{"id":"https://openalex.org/W4317502677","doi":"https://doi.org/10.3390/sym15020285","title":"Extinction and Ergodic Stationary Distribution of COVID-19 Epidemic Model with Vaccination Effects","display_name":"Extinction and Ergodic Stationary Distribution of COVID-19 Epidemic Model with Vaccination Effects","publication_year":2023,"publication_date":"2023-01-19","ids":{"openalex":"https://openalex.org/W4317502677","doi":"https://doi.org/10.3390/sym15020285"},"language":"en","primary_location":{"id":"doi:10.3390/sym15020285","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym15020285","pdf_url":"https://www.mdpi.com/2073-8994/15/2/285/pdf?version=1675303438","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"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":"Symmetry","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2073-8994/15/2/285/pdf?version=1675303438","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":null,"display_name":"Humera Batool","orcid":null},"institutions":[{"id":"https://openalex.org/I196934937","display_name":"Liaocheng University","ror":"https://ror.org/03yh0n709","country_code":"CN","type":"education","lineage":["https://openalex.org/I196934937"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Humera Batool","raw_affiliation_strings":["School of Mathematics, Liaocheng University, Liaocheng 252000, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics, Liaocheng University, Liaocheng 252000, China","institution_ids":["https://openalex.org/I196934937"]}]},{"author_position":"middle","author":{"id":null,"display_name":"Weiyu Li","orcid":null},"institutions":[{"id":"https://openalex.org/I308837","display_name":"Suzhou University of Science and Technology","ror":"https://ror.org/04en8wb91","country_code":"CN","type":"education","lineage":["https://openalex.org/I308837"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Weiyu Li","raw_affiliation_strings":["School of Mathematical Sciences, Suzhou University of Science and Technology, Suzhou 215000, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematical Sciences, Suzhou University of Science and Technology, Suzhou 215000, China","institution_ids":["https://openalex.org/I308837"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5083760199","display_name":"Zhonggui Sun","orcid":"https://orcid.org/0000-0003-0657-0103"},"institutions":[{"id":"https://openalex.org/I196934937","display_name":"Liaocheng University","ror":"https://ror.org/03yh0n709","country_code":"CN","type":"education","lineage":["https://openalex.org/I196934937"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Zhonggui Sun","raw_affiliation_strings":["School of Mathematics, Liaocheng University, Liaocheng 252000, China"],"raw_orcid":"https://orcid.org/0000-0003-0657-0103","affiliations":[{"raw_affiliation_string":"School of Mathematics, Liaocheng University, Liaocheng 252000, China","institution_ids":["https://openalex.org/I196934937"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5083760199"],"corresponding_institution_ids":["https://openalex.org/I196934937"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2227},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2227},"fwci":0.8983,"has_fulltext":false,"cited_by_count":5,"citation_normalized_percentile":{"value":0.66503497,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":97},"biblio":{"volume":"15","issue":"2","first_page":"285","last_page":"285"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10410","display_name":"COVID-19 epidemiological studies","score":0.9987999796867371,"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/T10410","display_name":"COVID-19 epidemiological studies","score":0.9987999796867371,"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/T10482","display_name":"Mathematical and Theoretical Epidemiology and Ecology Models","score":0.9958000183105469,"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/T10118","display_name":"SARS-CoV-2 and COVID-19 Research","score":0.9837999939918518,"subfield":{"id":"https://openalex.org/subfields/2725","display_name":"Infectious Diseases"},"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/stationary-distribution","display_name":"Stationary distribution","score":0.7229650020599365},{"id":"https://openalex.org/keywords/uniqueness","display_name":"Uniqueness","score":0.7189381122589111},{"id":"https://openalex.org/keywords/epidemic-model","display_name":"Epidemic model","score":0.7179297208786011},{"id":"https://openalex.org/keywords/ergodic-theory","display_name":"Ergodic theory","score":0.7129696607589722},{"id":"https://openalex.org/keywords/basic-reproduction-number","display_name":"Basic reproduction number","score":0.6702429056167603},{"id":"https://openalex.org/keywords/coronavirus-disease-2019","display_name":"Coronavirus disease 2019 (COVID-19)","score":0.5350521802902222},{"id":"https://openalex.org/keywords/vaccination","display_name":"Vaccination","score":0.5217496156692505},{"id":"https://openalex.org/keywords/extinction","display_name":"Extinction (optical mineralogy)","score":0.5006701946258545},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.47121208906173706},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.4222766160964966},{"id":"https://openalex.org/keywords/statistical-physics","display_name":"Statistical physics","score":0.4088466763496399},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.3987932801246643},{"id":"https://openalex.org/keywords/mathematical-economics","display_name":"Mathematical economics","score":0.3587048649787903},{"id":"https://openalex.org/keywords/econometrics","display_name":"Econometrics","score":0.3418717086315155},{"id":"https://openalex.org/keywords/disease","display_name":"Disease","score":0.301385760307312},{"id":"https://openalex.org/keywords/infectious-disease","display_name":"Infectious disease (medical specialty)","score":0.2753271758556366},{"id":"https://openalex.org/keywords/markov-chain","display_name":"Markov chain","score":0.23657169938087463},{"id":"https://openalex.org/keywords/virology","display_name":"Virology","score":0.22032427787780762},{"id":"https://openalex.org/keywords/biology","display_name":"Biology","score":0.17631122469902039},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.17146113514900208},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.15896883606910706},{"id":"https://openalex.org/keywords/medicine","display_name":"Medicine","score":0.15876993536949158},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.12366107106208801}],"concepts":[{"id":"https://openalex.org/C98951983","wikidata":"https://www.wikidata.org/wiki/Q7604341","display_name":"Stationary distribution","level":3,"score":0.7229650020599365},{"id":"https://openalex.org/C2777021972","wikidata":"https://www.wikidata.org/wiki/Q22976830","display_name":"Uniqueness","level":2,"score":0.7189381122589111},{"id":"https://openalex.org/C1627819","wikidata":"https://www.wikidata.org/wiki/Q2572354","display_name":"Epidemic model","level":3,"score":0.7179297208786011},{"id":"https://openalex.org/C122044880","wikidata":"https://www.wikidata.org/wiki/Q5498822","display_name":"Ergodic theory","level":2,"score":0.7129696607589722},{"id":"https://openalex.org/C204264503","wikidata":"https://www.wikidata.org/wiki/Q901464","display_name":"Basic reproduction number","level":3,"score":0.6702429056167603},{"id":"https://openalex.org/C3008058167","wikidata":"https://www.wikidata.org/wiki/Q84263196","display_name":"Coronavirus disease 2019 (COVID-19)","level":4,"score":0.5350521802902222},{"id":"https://openalex.org/C22070199","wikidata":"https://www.wikidata.org/wiki/Q192995","display_name":"Vaccination","level":2,"score":0.5217496156692505},{"id":"https://openalex.org/C101991246","wikidata":"https://www.wikidata.org/wiki/Q5422078","display_name":"Extinction (optical mineralogy)","level":2,"score":0.5006701946258545},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.47121208906173706},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4222766160964966},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.4088466763496399},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3987932801246643},{"id":"https://openalex.org/C144237770","wikidata":"https://www.wikidata.org/wiki/Q747534","display_name":"Mathematical economics","level":1,"score":0.3587048649787903},{"id":"https://openalex.org/C149782125","wikidata":"https://www.wikidata.org/wiki/Q160039","display_name":"Econometrics","level":1,"score":0.3418717086315155},{"id":"https://openalex.org/C2779134260","wikidata":"https://www.wikidata.org/wiki/Q12136","display_name":"Disease","level":2,"score":0.301385760307312},{"id":"https://openalex.org/C524204448","wikidata":"https://www.wikidata.org/wiki/Q788926","display_name":"Infectious disease (medical specialty)","level":3,"score":0.2753271758556366},{"id":"https://openalex.org/C98763669","wikidata":"https://www.wikidata.org/wiki/Q176645","display_name":"Markov chain","level":2,"score":0.23657169938087463},{"id":"https://openalex.org/C159047783","wikidata":"https://www.wikidata.org/wiki/Q7215","display_name":"Virology","level":1,"score":0.22032427787780762},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.17631122469902039},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.17146113514900208},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.15896883606910706},{"id":"https://openalex.org/C71924100","wikidata":"https://www.wikidata.org/wiki/Q11190","display_name":"Medicine","level":0,"score":0.15876993536949158},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.12366107106208801},{"id":"https://openalex.org/C2908647359","wikidata":"https://www.wikidata.org/wiki/Q2625603","display_name":"Population","level":2,"score":0.0},{"id":"https://openalex.org/C99454951","wikidata":"https://www.wikidata.org/wiki/Q932068","display_name":"Environmental health","level":1,"score":0.0},{"id":"https://openalex.org/C142724271","wikidata":"https://www.wikidata.org/wiki/Q7208","display_name":"Pathology","level":1,"score":0.0},{"id":"https://openalex.org/C151730666","wikidata":"https://www.wikidata.org/wiki/Q7205","display_name":"Paleontology","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/sym15020285","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym15020285","pdf_url":"https://www.mdpi.com/2073-8994/15/2/285/pdf?version=1675303438","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"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":"Symmetry","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:5cd109cbd1f94f3c9a70999d16c9a125","is_oa":false,"landing_page_url":"https://doaj.org/article/5cd109cbd1f94f3c9a70999d16c9a125","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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry, Vol 15, Iss 2, p 285 (2023)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/15/2/285/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/sym15020285","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":"Symmetry","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/sym15020285","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym15020285","pdf_url":"https://www.mdpi.com/2073-8994/15/2/285/pdf?version=1675303438","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"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":"Symmetry","raw_type":"journal-article"},"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/3","score":0.8500000238418579,"display_name":"Good health and well-being"}],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":false},"content_urls":{"pdf":"https://content.openalex.org/works/W4317502677.pdf"},"referenced_works_count":22,"referenced_works":["https://openalex.org/W207440351","https://openalex.org/W1579485993","https://openalex.org/W2038997661","https://openalex.org/W2098983755","https://openalex.org/W2895052910","https://openalex.org/W2951677918","https://openalex.org/W2972632506","https://openalex.org/W3001118548","https://openalex.org/W3023727013","https://openalex.org/W3030631368","https://openalex.org/W3087667739","https://openalex.org/W3127671850","https://openalex.org/W3155581831","https://openalex.org/W3173394315","https://openalex.org/W3205580725","https://openalex.org/W4230836784","https://openalex.org/W4289341491","https://openalex.org/W4294958189","https://openalex.org/W4296101251","https://openalex.org/W4297520985","https://openalex.org/W4297832403","https://openalex.org/W6842908155"],"related_works":["https://openalex.org/W4320351350","https://openalex.org/W2230721379","https://openalex.org/W2376250553","https://openalex.org/W3138234900","https://openalex.org/W2356839139","https://openalex.org/W4388177052","https://openalex.org/W3113377908","https://openalex.org/W2032448025","https://openalex.org/W4293088316","https://openalex.org/W4282960359"],"abstract_inverted_index":{"Human":[0],"society":[1],"always":[2],"wants":[3],"a":[4,29,70,118],"safe":[5],"environment":[6],"from":[7],"pollution":[8],"and":[9,41,73,86,115],"infectious":[10],"diseases,":[11],"such":[12],"as":[13],"COVID-19,":[14,18],"etc.":[15],"To":[16],"control":[17],"we":[19,58,106],"have":[20,36],"started":[21],"the":[22,26,37,49,60,67,75,78,81,97,101,109,113,126,133,137,141,144,149],"big":[23],"effort":[24],"for":[25,112,124],"discovery":[27],"of":[28,31,39,51,69,77,83,100,117,136,143,148],"vaccination":[30,71],"COVID-19.":[32],"Several":[33],"biological":[34,52],"problems":[35],"aspects":[38],"symmetry,":[40],"this":[42,55],"theory":[43,82],"has":[44],"many":[45],"applications":[46],"in":[47],"explaining":[48],"dynamics":[50,76,85],"models.":[53],"In":[54],"research":[56],"article,":[57],"developed":[59],"stochastic":[61],"COVID-19":[62],"mathematical":[63],"model,":[64],"along":[65,131],"with":[66,132],"inclusion":[68],"term,":[72],"studied":[74],"disease":[79,127],"through":[80],"symmetric":[84],"ergodic":[87],"stationary":[88,134],"distribution.":[89],"The":[90,121],"basic":[91],"reproduction":[92],"number":[93],"is":[94],"evaluated":[95],"using":[96],"equilibrium":[98],"points":[99],"proposed":[102,138],"model.":[103,139],"For":[104,140],"well-posedness,":[105],"also":[107,129],"test":[108],"given":[110],"problem":[111],"existence":[114],"uniqueness":[116],"non-negative":[119],"solution.":[120],"necessary":[122],"conditions":[123],"eradicating":[125],"are":[128,151],"analyzed":[130],"distribution":[135],"verification":[142],"obtained":[145],"result,":[146],"simulations":[147],"model":[150],"performed.":[152]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":2}],"updated_date":"2026-07-22T07:51:19.307946","created_date":"2025-10-10T00:00:00"}
