{"id":"https://openalex.org/W1969957193","doi":"https://doi.org/10.4236/am.2013.410a2006","title":"Analysis of a Delayed SIR Model with Exponential Birth and Saturated Incidence Rate","display_name":"Analysis of a Delayed SIR Model with Exponential Birth and Saturated Incidence Rate","publication_year":2013,"publication_date":"2013-01-01","ids":{"openalex":"https://openalex.org/W1969957193","doi":"https://doi.org/10.4236/am.2013.410a2006","mag":"1969957193"},"language":"en","primary_location":{"id":"doi:10.4236/am.2013.410a2006","doi":"https://doi.org/10.4236/am.2013.410a2006","is_oa":true,"landing_page_url":"https://doi.org/10.4236/am.2013.410a2006","pdf_url":"http://www.scirp.org/journal/PaperDownload.aspx?paperID=37459","source":{"id":"https://openalex.org/S4210211746","display_name":"Applied Mathematics","issn_l":"2152-7385","issn":["2152-7385","2152-7393"],"is_oa":true,"is_in_doaj":false,"is_core":true,"listed_in":["anvur-scientific","cwts-core"],"host_organization":"https://openalex.org/P4310320480","host_organization_name":"Scientific Research Publishing","host_organization_lineage":["https://openalex.org/P4310320480"],"host_organization_lineage_names":["Scientific Research Publishing"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Applied Mathematics","raw_type":"journal-article"},"volume":null,"type":"article","is_traditional":true,"indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"http://www.scirp.org/journal/PaperDownload.aspx?paperID=37459","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5101845319","display_name":"Wanwan Wang","orcid":"https://orcid.org/0000-0001-8991-3575","observed_orcids":["https://orcid.org/0000-0001-8991-3575"]},"institutions":[{"id":"https://openalex.org/I135714990","display_name":"North University of China","ror":"https://ror.org/047bp1713","country_code":"CN","type":"education","lineage":["https://openalex.org/I135714990"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Wanwan Wang","raw_affiliation_strings":["Department of Mathematics, North University of China, Taiyuan, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, North University of China, Taiyuan, China","institution_ids":["https://openalex.org/I135714990"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5049411283","display_name":"Maoxing Liu","orcid":"https://orcid.org/0000-0002-3342-7209","observed_orcids":["https://orcid.org/0000-0002-3342-7209"]},"institutions":[{"id":"https://openalex.org/I135714990","display_name":"North University of China","ror":"https://ror.org/047bp1713","country_code":"CN","type":"education","lineage":["https://openalex.org/I135714990"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Maoxing Liu","raw_affiliation_strings":["Department of Mathematics, North University of China, Taiyuan, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, North University of China, Taiyuan, China","institution_ids":["https://openalex.org/I135714990"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5028184836","display_name":"Jinqing Zhao","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I135714990","display_name":"North University of China","ror":"https://ror.org/047bp1713","country_code":"CN","type":"education","lineage":["https://openalex.org/I135714990"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Jinqing Zhao","raw_affiliation_strings":["Department of Mathematics, North University of China, Taiyuan, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, North University of China, Taiyuan, China","institution_ids":["https://openalex.org/I135714990"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5049411283"],"corresponding_institution_ids":["https://openalex.org/I135714990"],"apc_list":null,"apc_paid":null,"fwci":0.349,"has_fulltext":true,"cited_by_count":3,"citation_normalized_percentile":{"value":0.5032691,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":87,"max":93},"biblio":{"volume":"04","issue":"10","first_page":"60","last_page":"67"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10482","display_name":"Mathematical and Theoretical Epidemiology and Ecology Models","score":1.0,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10482","display_name":"Mathematical and Theoretical Epidemiology and Ecology Models","score":1.0,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10541","display_name":"Nonlinear Differential Equations Analysis","score":0.0,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10410","display_name":"COVID-19 epidemiological studies","score":0.0,"subfield":{"id":"https://openalex.org/subfields/2611","display_name":"Modeling and Simulation"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/saturated-incidence-rate","display_name":"saturated incidence rate","score":0.984000027179718},{"id":"https://openalex.org/keywords/demographic-structure","display_name":"demographic structure","score":0.7400000095367432},{"id":"https://openalex.org/keywords/endemic-equilibrium","display_name":"endemic equilibrium","score":0.8939999938011169},{"id":"https://openalex.org/keywords/local-stability","display_name":"local stability","score":0.9070000052452087},{"id":"https://openalex.org/keywords/hopf-bifurcation","display_name":"Hopf bifurcation","score":0.8450000286102295},{"id":"https://openalex.org/keywords/time-delay","display_name":"time delay","score":0.6000000238418579},{"id":"https://openalex.org/keywords/epidemic-dynamics","display_name":"epidemic dynamics","score":0.7490000128746033}],"concepts":[{"id":"https://openalex.org/C167964875","wikidata":"https://www.wikidata.org/wiki/Q17011487","display_name":"Exponential stability","level":3,"score":0.6300652027130127},{"id":"https://openalex.org/C207821765","wikidata":"https://www.wikidata.org/wiki/Q405372","display_name":"Instability","level":2,"score":0.6085195541381836},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6079409718513489},{"id":"https://openalex.org/C151376022","wikidata":"https://www.wikidata.org/wiki/Q168698","display_name":"Exponential function","level":2,"score":0.5565848350524902},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.4904195964336395},{"id":"https://openalex.org/C75235859","wikidata":"https://www.wikidata.org/wiki/Q582659","display_name":"Exponential growth","level":2,"score":0.4893264174461365},{"id":"https://openalex.org/C61511704","wikidata":"https://www.wikidata.org/wiki/Q1671857","display_name":"Incidence (geometry)","level":2,"score":0.47254276275634766},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.43333035707473755},{"id":"https://openalex.org/C47446073","wikidata":"https://www.wikidata.org/wiki/Q5165890","display_name":"Control theory (sociology)","level":3,"score":0.4266815185546875},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.32558169960975647},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.24287036061286926},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.23325815796852112},{"id":"https://openalex.org/C57879066","wikidata":"https://www.wikidata.org/wiki/Q41217","display_name":"Mechanics","level":1,"score":0.2188417911529541},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.21074345707893372},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.14281976222991943},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.1100875735282898},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.09198415279388428},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C2775924081","wikidata":"https://www.wikidata.org/wiki/Q55608371","display_name":"Control (management)","level":2,"score":0.0},{"id":"https://openalex.org/C187736073","wikidata":"https://www.wikidata.org/wiki/Q2920921","display_name":"Management","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.4236/am.2013.410a2006","doi":"https://doi.org/10.4236/am.2013.410a2006","is_oa":true,"landing_page_url":"https://doi.org/10.4236/am.2013.410a2006","pdf_url":"http://www.scirp.org/journal/PaperDownload.aspx?paperID=37459","source":{"id":"https://openalex.org/S4210211746","display_name":"Applied Mathematics","issn_l":"2152-7385","issn":["2152-7385","2152-7393"],"is_oa":true,"is_in_doaj":false,"is_core":true,"listed_in":["anvur-scientific","cwts-core"],"host_organization":"https://openalex.org/P4310320480","host_organization_name":"Scientific Research Publishing","host_organization_lineage":["https://openalex.org/P4310320480"],"host_organization_lineage_names":["Scientific Research Publishing"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Applied Mathematics","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.4236/am.2013.410a2006","doi":"https://doi.org/10.4236/am.2013.410a2006","is_oa":true,"landing_page_url":"https://doi.org/10.4236/am.2013.410a2006","pdf_url":"http://www.scirp.org/journal/PaperDownload.aspx?paperID=37459","source":{"id":"https://openalex.org/S4210211746","display_name":"Applied Mathematics","issn_l":"2152-7385","issn":["2152-7385","2152-7393"],"is_oa":true,"is_in_doaj":false,"is_core":true,"listed_in":["anvur-scientific","cwts-core"],"host_organization":"https://openalex.org/P4310320480","host_organization_name":"Scientific Research Publishing","host_organization_lineage":["https://openalex.org/P4310320480"],"host_organization_lineage_names":["Scientific Research Publishing"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Applied Mathematics","raw_type":"journal-article"},"sustainable_development_goals":[],"study_designs":[],"awards":[{"id":"https://openalex.org/G1321282689","display_name":"\u57fa\u4e8e\u590d\u6742\u7f51\u7edc\u7684\u4f20\u67d3\u75c5\u6a21\u578b\u7814\u7a76","funder_award_id":"10901145","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G5503175958","display_name":null,"funder_award_id":"2012011002-1","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W1969957193.pdf","grobid_xml":"https://content.openalex.org/works/W1969957193.grobid-xml"},"referenced_works_count":22,"referenced_works":["https://openalex.org/W1968567447","https://openalex.org/W1969197664","https://openalex.org/W1980421658","https://openalex.org/W1989861394","https://openalex.org/W1991038590","https://openalex.org/W1997498552","https://openalex.org/W2018296903","https://openalex.org/W2023067988","https://openalex.org/W2028378134","https://openalex.org/W2042391826","https://openalex.org/W2043444618","https://openalex.org/W2049640908","https://openalex.org/W2054353417","https://openalex.org/W2059655508","https://openalex.org/W2080967410","https://openalex.org/W2090620943","https://openalex.org/W2091419790","https://openalex.org/W2156285549","https://openalex.org/W2291447000","https://openalex.org/W2600848460","https://openalex.org/W3157204008","https://openalex.org/W4292948660"],"related_works":["https://openalex.org/W2291447000","https://openalex.org/W2299895035","https://openalex.org/W2144803946","https://openalex.org/W2120145525","https://openalex.org/W7101140665","https://openalex.org/W1972735423","https://openalex.org/W2600848460","https://openalex.org/W2081727883","https://openalex.org/W3157204008","https://openalex.org/W2051622558"],"abstract_inverted_index":{"In":[0],"this":[1],"paper,":[2],"a":[3],"delayed":[4],"SIR":[5],"model":[6,70],"with":[7,25,58],"exponential":[8],"demographic":[9],"structure":[10],"and":[11,35],"the":[12,21,27,36,42,49,69,75],"saturated":[13],"incidence":[14],"rate":[15],"is":[16,23,30,39,44],"formulated.":[17],"The":[18],"stability":[19,53],"of":[20,68],"equilibria":[22],"analyzed":[24],"delay:":[26],"endemic":[28,37],"equilibrium":[29,38],"locally":[31],"stable":[32,40],"without":[33],"delay;":[34],"if":[41],"delay":[43],"under":[45],"some":[46,65],"condition.":[47],"Moreover":[48],"dynamical":[50],"behaviors":[51],"from":[52],"to":[54,73],"instability":[55],"will":[56],"change":[57],"an":[59],"appropriate":[60],"critical":[61],"value.":[62],"At":[63],"last,":[64],"numerical":[66],"simulations":[67],"are":[71],"given":[72],"illustrate":[74],"main":[76],"theoretical":[77],"results.":[78]},"counts_by_year":[{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2015,"cited_by_count":1}],"updated_date":"2026-10-30T12:39:25.321706","created_date":"2016-06-24T00:00:00"}