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{"work":{"id":29627601,"created_at":"2016-11-02T14:59:39.498-07:00","from_world_paper_id":158840017,"updated_at":"2025-02-01T20:32:28.561-08:00","_data":{"ai_title_tag":"Epidemic Threshold in Interconnected Networks","grobid_abstract":"Most real-world networks are not isolated. In order to function fully, they are interconnected with other networks, and this interconnection influences their dynamic processes. For example, when the spread of a disease involves two species, the dynamics of the spread within each species (the contact network) differs from that of the spread between the two species (the interconnected network). We model two generic interconnected networks using two adjacency matrices, A and B, in which A is a 2N × 2N matrix that depicts the connectivity within each of two networks of size N , and B a 2N × 2N matrix that depicts the interconnections between the two. Using an N-intertwined mean-field approximation, we determine that a critical susceptable-infected-susceptable (SIS) epidemic threshold in two interconnected networks is 1/λ1(A + αB), where the infection rate is β within each of the two individual networks and αβ in the interconnected links between the two networks and λ1(A + αB) is the largest eigenvalue of the matrix A+αB. In order to determine how the epidemic threshold is dependent upon the structure of interconnected networks, we analytically derive λ1(A + αB) using perturbation approximation for small and large α, the lower and upper bound for any α as a function of the adjacency matrix of the two individual networks, and the interconnections between the two and their largest eigenvalues/eigenvectors. We verify these approximation and boundary values for λ1(A + αB) using numerical simulations, and determine how component network features affect λ1(A + αB). We note that, given two isolated networks G1 and G2 with principle eigenvectors x and y respectively, λ1(A + αB) tends to be higher when nodes i and j with a higher eigenvector component product xiyj are interconnected. This finding suggests essential insights into ways of designing interconnected networks to be robust against epidemics.","publication_date":"2013,,","publication_name":"Physical Review E","grobid_abstract_attachment_id":"50066618"},"document_type":"paper","pre_hit_view_count_baseline":null,"quality":"high","language":"en","title":"Effect of the interconnected network structure on the epidemic threshold","broadcastable":true,"draft":null,"has_indexable_attachment":true,"indexable":true}}["work"]; window.loswp.workCoauthors = [36407694,55534402]; 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":50066618,"attachmentType":"pdf"}"><img alt="First page of “Effect of the interconnected network structure on the epidemic threshold”" class="ds-work-cover--cover-thumbnail" src="https://0.academia-photos.com/attachment_thumbnails/50066618/mini_magick20190129-4195-1l4buej.png?1548830913" /><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">Effect of the interconnected network structure on the epidemic threshold</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="36407694" href="https://biu.academia.edu/SHavlin"><img alt="Profile image of S. Havlin" class="ds-work-card--author-avatar" src="//a.academia-assets.com/images/s65_no_pic.png" />S. Havlin</a><a class="ds-work-card--author js-wsj-grid-card-author ds2-5-body-md ds2-5-body-link" data-author-id="55534402" href="https://independent.academia.edu/DAgostinoGregorio"><img alt="Profile image of Gregorio D'Agostino" class="ds-work-card--author-avatar" src="https://0.academia-photos.com/55534402/18281148/18251545/s65_gregorio.d_agostino.jpg" />Gregorio D'Agostino</a></div><div class="ds-work-card--detail"><p class="ds-work-card--detail ds2-5-body-sm">2013, Physical Review E</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">21 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 = 29627601; const worksViewsPath = "/v0/works/views?subdomain_param=api&work_ids%5B%5D=29627601"; 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">Most real-world networks are not isolated. In order to function fully, they are interconnected with other networks, and this interconnection influences their dynamic processes. For example, when the spread of a disease involves two species, the dynamics of the spread within each species (the contact network) differs from that of the spread between the two species (the interconnected network). We model two generic interconnected networks using two adjacency matrices, A and B, in which A is a 2N × 2N matrix that depicts the connectivity within each of two networks of size N , and B a 2N × 2N matrix that depicts the interconnections between the two. Using an N-intertwined mean-field approximation, we determine that a critical susceptable-infected-susceptable (SIS) epidemic threshold in two interconnected networks is 1/λ1(A + αB), where the infection rate is β within each of the two individual networks and αβ in the interconnected links between the two networks and λ1(A + αB) is the largest eigenvalue of the matrix A+αB. In order to determine how the epidemic threshold is dependent upon the structure of interconnected networks, we analytically derive λ1(A + αB) using perturbation approximation for small and large α, the lower and upper bound for any α as a function of the adjacency matrix of the two individual networks, and the interconnections between the two and their largest eigenvalues/eigenvectors. We verify these approximation and boundary values for λ1(A + αB) using numerical simulations, and determine how component network features affect λ1(A + αB). We note that, given two isolated networks G1 and G2 with principle eigenvectors x and y respectively, λ1(A + αB) tends to be higher when nodes i and j with a higher eigenvector component product xiyj are interconnected. This finding suggests essential insights into ways of designing interconnected networks to be robust against epidemics.</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":50066618,"attachmentType":"pdf","workUrl":"https://www.academia.edu/29627601/Effect_of_the_interconnected_network_structure_on_the_epidemic_threshold"}">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":50066618,"attachmentType":"pdf","workUrl":"https://www.academia.edu/29627601/Effect_of_the_interconnected_network_structure_on_the_epidemic_threshold"}"><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"><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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If the infection strength is higher than a certain threshold -which we define as the epidemic threshold -then the epidemic spreads through the population and persists in the long run. For a single generic graph representing the contact network of the population under consideration, the epidemic threshold turns out to be equal to the inverse of the spectral radius of the contact graph. However, in a real world scenario it is not possible to isolate a population completely: there is always some interconnection with another network, which partially overlaps with the contact network. Results for epidemic threshold in interconnected networks are limited to homogeneous mixing populations and degree distribution arguments. In this paper, we adopt a spectral approach. We show how the epidemic threshold in a given network changes as a result of being coupled with another network with fixed infection strength. In our model, the contact network and the interconnections are generic. Using bifurcation theory and algebraic graph theory, we rigorously derive the epidemic threshold in interconnected networks. These results have implications for the broad field of epidemic modeling and control. Our analytical results are supported by numerical simulations.</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":"Effect of coupling on the epidemic threshold in interconnected complex networks: A spectral analysis","attachmentId":45549336,"attachmentType":"pdf","work_url":"https://www.academia.edu/13250613/Effect_of_coupling_on_the_epidemic_threshold_in_interconnected_complex_networks_A_spectral_analysis","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/13250613/Effect_of_coupling_on_the_epidemic_threshold_in_interconnected_complex_networks_A_spectral_analysis"><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="100494307" 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/100494307/Interconnected_Complex_Networks_A_Spectral_Analysis">Interconnected Complex Networks: A Spectral Analysis</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="32507196" href="https://ksu.academia.edu/CaterinaScoglio">Caterina Scoglio</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2016</p><p class="ds-related-work--abstract ds2-5-body-sm">In epidemic modeling, the term infection strength indicates the ratio of infection rate and cure rate. If the infection strength is higher than a certain threshold-which we define as the epidemic threshold-then the epidemic spreads through the population and persists in the long run. For a single generic graph representing the contact network of the population under consideration, the epidemic threshold turns out to be equal to the inverse of the spectral radius of the contact graph. However, in a real world scenario it is not possible to isolate a population completely: there is always some interconnection with another network, which partially overlaps with the contact network. Results for epidemic threshold in interconnected networks are limited to homogeneous mixing populations and degree distribution arguments. In this paper, we adopt a spectral approach. We show how the epidemic threshold in a given network changes as a result of being coupled with another network with fixed infection strength. In our model, the contact network and the interconnections are generic. Using bifurcation theory and algebraic graph theory, we rigorously derive the epidemic threshold in interconnected networks. These results have implications for the broad field of epidemic modeling and control. Our analytical results are supported by numerical simulations.</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":"Interconnected Complex Networks: A Spectral Analysis","attachmentId":101302880,"attachmentType":"pdf","work_url":"https://www.academia.edu/100494307/Interconnected_Complex_Networks_A_Spectral_Analysis","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/100494307/Interconnected_Complex_Networks_A_Spectral_Analysis"><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="51024608" 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/51024608/Analysis_of_the_susceptible_infected_susceptible_epidemic_dynamics_in_networks_via_the_non_backtracking_matrix">Analysis of the susceptible-infected-susceptible epidemic dynamics in networks via the non-backtracking matrix</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="138297164" href="https://osaka-u.academia.edu/MasakiOgura">Masaki Ogura</a></div><p class="ds-related-work--metadata ds2-5-body-xs">IMA Journal of Applied Mathematics</p><p class="ds-related-work--abstract ds2-5-body-sm">We study the stochastic susceptible-infected-susceptible model of epidemic processes on finite directed and weighted networks with arbitrary structure. We present a new lower bound on the exponential rate at which the probabilities of nodes being infected decay over time. This bound is directly related to the leading eigenvalue of a matrix that depends on the non-backtracking and incidence matrices of the network. The dimension of this matrix is $N+M$, where $N$ and $M$ are the number of nodes and edges, respectively. We show that this new lower bound improves on an existing bound corresponding to the so-called quenched mean-field theory. Although the bound obtained from a recently developed second-order moment-closure technique requires the computation of the leading eigenvalue of an $N^2\times N^2$ matrix, we illustrate in our numerical simulations that the new bound is tighter, while being computationally less expensive for sparse networks. We also present the expression for the ...</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":"Analysis of the susceptible-infected-susceptible epidemic dynamics in networks via the non-backtracking matrix","attachmentId":68895658,"attachmentType":"pdf","work_url":"https://www.academia.edu/51024608/Analysis_of_the_susceptible_infected_susceptible_epidemic_dynamics_in_networks_via_the_non_backtracking_matrix","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/51024608/Analysis_of_the_susceptible_infected_susceptible_epidemic_dynamics_in_networks_via_the_non_backtracking_matrix"><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="30169123" 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/30169123/Multiple_transitions_of_the_susceptible_infected_susceptible_epidemic_model_on_complex_networks">Multiple transitions of the susceptible-infected-susceptible epidemic model on complex networks</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="15839788" href="https://dwcc.academia.edu/AngelicaMata">Angelica Mata</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physical Review E, 2015</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":"Multiple transitions of the susceptible-infected-susceptible epidemic model on complex networks","attachmentId":50626383,"attachmentType":"pdf","work_url":"https://www.academia.edu/30169123/Multiple_transitions_of_the_susceptible_infected_susceptible_epidemic_model_on_complex_networks","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/30169123/Multiple_transitions_of_the_susceptible_infected_susceptible_epidemic_model_on_complex_networks"><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="14719463" 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/14719463/Epidemic_thresholds_of_the_susceptible_infected_susceptible_model_on_networks_A_comparison_of_numerical_and_theoretical_results">Epidemic thresholds of the susceptible-infected-susceptible model on networks: A comparison of numerical and theoretical results</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="33676389" href="https://ufv-br.academia.edu/SilvioFerreira">Silvio Ferreira</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physical Review E, 2012</p><p class="ds-related-work--abstract ds2-5-body-sm">Recent work has shown that different theoretical approaches to the dynamics of the susceptible-infectedsusceptible (SIS) model for epidemics lead to qualitatively different estimates for the position of the epidemic threshold in networks. Here we present large-scale numerical simulations of the SIS dynamics on various types of networks, allowing the precise determination of the effective threshold for systems of finite size N . We compare quantitatively the numerical thresholds with theoretical predictions of the heterogeneous mean-field theory and of the quenched mean-field theory. We show that the latter is in general more accurate, scaling with N with the correct exponent, but often failing to capture the correct prefactor.</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 thresholds of the susceptible-infected-susceptible model on networks: A comparison of numerical and theoretical results","attachmentId":43944205,"attachmentType":"pdf","work_url":"https://www.academia.edu/14719463/Epidemic_thresholds_of_the_susceptible_infected_susceptible_model_on_networks_A_comparison_of_numerical_and_theoretical_results","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/14719463/Epidemic_thresholds_of_the_susceptible_infected_susceptible_model_on_networks_A_comparison_of_numerical_and_theoretical_results"><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="6216076" 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/6216076/Epidemic_Spreading_in_Real_Networks_An_Eigenvalue_Viewpoint">Epidemic Spreading in Real Networks: An Eigenvalue Viewpoint</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="9524897" href="https://lehman.academia.edu/YangWang">Yang Wang</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2003</p><p class="ds-related-work--abstract ds2-5-body-sm">How will a virus propagate in a real network? Does an epidemic threshold exist for a finite powerlaw graph, or any finite graph? How long does it take to disinfect a network given particular values of infection rate and virus death rate?</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 Spreading in Real Networks: An Eigenvalue Viewpoint","attachmentId":33086790,"attachmentType":"pdf","work_url":"https://www.academia.edu/6216076/Epidemic_Spreading_in_Real_Networks_An_Eigenvalue_Viewpoint","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/6216076/Epidemic_Spreading_in_Real_Networks_An_Eigenvalue_Viewpoint"><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="811007" 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/811007/Epidemic_incidence_in_correlated_complex_networks">Epidemic incidence in correlated complex networks</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="132586" href="https://unizar.academia.edu/yamirmoreno">yamir moreno</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physical Review E, 2003</p><p class="ds-related-work--abstract ds2-5-body-sm">We introduce a numerical method to solve epidemic models on the underlying topology of complex networks. The approach exploits the mean-field like rate equations describing the system and allows to work with very large system sizes, where Monte Carlo simulations are useless due to memory needs. We then study the SIR epidemiological model on assortative networks, providing numerical evidence of the absence of epidemic thresholds. Besides, the time profiles of the populations are analyzed. Finally, we stress that the present method would allow to solve arbitrary epidemic-like models provided that they can be described by mean-field rate equations. 89.75.Fb, 05.70.Jk, 05.40.a A few years ago, Watts and Strogatz [1] introduced a model able to produce networks with properties of both regular lattices and random graphs with small diameter. Their model soon led to a burst of activity in the field [2, 3], further spurred by Barabasi and collaborators who found that many seemingly diverse systems share several topological properties such as a power law behavior in their connectivity distributions when represented as networks . These complex networks are formed by a set of many elements (or nodes) that are linked together through edges (or links) if they interact directly. Empirical evidence supports that in notable networks, such as metabolic or communication webs, the probability P (k) that any node has k links to other nodes is distributed accordingly to a power law P (k) ∼ k −γ [5, 6, 7], with γ ≤ 3 in most cases.</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 incidence in correlated complex networks","attachmentId":4860097,"attachmentType":"pdf","work_url":"https://www.academia.edu/811007/Epidemic_incidence_in_correlated_complex_networks","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/811007/Epidemic_incidence_in_correlated_complex_networks"><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="5447214" 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/5447214/Epidemics_on_Interconnected_Networks">Epidemics on Interconnected Networks</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="7630487" href="https://bu.academia.edu/EugeneStanley">Eugene Stanley</a></div><p class="ds-related-work--abstract ds2-5-body-sm">Populations are seldom completely isolated from their environment. Individuals in a particular geographic or social region may be considered a distinct network due to strong local ties but will also interact with individuals in other networks. We study the susceptible-infected-recovered process on interconnected network systems and find two distinct regimes. In strongly coupled network systems, epidemics occur simultaneously across the entire system at a critical infection strength β c , below which the disease does not spread. In contrast, in weakly coupled network systems, a mixed phase exists below β c of the coupled network system, where an epidemic occurs in one network but does not spread to the coupled network. We derive an expression for the network and disease parameters that allow this mixed phase and verify it numerically. Public health implications of communities comprising these two classes of network systems are also mentioned.</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":"Epidemics on Interconnected Networks","attachmentId":32570476,"attachmentType":"pdf","work_url":"https://www.academia.edu/5447214/Epidemics_on_Interconnected_Networks","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/5447214/Epidemics_on_Interconnected_Networks"><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="810998" 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/810998/Epidemic_outbreaks_in_complex_heterogeneous_networks">Epidemic outbreaks in complex heterogeneous networks</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="132586" href="https://unizar.academia.edu/yamirmoreno">yamir moreno</a></div><p class="ds-related-work--metadata ds2-5-body-xs">The European Physical …, 2002</p><p class="ds-related-work--abstract ds2-5-body-sm">We present a detailed analytical and numerical study for the spreading of infections in complex population networks with acquired immunity. We show that the large connectivity fluctuations usually found in these networks strengthen considerably the incidence of epidemic outbreaks. Scale-free networks, which are characterized by diverging connectivity fluctuations, exhibit the lack of an epidemic threshold and always show a finite fraction of infected individuals. This particular weakness, observed also in models without immunity, defines a new epidemiological framework characterized by a highly heterogeneous response of the system to the introduction of infected individuals with different connectivity. The understanding of epidemics in complex networks might deliver new insights in the spread of information and diseases in biological and technological networks that often appear to be characterized by complex heterogeneous architectures.</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 outbreaks in complex heterogeneous networks","attachmentId":4860088,"attachmentType":"pdf","work_url":"https://www.academia.edu/810998/Epidemic_outbreaks_in_complex_heterogeneous_networks","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/810998/Epidemic_outbreaks_in_complex_heterogeneous_networks"><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="55225889" 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/55225889/Epidemic_threshold_and_topological_structure_of_susceptible_infectious_susceptible_epidemics_in_adaptive_networks">Epidemic threshold and topological structure of susceptible-infectious-susceptible epidemics in adaptive networks</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="37393017" href="https://independent.academia.edu/StojanTrajanovski">Stojan Trajanovski</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physical Review E, 2013</p><p class="ds-related-work--abstract ds2-5-body-sm">The interplay between disease dynamics on a network and the dynamics of the structure of that network characterizes many real-world systems of contacts. A continuous-time adaptive Susceptible-Infectious-Susceptible (ASIS) model is introduced in order to investigate this interaction, where a susceptible node avoids infections by breaking its links to its infected neighbors while it enhances the connections with other susceptible nodes by creating links to them. When the initial topology of the network is a complete graph, an exact solution to the average metastable state fraction of infected nodes is derived without resorting to any mean-field approximation. A linear scaling law of the epidemic threshold τc as a function of the effective link-breaking rate ω is found. Furthermore, the bifurcation nature of the metastable fraction of infected nodes of the ASIS model is explained. The metastable state topology shows high connectivity and low modularity in two regions of the τ, ω-plane for any effective infection rate τ > τc: (i) a "strongly adaptive" region with very high ω, and (ii) a "weakly adaptive" region with very low ω. These two regions are separated from the other half-open elliptical-like regions of low connectivity and high modularity in a contour-line-like way. Our results indicate that the adaptation of the topology in response to disease dynamics suppresses the infection, while it promotes the network evolution towards a topology that exhibits assortative-mixing, modularity and a binomial-like degree distribution.</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 threshold and topological structure of susceptible-infectious-susceptible epidemics in adaptive networks","attachmentId":71197089,"attachmentType":"pdf","work_url":"https://www.academia.edu/55225889/Epidemic_threshold_and_topological_structure_of_susceptible_infectious_susceptible_epidemics_in_adaptive_networks","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/55225889/Epidemic_threshold_and_topological_structure_of_susceptible_infectious_susceptible_epidemics_in_adaptive_networks"><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":50066618,"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":50066618,"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_50066618" 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. You can download the paper by clicking the button above.</p></div></div></div></div><div class="ds-sidebar--container js-work-sidebar"><div class="ds-related-content--container"><h2 class="ds-related-content--heading">Related papers</h2><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="0" data-entity-id="11471669" 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/11471669/Different_epidemic_models_on_complex_networks">Different epidemic models on complex networks</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="26643484" href="https://independent.academia.edu/MichaelSmall1">Michael Small</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2009</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline 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