{"id":"https://openalex.org/W4283759218","doi":"https://doi.org/10.1016/j.rinp.2022.105777","title":"Dynamic of a two-strain COVID-19 model with vaccination","display_name":"Dynamic of a two-strain COVID-19 model with vaccination","publication_year":2022,"publication_date":"2022-06-29","ids":{"openalex":"https://openalex.org/W4283759218","doi":"https://doi.org/10.1016/j.rinp.2022.105777","pmid":"https://pubmed.ncbi.nlm.nih.gov/35791392","pmcid":"https://www.ncbi.nlm.nih.gov/pmc/articles/9242689"},"language":"en","primary_location":{"id":"doi:10.1016/j.rinp.2022.105777","doi":"https://doi.org/10.1016/j.rinp.2022.105777","is_oa":true,"landing_page_url":"https://doi.org/10.1016/j.rinp.2022.105777","pdf_url":"https://www.sciencedirect.com/science/article/pii/S2211379722004430/pdf","source":{"id":"https://openalex.org/S2764941219","display_name":"Results in Physics","issn_l":"2211-3797","issn":["2211-3797"],"is_oa":true,"is_in_doaj":true,"is_core":true,"listed_in":["cwts-core","doaj","jufo-1","norway-1","oarl","poland-70"],"host_organization":"https://openalex.org/P4310320990","host_organization_name":"Elsevier BV","host_organization_lineage":["https://openalex.org/P4310320990"],"host_organization_lineage_names":["Elsevier BV"],"type":"journal"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Results in Physics","raw_type":"journal-article"},"volume":null,"type":"article","is_traditional":true,"indexed_in":["crossref","doaj","pubmed"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.sciencedirect.com/science/article/pii/S2211379722004430/pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5013931862","display_name":"S.Y. Tchoumi","orcid":"https://orcid.org/0000-0002-6704-2745","observed_orcids":["https://orcid.org/0000-0002-6704-2745"]},"institutions":[{"id":"https://openalex.org/I59196294","display_name":"University of Ngaound\u00e9r\u00e9","ror":"https://ror.org/03gq1d339","country_code":"CM","type":"education","lineage":["https://openalex.org/I59196294"]}],"countries":["CM"],"is_corresponding":true,"raw_author_name":"S.Y. Tchoumi","raw_affiliation_strings":["Department of Mathematics and Computer Sciences ENSAI, University of Ngaoundere, P.O. Box 455 Ngaoundere, Cameroon"],"raw_orcid":"https://orcid.org/0000-0002-6704-2745","affiliations":[{"raw_affiliation_string":"Department of Mathematics and Computer Sciences ENSAI, University of Ngaoundere, P.O. Box 455 Ngaoundere, Cameroon","institution_ids":["https://openalex.org/I59196294"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5007070447","display_name":"Herieth Rwezaura","orcid":"https://orcid.org/0000-0003-1950-3284","observed_orcids":["https://orcid.org/0000-0003-1950-3284"]},"institutions":[{"id":"https://openalex.org/I182078086","display_name":"University of Dar es Salaam","ror":"https://ror.org/0479aed98","country_code":"TZ","type":"education","lineage":["https://openalex.org/I182078086"]}],"countries":["TZ"],"is_corresponding":false,"raw_author_name":"H. Rwezaura","raw_affiliation_strings":["Mathematics Department, University of Dar es Salaam, P.O. Box 35062, Dar es Salaam, Tanzania"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Mathematics Department, University of Dar es Salaam, P.O. Box 35062, Dar es Salaam, Tanzania","institution_ids":["https://openalex.org/I182078086"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5088951208","display_name":"Jean M. Tchuenche","orcid":"https://orcid.org/0000-0001-8655-6154","observed_orcids":["https://orcid.org/0000-0001-8655-6154"]},"institutions":[{"id":"https://openalex.org/I192619145","display_name":"University of the Witwatersrand","ror":"https://ror.org/03rp50x72","country_code":"ZA","type":"education","lineage":["https://openalex.org/I192619145"]},{"id":"https://openalex.org/I97231232","display_name":"Nelson Mandela African Institution of Science and Technology","ror":"https://ror.org/041vsn055","country_code":"TZ","type":"education","lineage":["https://openalex.org/I97231232"]}],"countries":["TZ","ZA"],"is_corresponding":false,"raw_author_name":"J.M. Tchuenche","raw_affiliation_strings":["School of Computational and Applied Mathematics, University of the Witwatersrand, Johannesburg, Private Bag 3, Wits 2050, South Africa","School of Computational and Communication Sciences and Engineering, Nelson Mandela African Institute of Science and Technology, Arusha, Tanzania"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Computational and Applied Mathematics, University of the Witwatersrand, Johannesburg, Private Bag 3, Wits 2050, South Africa","institution_ids":["https://openalex.org/I192619145"]},{"raw_affiliation_string":"School of Computational and Communication Sciences and Engineering, Nelson Mandela African Institute of Science and Technology, Arusha, Tanzania","institution_ids":["https://openalex.org/I97231232"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5013931862"],"corresponding_institution_ids":["https://openalex.org/I59196294"],"apc_list":{"value":1550,"currency":"USD","value_usd":1550},"apc_paid":{"value":1550,"currency":"USD","value_usd":1550},"fwci":7.8156,"has_fulltext":true,"cited_by_count":47,"citation_normalized_percentile":{"value":0.98796184,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":87,"max":99},"biblio":{"volume":"39","issue":null,"first_page":"105777","last_page":"105777"},"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":0.8598999977111816,"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":0.8598999977111816,"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":0.13950000703334808,"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/T11560","display_name":"Animal Disease Management and Epidemiology","score":0.0,"subfield":{"id":"https://openalex.org/subfields/1103","display_name":"Animal Science and Zoology"},"field":{"id":"https://openalex.org/fields/11","display_name":"Agricultural and Biological Sciences"},"domain":{"id":"https://openalex.org/domains/1","display_name":"Life Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/covid-19","display_name":"COVID-19 pandemic","score":0.9900000095367432},{"id":"https://openalex.org/keywords/two-strain-epidemic-model","display_name":"two-strain epidemic model","score":0.9470000267028809},{"id":"https://openalex.org/keywords/transmission-dynamics","display_name":"transmission dynamics","score":0.6269999742507935},{"id":"https://openalex.org/keywords/basic-reproduction-number","display_name":"basic reproduction number","score":0.8740000128746033},{"id":"https://openalex.org/keywords/global-stability","display_name":"global stability","score":0.8920000195503235},{"id":"https://openalex.org/keywords/disease-free-equilibrium","display_name":"disease-free equilibrium","score":0.7540000081062317},{"id":"https://openalex.org/keywords/endemic-equilibrium","display_name":"endemic equilibrium","score":0.9639999866485596},{"id":"https://openalex.org/keywords/backward-bifurcation","display_name":"backward bifurcation","score":0.8730000257492065},{"id":"https://openalex.org/keywords/center-manifold-theory","display_name":"center manifold theory","score":0.8939999938011169},{"id":"https://openalex.org/keywords/sars-cov-2-variants","display_name":"SARS-CoV-2 variants","score":0.7789999842643738},{"id":"https://openalex.org/keywords/strain-dominance","display_name":"strain dominance","score":0.35899999737739563}],"concepts":[{"id":"https://openalex.org/C3008058167","wikidata":"https://www.wikidata.org/wiki/Q84263196","display_name":"Coronavirus disease 2019 (COVID-19)","level":4,"score":0.7700433731079102},{"id":"https://openalex.org/C22070199","wikidata":"https://www.wikidata.org/wiki/Q192995","display_name":"Vaccination","level":2,"score":0.6562795042991638},{"id":"https://openalex.org/C2778022156","wikidata":"https://www.wikidata.org/wiki/Q576145","display_name":"Strain (injury)","level":2,"score":0.6502573490142822},{"id":"https://openalex.org/C159047783","wikidata":"https://www.wikidata.org/wiki/Q7215","display_name":"Virology","level":1,"score":0.5965203642845154},{"id":"https://openalex.org/C3006700255","wikidata":"https://www.wikidata.org/wiki/Q81068910","display_name":"2019-20 coronavirus outbreak","level":3,"score":0.5387552380561829},{"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":0.5109649896621704},{"id":"https://openalex.org/C1627819","wikidata":"https://www.wikidata.org/wiki/Q2572354","display_name":"Epidemic model","level":3,"score":0.43007558584213257},{"id":"https://openalex.org/C71924100","wikidata":"https://www.wikidata.org/wiki/Q11190","display_name":"Medicine","level":0,"score":0.387907475233078},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.3574772775173187},{"id":"https://openalex.org/C116675565","wikidata":"https://www.wikidata.org/wiki/Q3241045","display_name":"Outbreak","level":2,"score":0.2733824551105499},{"id":"https://openalex.org/C126322002","wikidata":"https://www.wikidata.org/wiki/Q11180","display_name":"Internal medicine","level":1,"score":0.09414157271385193},{"id":"https://openalex.org/C99454951","wikidata":"https://www.wikidata.org/wiki/Q932068","display_name":"Environmental health","level":1,"score":0.08178821206092834},{"id":"https://openalex.org/C524204448","wikidata":"https://www.wikidata.org/wiki/Q788926","display_name":"Infectious disease (medical specialty)","level":3,"score":0.0699705183506012},{"id":"https://openalex.org/C2908647359","wikidata":"https://www.wikidata.org/wiki/Q2625603","display_name":"Population","level":2,"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":6,"locations":[{"id":"doi:10.1016/j.rinp.2022.105777","doi":"https://doi.org/10.1016/j.rinp.2022.105777","is_oa":true,"landing_page_url":"https://doi.org/10.1016/j.rinp.2022.105777","pdf_url":"https://www.sciencedirect.com/science/article/pii/S2211379722004430/pdf","source":{"id":"https://openalex.org/S2764941219","display_name":"Results in Physics","issn_l":"2211-3797","issn":["2211-3797"],"is_oa":true,"is_in_doaj":true,"is_core":true,"listed_in":["cwts-core","doaj","jufo-1","norway-1","oarl","poland-70"],"host_organization":"https://openalex.org/P4310320990","host_organization_name":"Elsevier BV","host_organization_lineage":["https://openalex.org/P4310320990"],"host_organization_lineage_names":["Elsevier BV"],"type":"journal"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Results in Physics","raw_type":"journal-article"},{"id":"doi:10.22541/au.163500462.24575457/v1","doi":"https://doi.org/10.22541/au.163500462.24575457/v1","is_oa":true,"landing_page_url":"https://doi.org/10.22541/au.163500462.24575457/v1","pdf_url":"https://www.authorea.com/doi/pdf/10.22541/au.163500462.24575457","source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"posted-content"},{"id":"doi:10.21203/rs.3.rs-553546/v1","doi":"https://doi.org/10.21203/rs.3.rs-553546/v1","is_oa":true,"landing_page_url":"https://doi.org/10.21203/rs.3.rs-553546/v1","pdf_url":"https://www.researchsquare.com/article/rs-553546/latest.pdf","source":{"id":"https://openalex.org/S4306525896","display_name":"Research Square","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","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Research Square","raw_type":"posted-content"},{"id":"pmid:35791392","doi":null,"is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/35791392","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":"Results in physics","raw_type":"Journal Article"},{"id":"pmh:oai:pubmedcentral.nih.gov:9242689","doi":null,"is_oa":true,"landing_page_url":"https://pmc.ncbi.nlm.nih.gov/articles/PMC9242689/","pdf_url":null,"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":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Results Phys","raw_type":"Text"},{"id":"pmh:oai:doaj.org/article:96179c2fba5f41f1a0c0269b764ef2f3","doi":null,"is_oa":true,"landing_page_url":"https://doaj.org/article/96179c2fba5f41f1a0c0269b764ef2f3","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":"Results in Physics, Vol 39, Iss , Pp 105777- (2022)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.1016/j.rinp.2022.105777","doi":"https://doi.org/10.1016/j.rinp.2022.105777","is_oa":true,"landing_page_url":"https://doi.org/10.1016/j.rinp.2022.105777","pdf_url":"https://www.sciencedirect.com/science/article/pii/S2211379722004430/pdf","source":{"id":"https://openalex.org/S2764941219","display_name":"Results in Physics","issn_l":"2211-3797","issn":["2211-3797"],"is_oa":true,"is_in_doaj":true,"is_core":true,"listed_in":["cwts-core","doaj","jufo-1","norway-1","oarl","poland-70"],"host_organization":"https://openalex.org/P4310320990","host_organization_name":"Elsevier BV","host_organization_lineage":["https://openalex.org/P4310320990"],"host_organization_lineage_names":["Elsevier BV"],"type":"journal"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Results in Physics","raw_type":"journal-article"},"sustainable_development_goals":[{"display_name":"Good health and well-being","id":"https://metadata.un.org/sdg/3","score":0.6240000128746033}],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4283759218.pdf","grobid_xml":"https://content.openalex.org/works/W4283759218.grobid-xml"},"referenced_works_count":69,"referenced_works":["https://openalex.org/W76494959","https://openalex.org/W92765303","https://openalex.org/W1974414567","https://openalex.org/W2031877307","https://openalex.org/W2039334331","https://openalex.org/W2042743172","https://openalex.org/W2057584364","https://openalex.org/W2059350891","https://openalex.org/W2075336720","https://openalex.org/W2084640233","https://openalex.org/W2085451305","https://openalex.org/W2094136517","https://openalex.org/W2105068802","https://openalex.org/W2107555870","https://openalex.org/W2112680994","https://openalex.org/W2115414622","https://openalex.org/W2121941395","https://openalex.org/W2133060079","https://openalex.org/W2148301044","https://openalex.org/W2195009776","https://openalex.org/W2326653536","https://openalex.org/W2331046599","https://openalex.org/W2921775843","https://openalex.org/W2996019972","https://openalex.org/W3003668884","https://openalex.org/W3008028633","https://openalex.org/W3009351032","https://openalex.org/W3009509357","https://openalex.org/W3010096538","https://openalex.org/W3016443614","https://openalex.org/W3017787557","https://openalex.org/W3035011439","https://openalex.org/W3044914835","https://openalex.org/W3044916830","https://openalex.org/W3045835659","https://openalex.org/W3046315242","https://openalex.org/W3084273843","https://openalex.org/W3087402740","https://openalex.org/W3092054641","https://openalex.org/W3092604180","https://openalex.org/W3103548139","https://openalex.org/W3109175407","https://openalex.org/W3114795912","https://openalex.org/W3116712204","https://openalex.org/W3121336634","https://openalex.org/W3128463352","https://openalex.org/W3133833230","https://openalex.org/W3140873963","https://openalex.org/W3141986261","https://openalex.org/W3145677695","https://openalex.org/W3145739395","https://openalex.org/W3150171980","https://openalex.org/W3151466979","https://openalex.org/W3157323802","https://openalex.org/W3161861537","https://openalex.org/W3172847984","https://openalex.org/W3179296235","https://openalex.org/W3213309010","https://openalex.org/W3217247647","https://openalex.org/W4225282881","https://openalex.org/W4226196963","https://openalex.org/W4239975174","https://openalex.org/W4377941071","https://openalex.org/W6774316164","https://openalex.org/W6777149124","https://openalex.org/W6782411875","https://openalex.org/W6786489126","https://openalex.org/W6791389393","https://openalex.org/W6979026940"],"related_works":["https://openalex.org/W4383215515","https://openalex.org/W4403243897","https://openalex.org/W4407341234","https://openalex.org/W4412167069","https://openalex.org/W4293156014","https://openalex.org/W6957796549","https://openalex.org/W4377097425","https://openalex.org/W4390462261","https://openalex.org/W4310616840","https://openalex.org/W4392990345"],"abstract_inverted_index":{"COVID-19":[0,48],"is":[1,57,96,195],"a":[2,42,46],"respiratory":[3],"illness":[4],"caused":[5],"by":[6],"an":[7],"ribonucleic":[8],"acid":[9],"(RNA)":[10],"virus":[11],"prone":[12],"to":[13,73,122,180,197],"mutations.":[14],"In":[15,71],"December":[16],"2020,":[17],"variants":[18],"with":[19,51],"different":[20],"characteristics":[21],"that":[22,98,145],"could":[23,158],"affect":[24],"transmissibility":[25],"emerged":[26],"around":[27],"the":[28,36,64,74,79,91,99,104,113,124,132,162,187,208],"world.":[29],"To":[30],"address":[31],"this":[32],"new":[33],"dynamic":[34],"of":[35,45,66,106,112,174],"disease,":[37],"we":[38],"formulate":[39],"and":[40,60,76,85,109,137,153,167,185],"analyze":[41],"mathematical":[43],"model":[44,56,80,100,114,125],"two-strain":[47],"transmission":[49,191],"dynamics":[50],"strain":[52,86,164,181,201],"1":[53,84,182],"vaccination.":[54],"The":[55],"theoretically":[58],"analyzed":[59],"sufficient":[61],"conditions":[62],"for":[63,139],"stability":[65,111],"its":[67],"equilibria":[68,115],"are":[69,116,128],"derived.":[70],"addition":[72],"disease-free":[75],"endemic":[77,88],"equilibria,":[78],"also":[81],"has":[82],"single-strain":[83],"2":[87,157,190,202],"equilibria.":[89],"Using":[90],"center":[92],"manifold":[93],"theory,":[94],"it":[95],"shown":[97],"does":[101],"not":[102,204],"exhibit":[103],"phenomenon":[105],"backward":[107],"bifurcation,":[108],"global":[110],"proved":[117],"using":[118],"various":[119],"approaches.":[120],"Simulations":[121],"support":[123],"theoretical":[126],"results":[127],"provided.":[129],"We":[130],"calculate":[131],"basic":[133],"reproductive":[134],"number":[135],"R1":[136],"R2":[138,194],"both":[140,147],"strains":[141,148],"independently.":[142],"Results":[143],"indicate":[144],"\u2013":[146,155],"will":[149,203],"persist":[150],"when":[151,170],"R1>1":[152],"R2>1":[154],"Stain":[156],"establish":[159,205],"itself":[160],"as":[161],"dominant":[163],"if":[165],"R1<1":[166],"R2>1,":[168],"or":[169],"R2>R1>1.":[171],"However,":[172],"because":[173],"de":[175],"novo":[176],"herd":[177],"immunity":[178],"due":[179],"vaccine":[183],"efficacy":[184],"provided":[186],"initial":[188],"stain":[189],"threshold":[192],"parameter":[193],"controlled":[196],"remain":[198],"below":[199],"unity,":[200],"itself/persist":[206],"in":[207],"community.":[209]},"counts_by_year":[{"year":2026,"cited_by_count":8},{"year":2025,"cited_by_count":9},{"year":2024,"cited_by_count":11},{"year":2023,"cited_by_count":15},{"year":2022,"cited_by_count":3},{"year":2021,"cited_by_count":1}],"updated_date":"2026-11-11T12:16:26.678958","created_date":"2022-07-02T00:00:00"}
