{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,14]],"date-time":"2026-09-14T15:03:31Z","timestamp":1789398211517,"version":"build-2803163510"},"reference-count":45,"publisher":"AIP Publishing","issue":"9","funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["U23A20331"],"award-info":[{"award-number":["U23A20331"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12571537"],"award-info":[{"award-number":["12571537"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12571592"],"award-info":[{"award-number":["12571592"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12271314"],"award-info":[{"award-number":["12271314"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100018685","name":"Shanxi Province Foundation for Returness","doi-asserted-by":"publisher","award":["20230013"],"award-info":[{"award-number":["20230013"]}],"id":[{"id":"10.13039\/501100018685","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004480","name":"Natural Science Foundation of Shanxi Province","doi-asserted-by":"publisher","award":["202303021222104"],"award-info":[{"award-number":["202303021222104"]}],"id":[{"id":"10.13039\/501100004480","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003392","name":"Natural Science Foundation of Fujian Province","doi-asserted-by":"publisher","award":["2023J06003"],"award-info":[{"award-number":["2023J06003"]}],"id":[{"id":"10.13039\/501100003392","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["pubs.aip.org"],"crossmark-restriction":true},"short-container-title":[],"published-print":{"date-parts":[[2026,9,1]]},"abstract":"<jats:p>Understanding disease spread requires coupling behavioral dynamics with contact network structure, yet their interaction is often overlooked or treated separately. We address this by formulating a susceptible-infected-recovered model on clustered networks with stochastic behavioral adaptation. In our model, individuals are characterized by two vigilance states: non-vigilant and vigilant, and update their states upon contact with infected neighbors according to a stochastic transition rule. Using a multitype branching process combined with percolation theory, we establish a theoretical framework for calculating the epidemic invasion probability, epidemic threshold, and final epidemic size and validate them through extensive stochastic simulations. Our results show that behavioral adaptation and transmission probabilities influence the epidemic threshold through distinct mechanisms: behavioral effects act in an approximately linear and additive manner, whereas transmission introduces nonlinear coupling between infection channels. Moreover, network clustering plays a dual role in epidemic invasion: it enhances outbreak probability in low-transmissibility regimes via local reinforcement of infection pathways, but suppresses large-scale spreading under high transmissibility due to local saturation and reduced effective branching. In addition, behavioral adaptation not only modulates the epidemic threshold but also reorganizes outbreak composition: non-vigilant infections dominate the epidemic burden, whereas vigilant infections exhibit a pronounced ridge-like maximum arising from a flux balance between vigilance induction and relaxation. Overall, our results reveal a rich interplay between behavioral adaptation, transmission heterogeneity, and network clustering in shaping both the onset and structure of epidemic outbreaks.<\/jats:p>","DOI":"10.1063\/5.0342298","type":"journal-article","created":{"date-parts":[[2026,9,14]],"date-time":"2026-09-14T14:09:20Z","timestamp":1789394960000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":["Epidemic spreading on clustered networks with behavioral adaptation"],"prefix":"10.1063","volume":"36","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1990-7986","authenticated-orcid":false,"given":"Xiao-Long","family":"Peng","sequence":"first","affiliation":[{"name":"Complex Systems Research Center, Shanxi University 1 , Taiyuan 030006,","place":["China"]},{"name":"Key Laboratory of Complex Systems and Data Science of Ministry of Education, Shanxi University 2 , Taiyuan 030006,","place":["China"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Shu-Yan","family":"Chang","sequence":"additional","affiliation":[{"name":"Complex Systems Research Center, Shanxi University 1 , Taiyuan 030006,","place":["China"]},{"name":"Key Laboratory of Complex Systems and Data Science of Ministry of Education, Shanxi University 2 , Taiyuan 030006,","place":["China"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6140-1479","authenticated-orcid":false,"given":"Li","family":"Li","sequence":"additional","affiliation":[{"name":"School of Computer and Information Technology, Shanxi University 3 , Taiyuan 030006,","place":["China"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yi-Cheng","family":"Zhang","sequence":"additional","affiliation":[{"name":"Department of Physics, University of Fribourg 4 , Fribourg 1700,","place":["Switzerland"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xin","family":"Chen","sequence":"additional","affiliation":[{"name":"Key Laboratory of Cellular Stress Biology, School of Life Sciences, Faculty of Medicine and Life Sciences, Xiamen University 5 , Xiamen 361005,","place":["China"]},{"name":"Xiamen Cardiovascular Hospital of Xiamen University, School of Medicine, Fujian Branch of National Clinical Research Center for Cardiovascular Diseases 6 , Xiamen 361005,","place":["China"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1831-1603","authenticated-orcid":false,"given":"Gui-Quan","family":"Sun","sequence":"additional","affiliation":[{"name":"Complex Systems Research Center, Shanxi University 1 , Taiyuan 030006,","place":["China"]},{"name":"School of Mathematics, North University of China 7 , Taiyuan 030051,","place":["China"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"317","published-online":{"date-parts":[[2026,9,14]]},"reference":[{"key":"2026091410091054800_c1","doi-asserted-by":"publisher","first-page":"860","DOI":"10.1126\/science.abb5793","article-title":"Projecting the transmission dynamics of SARS-CoV-2 through the postpandemic period","volume":"368","year":"2020","journal-title":"Science"},{"key":"2026091410091054800_c2","doi-asserted-by":"publisher","first-page":"e2304750120","DOI":"10.1073\/pnas.2304750120","article-title":"Reconstructing household transmission dynamics to estimate the infectiousness of asymptomatic influenza virus infections","volume":"120","year":"2023","journal-title":"Proc. Natl. Acad. Sci. U.S.A."},{"key":"2026091410091054800_c3","doi-asserted-by":"publisher","first-page":"1205","DOI":"10.1038\/s41591-020-0962-9","article-title":"Age-dependent effects in the transmission and control of COVID-19 epidemics","volume":"26","year":"2020","journal-title":"Nat. Med."},{"key":"2026091410091054800_c4","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.physrep.2021.02.001","article-title":"Non-pharmaceutical interventions during the COVID-19 pandemic: A review","volume":"913","year":"2021","journal-title":"Phys. Rep."},{"key":"2026091410091054800_c5","doi-asserted-by":"publisher","first-page":"667","DOI":"10.1126\/science.287.5453.667","article-title":"A simple model for complex dynamical transitions in epidemics","volume":"287","year":"2000","journal-title":"Science"},{"key":"2026091410091054800_c6","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.physrep.2024.01.003","article-title":"Epidemic spreading on higher-order networks","volume":"1056","year":"2024","journal-title":"Phys. Rep."},{"key":"2026091410091054800_c7","doi-asserted-by":"publisher","first-page":"1025","DOI":"10.1038\/nature08932","article-title":"Catastrophic cascade of failures in interdependent networks","volume":"464","year":"2010","journal-title":"Nature"},{"key":"2026091410091054800_c8","doi-asserted-by":"publisher","first-page":"509","DOI":"10.1126\/science.286.5439.509","article-title":"Emergence of scaling in random networks","volume":"286","year":"1999","journal-title":"Science"},{"key":"2026091410091054800_c9","doi-asserted-by":"publisher","first-page":"175","DOI":"10.1016\/j.physrep.2005.10.009","article-title":"Complex networks: Structure and dynamics","volume":"424","year":"2006","journal-title":"Phys. Rep."},{"key":"2026091410091054800_c10","doi-asserted-by":"publisher","first-page":"3200","DOI":"10.1103\/PhysRevLett.86.3200","article-title":"Epidemic spreading in scale-free networks","volume":"86","year":"2001","journal-title":"Phys. Rev. Lett."},{"key":"2026091410091054800_c11","doi-asserted-by":"publisher","first-page":"65","DOI":"10.1038\/nature14604","article-title":"Influence maximization in complex networks through optimal percolation","volume":"524","year":"2015","journal-title":"Nature"},{"key":"2026091410091054800_c12","doi-asserted-by":"publisher","first-page":"026106","DOI":"10.1103\/PhysRevE.86.026106","article-title":"Epidemic spreading on interconnected networks","volume":"86","year":"2012","journal-title":"Phys. Rev. E"},{"key":"2026091410091054800_c13","doi-asserted-by":"publisher","first-page":"29748","DOI":"10.1038\/srep29748","article-title":"Epidemic spreading on complex networks with community structures","volume":"6","year":"2016","journal-title":"Sci. Rep."},{"key":"2026091410091054800_c14","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.physrep.2025.01.005","article-title":"Optimal control of spatial diseases spreading in networked reaction\u2013diffusion systems","volume":"1111","year":"2025","journal-title":"Phys. Rep."},{"key":"2026091410091054800_c15","doi-asserted-by":"publisher","first-page":"208701","DOI":"10.1103\/PhysRevLett.96.208701","article-title":"Epidemic dynamics on an adaptive network","volume":"96","year":"2006","journal-title":"Phys. Rev. Lett."},{"key":"2026091410091054800_c16","doi-asserted-by":"publisher","first-page":"167","DOI":"10.1137\/S003614450342480","article-title":"The structure and function of complex networks","volume":"45","year":"2003","journal-title":"SIAM Rev."},{"key":"2026091410091054800_c17","doi-asserted-by":"publisher","first-page":"440","DOI":"10.1038\/30918","article-title":"Collective dynamics of \u2018small-world\u2019 networks","volume":"393","year":"1998","journal-title":"Nature"},{"key":"2026091410091054800_c18","doi-asserted-by":"publisher","first-page":"088701","DOI":"10.1103\/PhysRevLett.97.088701","article-title":"Percolation and epidemic thresholds in clustered networks","volume":"97","year":"2006","journal-title":"Phys. Rev. Lett."},{"key":"2026091410091054800_c19","doi-asserted-by":"publisher","first-page":"2485","DOI":"10.1038\/s41467-019-10431-6","article-title":"Simplicial models of social contagion","volume":"10","year":"2019","journal-title":"Nat. Commun."},{"key":"2026091410091054800_c20","doi-asserted-by":"publisher","first-page":"183","DOI":"10.1140\/epjb\/e2004-00020-6","article-title":"Structure of cycles and local ordering in complex networks","volume":"38","year":"2004","journal-title":"Eur. Phys. J. B"},{"key":"2026091410091054800_c21","doi-asserted-by":"publisher","first-page":"026121","DOI":"10.1103\/PhysRevE.68.026121","article-title":"Properties of highly clustered networks","volume":"68","year":"2003","journal-title":"Phys. Rev. E"},{"key":"2026091410091054800_c22","doi-asserted-by":"publisher","first-page":"058701","DOI":"10.1103\/PhysRevLett.103.058701","article-title":"Random graphs with clustering","volume":"103","year":"2009","journal-title":"Phys. Rev. Lett."},{"key":"2026091410091054800_c23","doi-asserted-by":"publisher","first-page":"034306","DOI":"10.1103\/PhysRevE.105.034306","article-title":"Multitype branching process method for modeling complex contagion on clustered networks","volume":"105","year":"2022","journal-title":"Phys. Rev. E"},{"key":"2026091410091054800_c24","doi-asserted-by":"publisher","first-page":"10551","DOI":"10.1073\/pnas.1507820112","article-title":"Complex dynamics of synergistic coinfections on realistically clustered networks","volume":"112","year":"2015","journal-title":"Proc. Natl. Acad. Sci. U.S.A."},{"key":"2026091410091054800_c25","doi-asserted-by":"publisher","first-page":"823","DOI":"10.1007\/s00285-019-01380-1","article-title":"Epidemic threshold in pairwise models for clustered networks: Closures and fast correlations","volume":"79","year":"2019","journal-title":"J. Math. Biol."},{"key":"2026091410091054800_c26","doi-asserted-by":"publisher","first-page":"1605","DOI":"10.1038\/s41467-023-37190-9","article-title":"Higher-order interactions shape collective dynamics differently in hypergraphs and simplicial complexes","volume":"14","year":"2023","journal-title":"Nat. Commun."},{"key":"2026091410091054800_c27","doi-asserted-by":"publisher","first-page":"e2315931121","DOI":"10.1073\/pnas.2315931121","article-title":"Higher-order homophily on simplicial complexes","volume":"121","year":"2024","journal-title":"Proc. Natl. Acad. Sci. U.S.A."},{"key":"2026091410091054800_c28","doi-asserted-by":"publisher","first-page":"555","DOI":"10.1038\/s41467-024-55506-1","article-title":"Hyperedge overlap drives explosive transitions in systems with higher-order interactions","volume":"16","year":"2025","journal-title":"Nat. Commun."},{"key":"2026091410091054800_c29","doi-asserted-by":"publisher","first-page":"6872","DOI":"10.1073\/pnas.0810762106","article-title":"The spread of awareness and its impact on epidemic outbreaks","volume":"106","year":"2009","journal-title":"Proc. Natl. Acad. Sci. U.S.A."},{"key":"2026091410091054800_c30","doi-asserted-by":"publisher","first-page":"218702","DOI":"10.1103\/PhysRevLett.115.218702","article-title":"Kinetics of social contagion","volume":"115","year":"2015","journal-title":"Phys. Rev. Lett."},{"key":"2026091410091054800_c31","doi-asserted-by":"publisher","first-page":"1308","DOI":"10.1038\/s41467-022-28964-8","article-title":"Universality, criticality and complexity of information propagation in social media","volume":"13","year":"2022","journal-title":"Nat. Commun."},{"key":"2026091410091054800_c32","doi-asserted-by":"publisher","first-page":"012313","DOI":"10.1103\/PhysRevE.97.012313","article-title":"Effect of risk perception on epidemic spreading in temporal networks","volume":"97","year":"2018","journal-title":"Phys. Rev. E"},{"key":"2026091410091054800_c33","doi-asserted-by":"publisher","first-page":"5097","DOI":"10.1038\/srep05097","article-title":"Asymmetrically interacting spreading dynamics on complex layered networks","volume":"4","year":"2014","journal-title":"Sci. Rep."},{"key":"2026091410091054800_c34","doi-asserted-by":"publisher","first-page":"036117","DOI":"10.1103\/PhysRevE.86.036117","article-title":"Epidemic spreading with information-driven vaccination","volume":"86","year":"2012","journal-title":"Phys. Rev. E"},{"key":"2026091410091054800_c35","doi-asserted-by":"publisher","first-page":"128701","DOI":"10.1103\/PhysRevLett.111.128701","article-title":"Dynamical interplay between awareness and epidemic spreading in multiplex networks","volume":"111","year":"2013","journal-title":"Phys. Rev. Lett."},{"key":"2026091410091054800_c36","doi-asserted-by":"publisher","first-page":"1561","DOI":"10.1007\/s00285-016-1007-3","article-title":"Prevention of infectious diseases by public vaccination and individual protection","volume":"73","year":"2016","journal-title":"J. Math. Biol."},{"key":"2026091410091054800_c37","doi-asserted-by":"publisher","first-page":"632","DOI":"10.1038\/srep00632","article-title":"On the existence of a threshold for preventive behavioral responses to suppress epidemic spreading","volume":"2","year":"2012","journal-title":"Sci. Rep."},{"key":"2026091410091054800_c38","doi-asserted-by":"publisher","first-page":"12621","DOI":"10.1038\/s41598-021-92094-2","article-title":"Prosocial behavior of wearing a mask during an epidemic: An evolutionary explanation","volume":"11","year":"2021","journal-title":"Sci. Rep."},{"key":"2026091410091054800_c39","doi-asserted-by":"publisher","first-page":"20200412","DOI":"10.1098\/rsta.2020.0412","article-title":"Emergence of protective behaviour under different risk perceptions to disease spreading","volume":"380","year":"2022","journal-title":"Phil. Trans. R. Soc. A"},{"key":"2026091410091054800_c40","doi-asserted-by":"publisher","first-page":"e2302245120","DOI":"10.1073\/pnas.2302245120","article-title":"Spreading processes with mutations over multilayer networks","volume":"120","year":"2023","journal-title":"Proc. Natl. Acad. Sci. U.S.A."},{"key":"2026091410091054800_c41","doi-asserted-by":"publisher","first-page":"293","DOI":"10.1007\/s00285-007-0116-4","article-title":"SIR dynamics in random networks with heterogeneous connectivity","volume":"56","year":"2008","journal-title":"J. Math. Biol."},{"key":"2026091410091054800_c42","doi-asserted-by":"publisher","first-page":"016128","DOI":"10.1103\/PhysRevE.66.016128","article-title":"Spread of epidemic disease on networks","volume":"66","year":"2002","journal-title":"Phys. Rev. E"},{"key":"2026091410091054800_c43","doi-asserted-by":"publisher","first-page":"034306","DOI":"10.1103\/PhysRevE.104.034306","article-title":"Higher-order percolation processes on multiplex hypergraphs","volume":"104","year":"2021","journal-title":"Phys. Rev. E"},{"key":"2026091410091054800_c44","doi-asserted-by":"publisher","first-page":"5664","DOI":"10.1073\/pnas.1918529117","article-title":"The effects of evolutionary adaptations on spreading processes in complex networks","volume":"117","year":"2020","journal-title":"Proc. Natl. Acad. Sci. U.S.A."},{"key":"2026091410091054800_c45","doi-asserted-by":"publisher","first-page":"192","DOI":"10.1016\/j.matcom.2026.01.011","article-title":"Percolation thresholds and epidemic spreading on small-world networks: Exact results and critical dynamics","volume":"245","year":"2026","journal-title":"Math. Comput. Simul."}],"container-title":["Chaos: An Interdisciplinary Journal of Nonlinear Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/pubs.aip.org\/aip\/cha\/article-pdf\/doi\/10.1063\/5.0342298\/21139039\/093129_1_5.0342298.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/pubs.aip.org\/aip\/cha\/article-pdf\/doi\/10.1063\/5.0342298\/21139039\/093129_1_5.0342298.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,9,14]],"date-time":"2026-09-14T14:09:22Z","timestamp":1789394962000},"score":1,"resource":{"primary":{"URL":"https:\/\/pubs.aip.org\/cha\/article\/36\/9\/093129\/3404459\/Epidemic-spreading-on-clustered-networks-with"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,9,1]]},"references-count":45,"journal-issue":{"issue":"9","published-print":{"date-parts":[[2026,9,1]]}},"URL":"https:\/\/doi.org\/10.1063\/5.0342298","relation":{},"ISSN":["1054-1500","1089-7682"],"issn-type":[{"value":"1054-1500","type":"print"},{"value":"1089-7682","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2026,9]]},"published":{"date-parts":[[2026,9,1]]},"article-number":"093129"}}