{"id":"https://openalex.org/W7203770075","doi":"https://doi.org/10.52280/pujm.2026.58(03)04","title":"Impact of Vaccination on Influenza Disease Transmission in Humans and Birds: Sensitivity and Bifurcation Analysis within Fractional Framework","display_name":"Impact of Vaccination on Influenza Disease Transmission in Humans and Birds: Sensitivity and Bifurcation Analysis within Fractional Framework","publication_year":2026,"publication_date":"2026-08-17","ids":{"openalex":"https://openalex.org/W7203770075","doi":"https://doi.org/10.52280/pujm.2026.58(03)04"},"language":"en","primary_location":{"id":"doi:10.52280/pujm.2026.58(03)04","doi":"https://doi.org/10.52280/pujm.2026.58(03)04","is_oa":true,"landing_page_url":"https://doi.org/10.52280/pujm.2026.58(03)04","pdf_url":"https://doi.org/10.52280/pujm.2026.58(03)04","source":{"id":"https://openalex.org/S4210188119","display_name":"Punjab University Journal of Mathematics","issn_l":"1016-2526","issn":["1016-2526"],"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":["dongbi-c","poland-20","russia-white-list-2"],"host_organization":"https://openalex.org/P4310319273","host_organization_name":"University of the Punjab","host_organization_lineage":["https://openalex.org/P4310319273"],"host_organization_lineage_names":["University of the Punjab"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Punjab University Journal of Mathematics","raw_type":"journal-article"},"volume":null,"type":"article","is_traditional":true,"indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"bronze","oa_url":"https://doi.org/10.52280/pujm.2026.58(03)04","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5009404406","display_name":"Usama Atta","orcid":"https://orcid.org/0000-0001-8827-8450","observed_orcids":["https://orcid.org/0000-0001-8827-8450"]},"institutions":[{"id":"https://openalex.org/I4210116806","display_name":"Ghazi University","ror":"https://ror.org/023a7t361","country_code":"PK","type":"education","lineage":["https://openalex.org/I4210116806"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Usama Atta","raw_affiliation_strings":["Ghazi University"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Ghazi University","institution_ids":["https://openalex.org/I4210116806"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5147485022","display_name":"Aqeel Ahmad","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I4210116806","display_name":"Ghazi University","ror":"https://ror.org/023a7t361","country_code":"PK","type":"education","lineage":["https://openalex.org/I4210116806"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Aqeel Ahmad","raw_affiliation_strings":["Ghazi University"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Ghazi University","institution_ids":["https://openalex.org/I4210116806"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5025877060","display_name":"Sadia Sattar","orcid":"https://orcid.org/0009-0005-0380-9117","observed_orcids":["https://orcid.org/0009-0005-0380-9117"]},"institutions":[{"id":"https://openalex.org/I4210116806","display_name":"Ghazi University","ror":"https://ror.org/023a7t361","country_code":"PK","type":"education","lineage":["https://openalex.org/I4210116806"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Sadia Sattar","raw_affiliation_strings":["Ghazi University"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Ghazi University","institution_ids":["https://openalex.org/I4210116806"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I4210116806"],"apc_list":null,"apc_paid":null,"fwci":2.0939,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":{"value":0.77711632,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":91,"max":95},"biblio":{"volume":"58","issue":"03","first_page":"331","last_page":"331"},"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":0.9970999956130981,"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":0.9970999956130981,"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/T10167","display_name":"Influenza Virus Research Studies","score":0.002400000113993883,"subfield":{"id":"https://openalex.org/subfields/2713","display_name":"Epidemiology"},"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/T11560","display_name":"Animal Disease Management and Epidemiology","score":0.00019999999494757503,"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/avian-influenza","display_name":"avian influenza","score":0.9990000128746033},{"id":"https://openalex.org/keywords/influenza-vaccination","display_name":"influenza vaccination","score":0.6899999976158142},{"id":"https://openalex.org/keywords/zoonotic-transmission","display_name":"zoonotic transmission","score":0.9179999828338623},{"id":"https://openalex.org/keywords/fractal-fractional-model","display_name":"fractal-fractional model","score":0.9679999947547913},{"id":"https://openalex.org/keywords/mittag-leffler-kernel","display_name":"Mittag-Leffler kernel","score":0.878000020980835},{"id":"https://openalex.org/keywords/basic-reproduction-number","display_name":"basic reproduction number","score":0.8740000128746033},{"id":"https://openalex.org/keywords/transcritical-bifurcation","display_name":"transcritical bifurcation","score":0.9340000152587891},{"id":"https://openalex.org/keywords/sensitivity-analysis","display_name":"sensitivity analysis","score":0.8610000014305115},{"id":"https://openalex.org/keywords/poultry-industry","display_name":"poultry industry","score":0.5239999890327454}],"concepts":[{"id":"https://openalex.org/C22070199","wikidata":"https://www.wikidata.org/wiki/Q192995","display_name":"Vaccination","level":2,"score":0.6802999973297119},{"id":"https://openalex.org/C21200559","wikidata":"https://www.wikidata.org/wiki/Q7451068","display_name":"Sensitivity (control systems)","level":2,"score":0.6762999892234802},{"id":"https://openalex.org/C519248777","wikidata":"https://www.wikidata.org/wiki/Q160122","display_name":"Influenza A virus subtype H5N1","level":3,"score":0.6482999920845032},{"id":"https://openalex.org/C761482","wikidata":"https://www.wikidata.org/wiki/Q118093","display_name":"Transmission (telecommunications)","level":2,"score":0.5777999758720398},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.5116999745368958},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.4578999876976013},{"id":"https://openalex.org/C116675565","wikidata":"https://www.wikidata.org/wiki/Q3241045","display_name":"Outbreak","level":2,"score":0.4316999912261963},{"id":"https://openalex.org/C89623803","wikidata":"https://www.wikidata.org/wiki/Q12184","display_name":"Pandemic","level":5,"score":0.4311999976634979},{"id":"https://openalex.org/C159047783","wikidata":"https://www.wikidata.org/wiki/Q7215","display_name":"Virology","level":1,"score":0.4187999963760376},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4106000065803528},{"id":"https://openalex.org/C204264503","wikidata":"https://www.wikidata.org/wiki/Q901464","display_name":"Basic reproduction number","level":3,"score":0.40049999952316284},{"id":"https://openalex.org/C2781349735","wikidata":"https://www.wikidata.org/wiki/Q4904824","display_name":"Bifurcation","level":3,"score":0.3995000123977661},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.3749000132083893},{"id":"https://openalex.org/C2992243407","wikidata":"https://www.wikidata.org/wiki/Q525512","display_name":"Disease transmission","level":2,"score":0.3538999855518341},{"id":"https://openalex.org/C1627819","wikidata":"https://www.wikidata.org/wiki/Q2572354","display_name":"Epidemic model","level":3,"score":0.34450000524520874},{"id":"https://openalex.org/C2779134260","wikidata":"https://www.wikidata.org/wiki/Q12136","display_name":"Disease","level":2,"score":0.3319999873638153},{"id":"https://openalex.org/C47446073","wikidata":"https://www.wikidata.org/wiki/Q5165890","display_name":"Control theory (sociology)","level":3,"score":0.33180001378059387},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.32749998569488525},{"id":"https://openalex.org/C149782125","wikidata":"https://www.wikidata.org/wiki/Q160039","display_name":"Econometrics","level":1,"score":0.3239000141620636},{"id":"https://openalex.org/C46578552","wikidata":"https://www.wikidata.org/wiki/Q2725393","display_name":"Global health","level":3,"score":0.31839999556541443},{"id":"https://openalex.org/C138816342","wikidata":"https://www.wikidata.org/wiki/Q189603","display_name":"Public health","level":2,"score":0.3160000145435333},{"id":"https://openalex.org/C2777546802","wikidata":"https://www.wikidata.org/wiki/Q834390","display_name":"Influenza A virus","level":3,"score":0.3057999908924103},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.29339998960494995},{"id":"https://openalex.org/C71924100","wikidata":"https://www.wikidata.org/wiki/Q11190","display_name":"Medicine","level":0,"score":0.29249998927116394},{"id":"https://openalex.org/C41949839","wikidata":"https://www.wikidata.org/wiki/Q1756677","display_name":"Stability theory","level":3,"score":0.2888000011444092},{"id":"https://openalex.org/C203479927","wikidata":"https://www.wikidata.org/wiki/Q5165939","display_name":"Controller (irrigation)","level":2,"score":0.2809000015258789},{"id":"https://openalex.org/C99454951","wikidata":"https://www.wikidata.org/wiki/Q932068","display_name":"Environmental health","level":1,"score":0.28029999136924744},{"id":"https://openalex.org/C2780441014","wikidata":"https://www.wikidata.org/wiki/Q7907952","display_name":"Vaccine efficacy","level":3,"score":0.2793000042438507},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.27410000562667847},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.27300000190734863},{"id":"https://openalex.org/C55610538","wikidata":"https://www.wikidata.org/wiki/Q1502581","display_name":"Transcritical bifurcation","level":5,"score":0.2678999900817871},{"id":"https://openalex.org/C2987315675","wikidata":"https://www.wikidata.org/wiki/Q1773974","display_name":"Disease control","level":2,"score":0.26249998807907104},{"id":"https://openalex.org/C2780689484","wikidata":"https://www.wikidata.org/wiki/Q383260","display_name":"Influenza vaccine","level":3,"score":0.2605000138282776},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.2535000145435333},{"id":"https://openalex.org/C2775924081","wikidata":"https://www.wikidata.org/wiki/Q55608371","display_name":"Control (management)","level":2,"score":0.25270000100135803}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.52280/pujm.2026.58(03)04","doi":"https://doi.org/10.52280/pujm.2026.58(03)04","is_oa":true,"landing_page_url":"https://doi.org/10.52280/pujm.2026.58(03)04","pdf_url":"https://doi.org/10.52280/pujm.2026.58(03)04","source":{"id":"https://openalex.org/S4210188119","display_name":"Punjab University Journal of Mathematics","issn_l":"1016-2526","issn":["1016-2526"],"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":["dongbi-c","poland-20","russia-white-list-2"],"host_organization":"https://openalex.org/P4310319273","host_organization_name":"University of the Punjab","host_organization_lineage":["https://openalex.org/P4310319273"],"host_organization_lineage_names":["University of the Punjab"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Punjab University Journal of Mathematics","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.52280/pujm.2026.58(03)04","doi":"https://doi.org/10.52280/pujm.2026.58(03)04","is_oa":true,"landing_page_url":"https://doi.org/10.52280/pujm.2026.58(03)04","pdf_url":"https://doi.org/10.52280/pujm.2026.58(03)04","source":{"id":"https://openalex.org/S4210188119","display_name":"Punjab University Journal of Mathematics","issn_l":"1016-2526","issn":["1016-2526"],"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":["dongbi-c","poland-20","russia-white-list-2"],"host_organization":"https://openalex.org/P4310319273","host_organization_name":"University of the Punjab","host_organization_lineage":["https://openalex.org/P4310319273"],"host_organization_lineage_names":["University of the Punjab"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Punjab University Journal of Mathematics","raw_type":"journal-article"},"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/3","display_name":"Good health and well-being","score":0.5683000087738037}],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7203770075.pdf","grobid_xml":"https://content.openalex.org/works/W7203770075.grobid-xml"},"referenced_works_count":35,"referenced_works":["https://openalex.org/W2266990400","https://openalex.org/W2609978083","https://openalex.org/W2765500405","https://openalex.org/W2795604327","https://openalex.org/W3001468641","https://openalex.org/W4307701268","https://openalex.org/W4319791808","https://openalex.org/W4361233677","https://openalex.org/W4377016705","https://openalex.org/W4386647898","https://openalex.org/W4389060208","https://openalex.org/W4389586755","https://openalex.org/W4390583768","https://openalex.org/W4392179360","https://openalex.org/W4392394435","https://openalex.org/W4395106365","https://openalex.org/W4396695271","https://openalex.org/W4399119178","https://openalex.org/W4401034570","https://openalex.org/W4402050970","https://openalex.org/W4403781809","https://openalex.org/W4403877336","https://openalex.org/W4405814506","https://openalex.org/W4405909571","https://openalex.org/W4408559458","https://openalex.org/W4409318764","https://openalex.org/W4409903344","https://openalex.org/W4409953953","https://openalex.org/W4410426224","https://openalex.org/W4413811671","https://openalex.org/W4417415380","https://openalex.org/W7132844790","https://openalex.org/W7138850297","https://openalex.org/W7161640844","https://openalex.org/W7164661043"],"related_works":["https://openalex.org/W4404636004","https://openalex.org/W7128178691","https://openalex.org/W2596914350","https://openalex.org/W7155579776","https://openalex.org/W6893233159","https://openalex.org/W2042743172","https://openalex.org/W2807812757","https://openalex.org/W7223176558","https://openalex.org/W4409042504","https://openalex.org/W2796306135"],"abstract_inverted_index":{"Avian":[0],"influenza":[1,77],"is":[2],"a":[3,17,73,96,209],"zoonotic":[4,251],"disease":[5],"that":[6,147],"spreads":[7],"from":[8],"birds":[9,150],"to":[10,20,58,178,190,203,267],"humans":[11,67],"through":[12],"viral":[13],"infection":[14,148],"and":[15,68,87,100,105,115,129,132,138,152,165,186,227,238],"poses":[16],"substantial":[18],"threat":[19],"the":[21,39,43,60,63,80,93,109,113,119,123,130,153,180,184,187,192,199,218,222,225,241,250,259,268],"poultry":[22],"industry,":[23],"which":[24],"can":[25],"result":[26],"in":[27,79,149,245],"severe":[28],"financial":[29],"consequences":[30],"as":[31,33,52,240],"well":[32],"serious":[34],"public":[35],"health":[36],"concerns":[37],"for":[38],"community.":[40],"To":[41],"control":[42,49],"spread":[44,157],"of":[45,62,92,95,102,118,125,141,155,183,195,198],"avian":[46,76,247],"influenza,":[47],"effective":[48],"measures":[50,255],"such":[51],"vaccination":[53],"are":[54,168],"important.":[55],"In":[56,201],"order":[57,202],"highlight":[59],"influence":[61],"vaccine":[64],"on":[65,213],"infected":[66],"birds,":[69],"this":[70],"study":[71],"formulates":[72],"novel":[74],"fractal-fractional":[75],"model":[78],"Mittag-Leffler":[81],"sense.":[82],"We":[83,174,216],"perform":[84],"comprehensive":[85],"qualitative":[86],"quantitative":[88],"analysis,":[89],"including":[90],"verification":[91],"presence":[94,124],"single":[97],"solution,":[98],"positivity":[99],"boundedness":[101],"solutions,":[103,206],"influenzafree":[104],"existing":[106],"equilibrium":[107,121,143],"points,":[108,122,144],"threshold":[110],"parameter":[111],"R0,":[112],"local":[114,131,137],"global":[116,133,139],"stability":[117,140],"influenza-free":[120],"forward":[126],"transcritical":[127],"bifurcation,":[128],"sensitivity":[134,231],"analysis.":[135],"Through":[136,249],"Influenza-free":[142],"we":[145,207],"verify":[146],"stops":[151],"chances":[154],"Influenza":[156],"decrease":[158],"when":[159,170],"R0":[160,171],"<":[161],"1,":[162],"while":[163],"outbreaks":[164],"human":[166,226,269],"transmission":[167,234,252],"possible":[169],">":[172],"1.":[173],"provide":[175,217,264],"statistical":[176],"analysis":[177,232],"show":[179],"symmetric":[181],"behaviour":[182],"compartments":[185,197],"chaos":[188],"test":[189],"reveal":[191],"bounded":[193],"nature":[194],"all":[196],"model.":[200],"obtain":[204],"numerical":[205,210],"derive":[208],"technique":[211],"based":[212],"Newton":[214],"polynomials.":[215],"phase":[219],"plots":[220],"illustrating":[221],"interactions":[223],"between":[224],"bird":[228,260],"compartments.":[229],"The":[230],"identifies":[233],"rate,":[235],"recruitment,":[236],"vaccination,":[237],"recovery":[239],"most":[242],"influential":[243],"factors":[244],"controlling":[246],"influenza.":[248],"pathway,":[253],"these":[254],"not":[256],"only":[257],"stabilize":[258],"population":[261],"but":[262],"also":[263],"indirect":[265],"protection":[266],"population.":[270]},"counts_by_year":[{"year":2026,"cited_by_count":1}],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-08-20T00:00:00"}