{"id":"https://openalex.org/W2578148890","doi":"https://doi.org/10.1142/s0218127416502345","title":"Dynamics of a Delayed HIV-1 Infection Model with Saturation Incidence Rate and CTL Immune Response","display_name":"Dynamics of a Delayed HIV-1 Infection Model with Saturation Incidence Rate and CTL Immune Response","publication_year":2016,"publication_date":"2016-12-30","ids":{"openalex":"https://openalex.org/W2578148890","doi":"https://doi.org/10.1142/s0218127416502345","mag":"2578148890"},"language":"en","primary_location":{"id":"doi:10.1142/s0218127416502345","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218127416502345","pdf_url":null,"source":{"id":"https://openalex.org/S81011612","display_name":"International Journal of Bifurcation and Chaos","issn_l":"0218-1274","issn":["0218-1274","1793-6551"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319815","host_organization_name":"World Scientific","host_organization_lineage":["https://openalex.org/P4310319815"],"host_organization_lineage_names":["World Scientific"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Bifurcation and Chaos","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5013913244","display_name":"Ting Guo","orcid":"https://orcid.org/0000-0003-4109-4755"},"institutions":[{"id":"https://openalex.org/I120825670","display_name":"Yunnan Normal University","ror":"https://ror.org/00sc9n023","country_code":"CN","type":"education","lineage":["https://openalex.org/I120825670"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Ting Guo","raw_affiliation_strings":["Department of Mathematical Sciences, Yunnan Normal University, Kunming 650092, P. R. China"],"affiliations":[{"raw_affiliation_string":"Department of Mathematical Sciences, Yunnan Normal University, Kunming 650092, P. R. China","institution_ids":["https://openalex.org/I120825670"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5073237695","display_name":"Haihong Liu","orcid":"https://orcid.org/0000-0002-2666-7933"},"institutions":[{"id":"https://openalex.org/I120825670","display_name":"Yunnan Normal University","ror":"https://ror.org/00sc9n023","country_code":"CN","type":"education","lineage":["https://openalex.org/I120825670"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Haihong Liu","raw_affiliation_strings":["Department of Mathematical Sciences, Yunnan Normal University, Kunming 650092, P. R. China"],"affiliations":[{"raw_affiliation_string":"Department of Mathematical Sciences, Yunnan Normal University, Kunming 650092, P. R. China","institution_ids":["https://openalex.org/I120825670"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5100861471","display_name":"Chenglin Xu","orcid":null},"institutions":[{"id":"https://openalex.org/I120825670","display_name":"Yunnan Normal University","ror":"https://ror.org/00sc9n023","country_code":"CN","type":"education","lineage":["https://openalex.org/I120825670"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Chenglin Xu","raw_affiliation_strings":["Department of Mathematical Sciences, Yunnan Normal University, Kunming 650092, P. R. China"],"affiliations":[{"raw_affiliation_string":"Department of Mathematical Sciences, Yunnan Normal University, Kunming 650092, P. R. China","institution_ids":["https://openalex.org/I120825670"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5039463187","display_name":"Fang Yan","orcid":"https://orcid.org/0000-0001-6624-9720"},"institutions":[{"id":"https://openalex.org/I120825670","display_name":"Yunnan Normal University","ror":"https://ror.org/00sc9n023","country_code":"CN","type":"education","lineage":["https://openalex.org/I120825670"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Fang Yan","raw_affiliation_strings":["Department of Mathematical Sciences, Yunnan Normal University, Kunming 650092, P. R. China"],"affiliations":[{"raw_affiliation_string":"Department of Mathematical Sciences, Yunnan Normal University, Kunming 650092, P. R. China","institution_ids":["https://openalex.org/I120825670"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5013913244"],"corresponding_institution_ids":["https://openalex.org/I120825670"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":11,"citation_normalized_percentile":{"value":0.21468803,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"26","issue":"14","first_page":"1650234","last_page":"1650234"},"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/T11764","display_name":"Evolution and Genetic Dynamics","score":0.9718999862670898,"subfield":{"id":"https://openalex.org/subfields/1311","display_name":"Genetics"},"field":{"id":"https://openalex.org/fields/13","display_name":"Biochemistry, Genetics and Molecular Biology"},"domain":{"id":"https://openalex.org/domains/1","display_name":"Life Sciences"}},{"id":"https://openalex.org/T11252","display_name":"Evolutionary Game Theory and Cooperation","score":0.9128999710083008,"subfield":{"id":"https://openalex.org/subfields/3312","display_name":"Sociology and Political Science"},"field":{"id":"https://openalex.org/fields/33","display_name":"Social Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/hopf-bifurcation","display_name":"Hopf bifurcation","score":0.6510334014892578},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6387799978256226},{"id":"https://openalex.org/keywords/equilibrium-point","display_name":"Equilibrium point","score":0.6190217733383179},{"id":"https://openalex.org/keywords/basic-reproduction-number","display_name":"Basic reproduction number","score":0.5853217840194702},{"id":"https://openalex.org/keywords/ctl*","display_name":"CTL*","score":0.5629514455795288},{"id":"https://openalex.org/keywords/stability-theory","display_name":"Stability theory","score":0.5480407476425171},{"id":"https://openalex.org/keywords/lemma","display_name":"Lemma (botany)","score":0.5416946411132812},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4345475435256958},{"id":"https://openalex.org/keywords/lyapunov-function","display_name":"Lyapunov function","score":0.4307400584220886},{"id":"https://openalex.org/keywords/saturation","display_name":"Saturation (graph theory)","score":0.4152528643608093},{"id":"https://openalex.org/keywords/immune-system","display_name":"Immune system","score":0.3952750563621521},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.28150343894958496},{"id":"https://openalex.org/keywords/bifurcation","display_name":"Bifurcation","score":0.26935601234436035},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.24584311246871948},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.22590205073356628},{"id":"https://openalex.org/keywords/biology","display_name":"Biology","score":0.14594358205795288},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.11508432030677795},{"id":"https://openalex.org/keywords/immunology","display_name":"Immunology","score":0.1126924455165863},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.08384701609611511},{"id":"https://openalex.org/keywords/population","display_name":"Population","score":0.08381295204162598}],"concepts":[{"id":"https://openalex.org/C172435161","wikidata":"https://www.wikidata.org/wiki/Q531126","display_name":"Hopf bifurcation","level":4,"score":0.6510334014892578},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6387799978256226},{"id":"https://openalex.org/C94766913","wikidata":"https://www.wikidata.org/wiki/Q1530271","display_name":"Equilibrium point","level":3,"score":0.6190217733383179},{"id":"https://openalex.org/C204264503","wikidata":"https://www.wikidata.org/wiki/Q901464","display_name":"Basic reproduction number","level":3,"score":0.5853217840194702},{"id":"https://openalex.org/C147969180","wikidata":"https://www.wikidata.org/wiki/Q5014579","display_name":"CTL*","level":4,"score":0.5629514455795288},{"id":"https://openalex.org/C41949839","wikidata":"https://www.wikidata.org/wiki/Q1756677","display_name":"Stability theory","level":3,"score":0.5480407476425171},{"id":"https://openalex.org/C2777759810","wikidata":"https://www.wikidata.org/wiki/Q149316","display_name":"Lemma (botany)","level":3,"score":0.5416946411132812},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4345475435256958},{"id":"https://openalex.org/C60640748","wikidata":"https://www.wikidata.org/wiki/Q2337858","display_name":"Lyapunov function","level":3,"score":0.4307400584220886},{"id":"https://openalex.org/C9930424","wikidata":"https://www.wikidata.org/wiki/Q7426587","display_name":"Saturation (graph theory)","level":2,"score":0.4152528643608093},{"id":"https://openalex.org/C8891405","wikidata":"https://www.wikidata.org/wiki/Q1059","display_name":"Immune system","level":2,"score":0.3952750563621521},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.28150343894958496},{"id":"https://openalex.org/C2781349735","wikidata":"https://www.wikidata.org/wiki/Q4904824","display_name":"Bifurcation","level":3,"score":0.26935601234436035},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.24584311246871948},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.22590205073356628},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.14594358205795288},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.11508432030677795},{"id":"https://openalex.org/C203014093","wikidata":"https://www.wikidata.org/wiki/Q101929","display_name":"Immunology","level":1,"score":0.1126924455165863},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.08384701609611511},{"id":"https://openalex.org/C2908647359","wikidata":"https://www.wikidata.org/wiki/Q2625603","display_name":"Population","level":2,"score":0.08381295204162598},{"id":"https://openalex.org/C46757340","wikidata":"https://www.wikidata.org/wiki/Q43238","display_name":"Poaceae","level":2,"score":0.0},{"id":"https://openalex.org/C149923435","wikidata":"https://www.wikidata.org/wiki/Q37732","display_name":"Demography","level":1,"score":0.0},{"id":"https://openalex.org/C144024400","wikidata":"https://www.wikidata.org/wiki/Q21201","display_name":"Sociology","level":0,"score":0.0},{"id":"https://openalex.org/C167672396","wikidata":"https://www.wikidata.org/wiki/Q417800","display_name":"CD8","level":3,"score":0.0},{"id":"https://openalex.org/C18903297","wikidata":"https://www.wikidata.org/wiki/Q7150","display_name":"Ecology","level":1,"score":0.0},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1142/s0218127416502345","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218127416502345","pdf_url":null,"source":{"id":"https://openalex.org/S81011612","display_name":"International Journal of Bifurcation and Chaos","issn_l":"0218-1274","issn":["0218-1274","1793-6551"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319815","host_organization_name":"World Scientific","host_organization_lineage":["https://openalex.org/P4310319815"],"host_organization_lineage_names":["World Scientific"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Bifurcation and Chaos","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[{"score":0.8799999952316284,"id":"https://metadata.un.org/sdg/3","display_name":"Good health and well-being"}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":40,"referenced_works":["https://openalex.org/W341845201","https://openalex.org/W1531715726","https://openalex.org/W1964154533","https://openalex.org/W1965601173","https://openalex.org/W1966560110","https://openalex.org/W1967853550","https://openalex.org/W1981054952","https://openalex.org/W1987514016","https://openalex.org/W1996469563","https://openalex.org/W2006556166","https://openalex.org/W2008054049","https://openalex.org/W2009710695","https://openalex.org/W2015207750","https://openalex.org/W2030181200","https://openalex.org/W2038304337","https://openalex.org/W2039742747","https://openalex.org/W2041092052","https://openalex.org/W2044403736","https://openalex.org/W2060208963","https://openalex.org/W2061241509","https://openalex.org/W2065237420","https://openalex.org/W2079871064","https://openalex.org/W2082583880","https://openalex.org/W2085722235","https://openalex.org/W2086865438","https://openalex.org/W2088503280","https://openalex.org/W2090487374","https://openalex.org/W2093089124","https://openalex.org/W2096008770","https://openalex.org/W2099533369","https://openalex.org/W2112680994","https://openalex.org/W2120104536","https://openalex.org/W2158652872","https://openalex.org/W2163284349","https://openalex.org/W2332172232","https://openalex.org/W2481048563","https://openalex.org/W2964117113","https://openalex.org/W3100061009","https://openalex.org/W4211202937","https://openalex.org/W4388364567"],"related_works":["https://openalex.org/W3198921492","https://openalex.org/W3095275161","https://openalex.org/W2388257287","https://openalex.org/W2071231746","https://openalex.org/W2911702025","https://openalex.org/W3119573913","https://openalex.org/W3134441753","https://openalex.org/W2020890117","https://openalex.org/W3207444811","https://openalex.org/W2606851158"],"abstract_inverted_index":{"In":[0],"this":[1,38],"paper,":[2],"we":[3],"investigate":[4],"the":[5,25,42,47,57,63,73,88,94,98,104,134,149,171,190],"dynamics":[6],"of":[7,46,91,93,100],"a":[8],"five-dimensional":[9],"virus":[10,28],"model":[11,39,50],"incorporating":[12],"saturation":[13],"incidence":[14],"rate,":[15],"CTL":[16],"immune":[17,32,67,77,154],"response":[18,33,68,78,155],"and":[19,31,44,72,97,118,163],"three":[20,52],"time":[21],"delays":[22],"which":[23],"represent":[24],"latent":[26],"period,":[27],"production":[29],"period":[30],"delay,":[34],"respectively.":[35,112],"We":[36],"begin":[37],"by":[40,83,114],"proving":[41],"positivity":[43],"boundedness":[45],"solutions.":[48],"Our":[49],"admits":[51],"possible":[53],"equilibrium":[54,59,65,75,105,172],"solutions,":[55],"namely":[56],"infection-free":[58],"[Formula:":[60,69,79,107,126,141,156,164,174],"see":[61,70,80,108,127,142,157,165,175],"text],":[62],"infectious":[64,74],"without":[66],"text]":[71,109,128,143,158,166,176],"with":[76],"text].":[81],"Moreover,":[82],"analyzing":[84],"corresponding":[85],"characteristic":[86],"equations,":[87],"local":[89],"stability":[90],"each":[92],"feasible":[95],"equilibria":[96],"existence":[99],"Hopf":[101],"bifurcation":[102],"at":[103],"point":[106,173],"are":[110,185],"established,":[111],"Further,":[113],"using":[115],"fluctuation":[116],"lemma":[117],"suitable":[119],"Lyapunov":[120],"functionals,":[121],"it":[122],"is":[123,129,144,159,167,177],"shown":[124],"that":[125],"globally":[130,178],"asymptotically":[131,179],"stable":[132],"when":[133],"basic":[135,150],"reproductive":[136,151],"numbers":[137,152],"for":[138,153,188],"viral":[139],"infection":[140],"less":[145,160],"than":[146,161,169],"unity.":[147],"When":[148],"unity":[162],"greater":[168],"unity,":[170],"stable.":[180],"Finally,":[181],"some":[182],"numerical":[183],"simulations":[184],"carried":[186],"out":[187],"illustrating":[189],"theoretical":[191],"results.":[192]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":2}],"updated_date":"2026-03-17T09:09:15.849793","created_date":"2025-10-10T00:00:00"}
