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具接种的随机SIQR流行病模型中平衡点的稳定性分析.docx

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具接种的随机SIQR流行病模型中平衡点的稳定性分析 Title:StabilityAnalysisoftheEquilibriumPointinRandomizedSIQREpidemiologicalModelforVaccination Abstract: Inthefaceoftheongoingglobalpandemic,understandingthedynamicsofinfectiousdiseasesandtheeffectsofvaccinationisofparamountimportance.TherandomizedSIQRmodeliswidelyusedtosimulatethespreadofinfectiousdiseases.ThispaperaimstoanalyzethestabilityoftheequilibriumpointintherandomizedSIQRmodel,focusingspecificallyontheimpactofvaccination.Thestabilityanalysisprovidesvaluableinsightsintothelong-termbehaviorofthediseaseandhelpspolicy-makersmakeinformeddecisionsregardingvaccinationstrategiestocontrolandmitigatethespreadofinfectiousdiseases. Introduction: TherandomizedSIQRmodelisavariantoftheclassicSIQRmodelthatincorporatesrandomnesstoaccountforvariousuncertaintiesindiseasetransmission.Itconsidersfourmaincompartments:susceptible(S),infected(I),quarantined(Q),andrecovered(R).Thismodelisespeciallyapplicablewhendealingwithvaccinationstrategies,asitallowsfortheconsiderationofrandomizedvaccinationcampaigns. EquilibriumPoint: Theequilibriumpointrepresentsastablestatewherethediseasereachesabalancebetweenitstransmissionandrecoveryrates.IntheSIQRmodel,theequilibriumpointoccurswhenthenumberofsusceptibleindividuals(S)iszero.Atthispoint,thediseaseisnolongerspreadinginthepopulation,andthenumberofinfectedindividuals(I)hasreacheditspeakbeforedecliningduetorecoveryorquarantine.Thenumberofquarantinedindividuals(Q)graduallydecreasesastheyrecover,contributingtotheincreaseinthenumberofrecoveredindividuals(R). StabilityAnalysis: Stabilityanalysishelpsusunderstandthelong-termbehaviorofthediseaseunderdifferentconditions,suchasvaccinationrates.Byexaminingthestabilityoftheequilibriumpoint,wecandeterminewhetherthediseasewillpersistoreventuallydieout.StabilityanalysisinvolvesinvestigatingtheeigenvaluesoftheJacobianmatrix,whichcharacterizesthelinearizationofthesystemofdifferentialequations.Theeigenvaluesprovidevaluableinformationaboutthesystem'sdynamicsanditsstability. ImpactofVac