{"id":"https://openalex.org/W4223534219","doi":"https://doi.org/10.3390/axioms11040170","title":"Fractional Modelling and Optimal Control of COVID-19 Transmission in Portugal","display_name":"Fractional Modelling and Optimal Control of COVID-19 Transmission in Portugal","publication_year":2022,"publication_date":"2022-04-11","ids":{"openalex":"https://openalex.org/W4223534219","doi":"https://doi.org/10.3390/axioms11040170"},"language":"en","primary_location":{"id":"doi:10.3390/axioms11040170","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms11040170","pdf_url":"https://www.mdpi.com/2075-1680/11/4/170/pdf?version=1649643399","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":["arxiv","crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2075-1680/11/4/170/pdf?version=1649643399","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":null,"display_name":"Silv\u00e9rio Rosa","orcid":"https://orcid.org/0000-0001-5153-4879"},"institutions":[{"id":"https://openalex.org/I161321875","display_name":"University of Beira Interior","ror":"https://ror.org/03nf36p02","country_code":"PT","type":"education","lineage":["https://openalex.org/I161321875"]},{"id":"https://openalex.org/I4210120471","display_name":"Instituto de Telecomunica\u00e7\u00f5es","ror":"https://ror.org/02ht4fk33","country_code":"PT","type":"nonprofit","lineage":["https://openalex.org/I4210120471"]}],"countries":["PT"],"is_corresponding":false,"raw_author_name":"Silv\u00e9rio Rosa","raw_affiliation_strings":["Department of Mathematics, Instituto de Telecomunica\u00e7\u00f5es (IT), Universidade da Beira Interior, 6201-001 Covilh\u00e3, Portugal"],"raw_orcid":"https://orcid.org/0000-0001-5153-4879","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Instituto de Telecomunica\u00e7\u00f5es (IT), Universidade da Beira Interior, 6201-001 Covilh\u00e3, Portugal","institution_ids":["https://openalex.org/I161321875","https://openalex.org/I4210120471"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5057131265","display_name":"Delfim F. M. Torres","orcid":"https://orcid.org/0000-0001-8641-2505"},"institutions":[{"id":"https://openalex.org/I60858718","display_name":"University of Aveiro","ror":"https://ror.org/00nt41z93","country_code":"PT","type":"education","lineage":["https://openalex.org/I60858718"]}],"countries":["PT"],"is_corresponding":true,"raw_author_name":"Delfim F. M. Torres","raw_affiliation_strings":["Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal"],"raw_orcid":"https://orcid.org/0000-0001-8641-2505","affiliations":[{"raw_affiliation_string":"Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal","institution_ids":["https://openalex.org/I60858718"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5057131265"],"corresponding_institution_ids":["https://openalex.org/I60858718"],"apc_list":{"value":1400,"currency":"CHF","value_usd":1515},"apc_paid":{"value":1400,"currency":"CHF","value_usd":1515},"fwci":1.9257,"has_fulltext":true,"cited_by_count":14,"citation_normalized_percentile":{"value":0.84569694,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"11","issue":"4","first_page":"170","last_page":"170"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10288","display_name":"Fractional Differential Equations Solutions","score":0.9995999932289124,"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/T10288","display_name":"Fractional Differential Equations Solutions","score":0.9995999932289124,"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/T10410","display_name":"COVID-19 epidemiological studies","score":0.9976000189781189,"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":0.9909999966621399,"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/coronavirus-disease-2019","display_name":"Coronavirus disease 2019 (COVID-19)","score":0.7562854886054993},{"id":"https://openalex.org/keywords/pandemic","display_name":"Pandemic","score":0.6127827167510986},{"id":"https://openalex.org/keywords/transmission","display_name":"Transmission (telecommunications)","score":0.5920132398605347},{"id":"https://openalex.org/keywords/2019-20-coronavirus-outbreak","display_name":"2019-20 coronavirus outbreak","score":0.5205110907554626},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.5122653841972351},{"id":"https://openalex.org/keywords/fractal","display_name":"Fractal","score":0.5082775950431824},{"id":"https://openalex.org/keywords/epidemic-model","display_name":"Epidemic model","score":0.4789210557937622},{"id":"https://openalex.org/keywords/severe-acute-respiratory-syndrome-coronavirus-2","display_name":"Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2)","score":0.47209542989730835},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.39381957054138184},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.38440290093421936},{"id":"https://openalex.org/keywords/econometrics","display_name":"Econometrics","score":0.36678752303123474},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.3647294044494629},{"id":"https://openalex.org/keywords/operations-research","display_name":"Operations research","score":0.3291168808937073},{"id":"https://openalex.org/keywords/virology","display_name":"Virology","score":0.21086624264717102},{"id":"https://openalex.org/keywords/medicine","display_name":"Medicine","score":0.1885102093219757},{"id":"https://openalex.org/keywords/demography","display_name":"Demography","score":0.18627870082855225},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.1806967854499817},{"id":"https://openalex.org/keywords/economics","display_name":"Economics","score":0.14976829290390015},{"id":"https://openalex.org/keywords/telecommunications","display_name":"Telecommunications","score":0.11658406257629395},{"id":"https://openalex.org/keywords/sociology","display_name":"Sociology","score":0.10600137710571289},{"id":"https://openalex.org/keywords/outbreak","display_name":"Outbreak","score":0.06711521744728088}],"concepts":[{"id":"https://openalex.org/C3008058167","wikidata":"https://www.wikidata.org/wiki/Q84263196","display_name":"Coronavirus disease 2019 (COVID-19)","level":4,"score":0.7562854886054993},{"id":"https://openalex.org/C89623803","wikidata":"https://www.wikidata.org/wiki/Q12184","display_name":"Pandemic","level":5,"score":0.6127827167510986},{"id":"https://openalex.org/C761482","wikidata":"https://www.wikidata.org/wiki/Q118093","display_name":"Transmission (telecommunications)","level":2,"score":0.5920132398605347},{"id":"https://openalex.org/C3006700255","wikidata":"https://www.wikidata.org/wiki/Q81068910","display_name":"2019-20 coronavirus outbreak","level":3,"score":0.5205110907554626},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.5122653841972351},{"id":"https://openalex.org/C40636538","wikidata":"https://www.wikidata.org/wiki/Q81392","display_name":"Fractal","level":2,"score":0.5082775950431824},{"id":"https://openalex.org/C1627819","wikidata":"https://www.wikidata.org/wiki/Q2572354","display_name":"Epidemic model","level":3,"score":0.4789210557937622},{"id":"https://openalex.org/C3007834351","wikidata":"https://www.wikidata.org/wiki/Q82069695","display_name":"Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2)","level":5,"score":0.47209542989730835},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.39381957054138184},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.38440290093421936},{"id":"https://openalex.org/C149782125","wikidata":"https://www.wikidata.org/wiki/Q160039","display_name":"Econometrics","level":1,"score":0.36678752303123474},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3647294044494629},{"id":"https://openalex.org/C42475967","wikidata":"https://www.wikidata.org/wiki/Q194292","display_name":"Operations research","level":1,"score":0.3291168808937073},{"id":"https://openalex.org/C159047783","wikidata":"https://www.wikidata.org/wiki/Q7215","display_name":"Virology","level":1,"score":0.21086624264717102},{"id":"https://openalex.org/C71924100","wikidata":"https://www.wikidata.org/wiki/Q11190","display_name":"Medicine","level":0,"score":0.1885102093219757},{"id":"https://openalex.org/C149923435","wikidata":"https://www.wikidata.org/wiki/Q37732","display_name":"Demography","level":1,"score":0.18627870082855225},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.1806967854499817},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.14976829290390015},{"id":"https://openalex.org/C76155785","wikidata":"https://www.wikidata.org/wiki/Q418","display_name":"Telecommunications","level":1,"score":0.11658406257629395},{"id":"https://openalex.org/C144024400","wikidata":"https://www.wikidata.org/wiki/Q21201","display_name":"Sociology","level":0,"score":0.10600137710571289},{"id":"https://openalex.org/C116675565","wikidata":"https://www.wikidata.org/wiki/Q3241045","display_name":"Outbreak","level":2,"score":0.06711521744728088},{"id":"https://openalex.org/C2908647359","wikidata":"https://www.wikidata.org/wiki/Q2625603","display_name":"Population","level":2,"score":0.0},{"id":"https://openalex.org/C524204448","wikidata":"https://www.wikidata.org/wiki/Q788926","display_name":"Infectious disease (medical specialty)","level":3,"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/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display_name":"Finance","level":1,"score":0.0},{"id":"https://openalex.org/C2779134260","wikidata":"https://www.wikidata.org/wiki/Q12136","display_name":"Disease","level":2,"score":0.0}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.3390/axioms11040170","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms11040170","pdf_url":"https://www.mdpi.com/2075-1680/11/4/170/pdf?version=1649643399","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:arXiv.org:2204.05055","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2204.05055","pdf_url":"https://arxiv.org/pdf/2204.05055","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"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":null,"raw_type":"text"},{"id":"pmh:oai:doaj.org/article:4f15d42bcd4d427c9a31370bfc026650","is_oa":true,"landing_page_url":"https://doaj.org/article/4f15d42bcd4d427c9a31370bfc026650","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":"Axioms, Vol 11, Iss 4, p 170 (2022)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2075-1680/11/4/170/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/axioms11040170","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 11; Issue 4; Pages: 170","raw_type":"Text"},{"id":"pmh:oai:ria.ua.pt:10773/34104","is_oa":true,"landing_page_url":"http://hdl.handle.net/10773/34104","pdf_url":null,"source":{"id":"https://openalex.org/S4306402433","display_name":"Portuguese National Funding Agency for Science, Research and Technology (RCAAP Project by FCT)","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","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"journal article"}],"best_oa_location":{"id":"doi:10.3390/axioms11040170","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms11040170","pdf_url":"https://www.mdpi.com/2075-1680/11/4/170/pdf?version=1649643399","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":[{"score":0.8299999833106995,"display_name":"Good health and well-being","id":"https://metadata.un.org/sdg/3"}],"awards":[{"id":"https://openalex.org/G2144570049","display_name":null,"funder_award_id":"UIDB/50008","funder_id":"https://openalex.org/F4320334779","funder_display_name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia"},{"id":"https://openalex.org/G2590986966","display_name":"Center for Research and Development in Mathematics and Applications","funder_award_id":"UIDB/04106/2020","funder_id":"https://openalex.org/F4320334779","funder_display_name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia"},{"id":"https://openalex.org/G3069120617","display_name":null,"funder_award_id":"04106","funder_id":"https://openalex.org/F4320334779","funder_display_name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia"},{"id":"https://openalex.org/G3375535518","display_name":"Instituto de Telecomunica\u00e7\u00f5es","funder_award_id":"UIDB/50008/2020","funder_id":"https://openalex.org/F4320334779","funder_display_name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia"},{"id":"https://openalex.org/G3850789590","display_name":null,"funder_award_id":"50008","funder_id":"https://openalex.org/F4320334779","funder_display_name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia"}],"funders":[{"id":"https://openalex.org/F4320334779","display_name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","ror":"https://ror.org/00snfqn58"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4223534219.pdf","grobid_xml":"https://content.openalex.org/works/W4223534219.grobid-xml"},"referenced_works_count":30,"referenced_works":["https://openalex.org/W861454831","https://openalex.org/W1993680495","https://openalex.org/W2015239928","https://openalex.org/W2050850927","https://openalex.org/W2102798763","https://openalex.org/W2109705085","https://openalex.org/W2112680994","https://openalex.org/W2785938440","https://openalex.org/W2801852471","https://openalex.org/W2896702456","https://openalex.org/W2922334671","https://openalex.org/W2923435848","https://openalex.org/W3019674250","https://openalex.org/W3024180951","https://openalex.org/W3088410715","https://openalex.org/W3094988683","https://openalex.org/W3102030039","https://openalex.org/W3102675908","https://openalex.org/W3102842140","https://openalex.org/W3103268002","https://openalex.org/W3107836368","https://openalex.org/W3120648879","https://openalex.org/W3126196578","https://openalex.org/W3131958821","https://openalex.org/W3132477113","https://openalex.org/W3184305081","https://openalex.org/W3196188801","https://openalex.org/W3203220925","https://openalex.org/W3207850185","https://openalex.org/W6748396876"],"related_works":["https://openalex.org/W4200329650","https://openalex.org/W3127156785","https://openalex.org/W4205754011","https://openalex.org/W3009669391","https://openalex.org/W4205215807","https://openalex.org/W3005417802","https://openalex.org/W4226296940","https://openalex.org/W3028835529","https://openalex.org/W3036314732","https://openalex.org/W3134376730"],"abstract_inverted_index":{"A":[0,71,89],"fractional-order":[1],"compartmental":[2],"model":[3,32],"was":[4],"recently":[5],"used":[6],"to":[7,35,42,56],"describe":[8],"real":[9],"data":[10],"of":[11,15,47],"the":[12,16,44,96],"first":[13],"wave":[14,46],"COVID-19":[17,48],"pandemic":[18],"in":[19,33,51,67],"Portugal":[20,52],"[Chaos":[21],"Solitons":[22],"Fractals":[23],"144":[24],"(2021),":[25],"Art.":[26],"110652].":[27],"Here,":[28],"we":[29],"modify":[30],"that":[31,49,60],"order":[34],"correct":[36],"time":[37],"dimensions":[38],"and":[39,59,69,80,84,95],"use":[40],"it":[41],"investigate":[43],"third":[45],"occurred":[50],"from":[53],"December":[54],"2020":[55],"February":[57],"2021,":[58],"has":[61],"surpassed":[62],"all":[63],"previous":[64],"waves,":[65],"both":[66],"number":[68],"consequences.":[70],"new":[72],"fractional":[73],"optimal":[74],"control":[75],"problem":[76],"is":[77,92],"then":[78],"formulated":[79],"solved,":[81],"with":[82],"vaccination":[83],"preventive":[85],"measures":[86],"as":[87],"controls.":[88],"cost-effectiveness":[90],"analysis":[91],"carried":[93],"out,":[94],"obtained":[97],"results":[98],"are":[99],"discussed.":[100]},"counts_by_year":[{"year":2025,"cited_by_count":3},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":7},{"year":2022,"cited_by_count":3}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}