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In this study, to further reveal the mathematical mechanism of spontaneous generation of periodic oscillation solution, we investigate a type of SEIR epidemic model with a small intrinsic growth rate. By utilizing the singular perturbation theory and center manifold theorem, we extend the relaxation oscillation of three\u2010dimensional SIR models to the four\u2010dimensional SEIR models and prove the existence of stable relaxation oscillation with a large amplitude in the model. Numerical simulations are performed to verify our theoretical results. The results presented in this study provide a new idea for the study of the intrinsic mechanism of periodic oscillation in epidemiology, enrich the dynamics of epidemic models, and deepen the understanding of the global dynamics of these models.<\/jats:p>","DOI":"10.1155\/2024\/5373794","type":"journal-article","created":{"date-parts":[[2024,6,26]],"date-time":"2024-06-26T12:55:09Z","timestamp":1719406509000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Relaxation Oscillation in SEIR Epidemic Models with the Intrinsic Growth Rate"],"prefix":"10.1155","volume":"2024","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9881-1538","authenticated-orcid":false,"given":"Yingying","family":"Zhang","sequence":"first","affiliation":[]},{"given":"Ruohan","family":"Wang","sequence":"additional","affiliation":[]},{"given":"Yanan","family":"Cai","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2024,6,26]]},"reference":[{"key":"e_1_2_10_1_2","doi-asserted-by":"publisher","DOI":"10.1126\/science.287.5453.667"},{"key":"e_1_2_10_2_2","doi-asserted-by":"publisher","DOI":"10.1098\/rspb.2006.0030"},{"key":"e_1_2_10_3_2","doi-asserted-by":"publisher","DOI":"10.1007\/s12064-018-0271-8"},{"key":"e_1_2_10_4_2","doi-asserted-by":"publisher","DOI":"10.1155\/2020\/3631923"},{"key":"e_1_2_10_5_2","doi-asserted-by":"publisher","DOI":"10.3201\/eid0703.010301"},{"key":"e_1_2_10_6_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.jtbi.2019.110027"},{"key":"e_1_2_10_7_2","doi-asserted-by":"publisher","DOI":"10.3934\/mbe.2019106"},{"volume-title":"Epidemiology and Prevention of Vaccine-Preventable Diseases","year":"2015","author":"Hamborsky J.","key":"e_1_2_10_8_2"},{"key":"e_1_2_10_9_2","doi-asserted-by":"publisher","DOI":"10.1093\/oxfordjournals.aje.a121575"},{"key":"e_1_2_10_10_2","doi-asserted-by":"publisher","DOI":"10.1093\/ije\/11.1.5"},{"key":"e_1_2_10_11_2","doi-asserted-by":"publisher","DOI":"10.1093\/ije\/12.3.332"},{"key":"e_1_2_10_12_2","doi-asserted-by":"publisher","DOI":"10.1086\/377694"},{"volume-title":"Population Dynamics of Infectious Diseases: Theory and Applications","year":"1982","author":"Anderson R.","key":"e_1_2_10_13_2"},{"key":"e_1_2_10_14_2","unstructured":"C.for Disease Control Prevention Measles cases and outbreaks 2020 https:\/\/www.cdc.gov\/measles\/cases-outbreaks.html."},{"key":"e_1_2_10_15_2","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-61317-3_8","volume-title":"Periodicity in Epidemiological Models","author":"Hethcote H.","year":"1989"},{"key":"e_1_2_10_16_2","doi-asserted-by":"publisher","DOI":"10.1137\/0140001"},{"key":"e_1_2_10_17_2","doi-asserted-by":"publisher","DOI":"10.1155\/2018\/1613709"},{"key":"e_1_2_10_18_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2014.06.035"},{"key":"e_1_2_10_19_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.mbs.2017.11.003"},{"key":"e_1_2_10_20_2","doi-asserted-by":"publisher","DOI":"10.1142\/s021812742050073x"},{"key":"e_1_2_10_21_2","doi-asserted-by":"publisher","DOI":"10.1155\/2020\/7541861"},{"key":"e_1_2_10_22_2","doi-asserted-by":"publisher","DOI":"10.1155\/2019\/3586508"},{"volume-title":"Infectious Diseases of Humans: Dynamics and Control","year":"1992","author":"Roy M. 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