{"id":"https://openalex.org/W2048930562","doi":"https://doi.org/10.1155/2013/875783","title":"Stability and Bifurcation Analysis for a Delay Differential Equation of Hepatitis B Virus Infection","display_name":"Stability and Bifurcation Analysis for a Delay Differential Equation of Hepatitis B Virus Infection","publication_year":2013,"publication_date":"2013-01-01","ids":{"openalex":"https://openalex.org/W2048930562","doi":"https://doi.org/10.1155/2013/875783","mag":"2048930562"},"language":"en","primary_location":{"id":"doi:10.1155/2013/875783","is_oa":true,"landing_page_url":"https://doi.org/10.1155/2013/875783","pdf_url":"https://downloads.hindawi.com/journals/jam/2013/875783.pdf","source":{"id":"https://openalex.org/S190082376","display_name":"Journal of Applied Mathematics","issn_l":"1110-757X","issn":["1110-757X","1687-0042"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310319869","host_organization_name":"Hindawi Publishing Corporation","host_organization_lineage":["https://openalex.org/P4310319869"],"host_organization_lineage_names":["Hindawi Publishing Corporation"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Applied Mathematics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://downloads.hindawi.com/journals/jam/2013/875783.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5102365736","display_name":"Xinchao Yang","orcid":null},"institutions":[{"id":"https://openalex.org/I34949971","display_name":"University of Jinan","ror":"https://ror.org/02mjz6f26","country_code":"CN","type":"education","lineage":["https://openalex.org/I34949971"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Xinchao Yang","raw_affiliation_strings":["School of Medical and Life Science, University of Jinan, Jinan, Shandong 250022, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Medical and Life Science, University of Jinan, Jinan, Shandong 250022, China","institution_ids":["https://openalex.org/I34949971"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5019346100","display_name":"Xiju Zong","orcid":"https://orcid.org/0000-0002-4179-1289"},"institutions":[{"id":"https://openalex.org/I34949971","display_name":"University of Jinan","ror":"https://ror.org/02mjz6f26","country_code":"CN","type":"education","lineage":["https://openalex.org/I34949971"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Xiju Zong","raw_affiliation_strings":["School of Electrical Engineering, University of Jinan, Jinan, Shandong 250022, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Electrical Engineering, University of Jinan, Jinan, Shandong 250022, China","institution_ids":["https://openalex.org/I34949971"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5100543018","display_name":"Xingong Cheng","orcid":null},"institutions":[{"id":"https://openalex.org/I34949971","display_name":"University of Jinan","ror":"https://ror.org/02mjz6f26","country_code":"CN","type":"education","lineage":["https://openalex.org/I34949971"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Xingong Cheng","raw_affiliation_strings":["School of Electrical Engineering, University of Jinan, Jinan, Shandong 250022, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Electrical Engineering, University of Jinan, Jinan, Shandong 250022, China","institution_ids":["https://openalex.org/I34949971"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5007740492","display_name":"Zhenlai Han","orcid":"https://orcid.org/0000-0002-0794-0797"},"institutions":[{"id":"https://openalex.org/I34949971","display_name":"University of Jinan","ror":"https://ror.org/02mjz6f26","country_code":"CN","type":"education","lineage":["https://openalex.org/I34949971"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Zhenlai Han","raw_affiliation_strings":["School of Science, University of Jinan, Jinan, Shandong 250022, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Science, University of Jinan, Jinan, Shandong 250022, China","institution_ids":["https://openalex.org/I34949971"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5019346100"],"corresponding_institution_ids":["https://openalex.org/I34949971"],"apc_list":{"value":1025,"currency":"USD","value_usd":1025},"apc_paid":{"value":1025,"currency":"USD","value_usd":1025},"fwci":0.0,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":{"value":0.14278709,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"2013","issue":null,"first_page":"1","last_page":"15"},"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":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2739","display_name":"Public Health, Environmental and Occupational Health"},"field":{"id":"https://openalex.org/fields/27","display_name":"Medicine"},"domain":{"id":"https://openalex.org/domains/4","display_name":"Health Sciences"}},"topics":[{"id":"https://openalex.org/T10482","display_name":"Mathematical and Theoretical Epidemiology and Ecology Models","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2739","display_name":"Public Health, Environmental and Occupational Health"},"field":{"id":"https://openalex.org/fields/27","display_name":"Medicine"},"domain":{"id":"https://openalex.org/domains/4","display_name":"Health Sciences"}},{"id":"https://openalex.org/T10288","display_name":"Fractional Differential Equations Solutions","score":0.9975000023841858,"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"}},{"id":"https://openalex.org/T10541","display_name":"Nonlinear Differential Equations Analysis","score":0.9907000064849854,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7049811482429504},{"id":"https://openalex.org/keywords/center-manifold","display_name":"Center manifold","score":0.700286865234375},{"id":"https://openalex.org/keywords/stability","display_name":"Stability (learning theory)","score":0.6860447525978088},{"id":"https://openalex.org/keywords/bifurcation","display_name":"Bifurcation","score":0.6438966989517212},{"id":"https://openalex.org/keywords/hopf-bifurcation","display_name":"Hopf bifurcation","score":0.5453677773475647},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4303187131881714},{"id":"https://openalex.org/keywords/differential-equation","display_name":"Differential equation","score":0.4274296164512634},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.41525474190711975},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.3758695721626282},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.16276204586029053},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.13318878412246704}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7049811482429504},{"id":"https://openalex.org/C74774103","wikidata":"https://www.wikidata.org/wiki/Q4504202","display_name":"Center manifold","level":5,"score":0.700286865234375},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.6860447525978088},{"id":"https://openalex.org/C2781349735","wikidata":"https://www.wikidata.org/wiki/Q4904824","display_name":"Bifurcation","level":3,"score":0.6438966989517212},{"id":"https://openalex.org/C172435161","wikidata":"https://www.wikidata.org/wiki/Q531126","display_name":"Hopf bifurcation","level":4,"score":0.5453677773475647},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4303187131881714},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.4274296164512634},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.41525474190711975},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3758695721626282},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.16276204586029053},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.13318878412246704},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"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/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1155/2013/875783","is_oa":true,"landing_page_url":"https://doi.org/10.1155/2013/875783","pdf_url":"https://downloads.hindawi.com/journals/jam/2013/875783.pdf","source":{"id":"https://openalex.org/S190082376","display_name":"Journal of Applied Mathematics","issn_l":"1110-757X","issn":["1110-757X","1687-0042"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310319869","host_organization_name":"Hindawi Publishing Corporation","host_organization_lineage":["https://openalex.org/P4310319869"],"host_organization_lineage_names":["Hindawi Publishing Corporation"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Applied Mathematics","raw_type":"journal-article"},{"id":"pmh:oai:CULeuclid:euclid.jam/1394807870","is_oa":false,"landing_page_url":"https://projecteuclid.org/euclid.jam/1394807870","pdf_url":null,"source":{"id":"https://openalex.org/S4306400787","display_name":"Project Euclid (Cornell University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Text"},{"id":"pmh:oai:RePEc:hin:jnljam:875783","is_oa":false,"landing_page_url":"http://downloads.hindawi.com/journals/JAM/2013/875783.xml","pdf_url":null,"source":{"id":"https://openalex.org/S4306401271","display_name":"RePEc: Research Papers in Economics","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I77793887","host_organization_name":"Federal Reserve Bank of St. Louis","host_organization_lineage":["https://openalex.org/I77793887"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article"},{"id":"pmh:oai:doaj.org/article:7a893114d61e419193578ed18eccee0f","is_oa":true,"landing_page_url":"https://doaj.org/article/7a893114d61e419193578ed18eccee0f","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Journal of Applied Mathematics, Vol 2013 (2013)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.1155/2013/875783","is_oa":true,"landing_page_url":"https://doi.org/10.1155/2013/875783","pdf_url":"https://downloads.hindawi.com/journals/jam/2013/875783.pdf","source":{"id":"https://openalex.org/S190082376","display_name":"Journal of Applied Mathematics","issn_l":"1110-757X","issn":["1110-757X","1687-0042"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310319869","host_organization_name":"Hindawi Publishing Corporation","host_organization_lineage":["https://openalex.org/P4310319869"],"host_organization_lineage_names":["Hindawi Publishing Corporation"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Applied Mathematics","raw_type":"journal-article"},"sustainable_development_goals":[{"display_name":"Good health and well-being","id":"https://metadata.un.org/sdg/3","score":0.6700000166893005}],"awards":[{"id":"https://openalex.org/G1160532259","display_name":null,"funder_award_id":"11026112","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G224886671","display_name":null,"funder_award_id":"BS2011SF001","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G5154784204","display_name":null,"funder_award_id":"XKY1133","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G618256116","display_name":null,"funder_award_id":"11026112","funder_id":"https://openalex.org/F4320322291","funder_display_name":"University of Jinan"},{"id":"https://openalex.org/G6554747031","display_name":null,"funder_award_id":"11201179","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"},{"id":"https://openalex.org/F4320322291","display_name":"University of Jinan","ror":"https://ror.org/02mjz6f26"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2048930562.pdf","grobid_xml":"https://content.openalex.org/works/W2048930562.grobid-xml"},"referenced_works_count":11,"referenced_works":["https://openalex.org/W1586658033","https://openalex.org/W1597176843","https://openalex.org/W1607633954","https://openalex.org/W1986896318","https://openalex.org/W1990326991","https://openalex.org/W2020975304","https://openalex.org/W2074212690","https://openalex.org/W2079871064","https://openalex.org/W2116636744","https://openalex.org/W2163284349","https://openalex.org/W2333739003"],"related_works":["https://openalex.org/W1987790493","https://openalex.org/W2153388945","https://openalex.org/W2038701038","https://openalex.org/W1975783007","https://openalex.org/W1996991127","https://openalex.org/W2382752082","https://openalex.org/W1988489950","https://openalex.org/W4312255936","https://openalex.org/W1986426368","https://openalex.org/W2363522146"],"abstract_inverted_index":{"The":[0,28,51],"stability":[1],"and":[2,62,79],"bifurcation":[3,33],"analysis":[4],"for":[5],"a":[6],"delay":[7],"differential":[8],"equation":[9],"of":[10,21,30,47,58,66],"hepatitis":[11],"B":[12],"virus":[13],"infection":[14],"is":[15,41],"investigated.":[16],"We":[17],"show":[18],"the":[19,31,38,45,48,56,59,63,67,75,80,87],"existence":[20,29],"nonnegative":[22],"equilibria":[23,40],"under":[24],"some":[25],"appropriated":[26],"conditions.":[27],"Hopf":[32],"with":[34],"delay<mml:math":[35],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[36],"id=\"M1\"><mml:mrow><mml:mi>\u03c4</mml:mi></mml:mrow></mml:math>at":[37],"endemic":[39],"established":[42],"by":[43,73],"analyzing":[44],"distribution":[46],"characteristic":[49],"values.":[50],"explicit":[52],"formulae":[53],"which":[54],"determine":[55],"direction":[57],"bifurcations,":[60],"stability,":[61],"other":[64],"properties":[65],"bifurcating":[68],"periodic":[69],"solutions":[70],"are":[71],"given":[72],"using":[74],"normal":[76],"form":[77],"theory":[78],"center":[81],"manifold":[82],"theorem.":[83],"Numerical":[84],"simulation":[85],"verifies":[86],"theoretical":[88],"results.":[89]},"counts_by_year":[{"year":2024,"cited_by_count":1}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}