{"id":"https://openalex.org/W3097092509","doi":"https://doi.org/10.3390/sym12111778","title":"An Approach for the Global Stability of Mathematical Model of an Infectious Disease","display_name":"An Approach for the Global Stability of Mathematical Model of an Infectious Disease","publication_year":2020,"publication_date":"2020-10-27","ids":{"openalex":"https://openalex.org/W3097092509","doi":"https://doi.org/10.3390/sym12111778","mag":"3097092509"},"language":"en","primary_location":{"id":"doi:10.3390/sym12111778","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym12111778","pdf_url":"https://www.mdpi.com/2073-8994/12/11/1778/pdf?version=1603791697","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2073-8994/12/11/1778/pdf?version=1603791697","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5005918189","display_name":"Mojtaba Masoumnezhad","orcid":"https://orcid.org/0000-0002-3395-8886"},"institutions":[{"id":"https://openalex.org/I4210089766","display_name":"Technical and Vocational University","ror":"https://ror.org/00854zy02","country_code":"IR","type":"education","lineage":["https://openalex.org/I4210089766"]}],"countries":["IR"],"is_corresponding":true,"raw_author_name":"Mojtaba Masoumnezhad","raw_affiliation_strings":["Department of Mechanical Engineering, Faculty of Chamran, Guilan Branch, Technical and Vocational University (TVU), Tehran 1435761137, Iran"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mechanical Engineering, Faculty of Chamran, Guilan Branch, Technical and Vocational University (TVU), Tehran 1435761137, Iran","institution_ids":["https://openalex.org/I4210089766"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5062638410","display_name":"Maziar Rajabi","orcid":"https://orcid.org/0000-0002-4747-0945"},"institutions":[{"id":"https://openalex.org/I170733385","display_name":"University of Applied Science and Technology","ror":"https://ror.org/04r58gt57","country_code":"IR","type":"education","lineage":["https://openalex.org/I170733385"]}],"countries":["IR"],"is_corresponding":false,"raw_author_name":"Maziar Rajabi","raw_affiliation_strings":["Department of Software Engineering, University of Applied Science and Technology, Shabestar, Tabriz 5164864763, Iran"],"raw_orcid":"https://orcid.org/0000-0002-4747-0945","affiliations":[{"raw_affiliation_string":"Department of Software Engineering, University of Applied Science and Technology, Shabestar, Tabriz 5164864763, Iran","institution_ids":["https://openalex.org/I170733385"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5034852125","display_name":"Amirahmad Chapnevis","orcid":"https://orcid.org/0000-0003-2258-3449"},"institutions":[{"id":"https://openalex.org/I158248296","display_name":"Amirkabir University of Technology","ror":"https://ror.org/04gzbav43","country_code":"IR","type":"education","lineage":["https://openalex.org/I158248296"]}],"countries":["IR"],"is_corresponding":false,"raw_author_name":"Amirahmad Chapnevis","raw_affiliation_strings":["Department of Computer Engineering and Information Technology, Amirkabir University of Technology, Tehran 1591634311, Iran"],"raw_orcid":"https://orcid.org/0000-0003-2258-3449","affiliations":[{"raw_affiliation_string":"Department of Computer Engineering and Information Technology, Amirkabir University of Technology, Tehran 1591634311, Iran","institution_ids":["https://openalex.org/I158248296"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5015434813","display_name":"A. E. Dorofeev","orcid":"https://orcid.org/0000-0002-0815-4472"},"institutions":[{"id":"https://openalex.org/I17877952","display_name":"Sechenov University","ror":"https://ror.org/02yqqv993","country_code":"RU","type":"education","lineage":["https://openalex.org/I17877952"]}],"countries":["RU"],"is_corresponding":false,"raw_author_name":"Aleksei Dorofeev","raw_affiliation_strings":["Head of Department of Propaedeutics of Dental Diseases, I.M. Sechenov First Moscow State Medical University, 119991 Moscow, Russia"],"raw_orcid":"https://orcid.org/0000-0002-0815-4472","affiliations":[{"raw_affiliation_string":"Head of Department of Propaedeutics of Dental Diseases, I.M. Sechenov First Moscow State Medical University, 119991 Moscow, Russia","institution_ids":["https://openalex.org/I17877952"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5057765710","display_name":"Stanford Shateyi","orcid":"https://orcid.org/0000-0002-5829-8634"},"institutions":[{"id":"https://openalex.org/I84929415","display_name":"University of Venda","ror":"https://ror.org/0338xea48","country_code":"ZA","type":"education","lineage":["https://openalex.org/I84929415"]}],"countries":["ZA"],"is_corresponding":true,"raw_author_name":"Stanford Shateyi","raw_affiliation_strings":["Department of Mathematics, University of Venda, Private Bag X5050, Thohoyandou 0950, South Africa"],"raw_orcid":"https://orcid.org/0000-0002-5829-8634","affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Venda, Private Bag X5050, Thohoyandou 0950, South Africa","institution_ids":["https://openalex.org/I84929415"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5080055790","display_name":"Narges Shayegh Kargar","orcid":null},"institutions":[{"id":"https://openalex.org/I55547365","display_name":"Payame Noor University","ror":"https://ror.org/031699d98","country_code":"IR","type":"education","lineage":["https://openalex.org/I55547365"]}],"countries":["IR"],"is_corresponding":false,"raw_author_name":"Narges Shayegh Kargar","raw_affiliation_strings":["Department of Mathematics, Payame Noor University (PNU), Tehran 193953697, Iran"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Payame Noor University (PNU), Tehran 193953697, Iran","institution_ids":["https://openalex.org/I55547365"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5002469157","display_name":"Hassan Saberi Nik","orcid":"https://orcid.org/0000-0003-2521-1325"},"institutions":[{"id":"https://openalex.org/I1312987727","display_name":"University of Neyshabur","ror":"https://ror.org/01x41eb05","country_code":"IR","type":"education","lineage":["https://openalex.org/I1312987727"]}],"countries":["IR"],"is_corresponding":false,"raw_author_name":"Hassan Saberi Nik","raw_affiliation_strings":["Department of Mathematics and Statistics, University of Neyshabur, Neyshabur 9319774446, Iran"],"raw_orcid":"https://orcid.org/0000-0003-2521-1325","affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics, University of Neyshabur, Neyshabur 9319774446, Iran","institution_ids":["https://openalex.org/I1312987727"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":7,"corresponding_author_ids":["https://openalex.org/A5005918189","https://openalex.org/A5057765710"],"corresponding_institution_ids":["https://openalex.org/I4210089766","https://openalex.org/I84929415"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":4.3162,"has_fulltext":true,"cited_by_count":34,"citation_normalized_percentile":{"value":0.95070712,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":91,"max":99},"biblio":{"volume":"12","issue":"11","first_page":"1778","last_page":"1778"},"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":0.9998000264167786,"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"}},"topics":[{"id":"https://openalex.org/T10482","display_name":"Mathematical and Theoretical Epidemiology and Ecology Models","score":0.9998000264167786,"subfield":{"id":"https://openalex.org/subfields/2739","display_name":"Public Health, Environmental and Occupational Health"},"field":{"id":"https://openalex.org/fields/27","display_name":"Medicine"},"domain":{"id":"https://openalex.org/domains/4","display_name":"Health Sciences"}},{"id":"https://openalex.org/T11764","display_name":"Evolution and Genetic Dynamics","score":0.9976999759674072,"subfield":{"id":"https://openalex.org/subfields/1311","display_name":"Genetics"},"field":{"id":"https://openalex.org/fields/13","display_name":"Biochemistry, Genetics and Molecular Biology"},"domain":{"id":"https://openalex.org/domains/1","display_name":"Life Sciences"}},{"id":"https://openalex.org/T10410","display_name":"COVID-19 epidemiological studies","score":0.9933000206947327,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/lyapunov-function","display_name":"Lyapunov function","score":0.8048174381256104},{"id":"https://openalex.org/keywords/stability","display_name":"Stability (learning theory)","score":0.5835540294647217},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5778948068618774},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.5706697106361389},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.569162905216217},{"id":"https://openalex.org/keywords/lyapunov-equation","display_name":"Lyapunov equation","score":0.5569456219673157},{"id":"https://openalex.org/keywords/stability-theory","display_name":"Stability theory","score":0.45563167333602905},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.44618669152259827},{"id":"https://openalex.org/keywords/lyapunov-exponent","display_name":"Lyapunov exponent","score":0.4412243366241455},{"id":"https://openalex.org/keywords/control-theory","display_name":"Control theory (sociology)","score":0.3347633481025696},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.28429222106933594},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.09630927443504333},{"id":"https://openalex.org/keywords/control","display_name":"Control (management)","score":0.05969071388244629}],"concepts":[{"id":"https://openalex.org/C60640748","wikidata":"https://www.wikidata.org/wiki/Q2337858","display_name":"Lyapunov function","level":3,"score":0.8048174381256104},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.5835540294647217},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5778948068618774},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.5706697106361389},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.569162905216217},{"id":"https://openalex.org/C25854792","wikidata":"https://www.wikidata.org/wiki/Q1028945","display_name":"Lyapunov equation","level":4,"score":0.5569456219673157},{"id":"https://openalex.org/C41949839","wikidata":"https://www.wikidata.org/wiki/Q1756677","display_name":"Stability theory","level":3,"score":0.45563167333602905},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.44618669152259827},{"id":"https://openalex.org/C191544260","wikidata":"https://www.wikidata.org/wiki/Q1238630","display_name":"Lyapunov exponent","level":3,"score":0.4412243366241455},{"id":"https://openalex.org/C47446073","wikidata":"https://www.wikidata.org/wiki/Q5165890","display_name":"Control theory (sociology)","level":3,"score":0.3347633481025696},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.28429222106933594},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.09630927443504333},{"id":"https://openalex.org/C2775924081","wikidata":"https://www.wikidata.org/wiki/Q55608371","display_name":"Control (management)","level":2,"score":0.05969071388244629},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","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/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.0},{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.0},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/sym12111778","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym12111778","pdf_url":"https://www.mdpi.com/2073-8994/12/11/1778/pdf?version=1603791697","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:6208826f1b6b4b53a0067bb3a5ed44f0","is_oa":true,"landing_page_url":"https://doaj.org/article/6208826f1b6b4b53a0067bb3a5ed44f0","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":"Symmetry, Vol 12, Iss 11, p 1778 (2020)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/12/11/1778/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/sym12111778","pdf_url":null,"source":{"id":"https://openalex.org/S4306400947","display_name":"MDPI (MDPI AG)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210097602","host_organization_name":"Multidisciplinary Digital Publishing Institute (Switzerland)","host_organization_lineage":["https://openalex.org/I4210097602"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/sym12111778","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym12111778","pdf_url":"https://www.mdpi.com/2073-8994/12/11/1778/pdf?version=1603791697","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3097092509.pdf","grobid_xml":"https://content.openalex.org/works/W3097092509.grobid-xml"},"referenced_works_count":38,"referenced_works":["https://openalex.org/W1606697907","https://openalex.org/W1963895250","https://openalex.org/W1966207826","https://openalex.org/W1980640538","https://openalex.org/W2004467084","https://openalex.org/W2008265020","https://openalex.org/W2016448831","https://openalex.org/W2027467474","https://openalex.org/W2038572450","https://openalex.org/W2053823258","https://openalex.org/W2063767038","https://openalex.org/W2068604088","https://openalex.org/W2082539696","https://openalex.org/W2111975293","https://openalex.org/W2112680994","https://openalex.org/W2161728228","https://openalex.org/W2171580218","https://openalex.org/W2275732351","https://openalex.org/W2309379662","https://openalex.org/W2398784811","https://openalex.org/W2400661904","https://openalex.org/W2523160158","https://openalex.org/W2589799985","https://openalex.org/W2597716599","https://openalex.org/W2610952273","https://openalex.org/W2744588392","https://openalex.org/W2763359948","https://openalex.org/W2765731421","https://openalex.org/W2897702875","https://openalex.org/W2903607556","https://openalex.org/W2908688458","https://openalex.org/W2927659276","https://openalex.org/W2934819644","https://openalex.org/W2974529677","https://openalex.org/W3104309680","https://openalex.org/W4234493213","https://openalex.org/W4388299012","https://openalex.org/W6733691774"],"related_works":["https://openalex.org/W2284157877","https://openalex.org/W2349796700","https://openalex.org/W2383986032","https://openalex.org/W2030955003","https://openalex.org/W1789972284","https://openalex.org/W2374426356","https://openalex.org/W2059628530","https://openalex.org/W2210722882","https://openalex.org/W1979419366","https://openalex.org/W2963043651"],"abstract_inverted_index":{"The":[0,15,33,88],"global":[1,64],"stability":[2,65,73],"analysis":[3],"for":[4],"the":[5,29,36,39,45,50,53,63,67,72,82,111],"mathematical":[6],"model":[7],"of":[8,28,35,41,52,66],"an":[9],"infectious":[10],"disease":[11],"is":[12,18,38],"discussed":[13],"here.":[14],"endemic":[16,68],"equilibrium":[17],"shown":[19],"to":[20,109],"be":[21,95],"globally":[22],"stable":[23],"by":[24],"using":[25],"a":[26],"modification":[27],"Volterra\u2013Lyapunov":[30,46,76],"matrix":[31],"method.":[32],"basis":[34],"method":[37],"combination":[40],"Lyapunov":[42],"functions":[43],"and":[44,55],"matrices.":[47],"By":[48],"reducing":[49],"dimensions":[51],"matrices":[54,80],"under":[56],"some":[57],"conditions,":[58],"we":[59,78],"can":[60,94],"easily":[61],"show":[62],"equilibrium.":[69],"To":[70],"prove":[71],"based":[74],"on":[75],"matrices,":[77],"use":[79],"with":[81,101],"symmetry":[83],"properties":[84],"(symmetric":[85],"positive":[86],"definite).":[87],"results":[89],"developed":[90],"in":[91,97],"this":[92],"paper":[93],"applied":[96],"more":[98],"complex":[99],"systems":[100],"nonlinear":[102],"incidence":[103],"rates.":[104],"Numerical":[105],"simulations":[106],"are":[107],"presented":[108],"illustrate":[110],"analytical":[112],"results.":[113]},"counts_by_year":[{"year":2026,"cited_by_count":3},{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":6},{"year":2023,"cited_by_count":5},{"year":2022,"cited_by_count":6},{"year":2021,"cited_by_count":13}],"updated_date":"2026-06-13T07:54:00.901334","created_date":"2025-10-10T00:00:00"}
