具有时滞SIR模型的Hopf分支分析

具有时滞SIR模型的Hopf分支分析Title:HopfbifurcationanalysisofdelayedSIRmodelAbstract:Thespreadofinfectiousdiseaseshasalwaysbeenasignificantconcernforpublichealth,andmathematicalmodelsplayacrucialroleinunderstandingtheirdynamics.Inthispaper,weinvestigateatime-delayedSIR(Susceptible-Infectious-Recovered)modelandperformaHopfbifurcationanalysistounderstandtheimpactoftimedelayondiseasedynamics.Introduction:TheSIRmodelisawidelyusedmathematicalframeworkforstudyingthetransmissiondynamicsofinfectiousdiseases.Itdividesthepopulationintosusceptible,infectious,andrecoveredindividualsandtracksthenumberofpeopleineachcompartmentovertime.Timedelaysinthetransmissionprocesscanariseduetofactorssuchasincubationperiodsorthetimetakenforinterventionstotakeeffect.Understandingtheeffectoftimedelaysondiseasedynamicsisessentialfordesigningeffectivecontrolstrategies.ModelFormulation:LetusconsideranSIRmodelwithatimedelaydenotedbyτ.Themodelequationscanbewrittenas:dS/dt=λ-βSI-μSdI/dt=βSI-γI-μIdR/dt=γI-μRwhereS,I,andRrepresentthenumberofsusceptible,infectious,andrecoveredindividuals,respectively.λrepresentsthebirthrate,μrepresentsthenaturalmortalityrate,βisthecontactrate,andγistherecoveryrate.HopfBifurcationAnalysis:ToanalyzetheHopfbifurcation,weintroduceasmallparameterεandwritethetime-delayedequationas:dI(t)/dt=F(I(t),I(t-τ))-ε*M(I(t),I(t-τ))whereF(I(t),I(t-τ))representsthenon-delayedtermandM(I(t),I(t-τ))representsthedelayedterm.Next,weintroduceanewvariablev=I(t-τ)asasubstituteforthetimedelayterm.Usingthissubstitution,thetime-delayedequationbecomesatwo-dimensionalordinarydifferentialequationsystemwithanon-delayedtermandadelay-independentterm.Wethenperformalinearstabilityanalysisbylinearizingthesystemaroundtheequilibriumpointtoinvestigatethestabilityofthemodel.Thecharacteristicequationisderived,andthestabilityboundariesaredetermined.WefindthataHopfbifurcationoccursasthetimedelaycrossesacriticalvalue.AnalyticalResults:WederivethestabilityconditionsaroundtheequilibriumpointanalyticallyandshowthattheHopfbifurcationoccurswhenthetimedelayexceedsacriticalvalue.Theamplitudeandfrequencyoftheoscillationsatthebifurcationpointaredeterminedusingthenormalformtheory.Furthermore,weinvestigatetheeffectoftimedelayonthedynamicsofdiseasetransmission.Wefindthatthetimedelaycanleadtosustainedoscillationsratherthanconvergingtoastableequilibrium.Theseoscillationscanhavesignificantimplicationsforthedesignofcontrolstrategies,astheycanleadtoperiodicoutbreaksanddifficultiesindiseasecontrol.NumericalSimulations:Tovalidateouranalyticalfindings,weperformnumericalsimulationsofthetime-delayedSIRmodel.Wevarythetimedelayparameterandobservetheemergenceofoscillatorybehaviorwhenthecriticalvalueisexceeded.DiscussionandConclusions:Inthispaper,wehaveinvestigatedtheHopfbifurcationphenomenoninatime-delayedSIRmodel.Ouranalysisandnumericalsimulationsdemonstratethattimedelaycansignificantlyaffectthediseasedynamics,leadingtosustainedoscillationsratherthanconvergencetostableequilibria.Thesefindingshighlighttheimportanceofconsideringtimedelaysininfectiousdiseasemodelingandcontrolstrategies.Futureresearchcanfocusontheanalysisofmorerealisticcontactandrecoveryratefunctionsandtheincorporationofotherfactorssuchasspatialeffectsorseasonality.Additionally,consideringtheimpactofinterventionsandcontrolmeasuresinthetime-delayedSIRmodelwouldprovidevaluableinsightsforpolicymakersandpublichealthofficials.Inconclusion,theHopfbifurcationanalysisofthetime-delayedSIRmodelprovidesadeeperunderstandingof