and_Math_Biology_Oscillations_in_epidemic_models_the_role_of_infection_and_recovery_times">Session 04 : Modelling and Math Biology Oscillations in epidemic models : the role of infection and recovery times</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><p class="ds-related-work--metadata ds2-5-body-xs">2012</p><p class="ds-related-work--abstract ds2-5-body-sm">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-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":"Session 04 : Modelling and Math Biology Oscillations in epidemic models : the role of infection and recovery times","attachmentId":100675305,"attachmentType":"pdf","work_url":"https://www.academia.edu/99639249/Session_04_Modelling_and_Math_Biology_Oscillations_in_epidemic_models_the_role_of_infection_and_recovery_times","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/99639249/Session_04_Modelling_and_Math_Biology_Oscillations_in_epidemic_models_the_role_of_infection_and_recovery_times"><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="7004606" 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/7004606/Stochastic_amplification_in_epidemics">Stochastic amplification in 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="11914239" href="https://independent.academia.edu/davidalonso11">david alonso</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of The Royal Society Interface, 2007</p><p class="ds-related-work--abstract ds2-5-body-sm">The role of stochasticity and its interplay with nonlinearity are central current issues in studies of the complex population patterns observed in nature, including the pronounced oscillations of wildlife and infectious diseases. The dynamics of childhood diseases have provided influential case studies to develop and test mathematical models with practical application to epidemiology, but are also of general relevance to the central question of whether simple nonlinear systems can explain and predict the complex temporal and spatial patterns observed in nature outside laboratory conditions. Here, we present a stochastic theory for the major dynamical transitions in epidemics from regular to irregular cycles, which relies on the discrete nature of disease transmission and low spatial coupling. The full spectrum of stochastic fluctuations is derived analytically to show how the amplification of noise varies across these transitions. The changes in noise amplification and coherence appear robust to seasonal forcing, questioning the role of seasonality and its interplay with deterministic components of epidemiological models. Childhood diseases are shown to fall into regions of parameter space of high noise amplification. This type of 'endogenous' stochastic resonance may be relevant to population oscillations in nonlinear ecological systems in general.</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":"Stochastic amplification in epidemics","attachmentId":48636719,"attachmentType":"pdf","work_url":"https://www.academia.edu/7004606/Stochastic_amplification_in_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/7004606/Stochastic_amplification_in_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="4" 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="5" data-entity-id="47492480" 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/47492480/Small_amplitude_long_period_outbreaks_in_seasonally_driven_epidemics">Small amplitude, long period outbreaks in seasonally driven 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="56135069" href="https://independent.academia.edu/IraSchwartz3">Ira Schwartz</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Mathematical Biology, 1992</p><p class="ds-related-work--abstract ds2-5-body-sm">It is now documented that childhood diseases such as measles, mumps, and chickenpox exhibit a wide range of recurrent behavior (periodic as well as chaotic) in large population centers in the first world. Mathematical models used in the past (such as the SEIR model with seasonal forcing) have been able to predict the onset of both periodic and chaotic sustained epidemics using parameters of childhood diseases. Although these models possess stable solutions which appear to have the correct frequency content, the corresponding outbreaks require extremely large populations to support the epidemic. This paper shows that by relaxing the assumption of uniformity in the supply of susceptibles, simple models predict stable long period oscillatory epidemics having small amplitude. Both coupled and single population models are considered.</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":"Small amplitude, long period outbreaks in seasonally driven epidemics","attachmentId":66557116,"attachmentType":"pdf","work_url":"https://www.academia.edu/47492480/Small_amplitude_long_period_outbreaks_in_seasonally_driven_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/47492480/Small_amplitude_long_period_outbreaks_in_seasonally_driven_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="6" 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="7" data-entity-id="55426202" 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/55426202/Epidemic_oscillations_Interaction_between_delays_and_seasonality">Epidemic oscillations: Interaction between delays and seasonality</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><p class="ds-related-work--abstract ds2-5-body-sm">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, 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="8" 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="9" data-entity-id="93279681" 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/93279681/On_the_stochastic_engine_of_transmittable_diseases_in_exponentially_growing_populations">On the stochastic engine of transmittable diseases in exponentially growing populations</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="147966780" href="https://linnaeus.academia.edu/TorstenLindstr%C3%B6m">Torsten Lindström</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2021</p><p class="ds-related-work--abstract ds2-5-body-sm">The purpose of this paper is to analyze the interplay of deterministic and stochastic models for epidemic diseases. Deterministic models for epidemic diseases are prone to predict global stability. If the natural birth and death rates are assumed small in comparison to disease parameters like the contact rate and the recovery rate, then the globally stable endemic equilibrium corresponds to a tiny proportion of infected individuals. Asymptotic equilibrium levels corresponding to low numbers of individuals invalidate the deterministic results. Diffusion effects force frequency functions of the stochastic model to possess similar stability properties as the deterministic model. Particular simulations of the stochastic model are, however, oscillatory and predict oscillatory patterns. Smaller or isolated populations show longer periods, more violent oscillations, and larger probabilities of extinction. We prove that evolution maximizes the infectiousness of the disease as measured by th...</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":"On the stochastic engine of transmittable diseases in exponentially growing populations","attachmentId":96060151,"attachmentType":"pdf","work_url":"https://www.academia.edu/93279681/On_the_stochastic_engine_of_transmittable_diseases_in_exponentially_growing_populations","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/93279681/On_the_stochastic_engine_of_transmittable_diseases_in_exponentially_growing_populations"><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":69510193,"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":69510193,"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_69510193" 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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js-related-work-sidebar-card" data-collection-position="2" data-entity-id="55603002" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/55603002/Fluctuations_in_a_SIS_epidemic_model_with_variable_size_population">Fluctuations in a SIS epidemic model with variable size population</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="38376177" href="https://independent.academia.edu/KhadijaNiri">Khadija Niri</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Applied Mathematics and Computation, 2010</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":"Fluctuations in a SIS epidemic model with variable size 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