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具有时滞影响的SIRC传染病模型的Hopf分支分析.docx

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具有时滞影响的SIRC传染病模型的Hopf分支分析 Title:HopfBifurcationAnalysisoftheDelayedSIRCEpidemicModel Abstract: Inrecentyears,thestudyofepidemicmodelshasgainedsignificantattentionduetotheoutbreakofvariousinfectiousdiseases.Toeffectivelycontrolandmitigatethespreadofinfectiousdiseases,itiscrucialtounderstandthedynamicsoftheepidemicsystems.ThispaperfocusesonthedelayedSusceptible-Infectious-Recovered-Constant(SIRC)epidemicmodel,incorporatingtimedelayasacriticalfactorinfluencingthespreadofdiseases.TheaimistoanalyzetheoccurrenceofHopfbifurcationinthemodelandshedlightontheunderstandingofitsdynamicbehavior. 1.Introduction Epidemicmodelsplayanessentialroleinpredictingandunderstandingthebehaviorofinfectiousdiseases.Manypreviousstudieshaveconsidereddifferentepidemicmodelswithvariouscomplexities.However,theinclusionoftimedelayinepidemicmodelshasbeenrelativelylessexplored,despiteitsprofoundimpactonthedynamicsofthesystem.ThispaperattemptstobridgethisgapbyinvestigatingthedelayedSIRCepidemicmodel. 2.TheBasicDelayedSIRCEpidemicModel ThissectionoutlinesthedelayedSIRCmodel,whichtakesintoaccountthebiologicaldelayencounteredintheprogressionofaninfectiousdisease.Thedifferentialequationsgoverningthemodelarepresented,alongwiththeinterpretationofeachparameterinvolved. 3.StabilityAnalysis TounderstandthestabilityanddynamicbehaviorofthedelayedSIRCmodel,weperformstabilityanalysisbyconsideringtheexistence,stability,anduniquenessoftheequilibriumpoints.Thecharacteristicequationisderived,andstabilityconditionsareobtainedtodeterminethestabilityofthemodel. 4.HopfBifurcationAnalysis Inthissection,weinvestigatetheoccurrenceoftheHopfbifurcationinthedelayedSIRCmodel.Byapplyingthenormalformtheoryandcentermanifoldreduction,theconditionsforHopfbifurcationarederived.Thecriticalvaluesofthebifurcationparameterarecalculated,indicatingtheoccurrenceofastablelimitcycle. 5.NumericalSimulations NumericalsimulationsareconductedtovalidatetheanalyticalfindingsoftheHopfbifurcationanalysis.ThedynamicbehaviorofthedelayedSIRCmodelwithtime