国际数学建模竞赛优秀论文英文模板.docx

国际数学建模竞赛优秀论文英文模板.docx

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国际数学建模竞赛优秀论文英文模板

TeamControlNumber

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38253

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ProblemChosen

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2015MathematicalContestinModeling（MCM）SummarySheet

EradicatingEbola

Abstract

ThispaperaimattheproblemwhichistoeradicateorinhibitthespreadofEbola,westartfromthreesubproblem,thatis:

thedemandfordrugs,drugsdeliveryrouteandthecarallocation.AndestablishthespreadingmodelofEbola,optimizationmodelofdrugstransportsystemandcarallocationmodelrespectivelybyusingthedifferentialequationmethodandsimulatedannealingalgorithm.Finally,dothemodelextensionandsensitivelyanalysis.

Thefirstissue,figureoutthedemandfordrugsindifferentregions.First,establishEbolaspreadSIRmodel.Andinthetimeoft,usingdifferentialequationtofindtheproportionofinfectedi（t）=1/Qln（s/s0）,thengetthedemandfordrugsinthisregionH=kNi（t）.

Thesecondissue,howtofindtheshortestroutetodeliverdrugs.UseGuinea,LiberiaandSierraLeonewhoseinfectionisrelativelyseriousastheinvestigationobject.AccordingtotheBinaryclassificationtofindtherulesofiteration,whichisusefultofindoutthenearestcitytoanyothercities,andtheresultisBombali.Soweputitasthecenterofdistribution.Thenusesimulatedannealingalgorithmandputforwardtwokindsofschemesforshortestpathbythedifferentwaysindrugsdelivery.

Schemesone,asynchronousmode:

putthreecountriesasaregionalcountries.UsingtheTSPmethodtosolvetheshortestrouteis54.8486,whichisstartfromBombalitodifferentregions.

Schemestwo,synchronizationmethod:

dividingthewholeareaintotwoareasaroundAandBbyusethelongitudecoordinatesofBombaliasastandard.Respectivelysolvetheshortestrouteis10.1739and29.8075,whichisstartfromBombaliandpassallcitiesinAandB,andsolvethesumofthetworouteis39.9814.

Accordingtothedifferentdrugdeliveryrequirements（suchastheshortestdistanceortransmissionsynchronization）,canchoosetheasynchronousorsynchronousway.

Thethirdissue,howtoallocatethenumberofcarsreasonable,andobtainthesuitablespeedofdrugproduction.Accordingtothepredictnumberwhichobtainedinmodelone,getthevehiclesanddrugdistributiontable（theresultsareshownTable4.6andTable4.7）.andobtainthespeedVofdrugsproductionis:

Atlast,theminimumspeedofdrugsproductionis56.14agent/daytomeettheneedinthreecountriesbycalculating.

Finally,usetheSIRmodelwhichwasoptimizedbyusingvaccinationcyclecontrol.Bydoingthiswecanknowthenumberofsusceptibleandinfectionsincrowdundertheconditionofthepulsevaccinationsignificantlylowerfasterthanwithoutpulsevaccination.Thus,usingpulsevaccinationcaneffectivelycontrolthespreadofEbola.

Keywords:

SIRmodel;SimulatedAnnealingAlgorithm;Pulsevaccination;Ebola

EradicatingEbola

Content

1RestatementoftheProblem

1.1Introduction

Ebolavirusisaveryrarekindofvirus.ItcancausehumansandprimatesproduceEbolahemorrhagicfevervirus,andhasahighmortalityrate.ThelargestandmostcomplexEbolaoutbreakappearedintheWestAfricancountryin2014.Thisoutbreakoccurredinguineafirst,thenthroughvariouswaystocountriessuchasSierraLeone,Liberia,NigeriaandSenegal.Thenumberofcasesanddeaths,whichoccurredinthisoutbreak,ismorethanthesumofalltheotherepidemic.Andoutbreakcontinuedtospreadbetweencountries.OnAugust8,2014,thegeneral-directoroftheworldhealthorganizationannouncedtheoutbreakofpublichealthemergencyofinternationalconcern.

Inthispaper,arealisticandreasonablemathematicmodel,whichconsidersseveralaspectssuchasvaccinemanufacturinganddrugdelivery,hasbeenbuilt.ThenoptimizingthemodeltoeliminateorsuppresstheharmdonebytheEbolavirus.

1.2TheProblem

Establishingamodeltosolvethespreadofthedisease,amountofdrugsneeded,possiblefeasibletransportationsystem,transportingposition,thespeedofavaccineordrugmanufacturingandanyotherkeyfactor.Thus,wedecomposetheproblemintothreesub-problem,modelingandfindingtheoptimizationmethodtofacetheEbolavirus.

♦Buildingamodel,whichcansolvethespreadofthediseaseandthedemandfordrugs.

♦Buildingamodeltofindthebestsolution.

♦Usingthegoalprogrammingtosolvetheproblemsofproductionanddistributionandoptimizationofotherfactors.

.

2GeneralAssumptions

Tosimplifytheproblem,wemakethefollowingbasicassumptions,eachofwhichisproperlyjustified.

♦Ourassumptionsisreasonableandeffective.

♦Vehiclesonlyruninthepathwhichwehavesimulated

♦Thisassumptiongreatlysimplifyourmodelandallowustofocusontheshortestpath.

♦Weconsiderthemodelthatareenclosed.

♦Peoplewhorecovered,willnotinfectedagain,andexitthetransmissionsystem

3VariablesandAbbreviations

ThevariablesandabbreviationsusedinthispaperarelistedinTable3.1.

Table3.1Assumingvariable

Symbol

Definition

S

thenumberofsusceptiblepeople

I

thenumberofinfectedpersons

R

thenumberofrecovered

T

avaccineordrugproductioncycle

H

theamountofdrugsneededbyRegion

A

acycleofavaccineordrugproduction

L

drugreserveareatotheshortestpathtoallaffectedareas

V

speedofvaccineorpharmaceuticalproduction

V’

vehiclespeed

λ

rateofpatientcontactperday

μ

daycurerateperday

αn

rightsofthoseinfectedregionsweight

4ModelingandSolving

4.1ModelI

4.1.1AnalysisoftheProblem

Accordingtotheliteraturethatdifferenttypesofvirushasitsowndifferentpropagationprocesscharacteristics,wedonotanalyzethespreadofvirusesfromamedicalpointofview,butfromthegeneraltoanalyzethepropagationmechanism.SowehavetoanalyzethespreadoftheEbolavirusandtherequirementsofdrugsthroughtheSIR[1]model.

4.1.2ModelDesign

Inthedynamicsofinfectiousdiseases,themainfollowKermackandMcKendrickSIRepidemicmodelwhichthedynamicsoftheestablishedmethodin1927.SIRmodeluntilnowisstillwidelyusedandcontinuetodevelop.SIRmodelofthetotalpopulationisdividedintothefollowingthreecategories:

susceptibles,theratioofthenumberdenotedbys（t）,attimetisnotlikelytobeinfected,butthenumberofinfectiousdiseasessuchproportionofthetotal;infectives,theratioofthenumberdenotedbyi（t）,attimetbecomeapatienthasbeeninfectedandhastheproportionofthetotalnumberofcontagious;recovered,theratioofthenumberdenotedbyr（t）,expressedthenumberofthoseinfectedattimetremovedfromthetotalproportion（ie,ithasquitinfectedsystems）.AssumingatotalpopulationofN（t）,thenthereareN（t）=s（t）+i（t）+r（t）.

SIRmodelisestablishedbasedonthefollowingtwoassumptions:

♦Intheinvestigatedregion-widespreadofthediseaseisnotconsideredduringthebirths,deaths,populationmobilityandotherdynamicfactors.TotalpopulationN（t）remainunchanged,thepopulationremainsaconstantN.

♦Thepatients’contactrate（theaveragenumberofeffectivecontactsperpatientperday）isconstantλ,thecurerate（patientsbecuredproportionofthetotalnumberofpatientsaday）isaconstantμ,clearlytheaverageinfectiousperiodof1/μ,infectiousperiodcontactnumberforQ=λ/μ.

Inthemodelbasedontheassumptionthatwedevelopasusceptiblepersontorecoverfromthesickpersonintheprocess,suchasFigure4.1:

Figure4.1SIRthemodelflowchart

SIRbasisdifferentialequationmodelcanbeexpressedas:

Butitcanseethats（t）,i（t）ismoredifficulttosolve,soweusethenumericalcalculationstoestimategeneralvariation.Assumingλ=1,μ=0.3,i（0）=0.02,s（0）=0.98（attheinitialtime）,thenweborrowMATLABsoftwareprogrammingtogetresults.AndaccordingtoTable4.1analyzedi（t）,s（t）ofthegeneralvariation.

Table4.1Assumingvariable

t

i（t）

s（t）

t

i（t）

s（t）

0

0.02

0.98

9

0.2863

0.1493

1

0.039

0.9525

10

0.2418

0.1145

2

0.0732

0.9019

15

0.0787

0.0543

3

0.1285

0.8169

20

0.0223

0.0434

4

0.2033

0.6927

25

0.0061

0.0408

5

0.2795

0.5438

30

0.0017

0.0401

6

0.3312

0.3995

35

0.0005

0.0399

7

0.3444

0.2839

40

0.0001

0.0399

8

0.3247

0.2027

45

0

0.0398

Figure4.2s（t）,i（t）Thepatientscalemap

Figure4.3i～sPhasetrackdiagram

FromTable4.1andFigure4.2,wecanseethati（t）increasedfromtheinitialvaluetoaboutt=7（maximum）,andthenbegantodecrease.

Basedonthecalculatingthenumericalandgraphicalobservation,useofphasetrajectoriesdiscussedi（t）,s（t）innature.Herei~splaneisphaseplane,thedomain（s,i）∈Dinphaseplanefor:

AccordingtoequationandcontactnumberoftheinfectiousperiodQ=λ/μ,wecaneliminatedt,get:

Calculatedusingintegralcharacteristics:

Curveinthedomainofdefinition,equationisaphasetrajectory.

Accordingtoequationandequation,havetoanalyzethechanges.Ifandonlyifthepatienti（t）forsomeperiodofgrowth,itthinkthatinthespreadofinfectiousdiseases,then1/Qisathreshold.Ifs0>1/Q,infectiousdiseaseswillspread,andreduceinfectiousperiodthenumberofcontactswithQ,namelyraisingthethreshold1/Qandwillmakes0≤1/Q,thenitwillnotspreaddiseases.

AndwenotethatQ=λ/μintheformula,thehigherthelevelofpeople'shealth,thesmallerpatients’contactrate;thehigherthelevelofmedical,thecurerateislargerandthesmallerQ.Therefore,toimprovethelevelofhygieneandmedicalhelptocontrolthespreadofinfectiousdiseases.Ofcourse,