{"id":"https://openalex.org/W4394572167","doi":"https://doi.org/10.1038/s41598-024-58192-7","title":"Fractional epidemic model of coronavirus disease with vaccination and crowding effects","display_name":"Fractional epidemic model of coronavirus disease with vaccination and crowding effects","publication_year":2024,"publication_date":"2024-04-08","ids":{"openalex":"https://openalex.org/W4394572167","doi":"https://doi.org/10.1038/s41598-024-58192-7","pmid":"https://pubmed.ncbi.nlm.nih.gov/38589475","pmcid":"https://www.ncbi.nlm.nih.gov/pmc/articles/11369089"},"language":"en","primary_location":{"id":"doi:10.1038/s41598-024-58192-7","doi":"https://doi.org/10.1038/s41598-024-58192-7","is_oa":true,"landing_page_url":"https://doi.org/10.1038/s41598-024-58192-7","pdf_url":"https://www.nature.com/articles/s41598-024-58192-7.pdf","source":{"id":"https://openalex.org/S196734849","display_name":"Scientific Reports","issn_l":"2045-2322","issn":["2045-2322"],"is_oa":true,"is_in_doaj":true,"is_core":true,"listed_in":["anvur-class-a","anvur-scientific","cwts-core","doaj","dongbi-c","doyens","jufo-1","ki-jl-1","medline","norway-1","poland-140","russia-white-list-1","vabb-shw"],"host_organization":"https://openalex.org/P4310319908","host_organization_name":"Nature Portfolio","host_organization_lineage":["https://openalex.org/P4310319908","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Nature Portfolio","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Scientific Reports","raw_type":"journal-article"},"volume":null,"type":"article","is_traditional":true,"indexed_in":["crossref","datacite","doaj","pubmed"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.nature.com/articles/s41598-024-58192-7.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5064110688","display_name":"Suhail Saleem","orcid":"https://orcid.org/0000-0002-6391-9124","observed_orcids":["https://orcid.org/0000-0002-6391-9124"]},"institutions":[{"id":"https://openalex.org/I899713450","display_name":"Air University","ror":"https://ror.org/03yfe9v83","country_code":"PK","type":"education","lineage":["https://openalex.org/I899713450"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Suhail Saleem","raw_affiliation_strings":["Department of Mathematics, Air University, PAF Complex E-9, Islamabad, 44000, Pakistan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Air University, PAF Complex E-9, Islamabad, 44000, Pakistan","institution_ids":["https://openalex.org/I899713450"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5005732217","display_name":"Muhammad Rafiq","orcid":"https://orcid.org/0000-0002-2165-3479","observed_orcids":["https://orcid.org/0000-0002-2165-3479"]},"institutions":[{"id":"https://openalex.org/I192392021","display_name":"University of Central Punjab","ror":"https://ror.org/04g0mqe67","country_code":"PK","type":"education","lineage":["https://openalex.org/I192392021"]},{"id":"https://openalex.org/I56306041","display_name":"Lebanese American University","ror":"https://ror.org/00hqkan37","country_code":"LB","type":"education","lineage":["https://openalex.org/I56306041"]}],"countries":["LB","PK"],"is_corresponding":false,"raw_author_name":"Muhammad Rafiq","raw_affiliation_strings":["Department of Computer Science and Mathematics, Lebanese American University, Beirut, 1102-2801, Lebanon","Department of Mathematics, Faculty of Science and Technology, University of Central Punjab, Lahore, Pakistan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Computer Science and Mathematics, Lebanese American University, Beirut, 1102-2801, Lebanon","institution_ids":["https://openalex.org/I56306041"]},{"raw_affiliation_string":"Department of Mathematics, Faculty of Science and Technology, University of Central Punjab, Lahore, Pakistan","institution_ids":["https://openalex.org/I192392021"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5032731280","display_name":"Nauman Ahmed","orcid":"https://orcid.org/0000-0003-1742-585X","observed_orcids":["https://orcid.org/0000-0003-1742-585X"]},"institutions":[{"id":"https://openalex.org/I142204992","display_name":"University of Lahore","ror":"https://ror.org/051jrjw38","country_code":"PK","type":"education","lineage":["https://openalex.org/I142204992"]},{"id":"https://openalex.org/I56306041","display_name":"Lebanese American University","ror":"https://ror.org/00hqkan37","country_code":"LB","type":"education","lineage":["https://openalex.org/I56306041"]}],"countries":["LB","PK"],"is_corresponding":false,"raw_author_name":"Nauman Ahmed","raw_affiliation_strings":["Department of Computer Science and Mathematics, Lebanese American University, Beirut, 1102-2801, Lebanon","Department of Mathematics and Statistics, The University of Lahore, Lahore, Pakistan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Computer Science and Mathematics, Lebanese American University, Beirut, 1102-2801, Lebanon","institution_ids":["https://openalex.org/I56306041"]},{"raw_affiliation_string":"Department of Mathematics and Statistics, The University of Lahore, Lahore, Pakistan","institution_ids":["https://openalex.org/I142204992"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5083618602","display_name":"Muhammad Shoaib Arif","orcid":"https://orcid.org/0000-0002-6009-5609","observed_orcids":["https://orcid.org/0000-0002-6009-5609"]},"institutions":[{"id":"https://openalex.org/I899713450","display_name":"Air University","ror":"https://ror.org/03yfe9v83","country_code":"PK","type":"education","lineage":["https://openalex.org/I899713450"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Muhammad Shoaib Arif","raw_affiliation_strings":["Department of Mathematics, Air University, PAF Complex E-9, Islamabad, 44000, Pakistan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Air University, PAF Complex E-9, Islamabad, 44000, Pakistan","institution_ids":["https://openalex.org/I899713450"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5059997764","display_name":"Ali Raza","orcid":"https://orcid.org/0000-0002-6443-9966","observed_orcids":["https://orcid.org/0000-0002-6443-9966"]},"institutions":[{"id":"https://openalex.org/I4396570470","display_name":"The University of Chenab, Gujrat","ror":"https://ror.org/01b009v28","country_code":"PK","type":"education","lineage":["https://openalex.org/I4396570470"]},{"id":"https://openalex.org/I69050122","display_name":"Near East University","ror":"https://ror.org/02x8svs93","country_code":"CY","type":"education","lineage":["https://openalex.org/I69050122"]}],"countries":["CY","PK"],"is_corresponding":false,"raw_author_name":"Ali Raza","raw_affiliation_strings":["Department of Mathematics, Mathematics Research Center, Near East University, Near East Boulevard, 99138, Nicosia/Mersin 10, Turkey","Department of Mathematics, University of Chanab, Gujrat, Pakistan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Mathematics Research Center, Near East University, Near East Boulevard, 99138, Nicosia/Mersin 10, Turkey","institution_ids":["https://openalex.org/I69050122"]},{"raw_affiliation_string":"Department of Mathematics, University of Chanab, Gujrat, Pakistan","institution_ids":["https://openalex.org/I4396570470"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5030597534","display_name":"Zafar Iqbal","orcid":"https://orcid.org/0000-0002-9206-7792","observed_orcids":["https://orcid.org/0000-0002-9206-7792","https://orcid.org/0009-0003-2004-2697","https://orcid.org/0009-0004-2958-3294"]},"institutions":[{"id":"https://openalex.org/I142204992","display_name":"University of Lahore","ror":"https://ror.org/051jrjw38","country_code":"PK","type":"education","lineage":["https://openalex.org/I142204992"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Zafar Iqbal","raw_affiliation_strings":["Department of Mathematics and Statistics, The University of Lahore, Lahore, Pakistan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics, The University of Lahore, Lahore, Pakistan","institution_ids":["https://openalex.org/I142204992"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5093048435","display_name":"Shafiullah Niazai","orcid":"https://orcid.org/0009-0004-5365-2229","observed_orcids":["https://orcid.org/0009-0004-5365-2229"]},"institutions":[{"id":"https://openalex.org/I4405254166","display_name":"Laghman University","ror":"https://ror.org/01gbjs041","country_code":"AF","type":"education","lineage":["https://openalex.org/I4405254166"]}],"countries":["AF"],"is_corresponding":true,"raw_author_name":"Shafiullah Niazai","raw_affiliation_strings":["Department of Mathematics, Education Faculty, Laghman University, Mehtarlam City, 2701, Laghman, Afghanistan. shafiullahniazai@lu.edu.af","Department of Mathematics, Education Faculty, Laghman University, Mehtarlam City, 2701, Laghman, Afghanistan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Education Faculty, Laghman University, Mehtarlam City, 2701, Laghman, Afghanistan. shafiullahniazai@lu.edu.af","institution_ids":["https://openalex.org/I4405254166"]},{"raw_affiliation_string":"Department of Mathematics, Education Faculty, Laghman University, Mehtarlam City, 2701, Laghman, Afghanistan","institution_ids":["https://openalex.org/I4405254166"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5011875207","display_name":"Ilyas Khan","orcid":"https://orcid.org/0000-0002-2056-9371","observed_orcids":["https://orcid.org/0000-0002-2056-9371","https://orcid.org/0009-0007-3847-3849"]},"institutions":[{"id":"https://openalex.org/I195631090","display_name":"Majmaah University","ror":"https://ror.org/01mcrnj60","country_code":"SA","type":"education","lineage":["https://openalex.org/I195631090"]},{"id":"https://openalex.org/I85461943","display_name":"Saveetha University","ror":"https://ror.org/0034me914","country_code":"IN","type":"education","lineage":["https://openalex.org/I85461943"]}],"countries":["IN","SA"],"is_corresponding":true,"raw_author_name":"Ilyas Khan","raw_affiliation_strings":["Department of Mathematics, College of Science Al-Zulfi Majmaah University, 11952, Al-Majmaah, Saudi Arabia. i.said@mu.edu.sa","Department of Mathematics, Saveetha School of Engineering, SIMATS, Chennai, Tamil Nadu, India. i.said@mu.edu.sa","Department of Mathematics, College of Science Al-Zulfi Majmaah University, 11952, Al-Majmaah, Saudi Arabia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, College of Science Al-Zulfi Majmaah University, 11952, Al-Majmaah, Saudi Arabia. i.said@mu.edu.sa","institution_ids":["https://openalex.org/I195631090"]},{"raw_affiliation_string":"Department of Mathematics, Saveetha School of Engineering, SIMATS, Chennai, Tamil Nadu, India. i.said@mu.edu.sa","institution_ids":["https://openalex.org/I85461943"]},{"raw_affiliation_string":"Department of Mathematics, College of Science Al-Zulfi Majmaah University, 11952, Al-Majmaah, Saudi Arabia","institution_ids":["https://openalex.org/I195631090"]}]}],"institutions":[],"countries_distinct_count":6,"institutions_distinct_count":9,"corresponding_author_ids":["https://openalex.org/A5011875207","https://openalex.org/A5093048435"],"corresponding_institution_ids":["https://openalex.org/I195631090","https://openalex.org/I4405254166","https://openalex.org/I85461943"],"apc_list":{"value":2590,"currency":"USD","value_usd":2590},"apc_paid":{"value":2590,"currency":"USD","value_usd":2590},"fwci":5.9142,"has_fulltext":true,"cited_by_count":19,"citation_normalized_percentile":{"value":0.97559621,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":96,"max":99},"biblio":{"volume":"14","issue":"1","first_page":"8157","last_page":"8157"},"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":0.9810000061988831,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/T10482","display_name":"Mathematical and Theoretical Epidemiology and Ecology Models","score":0.9810000061988831,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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":0.016499999910593033,"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/T10541","display_name":"Nonlinear Differential Equations Analysis","score":0.000699999975040555,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/covid-19","display_name":"COVID-19 pandemic","score":0.8999999761581421},{"id":"https://openalex.org/keywords/fractional-order-epidemic-model","display_name":"fractional-order epidemic model","score":0.777999997138977},{"id":"https://openalex.org/keywords/crowding-effects","display_name":"crowding effects","score":0.8360000252723694},{"id":"https://openalex.org/keywords/vaccination","display_name":"vaccination","score":0.8840000033378601},{"id":"https://openalex.org/keywords/nonlinear-incidence-rate","display_name":"nonlinear incidence rate","score":0.9860000014305115},{"id":"https://openalex.org/keywords/basic-reproduction-number","display_name":"basic reproduction number","score":0.8550000190734863},{"id":"https://openalex.org/keywords/disease-free-equilibrium","display_name":"disease-free equilibrium","score":0.8799999952316284},{"id":"https://openalex.org/keywords/endemic-equilibrium","display_name":"endemic equilibrium","score":0.9810000061988831},{"id":"https://openalex.org/keywords/global-stability","display_name":"global stability","score":0.6990000009536743},{"id":"https://openalex.org/keywords/nonstandard-finite-difference-scheme","display_name":"nonstandard finite difference scheme","score":0.8629999756813049},{"id":"https://openalex.org/keywords/mittag-leffler-function","display_name":"Mittag-Leffler function","score":0.8999999761581421},{"id":"https://openalex.org/keywords/lyapunov-function","display_name":"Lyapunov function","score":0.7379999756813049},{"id":"https://openalex.org/keywords/epidemic-control-measures","display_name":"epidemic control measures","score":0.5809999704360962}],"concepts":[{"id":"https://openalex.org/C41949839","wikidata":"https://www.wikidata.org/wiki/Q1756677","display_name":"Stability theory","level":3,"score":0.7293539643287659},{"id":"https://openalex.org/C1627819","wikidata":"https://www.wikidata.org/wiki/Q2572354","display_name":"Epidemic model","level":3,"score":0.6421445608139038},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5861080884933472},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5407095551490784},{"id":"https://openalex.org/C204264503","wikidata":"https://www.wikidata.org/wiki/Q901464","display_name":"Basic reproduction number","level":3,"score":0.511268138885498},{"id":"https://openalex.org/C94766913","wikidata":"https://www.wikidata.org/wiki/Q1530271","display_name":"Equilibrium point","level":3,"score":0.48984596133232117},{"id":"https://openalex.org/C60640748","wikidata":"https://www.wikidata.org/wiki/Q2337858","display_name":"Lyapunov function","level":3,"score":0.444301038980484},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.3331666886806488},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.22641262412071228},{"id":"https://openalex.org/C71924100","wikidata":"https://www.wikidata.org/wiki/Q11190","display_name":"Medicine","level":0,"score":0.15068399906158447},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.1439722180366516},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.1062394380569458},{"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/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","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}],"mesh":[{"descriptor_ui":"D000086382","descriptor_name":"COVID-19","qualifier_ui":"Q000453","qualifier_name":"epidemiology","is_major_topic":true},{"descriptor_ui":"D000086382","descriptor_name":"COVID-19","qualifier_ui":"Q000517","qualifier_name":"prevention & control","is_major_topic":true},{"descriptor_ui":"D006801","descriptor_name":"Humans","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D014611","descriptor_name":"Vaccination","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D034941","descriptor_name":"Upper Extremity","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D050936","descriptor_name":"Basic Reproduction Number","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D058872","descriptor_name":"Epidemics","qualifier_ui":"Q000517","qualifier_name":"prevention & control","is_major_topic":true}],"locations_count":6,"locations":[{"id":"doi:10.1038/s41598-024-58192-7","doi":"https://doi.org/10.1038/s41598-024-58192-7","is_oa":true,"landing_page_url":"https://doi.org/10.1038/s41598-024-58192-7","pdf_url":"https://www.nature.com/articles/s41598-024-58192-7.pdf","source":{"id":"https://openalex.org/S196734849","display_name":"Scientific Reports","issn_l":"2045-2322","issn":["2045-2322"],"is_oa":true,"is_in_doaj":true,"is_core":true,"listed_in":["anvur-class-a","anvur-scientific","cwts-core","doaj","dongbi-c","doyens","jufo-1","ki-jl-1","medline","norway-1","poland-140","russia-white-list-1","vabb-shw"],"host_organization":"https://openalex.org/P4310319908","host_organization_name":"Nature Portfolio","host_organization_lineage":["https://openalex.org/P4310319908","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Nature Portfolio","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Scientific Reports","raw_type":"journal-article"},{"id":"pmh:oai:pubmedcentral.nih.gov:11369089","doi":null,"is_oa":true,"landing_page_url":"https://www.ncbi.nlm.nih.gov/pmc/articles/11369089","pdf_url":"https://pmc.ncbi.nlm.nih.gov/articles/PMC11369089/pdf/41598_2024_Article_58192.pdf","source":{"id":"https://openalex.org/S2764455111","display_name":"PubMed Central","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Sci Rep","raw_type":"Text"},{"id":"pmid:38589475","doi":null,"is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/38589475","pdf_url":null,"source":{"id":"https://openalex.org/S4306525036","display_name":"PubMed","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Scientific reports","raw_type":"Journal Article"},{"id":"pmh:oai:doaj.org/article:d67b10a6829349c6a11af8224bf5ba14","doi":null,"is_oa":true,"landing_page_url":"https://doaj.org/article/d67b10a6829349c6a11af8224bf5ba14","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,"listed_in":[],"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":"Scientific Reports, Vol 14, Iss 1, Pp 1-23 (2024)","raw_type":"article"},{"id":"doi:10.60692/cf5g9-9bm58","doi":"https://doi.org/10.60692/cf5g9-9bm58","is_oa":true,"landing_page_url":"https://doi.org/10.60692/cf5g9-9bm58","pdf_url":null,"source":{"id":"https://openalex.org/S7407051682","display_name":"Greater South Information System","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article-journal"},{"id":"doi:10.60692/m9p73-n1126","doi":"https://doi.org/10.60692/m9p73-n1126","is_oa":true,"landing_page_url":"https://doi.org/10.60692/m9p73-n1126","pdf_url":null,"source":{"id":"https://openalex.org/S7407051682","display_name":"Greater South Information System","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article-journal"}],"best_oa_location":{"id":"doi:10.1038/s41598-024-58192-7","doi":"https://doi.org/10.1038/s41598-024-58192-7","is_oa":true,"landing_page_url":"https://doi.org/10.1038/s41598-024-58192-7","pdf_url":"https://www.nature.com/articles/s41598-024-58192-7.pdf","source":{"id":"https://openalex.org/S196734849","display_name":"Scientific Reports","issn_l":"2045-2322","issn":["2045-2322"],"is_oa":true,"is_in_doaj":true,"is_core":true,"listed_in":["anvur-class-a","anvur-scientific","cwts-core","doaj","dongbi-c","doyens","jufo-1","ki-jl-1","medline","norway-1","poland-140","russia-white-list-1","vabb-shw"],"host_organization":"https://openalex.org/P4310319908","host_organization_name":"Nature Portfolio","host_organization_lineage":["https://openalex.org/P4310319908","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Nature Portfolio","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Scientific Reports","raw_type":"journal-article"},"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/3","score":0.8048999905586243,"display_name":"Good health and well-being"}],"study_designs":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320324553","display_name":"Majmaah University","ror":"https://ror.org/01mcrnj60"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4394572167.pdf","grobid_xml":"https://content.openalex.org/works/W4394572167.grobid-xml"},"referenced_works_count":51,"referenced_works":["https://openalex.org/W2004007224","https://openalex.org/W2047140957","https://openalex.org/W2058860728","https://openalex.org/W2109958258","https://openalex.org/W2922726777","https://openalex.org/W2944712254","https://openalex.org/W2986593044","https://openalex.org/W3009041394","https://openalex.org/W3010839841","https://openalex.org/W3015292514","https://openalex.org/W3025509622","https://openalex.org/W3036284476","https://openalex.org/W3040435539","https://openalex.org/W3043564885","https://openalex.org/W3080171000","https://openalex.org/W3080185192","https://openalex.org/W3082071203","https://openalex.org/W3083620534","https://openalex.org/W3087861692","https://openalex.org/W3087996688","https://openalex.org/W3088268547","https://openalex.org/W3090027928","https://openalex.org/W3093186597","https://openalex.org/W3095818485","https://openalex.org/W3097385375","https://openalex.org/W3099401800","https://openalex.org/W3106246659","https://openalex.org/W3108214086","https://openalex.org/W3108559619","https://openalex.org/W3113302180","https://openalex.org/W3119800716","https://openalex.org/W3120252076","https://openalex.org/W3126996402","https://openalex.org/W3129313820","https://openalex.org/W3130853669","https://openalex.org/W3132820274","https://openalex.org/W3136178019","https://openalex.org/W3137665336","https://openalex.org/W3138000161","https://openalex.org/W3158895480","https://openalex.org/W3163283655","https://openalex.org/W3169606972","https://openalex.org/W3176955267","https://openalex.org/W3196505517","https://openalex.org/W3197937383","https://openalex.org/W4205655941","https://openalex.org/W4205875759","https://openalex.org/W4206520629","https://openalex.org/W4293385906","https://openalex.org/W4302290549","https://openalex.org/W4391317392"],"related_works":["https://openalex.org/W4400143008","https://openalex.org/W7167844506","https://openalex.org/W3145701861","https://openalex.org/W6920627599","https://openalex.org/W4385588887","https://openalex.org/W4312081807","https://openalex.org/W4384665937","https://openalex.org/W4281737107","https://openalex.org/W3213610225","https://openalex.org/W4375863411"],"abstract_inverted_index":{"Abstract":[0],"Most":[1],"of":[2,35,44,54,66,75,82,99,112,119,131,148,159,267,280,298,307,321,336],"the":[3,6,11,33,39,42,45,52,73,80,97,110,113,120,149,160,169,173,228,265,268,277,281,293,305,311,316,330,337,355],"countries":[4],"in":[5,17,32,51,354],"world":[7],"are":[8,141,273,340],"affected":[9],"by":[10,71,88,126,156,176,192,329,361],"coronavirus":[12,36,55,67],"epidemic":[13,63],"that":[14,200,287,304],"put":[15],"people":[16],"danger,":[18],"with":[19,96],"many":[20],"infected":[21,312],"cases":[22],"and":[23,77,108,117,135,146,183,213,241,295,314],"deaths.":[24],"The":[25,92,115,138,144,255,270,300,319,333,346,357],"crowding":[26,76],"factor":[27],"plays":[28],"a":[29,48,60,103,351],"significant":[30],"role":[31,50],"transmission":[34,81,339],"disease.":[37],"On":[38],"other":[40],"hand,":[41],"vaccines":[43],"covid-19":[46],"played":[47],"decisive":[49],"control":[53,308,344],"infection.":[56],"In":[57],"this":[58,83],"paper,":[59],"fractional":[61,121,322],"order":[62,122,151,323],"model":[64,123,152],"(SIVR)":[65],"disease":[68],"is":[69,85,102,124,153,165,190,198,203,216,231,244,261,285,327,359],"proposed":[70],"considering":[72],"effects":[74,101],"vaccination":[78],"because":[79],"infection":[84],"highly":[86],"influenced":[87],"these":[89,100],"two":[90],"factors.":[91],"nonlinear":[93],"incidence":[94],"rate":[95],"inclusion":[98],"better":[104],"approach":[105,291],"to":[106,263,275],"understand":[107],"analyse":[109],"dynamics":[111],"model.":[114,299],"positivity":[116,294],"boundedness":[118,296],"ensured":[125],"applying":[127],"some":[128,343,362],"standard":[129],"results":[130,158],"Mittag":[132],"Leffler":[133],"function":[134],"Laplace":[136],"transformation.":[137],"equilibrium":[139,202,215,230,243],"points":[140],"described":[142,328],"analytically.":[143],"existence":[145],"uniqueness":[147],"non-integer":[150],"also":[154],"confirmed":[155],"using":[157,177,193],"fixed-point":[161],"theory.":[162],"Stability":[163],"analysis":[164],"carried":[166],"out":[167],"for":[168],"system":[170],"at":[171],"both":[172],"steady":[174],"states":[175],"Jacobian":[178],"matrix":[179],"theory,":[180],"Routh\u2013Hurwitz":[181],"criterion":[182],"Volterra-type":[184],"Lyapunov":[185],"functions.":[186],"Basic":[187],"reproductive":[188],"number":[189],"calculated":[191],"next":[194],"generation":[195],"matrix.":[196],"It":[197,284],"verified":[199],"disease-free":[201,229],"locally":[204,217],"asymptotically":[205,218,233,246],"stable":[206,219,234,247],"if":[207,220,235,248],"$${R}_{0}<1$$":[208,236],"R":[209,222,237,250],"0":[210,223,238,251],"<":[211,239],"1":[212,225,240,253],"endemic":[214,242],"$${R}_{0}>1$$":[221,249],">":[224,252],".":[226,254],"Moreover,":[227],"globally":[232,245],"non-standard":[256,288],"finite":[257,289],"difference":[258,290],"(NSFD)":[259],"scheme":[260],"developed":[262],"approximate":[264],"solutions":[266],"system.":[269],"simulated":[271,301],"graphs":[272,302],"presented":[274],"show":[276,303],"key":[278],"features":[279],"NSFD":[282],"approach.":[283],"proved":[286],"preserves":[292],"properties":[297],"implementation":[306],"strategies":[309],"reduced":[310],"population":[313],"increase":[315],"recovered":[317],"population.":[318],"impact":[320],"parameter":[324],"$$\\alpha$$":[325],"\u03b1":[326],"graphical":[331],"templates.":[332],"future":[334],"trends":[335],"virus":[338],"predicted":[341],"under":[342],"measures.":[345],"current":[347],"work":[348],"will":[349],"be":[350],"value":[352],"addition":[353],"literature.":[356],"article":[358],"closed":[360],"useful":[363],"concluding":[364],"remarks.":[365]},"counts_by_year":[{"year":2026,"cited_by_count":8},{"year":2025,"cited_by_count":7},{"year":2024,"cited_by_count":4}],"updated_date":"2026-11-10T11:39:43.021295","created_date":"2024-04-09T00:00:00"}