具有双时滞的SEIR疾病模型的Hopf分支分析.docx

具有双时滞的SEIR疾病模型的Hopf分支分析
Title:HopfBifurcationAnalysisofSEIRDiseaseModelwithDoubleTimeDelays
Introduction:
Inrecenttimes,infectiousdiseaseshavecausedsignificantdisruptionsworldwide,emphasizingtheimportanceofstudyingepidemicmodels.TheSEIR(Susceptible-Exposed-Infectious-Recovered)model,awidely-usedepidemiologicalmodel,allowsforabetterunderstandingofdiseasedynamicsandenablestheformulationofeffectivecontrolstrategies.AddingtimedelaystotheSEIRmodelcancapturemorerealisticscenarios,wheretheincubationperiodandtheperiodofinfectiousnessarenotinstantaneous.Inthispaper,wefocusonaspecificvariantoftheSEIRmodel,incorporatingdoubletimedelays,andperformaHopfbifurcationanalysistoinvestigatetheemergenceofsustainedoscillatorybehaviorinthesystem.
SEIRModelwithDoubleTimeDelays:
TheSEIRmodelwithdoubletimedelaysisgivenbythefollowingsetofdifferentialequations:
dS/dt=λ-βS(t-τ₁)I(t-τ₂)-μS,
dE/dt=βS(t-τ₁)I(t-τ₂)-(σ+μ)E,
dI/dt=σE-(γ+μ)I,
dR/dt=γI-μR,
whereS,E,I,andRrepresentthepopulationsofsusceptible,exposed,infectious,andrecoveredindividuals,respectively.Theparametersλ,β,γ,σ,andμdenotethebirthrate,infectionrate,recoveryrate,incubationrate,anddeathrate,respectively.Thedoubletimedelaysτ₁andτ₂accountforthelagbetweenthetimeanindividualbecomesinfectedandthetimetheybecomeinfectious.
StabilityAnalysis:
ToinvestigatethestabilitypropertiesoftheSEIRmodelwithdoubletimedelays,wefirstcomputethebasicreproductionnumberR₀,whichrepresentstheaveragenumberofsecondaryinfectionscausedbyasingleinfectiousindividualintroducedintoacompletelysusceptiblepopulation.TheexpressionforR₀isderivedusingthenext-generationmatrixmethod,whichinvolveslinearizingthesystemequationsaroundthedisease-freeequilibrium.ThestabilityoftheequilibriumpointsisthendeterminedbasedonthesignofR₀:ifR₀<1,thedisease-freeequilibriumisgloballyasymptoticallystable,indicatingtheabsenceofanepidemic;ifR₀>1,thedisease-freeequilibriumisunstable,implyingthepossibilityofanepidemic.
HopfBifurcationAnalysis:
Beyondstabilityanalysis,theoccurrenceofHopfbifurcationr