{"id":"https://openalex.org/W2986413830","doi":"https://doi.org/10.1137/19m1243580","title":"Dynamical Behaviors of the Tumor-Immune System in a Stochastic Environment","display_name":"Dynamical Behaviors of the Tumor-Immune System in a Stochastic Environment","publication_year":2019,"publication_date":"2019-01-01","ids":{"openalex":"https://openalex.org/W2986413830","doi":"https://doi.org/10.1137/19m1243580","mag":"2986413830"},"language":"en","primary_location":{"id":"doi:10.1137/19m1243580","is_oa":false,"landing_page_url":"https://doi.org/10.1137/19m1243580","pdf_url":null,"source":{"id":"https://openalex.org/S77200987","display_name":"SIAM Journal on Applied Mathematics","issn_l":"0036-1399","issn":["0036-1399","1095-712X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Applied Mathematics","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/A5101658546","display_name":"Xiaoyue Li","orcid":"https://orcid.org/0000-0002-2367-7431"},"institutions":[{"id":"https://openalex.org/I184983240","display_name":"Northeast Normal University","ror":"https://ror.org/02rkvz144","country_code":"CN","type":"education","lineage":["https://openalex.org/I184983240"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Xiaoyue Li","raw_affiliation_strings":["School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin, 130024, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin, 130024, China","institution_ids":["https://openalex.org/I184983240"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5043828363","display_name":"Guoting Song","orcid":null},"institutions":[{"id":"https://openalex.org/I184983240","display_name":"Northeast Normal University","ror":"https://ror.org/02rkvz144","country_code":"CN","type":"education","lineage":["https://openalex.org/I184983240"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Guoting Song","raw_affiliation_strings":["School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin, 130024, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin, 130024, China","institution_ids":["https://openalex.org/I184983240"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5057366785","display_name":"Yang Xia","orcid":null},"institutions":[{"id":"https://openalex.org/I184983240","display_name":"Northeast Normal University","ror":"https://ror.org/02rkvz144","country_code":"CN","type":"education","lineage":["https://openalex.org/I184983240"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Yang Xia","raw_affiliation_strings":["School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin, 130024, China;","Shenzhen Foreign Languages School, Shenzhen, Guangdong, 518000, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin, 130024, China;","institution_ids":["https://openalex.org/I184983240"]},{"raw_affiliation_string":"Shenzhen Foreign Languages School, Shenzhen, Guangdong, 518000, China","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5100400931","display_name":"Chenggui Yuan","orcid":"https://orcid.org/0000-0003-0486-5450"},"institutions":[{"id":"https://openalex.org/I39586589","display_name":"Swansea University","ror":"https://ror.org/053fq8t95","country_code":"GB","type":"education","lineage":["https://openalex.org/I39586589"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"Chenggui Yuan","raw_affiliation_strings":["Department of Mathematics, Swansea University, Bay Campus, SA1 8EN, UK"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Swansea University, Bay Campus, SA1 8EN, UK","institution_ids":["https://openalex.org/I39586589"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":1.1254,"has_fulltext":false,"cited_by_count":33,"citation_normalized_percentile":{"value":0.76542137,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":94,"max":99},"biblio":{"volume":"79","issue":"6","first_page":"2193","last_page":"2217"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11829","display_name":"Mathematical Biology Tumor Growth","score":0.9998999834060669,"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"}},"topics":[{"id":"https://openalex.org/T11829","display_name":"Mathematical Biology Tumor Growth","score":0.9998999834060669,"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/T10621","display_name":"Gene Regulatory Network Analysis","score":0.9861000180244446,"subfield":{"id":"https://openalex.org/subfields/1312","display_name":"Molecular Biology"},"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/T10482","display_name":"Mathematical and Theoretical Epidemiology and Ecology Models","score":0.9585999846458435,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/uniqueness","display_name":"Uniqueness","score":0.7877509593963623},{"id":"https://openalex.org/keywords/ergodicity","display_name":"Ergodicity","score":0.7292001247406006},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6547605991363525},{"id":"https://openalex.org/keywords/stationary-distribution","display_name":"Stationary distribution","score":0.5953769683837891},{"id":"https://openalex.org/keywords/lyapunov-function","display_name":"Lyapunov function","score":0.5863317847251892},{"id":"https://openalex.org/keywords/stochastic-differential-equation","display_name":"Stochastic differential equation","score":0.5612902045249939},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.49688389897346497},{"id":"https://openalex.org/keywords/dynamical-systems-theory","display_name":"Dynamical systems theory","score":0.43749091029167175},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.4159315228462219},{"id":"https://openalex.org/keywords/exponential-stability","display_name":"Exponential stability","score":0.4108075499534607},{"id":"https://openalex.org/keywords/statistical-physics","display_name":"Statistical physics","score":0.32717663049697876},{"id":"https://openalex.org/keywords/markov-chain","display_name":"Markov chain","score":0.2382740080356598},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.156193345785141},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.13749486207962036},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.08881038427352905}],"concepts":[{"id":"https://openalex.org/C2777021972","wikidata":"https://www.wikidata.org/wiki/Q22976830","display_name":"Uniqueness","level":2,"score":0.7877509593963623},{"id":"https://openalex.org/C201779956","wikidata":"https://www.wikidata.org/wiki/Q5426803","display_name":"Ergodicity","level":2,"score":0.7292001247406006},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6547605991363525},{"id":"https://openalex.org/C98951983","wikidata":"https://www.wikidata.org/wiki/Q7604341","display_name":"Stationary distribution","level":3,"score":0.5953769683837891},{"id":"https://openalex.org/C60640748","wikidata":"https://www.wikidata.org/wiki/Q2337858","display_name":"Lyapunov function","level":3,"score":0.5863317847251892},{"id":"https://openalex.org/C51955184","wikidata":"https://www.wikidata.org/wiki/Q1545585","display_name":"Stochastic differential equation","level":2,"score":0.5612902045249939},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.49688389897346497},{"id":"https://openalex.org/C79379906","wikidata":"https://www.wikidata.org/wiki/Q3174497","display_name":"Dynamical systems theory","level":2,"score":0.43749091029167175},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4159315228462219},{"id":"https://openalex.org/C167964875","wikidata":"https://www.wikidata.org/wiki/Q17011487","display_name":"Exponential stability","level":3,"score":0.4108075499534607},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.32717663049697876},{"id":"https://openalex.org/C98763669","wikidata":"https://www.wikidata.org/wiki/Q176645","display_name":"Markov chain","level":2,"score":0.2382740080356598},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.156193345785141},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.13749486207962036},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.08881038427352905},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/19m1243580","is_oa":false,"landing_page_url":"https://doi.org/10.1137/19m1243580","pdf_url":null,"source":{"id":"https://openalex.org/S77200987","display_name":"SIAM Journal on Applied Mathematics","issn_l":"0036-1399","issn":["0036-1399","1095-712X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Applied Mathematics","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[{"score":0.7599999904632568,"display_name":"Life in Land","id":"https://metadata.un.org/sdg/15"}],"awards":[{"id":"https://openalex.org/G291184313","display_name":null,"funder_award_id":"11171056","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G5723352003","display_name":"\u6c34\u6bcd\u66b4\u53d1\u5f62\u6210\u673a\u5236\u4e0e\u751f\u6001\u6548\u5e94\u7684\u6a21\u578b\u7814\u7a76","funder_award_id":"11671072","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G5878380174","display_name":null,"funder_award_id":"20170101044JC","funder_id":"https://openalex.org/F4320310121","funder_display_name":"Natural Science Foundation of Jilin Province"},{"id":"https://openalex.org/G7924676889","display_name":null,"funder_award_id":"JJKH20170904KJ","funder_id":"https://openalex.org/F4320326270","funder_display_name":"Education Department of Jilin Province"},{"id":"https://openalex.org/G892097767","display_name":null,"funder_award_id":"11471071","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320310121","display_name":"Natural Science Foundation of Jilin Province","ror":null},{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320326270","display_name":"Education Department of Jilin Province","ror":"https://ror.org/05782e512"},{"id":"https://openalex.org/F4320335787","display_name":"Fundamental Research Funds for the Central Universities","ror":null}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":26,"referenced_works":["https://openalex.org/W847649018","https://openalex.org/W1141364993","https://openalex.org/W1966410158","https://openalex.org/W1971201122","https://openalex.org/W1972914572","https://openalex.org/W2014133785","https://openalex.org/W2018649712","https://openalex.org/W2026206905","https://openalex.org/W2056160862","https://openalex.org/W2061075226","https://openalex.org/W2079376474","https://openalex.org/W2093215206","https://openalex.org/W2107760337","https://openalex.org/W2111663010","https://openalex.org/W2145137020","https://openalex.org/W2147754310","https://openalex.org/W2158667119","https://openalex.org/W2169169493","https://openalex.org/W2170095111","https://openalex.org/W2434961035","https://openalex.org/W2602927034","https://openalex.org/W2962768007","https://openalex.org/W2962855395","https://openalex.org/W2963163821","https://openalex.org/W2963726397","https://openalex.org/W4241424808"],"related_works":["https://openalex.org/W4293402875","https://openalex.org/W142914808","https://openalex.org/W4225342163","https://openalex.org/W1560843892","https://openalex.org/W2054229853","https://openalex.org/W4300561179","https://openalex.org/W2794025271","https://openalex.org/W3048312975","https://openalex.org/W2390585706","https://openalex.org/W2964291893"],"abstract_inverted_index":{"This":[0],"paper":[1],"investigates":[2],"dynamical":[3],"behaviors":[4],"of":[5,18,26,55,74,98,104],"the":[6,16,19,24,51,56,61,72,83,87,94,105],"tumor-immune":[7,62,106],"system":[8],"perturbed":[9],"by":[10],"environmental":[11],"noise.":[12],"The":[13,30],"model":[14],"describes":[15],"response":[17],"cytotoxic":[20],"T":[21],"lymphocyte":[22],"to":[23,70,116],"growth":[25],"an":[27],"immunogenic":[28],"tumor.":[29],"main":[31],"methods":[32],"are":[33],"stochastic":[34,40,102],"Lyapunov":[35],"analysis,":[36],"comparison":[37],"theorem":[38],"for":[39,60,76],"differential":[41],"equations":[42],"(SDEs),":[43],"and":[44,53,79,82,96,101,113],"strong":[45],"ergodicity":[46],"theorem.":[47],"Firstly,":[48],"we":[49,65,92,109],"prove":[50],"existence":[52,95],"uniqueness":[54,97],"global":[57],"positive":[58],"solution":[59],"system.":[63,107],"Then":[64],"go":[66],"a":[67],"further":[68],"step":[69],"study":[71],"boundaries":[73],"moments":[75],"tumor":[77],"cells":[78,81],"effector":[80],"asymptotic":[84],"behavior":[85],"in":[86],"boundary":[88],"equilibrium":[89],"points.":[90],"Furthermore,":[91],"discuss":[93],"stationary":[99],"distribution":[100],"permanence":[103],"Finally,":[108],"give":[110],"several":[111],"examples":[112],"numerical":[114],"simulations":[115],"verify":[117],"our":[118],"results.":[119]},"counts_by_year":[{"year":2026,"cited_by_count":4},{"year":2025,"cited_by_count":7},{"year":2024,"cited_by_count":4},{"year":2023,"cited_by_count":7},{"year":2022,"cited_by_count":6},{"year":2021,"cited_by_count":3},{"year":2020,"cited_by_count":2}],"updated_date":"2026-07-18T07:39:51.176621","created_date":"2025-10-10T00:00:00"}
