流感传播和控制的数学模型研究课件

流感传播和控制的数学模型研究

Introduction

tomyself研究方向：

生物数学、传染病动力学（1）宏观艾滋病病毒传播与微观HIV的进展研究（2）流感（A/H1N1,H7N9）的模型研究（3）数学工具：ODE、DDE、IDE、Non-smoothODE3纲要流感病毒简介流感的预防与控制问题的提出传染病动力学简介流感的数学模型研究什么是H7N9禽流感病毒？流感病毒可分为甲（Ａ）、乙（Ｂ）、丙（Ｃ）三型。其中，甲型流感依据流感病毒血凝素蛋白（HA）的不同可分为1-16种亚型，根据病毒神经氨酸酶蛋白（NA）的不同可分为1-9种亚型，HA不同亚型可以与NA的不同亚型相互组合形成多达144种不同的流感病毒。而禽类特别是水禽是所有这些流感病毒的自然宿主，H7N9禽流感病毒是其中的一种。H7N9亚型流感病毒既往仅在禽间发现，在荷兰、日本及美国等地曾发生过禽间暴发疫情，但未发现过人的感染情况。H7N9禽流感感染病例的主要临床表现？目前3例确诊病例主要表现为典型的病毒性肺炎，起病急，病程早期均有高热（38℃以上），咳嗽等呼吸道感染症状。起病5-7天出现呼吸困难，重症肺炎并进行性加重，部分病例可迅速发展为急性呼吸窘迫综合症并死亡。全球人感染甲型H7流感病毒情况？1996年～2009年间，荷兰、意大利、加拿大、美国和英国曾报道人感染甲型H7流感病例，病毒亚型分别为H7N2、H7N3和H7N7，临床表现主要为结膜炎与轻型的上呼吸道感染。此前，我国从未发现过H7亚型流感病毒感染病例。H7N9病毒与既往H1N1、H5N1和其他季节性流感病例相比其毒力和传染性如何？由于目前只发现3例人感染H7N9禽流感病毒确诊病例，对该病毒及其所致疾病的研究资料十分有限，尚无法对该病毒的毒力和人际传播的能力做出准确判断。本次人感染H7N9禽流感的病例的感染来源？此次人感染的H7N9流感病毒从病毒生物学上属于禽源流感病毒，既往国际上所发现的人感染H7亚型的流感病毒也多来自于禽类，但截至目前，3例确诊病例的具体感染来源尚不清楚。H7N9禽流感病毒是否能够在人与人之间传播？目前尚未证实该病毒具有人传人的能力。香港在1997年发生禽流感，有6人死亡，当局捕杀上百万只家禽，避免疫情恶化。禽流感是一种主要流行于鸡群中的烈性传染病，一旦爆发，往往会造成家禽的大量死亡。潜伏期一般为3-5天。一般病程1～2天，症状变化很大。病鸡可能见有呼吸道症状，如打喷嚏、窦炎和结膜炎。病鸡头部常出现水肿，可能同时出现或不出现腹泻；病鸡体温升高，羽毛蓬松，鸡冠发绀。有的腿变红，鼻分泌物增多，呼吸极度困难，甩头，严重地可窒息死亡。产蛋率明显下降。然而近年来情况却悄悄地发生了变化，禽流感的传播已经跨越了原先的范围，开始侵袭人类社会。问题的提出

传染病能否在某个地方传播开来？能否形成地方病？

传染病高潮的什么时候来临？传染病的规模有多大？

什么预防与控制措施最为有效？

能否给公共卫生部门提供定量的建议？基于流感的传播机理建立数学模型，利用数据估计系统的参数，对疫情的发展趋势给予预测，分析人为的干预措施对疫情发展的影响，为制定控制方案提供定量的决策依据传染病模型简介方法：将人群分类，建立方程来描述每类人群的数量的变化

S

I

RS(t)：易感者的数量（或比例）I(t)：感染者的数量（或比例）R(t)：移除者的数量（或比例）SIR模型无法求出的解析解在相平面上研究解的性质模型4消去dtSIR模型相轨线的定义域相轨线11si0D在D内作相轨线的图形，进行分析si101D相轨线及其分析传染病蔓延传染病不蔓延s(t)单调减相轨线的方向P1s0imP1:s0>1/

i(t)先升后降至0P2:s0<1/

i(t)单调降至01/

阈值P3P4P2S0预防传染病蔓延的手段(日接触率)卫生水平(日治愈率)医疗水平传染病不蔓延的条件——s0<1/

降低s0提高r0

提高阈值1/

降低(=/),群体免疫Transmissionbetweenindividuals

R0BasicReproductionratio（基本再生数）Averagenumberofsecondarycasescausedby1infectiousindividualduringitsentireinfectiousperiodinafullysusceptiblepopulation禽流感的数学模型

S

I

R

A

E

Sp

Ip传播框图人群禽类X，Y：

denotethesusceptiblebirds,thebirdsinfectedwiththeavianinfluenzaS，B：thesusceptiblehumansandthehumansthatareinfectedwiththewildavianinfluenza.H：denotesthehumansinfectedwiththemutantavianinfluenza;以往简单模型模型建立WithAsofApril26,2013,theChinaMinistryofAgriculturereportedthat68,060birdandenvironmentalspecimenshavebeentested,46(0.07\%)wereconfirmedH7N9-positivebycultureWethusassumethat

Afact:onprevalenceinpoultry数据EffectofvariousInterventiontimingandintensityPredictionofthenextoutbreakPredictionofthenextoutbreakTheperiodicinfectionofpoultrymayinducethesecondoutbreakinhumanpopulation.Weestimatethereproductionnumberforhuman-to-humantransmissionas0.467(95\%CI0.387-0.651).Simulationresultsindicatethatapproximatetwofoldofthecurrenthuman-to-humantransmissionrateorperiodicoutbreaksofavianinfluenzainpoultrymayinduceanoutbreakinhuman.ThoughtherecentlimitedtransmissionpotentialofthenovelavianinfluenzaA(H7N9)virus,anewoutbreakmaybepossibleduetovirusmutationandadaptionorperiodicoutbreaksinpoultry,andhencecarefulsurveillanceandpersistentinterventionstrategiesinpoultryhavetoberequired.Conclusions

DatasourcesandNon-pharmaceuticalinterventions(Motivations)

Conclusionsanddiscussions

Modelwithinteractionsbetweentheuniversitycommunityandthegeneralpopulation

ModelamonguniversitycommunityFengxiaoformitigatingthe2009H1N1pandemicinXi’ancityA/H1N1outbreakinmainlandChinaUntil3/31/2010,0.127(800)millionconfirmedcases,with0.126millionlocalcasesand1228importedcases.

Thedataonlaboratory-confirmedcasesofpandemicA/H1N1influenzafrombeginningtotheendofNovemberwerequiteaccurate.H1N1caseswereunder-reportedsinceDecember.Note:BeforeOct2009,nodeathcaseswasreportedDailynumberofhospitalnotificationsofShaanxiProvince,ChinaDailyreportedcommunityandsporadiccasesofShaanxiProvince,ChinaNewlyreportedcasesfrom8thhospital

Dailynumberof8thhospitalnotificationsofXi’ancityThecountry'sexperiencewiththe2003SARSoutbreakenabledthecentralgovernmenttoquicklytakeasetofverystrictnonpharmaceuticalinterventions(NPIs)Intensivecontacttracingfollowed:Non-pharmaceuticalinterventions

byquarantineofsuspectedindividualswhohavethehighriskofhavingbeenexposedtothevirus

by

isolation

ofsymptomaticindividualsSchoolclosureandFengxiaoPrecaution:

travelorhygieneprecautionFengxiao

(封校)TherehasbeenoneA/H1N1confirmedcaseinourUniversity,universitystudents,faculty,andstaffmembersdonotallowtoleavetheircampuses,anddisallowon-campusvisitsModelamonguniversitycommunity

Ref:TangS.etal.(2010).Communitybasedmeasuresformitigatethe2009H1N1pandemicinChina.PLoSONE

IsolatedQuarantinedSusceptibleExposed,notyetinfectiousInfectiouswithsymptomsInfectiouswithsymptomsRecoveredInfectious,notyetsymptomaticInfectious,notyetsymptomaticExposed,notyetinfectiousFlowdiagramforthepandemicH1N1(Model1)Whenquarantineisimplemented,aproportionofE1isquarantined.TheseindividualsmovetothecompartmentQE1.ThoseintheQE1classthenprogresstotheQE2andwillbehospitalizedoncetheydevelopsymptoms.Wheneffectiveprecautionarymeasuresaretaken,aproportionoftheindividualsexposedtothevirusisprotectedfromtheinfection

TheModel1ControlreproductionnumberMeta-populationModel(orpatchmodel)Spatiallystratifiedcompartmentmodel

Extendourbaselinemodeltoameta-populationmodel,wherecouplingamongpatchesisthroughdispersalonadispersalnetwork

Usethismodelframework:

Thespreadamonganetworkofuniversities/collegeswithininacity(Xi’an)

ToevaluatetheeffectivenessofNPIsandinteractionsofdifferentspatiallyrelevantinterventions:

Fengxiao,quarantine,precautionandmobility

control.Dispersalnetworks

Therandomnetworkorsmall-worldnetworkintroducedbyWatts(WattsandStrogatz,1998)isemployedtogeneratematrixGwithanaveragenumberofconnectionspervertex(degree)offourNumericalintegrationformeta-populationmodel

NumericalintegrationsforthenetworkmodelswerecarriedoutusingtheRunge-KuttamethodinMatlab7.0.

Dispersalrates:

allsimulationswereinitiatedwithpseudorandomlygenerated

dispersalrates

independentlyandidenticallydistributed

amongallpatchesontheinterval

ImplementationofFengxiao:werandomlygeneratedthedispersalratesamongcommunitiesfromtheintervaltorepresenttheFengxiao,fromtheinterval()todescribeweak(strong)dispersal.ImplementationofFengxiaoImplementingFengxiaoaswellasstrengtheninglocalinterventionsinanyuniversity/collegeintermsofhospitalnotificationsHowtotriggerorsuspendFengxiaostrategy?

DefinetheupperthresholdofthehospitalnotificationssoastoswitchonFengxiaostrategy

DefinethelowthresholdofthehospitalnotificationssoastoswitchoffFengxiaostrategyh=4,nolocalcontrolmeasuresh=14,withorwithoutlocalcontrolmeasuresStrengtheningthelocalcontrolmeasures

Relaxingthelocalcontrolmeasures

500independentsimulationsarecarriedout,andmeansumofIclassandHclassareplotted

TheeffectivenessofFengxiaoandLocalmeasures(A-B)Fengxiao

alone.Magentacurve(withoutFengxiao);

Greencurve

(25,10);

Bluecurve(10,5)TheeffectivenessofFengxiaoandLocalmeasures(C-F)

Magentacurve(localcontrolonly);Green/blue

curve

(localandFengxiao)TheeffectivenessofFengxiaoandLocalmeasures(G-H)RelativelystronglocalcontrolmeasuresimplementedwhenFengxiaoissuspendedTheeffectivenessofFengxiaoandlocalmeasures

EarlyFengxiaocandelaytheepidemicpeaksignificantly

LateimplementationofFengxiaohaslittleeffectontheoutbreakThemagnitudesoftheoutbreaksbecomeweakerandweakeras

Fengxiaoandstrengtheningcontrolmeasuresareswitchedonandoff,andthesoonerthelocalcontrolmeasuresthelessseveretheoutbreaks

LocalcontrolstrategiesaffectsthepeakmagnitudeswhileFengxiaoinfluencesthepeaktimingandpreventsthediseasespreadtothegeneralpopulationModelwithinteractionsbetweentheuniversitycommunityandthegeneralpopulation

Hospitalnotificationsof8thhospitalofXi’ancityisemployedinthispartRef:Tang,S,Xiao,Y.,Yuan,L.,Cheke,R.A,Wu,J.,Campusquarantine(Fengxiao)forcurbingemergentinfectiousdiseases:LessonsfrommitigatingA/H1N1inXi'an,China,J.Theor.Biol.295(2012),47-58.ThemodelfortheUniversitypopulationUniversitypopulationUsingthenextgenerationmatrixmethodModelEquationsParameterdeterminationRecoveryrateforthesymptomaticclass

Recoveryrateforthehospitalizedclass

Estimationofthereproductionnumber

Model-basedestimateof

TheadaptiveMetropolis-HastingalgorithmisemployedtocarryouttheMCMCprocedure,andafteraburn-inperiodof500000iterationsthenext500000samplersgivesestimates(Haario2006,StatComput)Thelikelihood-basedmethodusingthefollowinglikelihood(WhiteandPagano(2008)),