{"id":"https://openalex.org/W4386067808","doi":"https://doi.org/10.3390/axioms12090805","title":"Global Dynamics of an Age-Structured Tuberculosis Model with Vaccine Failure and Nonlinear Infection Force","display_name":"Global Dynamics of an Age-Structured Tuberculosis Model with Vaccine Failure and Nonlinear Infection Force","publication_year":2023,"publication_date":"2023-08-22","ids":{"openalex":"https://openalex.org/W4386067808","doi":"https://doi.org/10.3390/axioms12090805"},"language":"en","primary_location":{"id":"doi:10.3390/axioms12090805","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms12090805","pdf_url":"https://www.mdpi.com/2075-1680/12/9/805/pdf?version=1692697313","source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Axioms","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2075-1680/12/9/805/pdf?version=1692697313","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5089327549","display_name":"Zhongkai Guo","orcid":"https://orcid.org/0000-0001-5822-1891"},"institutions":[{"id":"https://openalex.org/I3133134087","display_name":"Lanzhou Jiaotong University","ror":"https://ror.org/03144pv92","country_code":"CN","type":"education","lineage":["https://openalex.org/I3133134087"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Zhongkai Guo","raw_affiliation_strings":["School of Traffic and Transportation, Lanzhou Jiaotong University, Lanzhou 730070, China"],"affiliations":[{"raw_affiliation_string":"School of Traffic and Transportation, Lanzhou Jiaotong University, Lanzhou 730070, China","institution_ids":["https://openalex.org/I3133134087"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5100425301","display_name":"Liang Zhang","orcid":"https://orcid.org/0000-0003-4778-1361"},"institutions":[{"id":"https://openalex.org/I89652312","display_name":"Northwest A&F University","ror":"https://ror.org/0051rme32","country_code":"CN","type":"education","lineage":["https://openalex.org/I89652312"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Liang Zhang","raw_affiliation_strings":["College of Science, Northwest A&F University, Yangling 712100, China"],"affiliations":[{"raw_affiliation_string":"College of Science, Northwest A&F University, Yangling 712100, China","institution_ids":["https://openalex.org/I89652312"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5089327549","https://openalex.org/A5100425301"],"corresponding_institution_ids":["https://openalex.org/I3133134087","https://openalex.org/I89652312"],"apc_list":{"value":1400,"currency":"CHF","value_usd":1515},"apc_paid":{"value":1400,"currency":"CHF","value_usd":1515},"fwci":0.5884,"has_fulltext":true,"cited_by_count":2,"citation_normalized_percentile":{"value":0.69890009,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":95},"biblio":{"volume":"12","issue":"9","first_page":"805","last_page":"805"},"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":0.9993000030517578,"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":0.9993000030517578,"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/T10410","display_name":"COVID-19 epidemiological studies","score":0.9951000213623047,"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/T11764","display_name":"Evolution and Genetic Dynamics","score":0.991599977016449,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/stability-theory","display_name":"Stability theory","score":0.5907611846923828},{"id":"https://openalex.org/keywords/basic-reproduction-number","display_name":"Basic reproduction number","score":0.5855280160903931},{"id":"https://openalex.org/keywords/tuberculosis","display_name":"Tuberculosis","score":0.5412165522575378},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.4775969982147217},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.46611717343330383},{"id":"https://openalex.org/keywords/china","display_name":"China","score":0.45479869842529297},{"id":"https://openalex.org/keywords/transmission","display_name":"Transmission (telecommunications)","score":0.4247964322566986},{"id":"https://openalex.org/keywords/rank","display_name":"Rank (graph theory)","score":0.4218885898590088},{"id":"https://openalex.org/keywords/latency","display_name":"Latency (audio)","score":0.41840192675590515},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4050430655479431},{"id":"https://openalex.org/keywords/econometrics","display_name":"Econometrics","score":0.4040331542491913},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.3798302710056305},{"id":"https://openalex.org/keywords/medicine","display_name":"Medicine","score":0.3324492573738098},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.32459408044815063},{"id":"https://openalex.org/keywords/environmental-health","display_name":"Environmental health","score":0.12913477420806885},{"id":"https://openalex.org/keywords/geography","display_name":"Geography","score":0.11446544528007507}],"concepts":[{"id":"https://openalex.org/C41949839","wikidata":"https://www.wikidata.org/wiki/Q1756677","display_name":"Stability theory","level":3,"score":0.5907611846923828},{"id":"https://openalex.org/C204264503","wikidata":"https://www.wikidata.org/wiki/Q901464","display_name":"Basic reproduction number","level":3,"score":0.5855280160903931},{"id":"https://openalex.org/C2781069245","wikidata":"https://www.wikidata.org/wiki/Q12204","display_name":"Tuberculosis","level":2,"score":0.5412165522575378},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4775969982147217},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.46611717343330383},{"id":"https://openalex.org/C191935318","wikidata":"https://www.wikidata.org/wiki/Q148","display_name":"China","level":2,"score":0.45479869842529297},{"id":"https://openalex.org/C761482","wikidata":"https://www.wikidata.org/wiki/Q118093","display_name":"Transmission (telecommunications)","level":2,"score":0.4247964322566986},{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.4218885898590088},{"id":"https://openalex.org/C82876162","wikidata":"https://www.wikidata.org/wiki/Q17096504","display_name":"Latency (audio)","level":2,"score":0.41840192675590515},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4050430655479431},{"id":"https://openalex.org/C149782125","wikidata":"https://www.wikidata.org/wiki/Q160039","display_name":"Econometrics","level":1,"score":0.4040331542491913},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3798302710056305},{"id":"https://openalex.org/C71924100","wikidata":"https://www.wikidata.org/wiki/Q11190","display_name":"Medicine","level":0,"score":0.3324492573738098},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.32459408044815063},{"id":"https://openalex.org/C99454951","wikidata":"https://www.wikidata.org/wiki/Q932068","display_name":"Environmental health","level":1,"score":0.12913477420806885},{"id":"https://openalex.org/C205649164","wikidata":"https://www.wikidata.org/wiki/Q1071","display_name":"Geography","level":0,"score":0.11446544528007507},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","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/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C76155785","wikidata":"https://www.wikidata.org/wiki/Q418","display_name":"Telecommunications","level":1,"score":0.0},{"id":"https://openalex.org/C166957645","wikidata":"https://www.wikidata.org/wiki/Q23498","display_name":"Archaeology","level":1,"score":0.0},{"id":"https://openalex.org/C2908647359","wikidata":"https://www.wikidata.org/wiki/Q2625603","display_name":"Population","level":2,"score":0.0},{"id":"https://openalex.org/C142724271","wikidata":"https://www.wikidata.org/wiki/Q7208","display_name":"Pathology","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/axioms12090805","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms12090805","pdf_url":"https://www.mdpi.com/2075-1680/12/9/805/pdf?version=1692697313","source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Axioms","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:a901c45b02f740feaed6312ae9629d45","is_oa":true,"landing_page_url":"https://doaj.org/article/a901c45b02f740feaed6312ae9629d45","pdf_url":null,"source":{"id":"https://openalex.org/S112646816","display_name":"SHILAP Revista de lepidopterolog\u00eda","issn_l":"0300-5267","issn":["0300-5267","2340-4078"],"is_oa":true,"is_in_doaj":true,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Axioms, Vol 12, Iss 9, p 805 (2023)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2075-1680/12/9/805/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/axioms12090805","pdf_url":null,"source":{"id":"https://openalex.org/S4306400947","display_name":"MDPI (MDPI AG)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210097602","host_organization_name":"Multidisciplinary Digital Publishing Institute (Switzerland)","host_organization_lineage":["https://openalex.org/I4210097602"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Axioms; Volume 12; Issue 9; Pages: 805","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/axioms12090805","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms12090805","pdf_url":"https://www.mdpi.com/2075-1680/12/9/805/pdf?version=1692697313","source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Axioms","raw_type":"journal-article"},"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/3","display_name":"Good health and well-being","score":0.8299999833106995}],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4386067808.pdf","grobid_xml":"https://content.openalex.org/works/W4386067808.grobid-xml"},"referenced_works_count":36,"referenced_works":["https://openalex.org/W399052820","https://openalex.org/W637195664","https://openalex.org/W1964385385","https://openalex.org/W1975722775","https://openalex.org/W2028824623","https://openalex.org/W2037872618","https://openalex.org/W2055305897","https://openalex.org/W2059579326","https://openalex.org/W2071544114","https://openalex.org/W2090680333","https://openalex.org/W2123261528","https://openalex.org/W2191940372","https://openalex.org/W2329622328","https://openalex.org/W2381944992","https://openalex.org/W2475038746","https://openalex.org/W2592111873","https://openalex.org/W2595618202","https://openalex.org/W2619370078","https://openalex.org/W2749077978","https://openalex.org/W2895366291","https://openalex.org/W2917625172","https://openalex.org/W2942551575","https://openalex.org/W2966335801","https://openalex.org/W2973703033","https://openalex.org/W3125947980","https://openalex.org/W3148747445","https://openalex.org/W3173499730","https://openalex.org/W4206454833","https://openalex.org/W4210410413","https://openalex.org/W4225090553","https://openalex.org/W4282981782","https://openalex.org/W4284892142","https://openalex.org/W4291708333","https://openalex.org/W6665636276","https://openalex.org/W6702274633","https://openalex.org/W6807883888"],"related_works":["https://openalex.org/W2372335318","https://openalex.org/W2253278609","https://openalex.org/W2291742240","https://openalex.org/W2511412881","https://openalex.org/W4320351350","https://openalex.org/W2372947689","https://openalex.org/W2009249058","https://openalex.org/W3203592860","https://openalex.org/W4388177052","https://openalex.org/W2075737090"],"abstract_inverted_index":{"China":[0,124],"bears":[1],"a":[2,42],"heavy":[3],"burden":[4],"due":[5],"to":[6,26,108,127,144],"tuberculosis":[7],"(TB)":[8],"with":[9,17],"hundreds":[10],"of":[11,13,30,72,115,151],"thousands":[12],"people":[14],"falling":[15],"ill":[16],"the":[18,28,52,67,73,79,90,103,110,116,130,138,148,152,163],"disease":[19],"every":[20],"year.":[21],"Therefore,":[22],"it":[23],"is":[24,83,94],"necessary":[25],"understand":[27],"effectiveness":[29],"current":[31],"control":[32,160],"measures":[33,161],"in":[34,123,162],"China.":[35],"In":[36],"this":[37],"paper,":[38],"we":[39,136,154],"first":[40],"present":[41],"TB":[43,55,120,159],"model":[44,53,75,117],"that":[45,78,140],"incorporates":[46],"both":[47],"vaccination":[48],"and":[49,61,76,89,112],"treatment.":[50],"Additionally,":[51],"considers":[54],"transmission":[56],"characteristics":[57],"such":[58],"as":[59],"relapse":[60],"variable":[62],"latency.":[63],"We":[64,100],"then":[65,101],"define":[66],"basic":[68],"reproduction":[69],"number":[70],"R0":[71],"proposed":[74],"indicate":[77],"disease-free":[80],"equilibrium":[81,92],"state":[82,93],"globally":[84,95],"asymptotically":[85,96],"stable":[86,97],"if":[87,98],"R0<1,":[88],"endemic":[91],"R0>1.":[99],"apply":[102],"Grey":[104],"Wolf":[105],"Optimizer":[106],"algorithm":[107],"obtain":[109],"parameters":[111,139],"initial":[113],"values":[114],"by":[118],"combining":[119],"data":[121],"collected":[122],"from":[125],"2007":[126],"2020.":[128],"Through":[129],"partial":[131],"rank":[132],"correlation":[133],"coefficient":[134],"method,":[135],"identify":[137],"are":[141],"most":[142],"sensitive":[143],"R0.":[145],"Based":[146],"on":[147],"analysis":[149],"results":[150],"model,":[153],"propose":[155],"some":[156],"suggestions":[157],"for":[158],"conclusion":[164],"section.":[165]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":1}],"updated_date":"2026-02-28T09:26:25.869077","created_date":"2023-08-23T00:00:00"}