Modeling Heterogeneities in Malaria

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Title:
Modeling Heterogeneities in Malaria
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1 online resource (138 p.)
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english
Creator:
Prosper, Olivia Felicia
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University of Florida
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Gainesville, Fla.
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Degree:
Doctorate ( Ph.D.)
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University of Florida
Degree Disciplines:
Mathematics
Committee Chair:
Martcheva-Drashanska, Maia
Committee Co-Chair:
Smith, David L
Committee Members:
Hager, William W
Chen, Yunmei
Ponciano, Jose M.

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Subjects / Keywords:
biology -- competition -- control -- differential -- epidemiology -- equations -- estimation -- falciparum -- invasion -- malaria -- mathematical -- metapopulation -- model -- multi -- number -- optimal -- ordinary -- parameter -- reproduction -- species -- vaccination -- vivax
Mathematics -- Dissertations, Academic -- UF
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Mathematics thesis, Ph.D.
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Abstract:
Malaria is an extremely complex disease that continues to thwart the efforts to control it in many countries around the globe. This dissertation presents three mathematical models developed to address two types of heterogeneities inherent to malaria ecology and their implications for malaria dynamics and malaria control: heterogeneity in the parasite species within a population, and heterogeneity in the transmission landscape. First, we developed a mathematical model describing the dynamics of P. vivax and P. falciparum in the human and mosquito populations and fit this model to clinical case data to understand how improving control measures affects the competition between the two Plasmodium species. The predictions of our model are counter to what one expects based on the case data alone. Although the proportion of cases due to falciparum has been increasing, our model reveals that this observation is insufficient to draw conclusions about the long-term competitive outcome of the two species. Next, we considered a simple malaria metapopulation model incorporating two regions connected by human movement, with different degrees of malaria transmission in each region. If without migration, the disease is endemic in one region but not in the other, then the addition of human migration can cause the disease to persist in both regions. The advent of a malaria vaccine available for distribution is potentially around the corner. This exciting possibility highlights the need to study the effect of implementing a malaria vaccination program on malaria dynamics, and to determine optimal ways of distributing this limited resource. To that end, we developed an n-patch malaria metapopulation model with vaccination and short-term human movement. We formulated an optimal control problem based on this model and the goal of jointly minimizing the number of infected individuals, the number of vaccines distributed, and the vaccination rate. A qualitative comparison of the optimal control strategies for different vaccine efficacies and different short-term movement patterns confirms our intuition for how the optimal strategy changes. In particular, we should sustain the maximum allowable vaccination rate longer in that patch with higher transmission intensity.
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In the series University of Florida Digital Collections.
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Includes vita.
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This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility:
by Olivia Felicia Prosper.
Thesis:
Thesis (Ph.D.)--University of Florida, 2012.
Local:
Adviser: Martcheva-Drashanska, Maia.
Local:
Co-adviser: Smith, David L.
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RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2013-05-31

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MODELINGHETEROGENEITIESINMALARIAByOLIVIAF.PROSPERADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2012

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c2012OliviaF.Prosper 2

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Idedicatethiswork,rstandforemost,tomyparentswhohavealwaysbeensosupportiveandencouragingofme,whomIlovesoverymuch,andcouldn'thaveaccomplishedthiswithout;tomysisterandmostspecialfriend,whoIwillneverstopthinkingofasmylittlesister;tomypartner,whounderstandswhyIdowhatIdoandlovesmemoreforit;toMamie,whoIadmiresomuchforthestrongandindependentlifeshehaslived;toGrandmaandGrandad,twoofthemostcaringpeopleIknow;andinlovingmemoryofmyauntsandunclewhoImisssodearly.Aspecialthankyougoestomymother,Dr.Marie-FranceProsper-Chartier.Iamsoproudofwhatshehasaccomplished,havinggrownuponasmallfarminPlesderwithnotverymuch,andallwhileraisingafamily.Shehasinspiredmetobestrong,hard-working,andindependent.Becauseofher,Ihadtheopportunitytoexcelinmyeducationandhavethecondencethatachievingthismilestonewaspossible.Itrulycouldnothavedonethiswithouther. 3

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ACKNOWLEDGMENTS Firstandforemost,Iwanttothankmydoctoraladvisor,Dr.MaiaMartcheva,forherguidanceandsupportoverthepastfewyears.ShewentbeyondthenecessaryresponsibilitiesofanadvisorandIfeelveryfortunatetohavehadherasamentor.Notonlyissheanexcellentmathematicianandexcitedtopassonthisknowledgetoherstudents,butshealsocaresaboutherstudentsbeyondouracademicworkandwillinglygiveshersupportandadvicewhenwemostneedit.IthankDr.CarlosCastillo-Chavezforhelpingmejump-startmyroleasaresearcherinmathematicalbiology.Despitehisbusyscheduleandmanyresponsibilities,heisalwaysquicktorespondtomyquestionsandofferhelpwhenIneedit.Ithankmyco-advisor,Dr.DavidSmith,forprovidinghonestfeedbackthatwillcertainlyhelpshapemyfuturework.Iadmirehisvastknowledgeaboutmalariaandhiscontributionstomalariaresearch.IthankDr.WilliamHager,Dr.YunmeiChen,andDr.JoseMiguelPoncianoCastellanosfortakingthetimetoserveonmydoctoralcommittee,andfortheirvaluablefeedbackaboutmyresearch.IwouldliketothankDr.ChristinaChiyakaforherinputduringtheearlystagesinthedevelopmentofthePlasmodiumfalciparum-PlasmodiumvivaxmodelpresentedinChapter 2 .IgivespecialthankstomyQSE3IGERTteammatesMiguelAcevedo,TrevorCaughlin,KennyLopiano,andNickRuktanonchaiforplayinganactiveroleinourweeklyresearchdiscussionsabouttheroleofhumanmovementinmalariadynamics,andtoDr.CraigOsenbergforhisguidanceandsupportthroughoutthisproject.WorkingwiththisgrouphasbeenawonderfulexperienceandIamsogratefulforwhatIhavelearnedfromthem.Inparticular,IwanttoacknowledgeNickRuktanonchai'sroleintheresearchabouttheroleofhumanmovementinmalariadynamicsandtheoptimalcontrolofmalariawithvaccines.Thisresearchbenetedgreatlyfromhisinsightintobiologicalproblems,creativity,andenthusiasmforworkingattheinterfaceofmathematicsandbiology.IwanttothankDr.BenBolkerforhisexcellentcoursesintroducingmetotheecologyofinfectiousdiseasesandstatisticalmethods.Inparticular,mysemester 4

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internshipthathementoredplayedabigroleinthedirectionofmydoctoralresearch.IthankmyamazingparentsHarrisonandMarie-France,mylittlesisterLeila,mypartnerMatthew,andmyfamily-to-befortheirlove,encouragement,andsupport.ThisresearchwaspartiallysupportedbytheNationalScienceFoundationunderGrantNo.0801544intheQuantitativeSpatialEcology,EvolutionandEnvironmentProgramattheUniversityofFloridaandbytheEmergingPathogensInstituteattheUniversityofFlorida. 5

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 8 LISTOFFIGURES ..................................... 9 ABSTRACT ......................................... 10 CHAPTER 1INTRODUCTION ................................... 12 1.1TheBeginningsofMalariaModeling ..................... 13 1.2HeterogeneitiesinMalariaEcology ...................... 17 2IMPACTOFENHANCEDMALARIACONTROLONTHECOMPETITIONBETWEENPLASMODIUMFALCIPARUMANDPLASMODIUMVIVAX ..... 20 2.1PlasmodiumvivaxandPlasmodiumfalciparumParasitesandObstaclesTheyPosetoMalariaControl ......................... 20 2.2ObjectivesofModelingP.vivax-P.falciparumDiseaseDynamics ..... 21 2.3P.falciparumandP.vivaxMalariaCo-infectionModel ........... 22 2.3.1ModelingtheDynamicsofPlasmodiumvivaxInfectionintheHumanPopulation ................................ 22 2.3.2ModelingtheDynamicsofPlasmodiumfalciparumInfectionintheHumanPopulation ........................... 24 2.3.3ModelingCo-Infection ......................... 24 2.3.4DiseaseDynamicsintheMosquitoPopulation ............ 26 2.3.5DerivationoftheDisease-FreeEquilibrium,BasicReproductiveNumberR0andControlReproductiveNumberRC .......... 30 2.3.6ExpressionofRCandR0DerivedfromNextGenerationApproach 32 2.3.7IsolatedEndemicEquilibriaandCoexistence ............ 39 2.3.8InvasionNumbersRfvandRvf ..................... 41 2.3.9FindingRvfUsingtheNext-GenerationApproach .......... 41 2.3.10FindingRfvUsingtheNext-GenerationApproach .......... 45 3PARAMETERESTIMATIONANDEVALUATINGUNCERTAINTY ........ 53 3.1DescriptionofModelParametersandChoiceofParameterValuesfortheYearsfrom1987to1996 ......................... 53 3.1.1EstimationofParametersfromtheLiterature ............. 53 3.1.1.1Timetoinfectiousness .................... 53 3.1.1.2Estimatingrecoveryrates .................. 55 3.1.1.3ParameterizingP.vivaxrelapse .............. 55 3.1.1.4Estimationof ........................ 56 6

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3.1.1.5Estimationof ........................ 57 3.1.2EstimationofPopulationGrowthandTransmissionParametersUsingPopulationandMalariaCaseDataforIndia .......... 57 3.1.3EstimationofParametersfortheEnhancedMalariaControlPeriod 60 3.2Discussion ................................... 67 4ASSESSINGTHEROLEOFSPATIALHETEROGENEITYANDHUMANMOVEMENTINMALARIADYNAMICSANDCONTROL ............. 70 4.1Ross-MacdonaldModel ............................ 73 4.1.1ModicationstoRoss-MacdonaldModel ............... 74 4.1.2Two-PatchMalariaModel ....................... 75 4.1.3TheBasicReproductiveNumberR0 ................. 77 4.2SensitivityAnalysis ............................... 82 4.2.1ElasticitiesforaSinglePatchwithoutMigration ........... 83 4.2.2ElasticitiesfortheTwo-PatchMetapopulationModel ........ 84 4.3NumericalResults ............................... 86 4.3.1ParameterEstimates .......................... 86 4.3.2EffectofMigrationonR0 ........................ 88 4.3.3EffectofMigrationonElasticity .................... 91 4.4ElasticityAnalysisofEndemicEquilibrium .................. 96 4.5Conclusion ................................... 101 5OPTIMALTEMPORALDISTRIBUTIONOFMALARIAVACCINESINAHETEROGENEOUSTRANSMISSIONLANDSCAPE ............... 104 5.1MalariaModelswithVaccination ....................... 104 5.2OptimalControlTheoryAppliedtoMalariaModels ............. 105 5.3MalariaModelwithShort-TermMovementandVaccination ........ 106 5.4OptimalControlProblem ............................ 111 5.5NumericalResults ............................... 117 5.5.1VaccinationModelParameters .................... 117 5.5.2NumericallySolvingtheOptimalControlProblem .......... 120 5.5.2.1Optimalstrategyinamalariaendemicsetting ....... 120 5.5.2.2Optimalstrategyinanon-endemicsetting ......... 123 5.6FutureDirections ................................ 126 REFERENCES ....................................... 129 BIOGRAPHICALSKETCH ................................ 138 7

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LISTOFTABLES Table page 2-1DescriptionofmodelStateVariablesattimet. .................. 27 2-2Descriptionofmodelparameters .......................... 31 3-1Descriptionofmodelparameterspertainingtomosquitopopulationdynamicsandtheirestimates .................................. 53 3-2Descriptionofmodelparameterspertainingtohumanpopulationdynamicsandtheirestimates .................................. 54 3-3EstimatesofrandK. ................................ 58 3-4Pre-1997estimatesofthetransmissionparameters. ............... 58 3-5Pre-1997mean,median,standarddeviations,andcondenceintervalsforRCvandRCf,derivedfromparametricbootstrap. ................. 59 3-6Post-1996modelsorderedby4AICcvalue(differencefrombestAICcvalue-123.9). ........................................ 64 3-71997-2010estimatesofRCvandRCfforeachcandidatemodel. ......... 64 3-8Likelihoodofcompetitiveoutcomespredictedbyparametricbootstrap ..... 66 4-1Descriptionofmodelparametersinpatchi .................... 76 4-2Wetanddryconditionestimatesofmodelparametersforlowandhightransmissionsettings ........................................ 87 4-3R0forthefourscenarios ............................... 88 4-4TargetpatchformalariacontrolintheR01>R02>1case ............ 96 4-5TargetpatchformalariacontrolintheR01>1>R02case ............ 96 5-1Descriptionofvaccinationmodelpatchistatevariables ............. 108 5-2Descriptionofvaccinationmodelpatchiparameters ............... 109 5-3Descriptionofvaccinationmodelpatchiparametervaluesindays)]TJ /F5 7.97 Tf 6.59 0 Td[(1 ...... 119 8

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LISTOFFIGURES Figure page 2-1P.falciparum-P.vivaxMalariaModelDiagram ................... 28 3-1Logisticmodelttedtopopulationdata ....................... 57 3-2HistogramsofestimatedReproductionandInvasionNumbersfortheyears1987-1996 ...................................... 60 3-3Fittingmodeltodata ................................. 62 3-4InvasionBoundaries ................................. 63 3-5Predictedoutcomeoftopmodel .......................... 65 3-6HistogramofestimatedReproductionandInvasionNumbersfortheyears1997-2010 ...................................... 67 4-1MalariaMetapopulationModelDiagram ...................... 75 4-2GlobalR0asafunctionofmigration ........................ 89 4-3MalariadynamicsinR01>R02>1settingwithandwithoutmigration ...... 90 4-4MalariadynamicsinR01=R02<1settingwithandwithoutmigration ...... 91 4-5MalariadynamicsinR01>1>R02settingwithandwithoutmigration ...... 92 4-6MalariadynamicsinR01>1>R02settingwithk21>>k12 ............ 92 4-7ElasticiyofR0inR01>1>R02scenario ...................... 93 4-8ElasticityofR0inR01>R02>1scenario ...................... 95 4-9ElasticityofEndemicEquilibriuminR01>1>R02andR01>R02>1settings 99 5-1Diagramofdiseasedynamicsinhumans. ..................... 110 5-2Comparisonofoptimalvaccinationstrategyinanendemicsettingunderdifferentmovementpatterns .................................. 123 5-3Comparisonofoptimalvaccinationstrategyinanendemicsettingunderdifferentmovementpatterns .................................. 124 5-4Comparisonofoptimalvaccinationstrategyinanon-endemicsettingunderdifferentmovementpatterns ............................. 126 5-5Comparisonofoptimalvaccinationstrategyinanon-endemicsettingunderdifferentmovementpatterns ............................. 127 9

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AbstractofdissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyMODELINGHETEROGENEITIESINMALARIAByOliviaF.ProsperMay2012Chair:MaiaMartchevaCochair:DavidSmithMajor:MathematicsMalariaisanextremelycomplexdiseasethatcontinuestothwarttheeffortstocontrolitinmanycountriesaroundtheglobe.Thisdissertationpresentsthreemathematicalmodelsdevelopedtoaddresstwotypesofheterogeneitiesinherenttomalariaecologyandtheirimplicationsformalariadynamicsandmalariacontrol:heterogeneityintheparasitespecieswithinapopulation,andheterogeneityinthetransmissionlandscape.First,wedevelopedamathematicalmodeldescribingthedynamicsofP.vivaxandP.falciparuminthehumanandmosquitopopulationsandtthismodeltoclinicalcasedatatounderstandhowimprovingcontrolmeasuresaffectsthecompetitionbetweenthetwoPlasmodiumspecies.Thepredictionsofourmodelarecountertowhatoneexpectsbasedonthecasedataalone.Althoughtheproportionofcasesduetofalciparumhasbeenincreasing,ourmodelrevealsthatthisobservationisinsufcienttodrawconclusionsaboutthelong-termcompetitiveoutcomeofthetwospecies.Next,weconsideredasimplemalariametapopulationmodelincorporatingtworegionsconnectedbyhumanmovement,withdifferentdegreesofmalariatransmissionineachregion.Ifwithoutmigration,thediseaseisendemicinoneregionbutnotintheother,thentheadditionofhumanmigrationcancausethediseasetopersistinbothregions.Theadventofamalariavaccineavailablefordistributionispotentiallyaroundthecorner.Thisexcitingpossibilityhighlightstheneedtostudytheeffectofimplementingamalariavaccinationprogramonmalariadynamics,andtodetermine 10

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optimalwaysofdistributingthislimitedresource.Tothatend,wedevelopedann-patchmalariametapopulationmodelwithvaccinationandshort-termhumanmovement.Weformulatedanoptimalcontrolproblembasedonthismodelandthegoalofjointlyminimizingthenumberofinfectedindividuals,thenumberofvaccinesdistributed,andthevaccinationrate.Aqualitativecomparisonoftheoptimalcontrolstrategiesfordifferentvaccineefcaciesanddifferentshort-termmovementpatternsconrmsourintuitionforhowtheoptimalstrategychanges.Inparticular,weshouldsustainthemaximumallowablevaccinationratelongerinthatpatchwithhighertransmissionintensity. 11

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CHAPTER1INTRODUCTIONDespitemanyeffortstoreducetheglobalburdenofmalaria,thisformidablediseasecontinuestoposeasubstantialpublichealthproblem.Roughly250millionpeoplesufferfrommalariainfectioneachyear,resultinginnearlyonemilliondeaths[ 108 ].Malariaisthefthleadingkilleramonginfectiousdiseasesworldwide,anditisthesecondleadingcauseofdeathinAfrica,behindHIV/AIDS[ 22 ].Severalfactors,includingthebiologyandepidemiologyofthedisease,environmentalconditions,differencesinmalariaparasitespecies,emergingdrug-resistanceofparasites,insecticideresistanceofmosquitoes,andsocio-economicbarriers,haveproventobedifcultobstaclestoovercomeintheongoingpursuitofmalariacontrol.Becausemalariapresentsthegreatestthreatinlessdevelopedcountries,determiningeffectivecontrolstrategiesthatmakeefcientuseoflimitedresourcesisessential.Mathematicalmodelingprovidesameanstostudymalariadynamicsandevaluatepotentialcontrolstrategies.Untilrecently,manymathematicalmodelsofmalariaconsideredasingle,homogeneouspopulationharboringasinglemalariaparasitespecies.Thesesimplemodelshavecontributedtoourcurrentknowledgeaboutmalaria.However,thecomplexityofmalariaecologyrequiresmodelingeffortsthataddresssomeoftheseintricacies.Weinvestigatedtherolesthatheterogeneityinmalariaparasitespeciesandspatialheterogeneityinmalariatransmissionplayinmalariadynamicsandtheimplicationstheseheterogeneitieshaveformalariacontrol.Webeginbypresentinginthischapterabriefhistoryofhowmalariamodelingbegananddiscussingafewextensionsoftheclassicalmalariamodel,followedbyasummaryofhowheterogeneitieshavebeenincorporatedintomalariamodels.InChapter 2 wepresentandanalyzeatwo-parasitemodelofmalaria.Chapter 3 demonstrateshowthismodelcanbeusedincombinationwithmalariacasedataandstatisticaltechniquestounderstandtheimpactofmalariacontrolonmalariaparasitecompetition.Theroleofhumanmigrationinaspatially 12

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heterogeneoustransmissionlandscapeisstudiedinChapter 4 .Finally,inChapter 5 weintroduceann-patchmalariamodelwithvaccinationandshort-termhumanmovement.Wepresentanoptimalcontrolprobleminwhichtheobjectiveistodeterminethebestwaytodistributemalariavaccinesovertimeinaspatiallyheterogeneoustransmissionlandscape. 1.1TheBeginningsofMalariaModelingDuringthelate19thcentury,SirRonaldRossconrmedearliertheoriesthatmosquitoesplayedanimportantroleinthetransmissionofmalaria.RossfoundmalariaparasitesinthegutoffemaleAnophelesmosquitoesthatpreviouslyfedonhumans.Knowingthatmosquitoeslaytheireggsinwater,Ross'mentor,Dr.PatrickManson,hypothesizedthathumansacquiredmalariabydrinkingwatertaintedwithmalariaparasites.However,Rossdiscoveredthatmalariaistransmitteddirectlyfromamosquitotoahumanthroughthebiteofthemosquito[ 1 ].Shortlyafterthisdiscovery,Rosswentontodeveloptherstmathematicalmodelformalaria,whichwaslatermodiedbyGeorgeMacdonaldin1957[ 62 ].Severalvariationsofthismodelappearinthemalariamodelingliterature,howeverallversionshavethreecommonassumptions[ 62 ]: malariadoesnotconferimmunitytofutureinfectionsinhumans mosquitobitesaredistributedrandomlyacrossthehumanpopulation humanandmosquitopopulationsarehomogeneousThemostcommonformulationoftheRoss-Macdonaldmodelisdescribedbythefollowingtwo-dimensionalsystemofordinarydifferentialequations:dx dt=maby(1)]TJ /F4 11.955 Tf 11.95 0 Td[(x))]TJ /F4 11.955 Tf 11.95 0 Td[(rxdz dt=acx(1)]TJ /F4 11.955 Tf 11.95 0 Td[(z))]TJ /F4 11.955 Tf 11.96 0 Td[(gz,wherexandzdenotethefractionofhumansandmosquitoes,respectively,infectedwithmalaria.Theratioofmosquitoestohumansisdenotedbyparametermandais 13

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thenumberofbitesonahuman,permosquito,perunitoftime.Consequently,maisthenumberofmosquitobitesperhuman,perunitoftime.Multiplyingmabyzyieldsthenumberofinfectedmosquitobitesperhuman,perunitoftime.Finally,bisthefractionofinfectiousbitesresultinginahumaninfection.Since(1)]TJ /F4 11.955 Tf 12.8 0 Td[(x)denotesthefractionofhumanssusceptibletomalaria,maby(1)]TJ /F4 11.955 Tf 12.84 0 Td[(x)describestherateatwhichtheproportionofhumancaseschanges.Humansthenrecoveratarecoveryrater.Therateofchangeintheproportionofinfectedmosquitoesisderivedsimilarly.Thenumberofmosquitobitesoninfectedhumansisax,cisthefractionofbitesresultinginaninfectioninthemosquito,ifthemosquitoissusceptible.Because(1)]TJ /F4 11.955 Tf 12.26 0 Td[(z)denotesthefractionofthemosquitopopulationthatremainssusceptible,thefractionofinfectedmosquitoeschangesattherateacx(1)]TJ /F4 11.955 Tf 12.82 0 Td[(z).Mosquitoeshaveveryshortlifespans,andconsequently,thismodelassumesthatmosquitoesdieduetonaturalcausesatamortalityrategbeforetheywouldhaveachancetorecoverordieasaconsequenceofthemalariainfection.Variationsonthismodelincorporatetheextrinsicincubationperiod(theamountoftimeittakesthemalariaparasitestodevelopinsidethemosquitobeforethemosquitobecomesinfectious),whichplaysanimportantroleinmalariadynamicsbyreducingthepoolofmosquitoesabletosurvivebeyondtheincubationperiodandtransmitmalariatohumans.WediscusssuchavariationontheRoss-MacdonaldmodelinChapter 4 .MathematicalanalysesoftheRoss-Macdonaldmodelrevealedseveralresultsrelevanttomalariacontrol.First,Rossdeterminedfromthemodelthatitisnotnecessarytoeliminateallmosquitoestoeliminatemalariafromapopulation;reducingthenumberofmosquitoesbelowacertainthresholdshouldbesufcient.Ameasureoftransmissionintensity,thebasicreproductionnumberR0canbederivedforthemodel.IfR0<1,thedisease-freeequilibriumisgloballystableandifR0>1,theendemicequilibriumisgloballystable.Moreover,R0hasaverynonlinearrelationshipwithmalariaprevalence:whenR0isclosetothethresholdone,prevalenceisverysensitiveto 14

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changesinR0,whereasprevalenceisveryrobusttochangesinR0whenR0islarge[ 62 ].TheimpactofRossandMacdonald'sworkextendsbeyondtheanalysisoftheirclassicalmodel.Theclassicalmodellaidastrongfoundationuponwhichmanymalariamodelshavebeenbuilt.OneextensionoftheRoss-Macdonaldmodeladdressesthephenomenonofnaturallyacquiredimmunitytomalaria.Malariainfectionsdonotconferlife-longimmunitytore-infection,howeverrepeatedexposuretoinfectiousmosquitobitesproducesanimmune-responseinhumans,resultingindistinctage-relatedpatternsofprevalenceinregionsofhighmalariaendemicity[ 7 101 ].Awell-knownmalariamodelincorporatingtemporarynaturally-acquiredimmunitywasdevelopedbyDietzetal.[ 41 ]inthe1970's.Thisdeterministicdifferenceequationmodelincludessevenstatevariablesassociatedwithhumandynamics,withmosquitomalariadynamicsincorporatedthroughasingletermrepresentingvectorialcapacity,thenumberofinfectivebitesasinglehostreceiveseachday.Dietzetal.derivedanexpressionforthehumanrecoveryrateasafunctionoftheinoculationratesothatindividualswithmanyexposurestomalariawouldhavelongerrecoverytimes.AcomparisonofthemodelsimulationstoelddatafromtheAfricansavannahrevealedthatthemodelisabletocapturesomeoftheobservedpatterns,suchastheparasiteratedistributionacrossdifferentagegroups.However,themodelfailedtoproduceseasonaluctuationsinprevalenceobservedinthedata[ 41 ].In[ 7 ],Aronincorporatedwaningimmunityusingadelaydifferentialequationmodel.Inthisdelaymodel,immunitytomalariaisboostedifre-exposureoccurswithintimeunitsofenteringtheimmunestage.Ifanindividualisnotre-exposedtomalariawithinthetimeperiod,thisindividuallosesthenaturallyacquiredimmunity.TheRoss-Macdonaldmodelhasalsobeenadaptedtoincludelatentperiods,theperiodittakesforaninfectedhost,mosquitoorhuman,tobecomeinfectious.Ratherthanexplicitlyincludingalatentstage,SharpeandLotka[ 98 ]rstintroduced 15

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latency,whichtheycalledtheincubationlag,totheclassicalmodelin1923usingadelay-differentialequation(DDE)formulation.InthisDDEsystem,thenumberofnewhumaninfectionsdependsonthenumberofmosquitoesthatbecameinfectedsomeutimeunitsago.Similarly,thenumberofnewmosquitoinfectionsdependsonthenumberofinfectedhumansthatbecameinfectedsomevtimeunitsago.Later,AndersonandMayincorporatedtheselatentperiodsbyexplicitlyaddinganincubatingstagetothemosquitoandhumandiseasedynamics[ 6 ].Ruanetal.[ 95 ]developedaslightlydifferentDDEmodelcomparedtotheDDEmodelofSharpeandLotka.Inparticular,Ruanetal.'smodelconsiderstheprobabilitythatamosquitosurvivestheincubationperiodandtheprobabilitythatahumandoesnotrecoverbeforeincubationiscomplete.TheresultsofRuanetal.'sworksuggestthatdrugsprolongingtheincubationperiodsineitherhumansormosquitoesmaybeaneffectivemalariacontrolstrategy[ 95 ].Manymalariamodels,includingtheclassicalRoss-Macdonaldmodel,assumethehumanandmosquitopopulationsizesareconstant.Thisassumption,whileappropriateforthestudyofmalariaovershorttimeintervals,doesnotadequatelyreectrealitywhenconsideringlong-termdynamicsorlong-termeffectsofmalariacontrol.Ngwaetal.[ 80 ]assumeaconstantbirthrateandadopttheideaofadensity-dependentdeathratefrompredator-preymodels.Functionalformsofthesedensitydependentdeathrates,inboththemosquitoandhumanpopulations,arechosensuchthateachpopulation'sgrowthismodeledbythelogisticequationintheabsenceofdisease.Analysisofthisvariablepopulationsizemodelrevealedthatreducingthemosquitopopulationsizewouldhavelittleeffectonmalariaprevalencewhentransmissionintensityishigh,sincethemosquitopopulationwilleventuallyrecover[ 80 ].Inotherwords,modelsignoringmosquitopopulationdynamicsmayunderestimatetheeffortrequiredtosuccessfullyreducemalariaprevalence.ThemodelsdescribedherecompriseonlyatinyfractionofpublishedmodelsthatexpandontheideasofRossandMacdonald.Thesemodelsillustratehowgradually 16

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introducingdifferentaspectsofmalariabiologytosimplemodelscaninformusabouthowtheseaspectsofthebiologycontributetotheoveralldiseasedynamics.Inthefollowingsection,wediscussaselectionofheterogeneitiesthatoccurnaturallyinmalariaecology,andhowmathematicalmodelshavebeenadaptedanddevelopedtoincorporatetheseadditionalcomplexities. 1.2HeterogeneitiesinMalariaEcologyMalariamodelingbecomesanincreasinglydifculttaskwhenweconsiderthemanywaysinwhichheterogeneityentersmalariaecology.Theseheterogeneities,tonameafew,includetemporalheterogeneitiesintemperatureandhumidity,environmentalspatialheterogeneitysuchasvariabletopographyandclimate,age-relatedheterogeneityinhumanhosts,andheterogeneityinvectorandparasitespecies.Tofurthercomplicatematters,theeffectsofthesedifferenttypesofheterogeneitiesareoftenintertwined.Forexample,temperatureandhumidityinuenceparasiteincubationandvectorcompetenceandsurvival[ 11 69 95 ].Thedependenceofmosquitopopulationdynamicsontheavailabilityofbreedingsites,whichinturndependsontemperatureandrainfall,resultsinseasonaluctuationsinmosquitoabundance.Aspartofthe1969-1976Garkiproject,astudyconductedbytheWHOandNigeriangovernmentonmalariacontrolintheAfricansavanna,eldmeasurementsofthevectorialcapacityduringdifferentseasonswereusedasinputstoadifferenceequationmalariamodel[ 77 ].IncorporatingthedifferencesinvectorialcapacityforthewetanddryseasonallowedacomparisonofthemathematicalmodeloutputtoprevalencedatafordifferentregionsinSub-SaharanAfrica.In1982,AronandMaymathematicallydescribedseasonalityinmalariausingsinusoidalforcing[ 9 ].Morerecently,Wyseetal.alsoused[ 111 ]trigonometricfunctionsrepresentingseasonalchangesintemperatureinamalariatransmissionmodeltostudytheeffectofseasonalityonmalariadynamics.Similarly,Chitnisetal.usedseasonalforcinginadifferenceequationmodelofmalariatostudytheeffectsofvectorcontrolinterventions 17

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[ 25 ].Modelswithperiodicseasonalforcingrelyonassumptionsaboutaveragemonthlytemperaturesinagivenregion.Recentstudiessuggest,however,thatevensmalldailyuctuationsintemperatureandvariationsinclimatecanhaveasignicantimpactonmalariadynamics[ 112 113 ].Ontheotherhand,Gethingetal.emphasizedthatsmallchangesinmalariadynamicsasaconsequenceofclimatechangecanbeoffsetbymoderateimprovementinexistinginterventionstrategies[ 46 ].Vectorheterogeneityarisesinmanyforms:heterogeneityinvectorbehavior,heterogeneityinvectorspecies,heterogeneityinvectorcompetence.MalariamodelscommonlyassumethatindividualsinapopulationhavethesamelikelihoodofbeingbittenbyanAnophelesmosquito.However,someindividualsaremorelikelytobebittenthanothers[ 100 ].DyeandHasibeder[ 42 54 ]incorporatedheterogeneousbitingofmosquitoesintomathematicalmodelsofmalariatransmission.Intheiranalysisofamodelincorporating,atrst,heterogeneityinonlythehostpopulation,Dyeetal.[ 42 ]discoveredthataddressingthisnon-homogeneouscontactbetweenthevectorandhumanpopulationincreasestheestimatesoftransmissionintensity,comparedwithmodelsthatassumehomogeneousmixingofthetwopopulations.Afterdeterminingthattheseestimatesoftransmissionintensitywerestilllowincomparisontowhatwasobservedintheeld,HasibederandDye[ 54 ]incorporatedheterogeneityinthevectorpopulation,intotheirpreviousframework.Consideringheterogeneityinboththevectorandhumanpopulationfurtherincreasedtheirestimatesoftransmissionintensity[ 54 ].Theresultthatheterogeneousbitingresultsinincreasedtransmissioninamalariamodelmirrorsanearlierndingthatheterogeneouscontactratesleadstoanincreaseinthenumberofparasiteoffspringinaschistosomiasismodel[ 12 40 ].Heterogeneousbitingisfurthercomplicatedbythefactthatthereareapproximately70speciesofAnophelesmosquitoescapableoftransmittinghumanmalaria,ofwhich41areconsideredimportantvectorsinmalariatransmission[ 56 ].Manyregionshave 18

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morethanonemalariavector,potentiallywithdifferentfeedinghabits,breedingsitepreferences,lifespans,andcompetenciesasamalariavector[ 11 ].FourspeciesofthePlasmodiumparasitearecommonlycitedasthecauseofmalariainfectioninhumans:Plasmodiumfalciparum,Plasmodiumvivax,Plasmod-iumovale,andPlasmodiummalariae[ 11 23 ].Thesefourspecieshavedifferentlifecyclesinthehumanandvectorhosts,differentsensitivitiestotemperature,andrequiredifferentmethodsoftreatmentininfectedhumans[ 11 ].Furthermore,withineachofthesespeciesofPlasmodiumparasite,thereexistmanystrains[ 11 51 ]withdifferentsensitivitiestotreatment.CombiningtheeffectsofthesensitivityofthesefourPlasmodiumparasitespeciesandmosquitoestoclimate,andthesensitivityofvectorpopulationstotheavailabilityofbreedingsitesandaccesstobloodmeals,resultsinaveryheterogeneoustransmissionlandscapeacrossspace[ 11 ].AfthPlasmodiumspeciesisnowrecognizedascausingdiseaseinhumans:Plasmodiumknowlesi.AlthoughtheprimaryhostofP.knowlesiaremacaques,itisnowrecognizedasbeingacauseofzoonotictransmissioninSoutheastAsia[ 23 34 ].Anotherlevelofcomplexityisaddedwhenweconsiderheterogeneityinsusceptibilitytomalariainfectioninhumansasaconsequenceofrepeatedexposuretomalariainendemicareas.Inthisdissertation,wefocusontwotypesofheterogeneitiesthatnaturallyariseinmalariaecology:heterogeneityinmalariaparasitespecies,andheterogeneityinthemalariatransmissionlandscape.Translatingtheseconceptsintomathematicalmodelsallowedustostudytheeffectstheseheterogeneitieshaveonmalariadynamicsandcontrol. 19

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CHAPTER2IMPACTOFENHANCEDMALARIACONTROLONTHECOMPETITIONBETWEENPLASMODIUMFALCIPARUMANDPLASMODIUMVIVAX 2.1PlasmodiumvivaxandPlasmodiumfalciparumParasitesandObstaclesTheyPosetoMalariaControlP.vivaxandP.falciparumhaveverysimilarlifecycles,withoneimportantexception.WhenahumanisinfectedbyamosquitowithP.vivax,someoftheparasitesbecomehypnozoites,whichcanremaindormantinthehumanlivercellsforsometime,thenreactivate.Consequently,individualsinfectedwithP.vivaxarepronetorelapses.Infact,P.vivaxinfectionsexhibitrelapsesroughly30%ofthetimeaftertheinitialclinicalepisode[ 3 ].Fortunately,P.falciparumparasitesdonothaveahypnozoitestage,andthusrelapsesdonotoccurinfalciparuminfections.Despitetheabsenceofrelapseinfalciparummalariainfections,P.falciparumisassociatedwiththehighestriskofmortalityforhumansamongthemalariaparasitespecies.Vivaxinfectionsareconsideredtobebenign,howeverthesymptomsarestilldebilitatinganddiminishbothaperson'squalityoflifeandtheirproductivity.Moreover,somerecentcasesofP.vivaxmalariahavebeenfarmoreseverethanistraditionallyexpectedofthisdisease,sometimesresultingindeath.TheliverstagesofP.vivaxcanalsobeextremelylong,uptothreeyears,allowingP.vivaxparasitestolaydormantandweatherthelowtransmissionseasonsuntilconditionshaveimproved,makingvivaxinsomerespects,amoreformidablefoethanP.falciparumintermsofmalariacontrol.Althoughsymptomsduetomalariainfectioncanbequitesevereandsometimesdeadly,itislikelythat,becausemultipleinfectionscantemporarilybuildaperson'simmunitytothedisease,alargeproportionofmalariacasesinIndiaareasymptomaticordisplayverymildsymptoms,particularlyinregionswithmeso-tohyper-endemicity[ 64 ].AstudyofmalariainfectioninpregnantwomeninJharkhandState,India,foundthatnearlyhalfofthewomeninthestudycarriedasymptomaticmalariainfection[ 52 ].Sinceasymptomatichumanmalariainfectionsarestillinfectioustomosquitoes,unlikely 20

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tobetreated,andconsequentlylonger-lived,asymptomatichumans,inadditiontoliverstagevivaxinfectedhumans,createareservoirformalariaparasites.Furthermore,thelongdurationofuntreatedorunsuccessfullytreatedinfectionsincreasesthelikelihoodofco-infectionwithP.vivaxandP.falciparumspecies.Itcanbeverydifculttoidentifymalariaco-infectionsbecauseitisnotyetverywellunderstoodhowthetwospeciesinteract.Co-infectedindividualscanalsobeverydifculttotreatbecauseadrugthatworksforoneinfectionmayconferresistanceintheother. 2.2ObjectivesofModelingP.vivax-P.falciparumDiseaseDynamicsInthisdissertation,wedevelopaP.vivaxP.falciparummalariamodelwithco-infectiontoaddressquestionsregardingcontrolmeasuresinthecontextofIndia.Wewanttondoutwhateffectcertaincontrolmeasureshaveonthecompetitionbetweenthetwoparasitespecies,howthepresenceoftwocirculatingparasitesaffectswhatcontrolmeasuresshouldbeimplementedandhowtheycanbestbeimplemented.ThecurrentliteratureonmalariainIndiadictatesthatthereisaneedtoaddressnotonlyPlasmodiumfalciparummalaria,whichismorecommonlystudiedandmodeled,butPlasmodiumvivaxaswell.Chiyakaetal.havepublishedthersttwo-speciesmalariamodel,incorporatingP.falciparumandP.malariae[ 27 ].However,thereisstillaneedtomodelP.falciparumandP.vivaxdiseasedynamics,particularlybecausetheepidemiologyofthesetwoparasitesissodifferent.Thesedifferences,whichareintrinsictotheparasitebiologyandarelikelytogreatlyenhancetheparasites'abilitytopersistinapopulationinthefaceofnumerouscontrolefforts,needtobeincludedinamathematicalmodelifwewanttoprovideinsightintoproblemsregardingcompetitionbetweenparasitespeciesinIndiaandhowtodevelopaneffectivecontrolpolicyforIndia.InSection 2.3 ,weintroducethetwo-parasiteordinarydifferentialequationmalariamodel.InSection 2.3.5 wepresentthedisease-freeequilibrium,thebasicreproductivenumberforthemodel,andthecontrolreproductivenumber.Wepresenttheisolatedendemicequilibriaofthesysteminsection 2.3.7 andacompletedescription 21

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andinterpretationoftheinvasionnumbersinSection 2.3.8 .Section 3.1 explainstheparametersusedinthemodelandthevalueschosenforeachparameter.InSection 3.1.2 ,forthe1987-1996period,weestimatetransmissionparametersbyttingtheODEmodeltoIndianmalariacasedata.Comparisonofseveralmodelsfortheenhancedcontrolperiod(1997-2010)inSection 3.1.3 allowedustodeterminewhichcontrolmeasurescontributedthemosttothesuccessofcontrolprograms.Inthesamesection,wealsopresentanuncertaintyanalysistodeterminethemostlikelyoutcomeofmalariainIndia. 2.3P.falciparumandP.vivaxMalariaCo-infectionModelInthetwo-parasitemalariamodelbelow,itisassumedthatthemosquitopopulationsize,Nm,isconstant,andthatthesizeofthehumanpopulation,N,exhibitslogisticgrowth.ThestatevariableMdenotesthenumberofmosquitoesthatarefullysusceptibletobothP.vivaxandP.falciparumparasites.Similarly,Sdenotesthenumberofhumanswhoarefullysusceptibletobothmalariaparasites.ThenumberofinfectedmosquitoesatagiventimeisJ,thesumofP.vivaxinfectedmosquitoes(Jv)andP.falciparuminfectedmosquitoes(Jf).HumandeathsduetoP.vivaxinfectionarerare,andarethusconsideredtobenegligible.AlthoughdeathsduetoP.falciparumdooccur,theassociatedmortalityrateinIndiaisverysmallcomparedwiththetotalmorbidityduetomalaria.Asaresult,inthistwo-parasitemodel,allhumansrecoverfrommalariainfection.Onceinfectiousindividualsrecoverfullyfrommalaria,theyagainbecomesusceptibletomalariainfectionandmovetoclassS.Thus,whenwerefertoanindividualsurvivingaparticularstage,wemeanthattheydidnotdieduetonaturalmortalitybeforetheendofthatstage. 2.3.1ModelingtheDynamicsofPlasmodiumvivaxInfectionintheHumanPopulationFirst,wedescribethedynamicsofP.vivaxmalariainthehumanpopulation.WhenaP.vivaxinfectedmosquitosuccessfullytransmitsamalariaparasitetoahuman,we 22

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assumethatthesehumansrstgothroughaliverstage,denotedbyL,inwhichthemalariaparasitesremainun-infectious.ThisliverstageactsasboththeinitialincubationperiodfortheP.vivaxmalariaparasitesinahumanandastheperiodbetweenrelapsesinwhichmalariaparasitesremaininadormantliverstageashypnozoites.Becauseasymptomaticindividualscreateareservoirformalaria,posingsignicantchallengestomalariacontrol,andbecausemalariainfectedindividualsdonotbecomeinfectiousuntiltheparasiteshavegonethroughthehumanliverstage[ 11 ],themodelallowsforafractionoftheindividualsintheliverstagetobi-passthesymptomaticstageandmovedirectlytotheP.vivaxinfectiousstageIv.Werefertoindividualswhobi-passthesymptomaticstageasasymptomatic,andthosewhodonotarereferredtoassymptomatic.AhumanpresentingsymptomsisconsideredaclinicalcaseandweletCvdenotetheP.vivaxclinicalcasesatanygiventime.AssuminganindividualinCvdoesnotdieofnaturalmortality,heorshewillbecomeinfectiousandmoveintotheIvstage.Onceinthisinfectiousstage,individualshavethepotentialtofullyrecover,returningtothesusceptibleclassSviaeithersuccessfultreatmentornaturalrecovery.Weassumethatalthoughindividualsmaybegintreatmentduringtheclinicalstage,thetreatmentdoesnotaffecttheperson'sprogressiontotheinfectiousstage.Consequently,weassumethatevenintreatedindividuals,gametocyteclearance(lossofinfectiousness)occursintheinfectiousstage.Asdescribedpreviously,becausesomeP.vivaxparasitesbecomehypnozoitesduringtheliverstage,remaindormantintheliverforsomeperiod,andarereactivatedatalatertime,vivaxmalariapatientswhoarenotsuccessfullytreatedarepronetorelapses.Thus,inourmodel,individualsinthevivaxinfectiousclassIvcanreturntotheliverstageLandrepeatthecycleofthevivaxinfection. 23

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2.3.2ModelingtheDynamicsofPlasmodiumfalciparumInfectionintheHumanPopulationPlasmodiumfalciparuminfections,whiletypicallymoreseverethanvivaxinfections,exhibitsimplerinfectiondynamicsthanvivaxinfections.Inparticular,P.falciparumparasitesdonothaveahypnozoitestage,andconsquently,individualsinfectedwithonlyP.falciparumdonotexperiencerelapses.InlightofthisdifferencebetweenP.falciparumandP.vivaxinfections,weomitthefalciparumincubationperiodwhichistypicallyshorterthanthatofP.vivax,meaningthatonceahumanisinfectedbyaP.falciparuminfectiousmosquito,thatindividualmovesdirectlyeithertothefalciparumclinicalstageCf,ormovestothefalciparuminfectiousstageIf.Asnotedinthedescriptionofvivaxinfectiondynamics,werefertotheindividualswhobi-passtheclinicalstageasasymptomaticindividuals.Thosewhopassthroughtheclinicalstagearereferredtoassymptomaticindividuals.OnceintheP.falciparuminfectiousstage,aswithP.vivaxinfection,individualscanfullyrecoverviaeithersuccessfultreatmentornaturalrecovery.ThestatevariablesIvandIfincludebothasymptomaticinfectiousindividualsandinfectiousindividualswhohaveshownsymptoms.Itwillbeassumedthatsymptomaticindividualsaretreatedandasymptomaticindividualsarenottreated.Thus,therecoveryratefromIvandIfwillbeafunctionofboththenaturalrecoveryrateandthetreatment-recoveryrate. 2.3.3ModelingCo-InfectionGuptaetal.used180samplesfromsixendemicregionsinIndiatoestimatetheproportionofmalariacasesthataremixedinfections.Thesamplesshowedthatroughly46%ofthemalariainfectionswereP.falciparumP.vivaxco-infections[ 50 ].Consequently,theabilityforhumanstoobtainconcurrentmalariainfectionsshouldplayanimportantroleinaP.falciparumP.vivaxmalariamodelforIndia.Mixedinfectionisincorporatedintothemodelbyintroducingtwomoreclinicalcasestatevariables,CvfandCfv.AP.vivaxinfectedindividualineithertheliverstageortheinfectious 24

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stagewhobecomesco-infectedwithfalciparumwillmovetoCvf.Atthisstageweassumethatindividualscomingfromtheliverstagebecomeinfectiouswithvivax,thosearrivingfromtheinfectiousstageremaininfectiouswithvivax,andallindividualsinCvfshowsymptomsofmalariainfection,althoughitmaynotbeclearwhichinfectioniscausingthesymptoms.AccordingtoSnounouetal.,theassumptionthatco-infectionwithP.falciparumcanreactivatehypnozoitesinthedormantliver-stage,producingP.vivaxblood-stageparasites,isplausible[ 103 ].Similarly,aP.falciparuminfectiousindividualcanbecomeco-infectedwithP.vivax.TheseindividualswillmovetotheCfvstageprovidedtheyhavenotsuccumbedtonaturalmortality.IndividualsinCfvarestillinfectiouswithP.falciparum,butpresentsymptomsassociatedwithP.vivaxinfection.WewillrefertoindividualswhohavebeeninstagesCvfandCfvasvivaxco-infectedandfalciparumco-infectedindividuals,respectively.Ifaco-infectedindividualsurvivestheclinicalstage,theybecomeinfectiouswithbothmalariaparasitesandmovetoIc.Inthistwo-parasitemodel,weassumethatallco-infectedindividualsaretreatedduringtheinfectiousco-infectedstageIc.Thisassumptionisreasonablesincemostco-infectedindividualsshowsymptoms[ 103 ].Thequestionis,whattreatmentdowegivetheseco-infectedindividuals?Accordingtothe2009malariadiagnosisandtreatmentguidelinesforIndia,P.falciparumP.vivaxco-infectedindividualsshouldbegiventhesametreatmentthatisgiventoP.falciparuminfectedpatients[ 81 ].However,malariadiagnostictestsoftenonlydetectoneofthetwoparasitespeciesinthehost,leadinghealth-careproviderstotreatonlytheobservedinfection[ 74 ].Whenonlyoneofthetwoinfectionsistreated,symptomsfortheothermalariainfectionemergeanywherefrom17to63dayspost-treatment[ 75 ].Themodelincorporatesthisemergenceofthehiddeninfectionbyallowingindividualsintheinfectiousco-infectedclassIctomoveintoeitherIvorIfafterrecoveryfromtheinitialobserved(andhencetreated)infection.IfP.falciparumistreatedrst,thentheP.vivaxinfectionwillemergeandindividualsmoveintotheinfectiousclassIv.Likewise,thosewhoaretreatedforP.vivaxrst 25

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movetotheinfectiousclassIfsometimepost-vivaxtreatment.Inourmodel,however,individualstreatedforP.vivaxrstwillnotshowsymptomsfollowingtreatmentoftheco-infectionsincefalciparumsymptomsdonotoccurfollowingfalciparuminfectiousness.Onlyco-infectedindividualstreatedforP.falciparumrsthavethepossibilityofdevelopingsymptoms,inparticularvivaxsymptoms,sinceindividualsinIvcanrelapse.Consequently,ourmodeldoesnotcapturethephenomenondescribedabovewhereP.falciparumsymptomsemergefollowingvivaxtreatment.ThisdiscrepancycanberesolvedbyaddingaP.falciparumincubationperiodtothemodel,howeverforsimplicity,andbecausethemajorityofco-infectionsaretreatedforP.falciparumrst,followedbytheonsetofP.vivaxsymptoms,wendthatincorporatingonlyavivaxincubation/liverstagesufcienttocapturethemostimportantfeaturesofthetwo-parasitespeciesdiseasedynamics. 2.3.4DiseaseDynamicsintheMosquitoPopulationThehumancomponentofthetwo-parasitemalariamodelincludesveinfectiousclasses:twoclassesareinfectiouswithvivaxonly(IvandCvf),twoclassesareinfectiouswithfalciparumonly(IfandCfv),andoneclassisinfectiouswithbothvi-vaxandfalciparum(Ic).Thus,mosquitoeshavevemeansbywhichtheycanbecomeinfected.AsusceptiblemosquitoinfectedbyahumaninclassIvorCvfwilldevelopaP.vivaxinfection.AsusceptiblemosquitoinfectedbyahumaninclassIforCfvwilldevelopaP.falciparuminfection.Intheeventthatamosquitobecomesinfectedbyaco-infectedinfectioushuman(Ic),themodelassumesthatthemosquitowillcontractonlyoneofthetwomalariaparasitespecies.Whichspeciesitcontractswilldependontheprobabilityofpickingupthatparticularspecies.Sincemosquitoeshaveashortlifespan,weassumethatallmosquitoesdienaturaldeathsratherthandisease-induceddeaths.AsummaryofallstatevariablesisgiveninTable 2-1 .Adescriptionofthemodelparameters,aswellastheestimatesusedinlatermodelsimulations,isgiveninTables 3-1 and 3-2 26

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Table2-1. DescriptionofmodelStateVariablesattimet. StateVariablesDescription NmMosquitopopulationsize-denedtobeconstant N(t)Humanpopulationsizeattimet M(t)Numberofsusceptiblemosquitoesattimet S(t)Numberofsusceptiblehumansattimet m(t)Proportionofmosquitoesthataresusceptibleattimet Jv(t)NumberofP.vivaxinfectedmosquitoesattimet Jf(t)NumberofP.falciparuminfectedmosquitoesattimet jv(t)ProportionofmosquitoesthatareP.vivaxinfectedattimet jf(t)ProportionofmosquitoesthatareP.falciparuminfectedattimet L(t)NumberofhumanP.vivaxliverstageinfectionsattimet Cv(t)NumberofhumanP.vivaxcasesattimet Cf(t)NumberofhumanP.falciparumcasesattimet Iv(t)NumberofP.vivaxinfectioushumansattimet If(t)NumberofP.falciparuminfectioushumansattimet Cvf(t)Numberofsymptomaticco-infectedcases,infectiouswithP.vivaxonly,attimet Cfv(t)Numberofsymptomaticco-infectedcases,infectiouswithP.falciparumonly,attimet Ic(t)Numberofco-infectedhumansinfectiouswithbothP.vivaxandP.falciparumattimet Thetwo-parasitemalariamodeldiagraminFigure 4-1 canbedescribedmathematicallyasfollows:MosquitoDynamics: dJv dt=bvIv+Cvf N(Nm)]TJ /F4 11.955 Tf 11.96 0 Td[(J)+bvIc N(Nm)]TJ /F4 11.955 Tf 11.96 0 Td[(J))]TJ /F4 11.955 Tf 11.96 0 Td[(dJv (2) dJf dt=bfIf+Cfv N(Nm)]TJ /F4 11.955 Tf 11.95 0 Td[(J)+(1)]TJ /F7 11.955 Tf 11.95 0 Td[()bfIc N(Nm)]TJ /F4 11.955 Tf 11.96 0 Td[(J))]TJ /F4 11.955 Tf 11.95 0 Td[(dJf (2) whereJ.=Jv+Jf,Nmisconstant,andM=Nm)]TJ /F4 11.955 Tf 11.95 0 Td[(J.Sinceitisdifculttoestimatehowlargethemosquitopopulationis,wemodifythemosquitodynamicsequationsbyconsideringtheproportionofmosquitoesinfectedratherthanthenumberofmosquitoesinfected.Thus,dividingequations( 2 )and( 2 )bythetotalmosquitopopulationsizeNm,wearriveatthefollowingsetofequations 27

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Figure2-1. MosquitoandHumanPopulationDynamicsDiagramundertheinuenceoftwocirculatingmalariaparasites.Boldarrowsindicatetheacquisitionofanewinfection,anddottedarrowsindicaterecoveryfromeitherP.vivaxorP.falciparuminfection. describingthemosquitoinfectiondynamics:djv dt=bvIv+Cvf N(1)]TJ /F4 11.955 Tf 11.95 0 Td[(j)+bvIc N(1)]TJ /F4 11.955 Tf 11.95 0 Td[(j))]TJ /F4 11.955 Tf 11.95 0 Td[(djvdjf dt=bfIf+Cfv N(1)]TJ /F4 11.955 Tf 11.96 0 Td[(j)+(1)]TJ /F7 11.955 Tf 11.95 0 Td[()bfIc N(1)]TJ /F4 11.955 Tf 11.96 0 Td[(j))]TJ /F4 11.955 Tf 11.95 0 Td[(djfwherenowj.=jv+jf=1 Nm(Jv+Jf),denotesthefractionofthemosquitopopulationthatisinfectedwithmalariaparasitesandhence,m=1)]TJ /F4 11.955 Tf 12.44 0 Td[(jrepresentsthefractionofmosquitoesthataresusceptibletomalariainfection.Notethat 28

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m0=)]TJ /F4 11.955 Tf 9.3 0 Td[(j0=)]TJ /F13 11.955 Tf 11.29 16.86 Td[(bvIv+Cvf N+bvIc N+bfIf+Cfv N+(1)]TJ /F7 11.955 Tf 11.96 0 Td[()bfIc Nm)]TJ /F4 11.955 Tf 11.96 0 Td[(d(1)]TJ /F4 11.955 Tf 11.96 0 Td[(m).Intheabovesystem,bvandbfarehuman-to-mosquitotransmissionrates,disthemosquitonaturalmortalityrate,andistheprobabilitythatifasusceptiblemosquitobitesanIchuman,themosquitowillcontractvivaxratherthanfalciparum.HumanDynamics:dS dt=dN dt)]TJ /F4 11.955 Tf 13.15 8.09 Td[(dLv dt)]TJ /F4 11.955 Tf 13.15 8.09 Td[(dC dt)]TJ /F4 11.955 Tf 13.49 8.09 Td[(dI dt=rN1)]TJ /F4 11.955 Tf 13.37 8.09 Td[(N K+vIv+fIf)]TJ /F8 11.955 Tf 11.96 0 Td[((vjv+fjf)S)]TJ /F7 11.955 Tf 11.95 0 Td[(SdL dt=vSjv+Iv)]TJ /F7 11.955 Tf 11.95 0 Td[(vfjfL)]TJ /F8 11.955 Tf 11.95 0 Td[((+)LdCv dt=vL)]TJ /F8 11.955 Tf 11.96 0 Td[((v+)CvdIv dt=(1)]TJ /F7 11.955 Tf 11.96 0 Td[(v)L+vCv+fIc)]TJ /F7 11.955 Tf 11.96 0 Td[(vfIvjf)]TJ /F8 11.955 Tf 11.95 0 Td[((+v+)IvdCvf dt=vf(Iv+L)jf)]TJ /F8 11.955 Tf 11.96 0 Td[((vf+)CvfdCf dt=ffSjf)]TJ /F8 11.955 Tf 11.95 0 Td[((f+)CfdIf dt=(1)]TJ /F7 11.955 Tf 11.96 0 Td[(f)fSjf+fCf+vIc)]TJ /F7 11.955 Tf 11.95 0 Td[(fvIfjv)]TJ /F8 11.955 Tf 11.96 0 Td[((f+)IfdCfv dt=fvIfjv)]TJ /F8 11.955 Tf 11.96 0 Td[((fv+)CfvdIc dt=vfCvf+fvCfv)]TJ /F8 11.955 Tf 11.96 0 Td[((v+f+)IcwhereC.=Cv+Cf+Cvf+Cfv,I.=Iv+If+Ic,andthetotalpopulationsizeisdescribedbythelogisticequationdN dt=rN)]TJ /F8 11.955 Tf 5.48 -9.68 Td[(1)]TJ /F6 7.97 Tf 13.32 4.71 Td[(N K)]TJ /F7 11.955 Tf 11.95 0 Td[(N.Themosquito-to-humantransmissionratesforvivaxandfalciparumaredenotedbyvandf,respectively.Thenaturalhumanmortalityrateisgivenby.Aproportionvofvivaxandaproportionfoffalciparumcasesaresymptomatic.Weassumesymptomaticindividualsgettreatedandclearblood-stageparasitesatarateifrominfectioniandasymptomaticindividualsclearblood-stageparasitesatarateri(i= 29

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v,f).Thus,theratesofreturningtothesusceptibleclass,denotedbyvandf,areafunctionofbothtreatmentandnaturalparasite-clearancerates:i=ii+(1)]TJ /F7 11.955 Tf 12.42 0 Td[(i)ri,fori=v,f.Vivax-infectedindividualsprogressataratefromtheliverstagetoeitherCvorIv.Vivax-symptomaticandfalciparum-symptomaticindividualsprogresstotheinfectiousstageatratevandf,respectively.Similarly,vivax-co-infectedandfalciparum-co-infectedindividualsenterIcatratesvfandfv,respectively.vandfarecross-immunitycoefcients.istherateatwhichvivax-infectedindividualsrelapse.Thisparameterisgivenby=prvf+(1)]TJ /F7 11.955 Tf 12.58 0 Td[(v)rv,wherepristheprobabilitythatatreatedvivaxpatientrelapses.Finally,vandfaretheprobabilitiesthatanIcindividualistreatedrstforvivaxand,respectively,forfalciparuminfection.AcompletelistofthemodelparametersandtheirdescriptionsispresentedinTable 2-2 2.3.5DerivationoftheDisease-FreeEquilibrium,BasicReproductiveNumberR0andControlReproductiveNumberRCNotethatdN dtcanberewrittenintheformdN dt=(r)]TJ /F7 11.955 Tf 12.33 0 Td[()N1)]TJ /F6 7.97 Tf 27.15 4.71 Td[(N K(1)]TJ /F16 5.978 Tf 7.79 3.69 Td[( r)sothattheintrinsicgrowthrateofthepopulation^risr)]TJ /F7 11.955 Tf 11.35 0 Td[(,andthecarryingcapacity^KisK)]TJ /F8 11.955 Tf 5.48 -9.69 Td[(1)]TJ /F17 7.97 Tf 13.15 5.26 Td[( r.Thedisease-freeequilibriumis(N,m,jv,jf,S,L,Cv,Cf,Iv,If,Cvf,Cfv,Ic)DFE=(^K,1,0,0,^K,0,0,0,0,0,0,0,0).NandSareeasilydeterminedbysettingtherighthandsideofdN dt=^rN1)]TJ /F6 7.97 Tf 13.32 4.71 Td[(N ^Kequaltozeroandnotingthatwhenthereisnodisease,S=N.Sincem=1)]TJ /F4 11.955 Tf 11.95 0 Td[(j,wehavethatm=1whenthereisnodisease.Thebasicreproductivenumber,R0,ofanepidemiologicalmodelistheaveragenumberofsecondarycasesproducedbyoneinfectiousindividualinanotherwisefullysusceptiblepopulationwherenocontrolisbeingimplemented.Thecontrolreproductivenumber,RC,isdenedsimilarly,withtheexceptionthatcontrolmeasuresareassumedtobeinplace.IfR0<1,thedisease-freeequilibriumislocallyasymptoticallystable,implyingthatthediseasewilleventuallybecomeextinct.Ontheotherhand,ifR0>1,thedisease-freeequilibriumisunstable[ 106 ].Consequently,determininganexpressionforthebasicreproductivenumberfromthemodelandestimatingitsvalueisakey 30

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Table2-2. Descriptionofmodelparameters ParametersDescription 1 DurationofP.vivaxliverstage 1 vTimeuntilinfectiousafterP.vivaxsymptomonset 1 fTimeuntilinfectiousafterP.falciparumsymptomonset 1 vfDurationofCvf 1 fvDurationofCfv Humannaturaldeathrate vP.vivaxblood-stageparasiteclearanceratewithtreatment fP.falciparumtreatmentrecoveryrate rvP.vivaxnaturalblood-stageparasiteclearancerate rfP.falciparumnaturalrecoveryrate vRecoveryratefromIvtoS fRecoveryratefromIftoS prProbabilityofpost-treatmentP.vivaxrelapse P.vivaxrelapserate vProbabilitythataP.vivaxinfectedhumanbecomessymptomatic fProbabilitythataP.falciparuminfectedhumanbecomessymptomatic vPv-inducedcross-immunitytoPf fPf-inducedcross-immunitytoPv Fractionofco-infectedinfectiousindividualsthatrecoverrstfromP.falciparum vRateofprogressionfromIctoIfduetoP.vivaxtreatment fRateofprogressionfromIctoIvduetoP.falciparumtreatment dMosquitonaturaldeathrate Probabilitythatasusceptiblemosquitothatgetsinfectedbyaco-infectedhumancontractsP.vivax 1)]TJ /F7 11.955 Tf 11.96 0 Td[(Probabilitythatasusceptiblemosquitothatgetsinfectedbyaco-infectedhumancontractsP.falciparum 31

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componenttounderstandinghowdifcultitwillbetocontroltransmissionofthediseaseandwhatcontrolmeasureswillbethemosteffective.Animportantgoalofanyinfectiousdiseasecontrolprogramistoimplementcontrolmeasuresinsuchawayastosuccessfullybringthecontrolreproductivenumberbelowone.Theisolationreproductivenumbersofamulti-parasitemodel,suchasthistwo-parasitemalariamodel,arethebasicreproductivenumbersforthemodelwhenonlyoneparasitespeciesispresentatatime. 2.3.6ExpressionofRCandR0DerivedfromNextGenerationApproachRCisathresholdcriterionthatdetermineswhetherP.vivaxorP.falciparumwillbeabletoinvadethedisease-freeequilibrium.FollowingtheapproachofDiekmannetal.[ 38 ],weconsiderasubsetofoursystemcomprisingonlyofequationsfortheinfectedstatevariables.Weordertheseequationsasfollows:fj0v,L0,C0v,I0v,j0f,C0f,I0f,C0vf,C0fv,I0cg.Next,wewritetheJacobian(evaluatedatthedisease-freeequilibrium)JofthesubsystemasthedifferenceoftwomatricesFandV(J=F)]TJ /F4 11.955 Tf 12.01 0 Td[(V).WechoosethesematricessuchthattheelementsofFincludeonlynewinfectionsandtheremainingtransitions(recovery,relapse,death,orprogressiontoanewdiseasestate)appearintheVmatrix,givingusF=0B@F1F2F301CAwhere,F1=0BBBBBBBBBB@000f1,40f2,1000.........0.........0001CCCCCCCCCCA,F2=0BBBBBBBBBB@00f1,80f1,100000.........00000f5,70f5,9f5,101CCCCCCCCCCA,F3=0BBBBBBBBBB@0000f6,50000.........0.........0001CCCCCCCCCCA,andV=0B@V1V20V31CAwhere, 32

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V1=0BBBBBBBBBB@v1,100000v2,20)]TJ /F4 11.955 Tf 9.3 0 Td[(v2,4...0)]TJ /F4 11.955 Tf 9.3 0 Td[(v3,2v3,30...0)]TJ /F4 11.955 Tf 9.3 0 Td[(v4,2)]TJ /F4 11.955 Tf 9.29 0 Td[(v4,3v4,4000v5,51CCCCCCCCCCA,V2=0BBBBBBBBBB@000000000......00......)]TJ /F4 11.955 Tf 9.29 0 Td[(v4,100001CCCCCCCCCCA,V3=0BBBBBBBBBB@v660000)]TJ /F4 11.955 Tf 9.3 0 Td[(v7,6v7,700)]TJ /F4 11.955 Tf 9.3 0 Td[(v7,1000v8,800000v9,9000)]TJ /F4 11.955 Tf 9.3 0 Td[(v10,8)]TJ /F4 11.955 Tf 9.3 0 Td[(v10,9v10,101CCCCCCCCCCA.ThenonzeroelementsofFaref1,4=bv=^K,f2,1=v=^K,f1,8=bv=^K,f1,10=bv=^K,f5,7=bf=^K,f5,9=bf=^K,f5,10=(1)]TJ /F7 11.955 Tf 11.95 0 Td[(bf=^K),f6,5=ff^K.ThenonzeroelementsofVarev1,1=d,v2,2=+,v2,4=,v3,2=v,v3,3=v+,v4,2=(1)]TJ /F7 11.955 Tf 12.66 0 Td[(v),v4,3=v,v4,4=+v+,v5,5=d,v4,10=f,v6,6=f+,v7,6=f,v7,7=f+,v7,10=v,v8,8=vf+,v9,9=fv+,v10,8=vf,v10,9=fv,v10,10=v+f+.IfFisnonnegativeandVisanonsingularM-matrix(aZ-matrixwhoseeigenvalueshavepositiverealpart),then(FV)]TJ /F5 7.97 Tf 6.59 0 Td[(1)<1ifandonlyifalleigenvaluesofJ=F)]TJ /F4 11.955 Tf 12.51 0 Td[(Vhavenegativerealpart(Lemma2in[ 37 ]).ThisisequivalenttosayingthatifFandVsatisfytheseproperties,thenthediseasefreeequilibriumislocallyasymptoticallystableonlywhenthespectralradius(ordominanteigenvalue)ofFV)]TJ /F5 7.97 Tf 6.58 0 Td[(1islessthanone.Furthermore,theinverseofanM-matrixisnonnegative[ 37 ],sothatFV)]TJ /F5 7.97 Tf 6.59 0 Td[(1isalsononnegative.FV)]TJ /F5 7.97 Tf 6.59 0 Td[(1nonnegativeimpliesthatFV)]TJ /F5 7.97 Tf 6.59 0 Td[(1hasapositiverealeigenvaluegreaterthanorequaltotheabsolutevalueofallothereigenvaluesofFV)]TJ /F5 7.97 Tf 6.59 0 Td[(1[ 16 ].Inotherwords, 33

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(FV)]TJ /F5 7.97 Tf 6.59 0 Td[(1)>0.Since(FV)]TJ /F5 7.97 Tf 6.59 0 Td[(1)ispositive,itmakessensetodeneRCtobeprecisely(FV)]TJ /F5 7.97 Tf 6.59 0 Td[(1).Thus,toderiveanexpressionforRC,wemustrstcheckthatFandVsatisfytheappropriateconditions.ClearlyFisanonnegativematrixandVisaZ-matrix,thatis,amatrixwithnonpositiveoff-diagonalelements.OnecanshowthataZ-matrixAisanM-matrixbyshowingthatthereexistsanonnegativevectorvsuchthatAvispositive[ 44 ].Weclaimthatforv=(1,,1)T,VTvispositive.SinceVisaZ-matrix,itisclearthatVTisalsoaZ-matrix.Furthermore,sinceVandVThavethesameeigenvalues,ifVTisanM-matrix,thensoisV.ShowingthatVTv>0isequivalenttoshowingthatallrowsumsofVTarepositive,orequivalentlythatallcolumnsumsofVarepositive.ItissimpletoshowthatSj:=10Pi=1vi,j>0foreachj2f1,2,,10g:S1=v1,1>0S2=v2,2)]TJ /F4 11.955 Tf 11.96 0 Td[(v3,2)]TJ /F4 11.955 Tf 11.96 0 Td[(v4,2=+)]TJ /F7 11.955 Tf 11.95 0 Td[(v)]TJ /F8 11.955 Tf 11.96 0 Td[((1)]TJ /F7 11.955 Tf 11.96 0 Td[(v)=>0S3=v3,3)]TJ /F4 11.955 Tf 11.96 0 Td[(v4,3=nuv+)]TJ /F7 11.955 Tf 11.96 0 Td[(v=>0S4=)]TJ /F4 11.955 Tf 9.3 0 Td[(v2,4+v4,4=)]TJ /F7 11.955 Tf 9.3 0 Td[(++v+=v+>0S5=v5,5>0S6=v6,6)]TJ /F4 11.955 Tf 11.96 0 Td[(v7,6=f+)]TJ /F7 11.955 Tf 11.96 0 Td[(f=>0S7=v7,7>0S8=v8,8)]TJ /F4 11.955 Tf 11.96 0 Td[(v10,8=vf+)]TJ /F7 11.955 Tf 11.96 0 Td[(vf=>0 34

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S9=v9,9)]TJ /F4 11.955 Tf 11.96 0 Td[(v10,9=fv+)]TJ /F7 11.955 Tf 11.96 0 Td[(fv=>0S10=)]TJ /F4 11.955 Tf 9.3 0 Td[(v4,10)]TJ /F4 11.955 Tf 11.96 0 Td[(v7,10+v10,10=)]TJ /F7 11.955 Tf 9.3 0 Td[(f)]TJ /F7 11.955 Tf 11.95 0 Td[(v+v+f+=>0.Thus,VisanM-matrix,andconsequently,RC=(FV)]TJ /F5 7.97 Tf 6.58 0 Td[(1).TodeterminetheexpressionforRC,werstcomputetheinverseofVusingtheformulaV)]TJ /F5 7.97 Tf 6.59 0 Td[(1=1 det(V)Adj(V),whereAdj(V)istheadjugateofV.ci,j:=()]TJ /F8 11.955 Tf 9.3 0 Td[(1)(i+j)Vi,jiscalledthe(i,j)cofactorofV.ThematrixCwhoseelementsarethecofactorsofViscalledthecofactormatrixofV.TheadjugateofVisdenedtobethetransposeofthecofactormatrixofV,thatis,Adj(V):=CT.WendthatC)]TJ /F5 7.97 Tf 6.59 0 Td[(1=1 det(V)0B@C1C20C31CA,whereCiaredenedasfollowsfori=1,2,3:C1=0BBBBBBBBBB@c1,100000c2,2c3,2c4,200c2,3c3,3c4,300c2,4c3,4c4,400000c5,51CCCCCCCCCCA,C2=0BBBBBBBBBB@0000000c8,2c9,2c10,200c8,3c9,3c10,300c8,4c9,4c10,4000001CCCCCCCCCCA,andC3=0BBBBBBBBBB@00000c6,7c7,7c8,7c9,7c10,700c8.800000c9,9000c8,10c9,10c10,101CCCCCCCCCCA.Thus,FV)]TJ /F5 7.97 Tf 6.58 0 Td[(1=1 det(V)0BBBB@K10K20K3K40001CCCCA,where 35

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K1=0BBBBBBB@0f1,4c2,4f1,4c3,4f1,4c4,4f2,1c1,1000000000001CCCCCCCA,K2=0BBBBBBB@f1,4c8,4+f1,8c8,8+f1,10c8,10f1,4c9,4+f1,10c9,10f1,4c10,4+f1,10c10,100000000001CCCCCCCAK3=0BBBB@0f5,7c6,7f5,7c7,7f6,5c5,500f7,5c5,5001CCCCA,andK4=0BBBB@f5,7c8,7+f5,10c8,10f5,7c9,7+f5,9c9,9+f5,10c9,10f5,7c10,7+f5,10c10,100000001CCCCA.ThenonzeroeigenvaluesofFV)]TJ /F5 7.97 Tf 6.58 0 Td[(1arepreciselytheeigenvaluesof\FV)]TJ /F5 7.97 Tf 6.58 0 Td[(1,where\FV)]TJ /F5 7.97 Tf 6.59 0 Td[(1=1 det(V)0B@^K100^K21CA,and^K1=0B@0f1,4c2,4f2,1c1,101CAand^K2=0BBBB@0f5,7c6,7f5,7c7,7f6,5c5,500f5,7c5,5001CCCCA.Since\FV)]TJ /F5 7.97 Tf 6.58 0 Td[(1isblocktriangular,itseigenvaluesaretheeigenvaluesof^K1=det(V)and^K2=det(V). 36

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Usingtheformulafortheelementsci,jofthecofactormatrixC,wendthatc1,1=[v2,2v3,3v4,4)]TJ /F4 11.955 Tf 11.96 0 Td[(v2,4(v3,2v4,3+v3,3v4,2)](v5,5v6,6v7,7v8,8v9,9v10,10)c2,4=v1,1(v32v43+v33v42)v5,5v6,6v10,10c5,5=v1,1[v2,2v3,3v4,4)]TJ /F4 11.955 Tf 11.96 0 Td[(v24(v32v43+v33v42)]v6,6v10,10c6,7=v1,1v5,5v7,6[v2,2v3,3v4,4)]TJ /F4 11.955 Tf 11.95 0 Td[(v24(v32v43+v33v42)]v8,8v10,10c7,7=v1,1v5,5v10,10[v2,2v3,3v4,4)]TJ /F4 11.955 Tf 11.95 0 Td[(v24(v32v43+v33v42)].ThedeterminantofVisdet(V)=v1,1v2,2v10,10[1)]TJ /F4 11.955 Tf 12.04 0 Td[(k2,4(k3,2k4,3+k4,2)],whereki,jdenotesvi,j=vj,j.Byndingtherootsofthecharacteristicpolynomialsof^K1and^K2,wearriveattheanalyticexpressionforRC:RC=maxfRCv,RCfg,whereRCvandRCfaredescribedbelow. RCf=s f d(1)]TJ /F7 11.955 Tf 11.95 0 Td[(f)+ff f+bf f+ (2) =p RaCf+RsCf, (2) where RaCf=f d(1)]TJ /F7 11.955 Tf 11.96 0 Td[(f)bf f+ (2) RsCf=f dff f+bf f+. (2) ObservethatRaCfisthecontributionofanasymptomaticinfectiousindividualtothebasicreproductivenumberandRsCfisthecontributionofasymptomaticinfectiousindividual.TheP.vivaxisolationcontrolreproductivenumberisgivenby RCv=vuut RaCv+RsCv 1)]TJ /F17 7.97 Tf 26.5 4.71 Td[( +v+~RaCv+~RsCv, (2) 37

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where RaCv=bv +v+(1)]TJ /F7 11.955 Tf 11.95 0 Td[(v) +v d (2) RsCv=bv +v+v +v v+v d,and (2) ~RiCv=RiCv=bvv d(+v+),fori=a,s.Notethatifwerezero,inotherwordsifP.vivaxpatientsneverrelapsed,RCv=p RaCv+RsCvwheretheinterpretationsofRaCvandRsCvareanalogoustothatofRaCfandRsCf,respectively.Thatis,RaCvwouldbethecontributionofanasymptomaticinfectiousindividualtothebasicreproductivenumberandRsCvthecontributionofasymptomaticinfectiousindividual.However,theinclusionofthepossibilityofrelapseinP.vivaxinfectedindividuals(>0)makestheexpressionforRCvmorecomplicatedanditsbiologicalinterpretationlessstraight-forward.ThenumeratorsquaredofRCvisthenumberofnewmosquitoinfectionsarisingfromasingleinfectedmosquito,withouttheintermediatehumanhostsrelapsing.TointerpretthedenominatorofRCv,rstnotethat~RaCv+~RsCv2[0,1)since~RaCv+~RsCv< +(1)]TJ /F7 11.955 Tf 11.95 0 Td[(v+v)= +<1andclearly~RaCv+~RsCvispositive.Letx= +v+~RaCv+~RsCv.Thenx<1impliesthat1 1)]TJ /F6 7.97 Tf 6.58 0 Td[(x=P1n=0xn.Because +v+istheprobabilitythatanindividualinIvrelapseswhentherearenofalciparum-infectedindividualsinthepopulation,xistheprobabilitythataliver-stagehumanwillrelapse.Thus,xnistheprobabilitythataliver-stagehumanwillrelapsentimes.So, R2Cv=1Xn=0(RaCv+RsCv)xn. (2) Theithterminthesumcanbeinterpretedasthenumberofnewmosquitoinfectionsgeneratedbyasinglemosquitowheretheintermediatehumanhostsrelapseexactlyitimes.ThisargumentfollowsthereasoningdevelopedbyvandenDriesscheandWatmoughin[ 106 ].Sincetheonlycontrolmeasureexplicitlyimplementedinthemodelistreatment,thebasicreproductivenumberforthemodelisgivenbythecontrolreproductivenumber 38

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evaluatedwiththetreatmentrecoveryrates(vandf)equaltothenaturalrecoveryrates(rvandrf,respectively).Usingourdenitionofv,f,andthisisequivalenttosettingv=rv,=rv,andf=rf.Thus,R0=maxfR0v,R0fg,where R0f=s f d(1)]TJ /F7 11.955 Tf 11.96 0 Td[(f)+ff f+bf rf+ (2) =p Ra0f+Rs0f, (2) where Ra0f=f d(1)]TJ /F7 11.955 Tf 11.95 0 Td[(f)bf rf+ (2) Rs0f=f dff f+bf rf+. (2) Similarly, R0v=vuuut bv 2rv+~Ra0v+~Rs0vv d 1)]TJ /F6 7.97 Tf 21.21 4.71 Td[(rv 2rv+~Ra0v+~Rs0v, (2) where ~Ra0v=(1)]TJ /F7 11.955 Tf 11.96 0 Td[(v) + (2) ~Rs0v=v +v v+. (2) 2.3.7IsolatedEndemicEquilibriaandCoexistenceDeterminingananalyticexpressionforthecoexistenceequilibriumcanbeadifcultproblemformorecomplicatedmodelssuchasthistwo-parasitemalariamodel.However,wecanstillgaininsightintotheconditionsunderwhichacoexistenceequilibriumoccursbystudyingthestabilityoftheisolatedendemicequilibria;thatis,theequilibriawhereonlyonepathogenispresentinapopulation.Linearizingthesystemabouttheseisolationequilibriaprovidesaconditionunderwhichtheabsentparasite 39

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speciescaninvadewhenintroducedtothepopulation.Thesethresholdquantitiesareknownastheinvasionreproductionnumbers.First,wendthevivax-onlyequilibrium,Evbyassumingallfalciparum-infectedvariablesarezeroandsettingeachequationintheresultingsystemequaltozero.Solvingthissystemofequationsforthenon-trivialequilibrium,wendthat jv=bvIv bvIv+d^K (2) S=^K+vIv vjv+ (2) L=vSjv+Iv + (2) Cv=v v+L, (2) where Iv=1)]TJ /F4 11.955 Tf 11.95 0 Td[(R2Cv R2Cv^K v+1)]TJ /F17 7.97 Tf 22.02 5.26 Td[(+v+ (RaCv+RsCv)1+ v (2) Itissimpletoshowthatthedenominatorinequation( 2 )isalwaysnegative.FirstrecallthatRaCv+RsCv<1sothat+v+ RaCv+RsCv>+v+.So,v+1)]TJ /F17 7.97 Tf 22.01 5.26 Td[(+v+ (RaCv+RsCv)1+ v1.Thus,IvispositiveonlyifR2Cv>1.Inotherwords,thevivax-boundaryequilibriumexistsonlywhenRCv>1.Now,wendthefalciparum-onlyequilibriumEfbysettingallvivax-infectedvariablesequaltozero,andndingthenon-trivialequilibriumoftheresultingsystem. jf=bfIf bfIf+d^K (2) S=^K+fIf fjf+ (2) Cf=ff f+Sjf, (2) 40

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where If=1)]TJ /F4 11.955 Tf 11.96 0 Td[(R2Cf R2Cf^K f)]TJ /F6 7.97 Tf 29.27 5.25 Td[(bff d(RaCf+RsCf)1+ f (2) UsingthedenitionofRaCf+RsCf,wehavethatbff d(RaCf+RsCf)=f+ (1)]TJ /F17 7.97 Tf 6.58 0 Td[(f)+ff f+1. 2.3.8InvasionNumbersRfvandRvfThebasicreproductionnumberisathresholdthatdetermineswhetheradiseasecaninvadethedisease-freeequilibriumornot.Likewise,invasionnumbersarethresholdquantitiesthatdetermineifadiseasecaninvadeanotherdisease'sendemicequilibrium.Thesequantitiesareveryusefulinunderstandingthecompetitionbetweenpathogensinamulti-strainmodel.Here,wendanalyticexpressionsforRfv,theinvasionnumberofP.vivaxwhenthesystemisattheP.falciparum-onlyequilibrium,andRvf,theinvasionnumberofP.falciparumattheP.vivax-onlyequilibrium.Typically,thefollowingresultcanbeestablished:ifRCf>1andRfv<1,thenthefalciparum-onlyequilibriumislocallyasymptoticallystableandunstableotherwise.Similarly,ifRCv>1andRvf<1,thevivax-onlyequilibriumislocallyasymptoticallystableandunstableotherwise.BothspeciescoexistwhenRfvandRvfaregreaterthanone.TheprocedureforndingananalyticexpressionfortheinvasionreproductionnumbersRfvandRvf,althoughmorechallengingtocarry-out,isidenticaltotheprocedureforderivingRC. 2.3.9FindingRvfUsingtheNext-GenerationApproachToderiveRvf,wendtheJacobianofthefalciparum-infectedsubsystem,withtheorder: 41

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fj0f,C0f,I0f,C0vf,C0fv,I0cg.TheJacobianevaluatedatthevivax-onlyequilibriumisgivenbyJ=F)]TJ /F4 11.955 Tf 11.96 0 Td[(V,whereF=0BBBBBBBBBBBBBB@00f1,30f1,5f1,6f2,100f3,1.........f4,1.........0.........0000001CCCCCCCCCCCCCCA,andV=0BBBBBBBBBBBBBB@v1,1000000v2,200000)]TJ /F4 11.955 Tf 9.3 0 Td[(v3,2v3,300)]TJ /F4 11.955 Tf 9.3 0 Td[(v3,6000v4,40000)]TJ /F4 11.955 Tf 9.29 0 Td[(v5,30v5,50000)]TJ /F4 11.955 Tf 9.3 0 Td[(v6,4)]TJ /F4 11.955 Tf 9.3 0 Td[(v6,5v6,61CCCCCCCCCCCCCCA,wheref1,3=bf(1)]TJ /F4 11.955 Tf 12.97 0 Td[(jv)^K,f1,5=f1,3,f1,6=(1)]TJ /F7 11.955 Tf 12.97 0 Td[()f1,3,f2,1=ffS,f3,1=(1)]TJ /F7 11.955 Tf 12.3 0 Td[(f)S,f4,1=vf(Iv+L),andv1,1=d,v2,2=f+,v3,2=f,v3,6=v,v4,4=vf+,v5,3=fvjv,v5,5fv+,v6,4=vf,v6,5=fv,v6,6=v+f+.FisnonnegativeandVisanonsingularZ-matrix.Weshow,aswedidinSection 2.3.6 ,thatVisalsoanM-matrixbyshowingthatthecolumnsumsofVarepositive.Sinceeachvi,j>0andv2,2>v3,2,v3,3>v5,3,v4,4>v6,4,v5,5>v6,5,andv6,6>v3,6,eachcolumnsumispositive,andhenceVisanonsingularM-matrix.Thus,V)]TJ /F5 7.97 Tf 6.59 -.01 Td[(1isnonnegative(henceFV)]TJ /F5 7.97 Tf 6.58 0 Td[(1isalsononnegative)andalleigenvaluesofJhavenegativerealpartifandonlyifRvf:=(FV)]TJ /F5 7.97 Tf 6.59 0 Td[(1)<1.UsingthenotationthatC:=(ci,j)isthecofactormatrixofV,wehavethatV)]TJ /F5 7.97 Tf 6.59 0 Td[(1=1 det(V)0BBBBBBBBBBBBBB@c1,1c2,2c2,3c3,3c4,3c5,3c6,3c4,4c2,5c3,5c4,5c5,5c6,5c2,6c3,6c4,6c5,6c6,61CCCCCCCCCCCCCCA. 42

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So,FV)]TJ /F5 7.97 Tf 6.59 0 Td[(1=1 det(V)0BBBBBBBBBBBBBBBBBBBB@f1,3c2,3f1,3c3,3f1,3c4,3f1,3c5,3f1,3c6,30+f1,5c2,5+f1,5c3,5+f1,5c4,5+f1,5c5,5+f1,5c6,5+f1,6c2,6+f1,6c3,6+f1,6c4,6+f1,6c5,6+f1,6c6,6f2,1c1,100f3,1c1,1.........f4,1c1,1.........0.........0000001CCCCCCCCCCCCCCCCCCCCAObservethatthenonzeroeigenvaluesofFV)]TJ /F5 7.97 Tf 6.59 0 Td[(1areexactlythenonzeroeigenvaluesof\FV)]TJ /F5 7.97 Tf 6.59 0 Td[(1=1 det(V)0BBBBBBBBBBBBBB@f1,3c2,3f1,3c3,3f1,3c4,30+f1,5c2,5+f1,5c3,5+f1,5c4,5+f1,6c2,6+f1,6c3,6+f1,6c4,6f2,1c1,100f3,1c1,1.........f4,1c1,1001CCCCCCCCCCCCCCA,wherec1,1=v2,2v6,6(1)]TJ /F4 11.955 Tf 11.96 0 Td[(k3,6k5,3k6,5)c2,3=v1,1v4,45,5v6,6v3,2c3,3=v1,1v2,2v4,4v5,5v6,6c4,3=v1,1v2,2v5,5v3,6v6,4c2,5=v1,1v4,4v6,6v3,2v5,3c3,5=v1,1v2,2v4,4v6,6v5,3c4,5=v1,1v2,2v5,3v3,6v6,4 43

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c2,6=v1,1v4,4v3,2v5,3v6,5c3,6=v1,1v2,2v4,4v5,3v6,5c4,6=v1,1v2,2v3,3v5,5v6,4,anddet(V)=v1,1v6,6(1)]TJ /F4 11.955 Tf 11.95 0 Td[(k3,6k5,3k6,5),wherewedeneki,j:=vi,j=vj,j.Theonlypositiverootofthecharacteristicpolynomialp()=j\FV)]TJ /F5 7.97 Tf 6.59 0 Td[(1)]TJ /F7 11.955 Tf 13.11 0 Td[(Ijis=p a1+a2+a3,wherea1=k2,1 1)]TJ /F4 11.955 Tf 11.95 0 Td[(k3,6k5,3k6,5(k3,2k1,3+k3,2k5,3k1,5+k3,2k5,3k6,5k1,6)a2=k3,1 1)]TJ /F4 11.955 Tf 11.95 0 Td[(k3,6k5,3k6,5(k1,3+k5,3k1,5+k5,3k6,5k1,6)a3=k4,1 1)]TJ /F4 11.955 Tf 11.95 0 Td[(k3,6k5,3k6,5(k6,4k1,6+k6,4k3,6k1,3+k6,4k3,6k5,3k1,5).SincetheinvasionnumberRvfisdenedtobethedominanteigenvalueofFV)]TJ /F5 7.97 Tf 6.58 0 Td[(1,Rvf==p a1+a2+a3,whichleadsto Rvf=1 1)]TJ /F4 11.955 Tf 11.95 0 Td[(k5,3k6,5k3,6(k2,1k3,2k1,3+k2,1k3,2k5,3k1,5+k2,1k3,2k5,3k6,5k1,6+k3,1k1,3+k3,1k5,3k1,5+k3,1k5,3k6,5k1,6 (2) +k4,1k6,4k1,6+k4,1k6,4k3,6k1,3+k4,1k6,4k3,6k5,3k1,5)1=2.Thefactor1=(1)]TJ /F4 11.955 Tf 10.38 0 Td[(k5,3k6,5k3,6)canbewrittenasthegeometricseries1Pn=0(k5,3k6,5k3,6)n,wherek5,3k6,5k3,6=fvjv fvjv+f+fv fv+v v+f+istheprobabilitythatafalciparum-onlyinfectedhumanwillloopthroughthepathIf!Cfv!Ic!Ifntimesbeforeinfectingamosquito.ThisloopariseswhenanIfindividualbecomesco-infected,progressestotheIcstage,andrecoversfromvivaxmalariainfectionrst,returningtotheIfstage.NotethatanIfindividualcanonlytransmitP.falciparumparasitesbyinfectingamosquitobeforeleavingthatstage,orbybecomingco-infectedandrecoveringrstfromvivaxinfection.Alsonotethatk5,3=fvjv fvjv+f+isthetransitionprobabilityforIf!Cfv, 44

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k6,5=fv fv+isthetransitionprobabilityCfv!Ic,andnallyk3,6=v v+f+representsthetransitionprobabilityIc!If.WecaninterprettheremainingtermsinRvfsimilarly.Insteadofapathrepresentingaloopthatasingleindividualtakes,eachpathbelowrepresentsthepathforhowonefalciparuminfectedmosquitocanleadtoanewmosquitoinfection.k2,1k3,2k1,3=jf!Cf!If!jfk2,1k3,2j5,3k1,5=jf!Cf!If!Cfv!jfk2,1k3,2k5,3k6,5k1,6=jf!Cf!If!Cfv!Ic!jfk3,1k1,3=jf!If!jfk3,1k5,3k1,5=jf!If!Cfv!jfk3,1k5,3k6,5k1,6=jf!If!Cfv!Ic!jfk4,1k6,4k1,6=jf!Cvf!Ic!jfk4,1k6,4k3,6k1,3=jf!Cvf!Ic!If!jfk4,1k6,4k3,6k5,3k1,5=jf!Cvf!Ic!If!Cfv!jfIfwemultiplyanyoneofthetermsaboveby1Pn=0(k5,3k6,5k3,6)n,thenthenthtermintheresultingsumwillhavethesamechainofeventsasabove,withtheexceptionthattheIfindividualtakestheIf!Cfv!Ic!Ifloopntimesbeforecontinuingtothenextstageinthechain.Thus,thenext-generationapproachleadstoanexpressionoftheinvasionnumberswhosesquarehasthebiologicalinterpretationwedesire:(Rvf)2isthenumberofsecondaryfalciparummosquitoinfectionscausedbyasinglefalciparum-infectedmosquitoinapopulationatthevivaxisolatedendemicequilibrium. 2.3.10FindingRfvUsingtheNext-GenerationApproachNowwederivetheinvasionnumberRfv,whoseexpressionismorecomplicatedthanthatofRvf. 45

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Theinfectedsubsystemwill,inthiscase,consistonlyofequationsforstatevariablesinfectedwithP.vivax,sincewewanttodeterminethestabilityofthefalci-parum-onlyequilibriumwhenvivaxattemptstoinvade.WerstndtheJacobianofourinfectedsubsystem,evaluatedatthefalciparum-onlyequilibrium,withtheequationsorderedasfollows:fj0v,L0,C0v,I0v,C0vf,C0fv,I0cg.WewriteJ=F)]TJ /F4 11.955 Tf 12.35 0 Td[(V,whereFandVare77squarematricesandF=0BBBBBBBBBBBBBBBBB@000f1,4f1,50f1,7f2,1000.........0.........0.........f6,1.........0001CCCCCCCCCCCCCCCCCAandV=0BBBBBBBBBBBBBBBBB@v1,10000v2,20)]TJ /F4 11.955 Tf 9.3 0 Td[(v2,4............)]TJ /F4 11.955 Tf 9.29 0 Td[(v3,2v3,3000...)]TJ /F4 11.955 Tf 9.29 0 Td[(v4,2)]TJ /F4 11.955 Tf 9.3 0 Td[(v4,3v4,400)]TJ /F4 11.955 Tf 9.3 0 Td[(v4,7...)]TJ /F4 11.955 Tf 9.29 0 Td[(v5,20)]TJ /F4 11.955 Tf 9.3 0 Td[(v5,4v5,50000v6,6000)]TJ /F4 11.955 Tf 9.3 0 Td[(v7,5)]TJ /F4 11.955 Tf 9.3 0 Td[(v7,6v7,71CCCCCCCCCCCCCCCCCA,wheretheelementsofFare:f1,4=bv(1)]TJ /F4 11.955 Tf 12.99 0 Td[(jv)=^K,f1,5=f1,4,f1,7=f1,4,f2,1=vS,andf6,1=fvIf.TheelementsofVare:v1,1=d,v2,2=vfjf++,v2,4=,v3,2=v,v3,3=v+,v4,2=(1)]TJ /F7 11.955 Tf 12.78 0 Td[(v),v4,3=v,v4,4=vfjf++v+,v4,7=f,v5,2=vfjf,v5,4=v5,2,v5,5=vf+,v7,5=vf,b7,6=fv,v7,7=v+f+. 46

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Again,itisclearthatVhastheZ-signpattern.So,asdescribedin 2.3.6 ,itisstraightforwardtoshowthatVisanM-matrixbyverifyingthatallcolumn-sumsofVarepositive.TheinverseofVhasthefollowingform,wherethedotsrepresentzeros:V)]TJ /F5 7.97 Tf 6.59 0 Td[(1=0BBBBBBBBBBBBBBBBB@dv1,1dv2,2dv2,3dv2,4dv2,5dv2,6dv2,7dv3,2dv3,3dv3,4dv3,5dv3,6dv2,7dv4,2dv4,3dv4,4dv4,5dv4,6dv4,7dv5,2dv5,3dv5,4dv5,5dv5,6dv5,7dv6,6dv7,2dv7,3dv7,4dv7,5dv7,6dv7,71CCCCCCCCCCCCCCCCCA.So,FV)]TJ /F5 7.97 Tf 6.58 0 Td[(1=0BBBBBBBBBBBBBBBBBBBBBBBB@f1,4dv4,2f1,4dv4,3f1,4dv4,4f1,4dv4,5f1,4dv4,6f1,4dv4,7+f1,5dv5,2+f1,5dv5,3+f1,5dv5,4+f1,5dv5,5+f1,5dv5,6+f1,5dv5,7+f1,7dv7,2+f1,7dv7,3+f1,7dv7,4+f1,7dv7,5+f1,7dv7,6+f1,7dv7,7f2,1dv1,1f6,1dv1,11CCCCCCCCCCCCCCCCCCCCCCCCAThenonzeroeigenvaluesofFV)]TJ /F5 7.97 Tf 6.58 0 Td[(1arepreciselythenonzeroeigenvaluesofthe3x3matrix 47

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\FV)]TJ /F5 7.97 Tf 6.59 0 Td[(1=0BBBBBBBBBB@f1,4dv4,2f1,4dv4,6+f1,5dv5,2+f1,5dv5,6+f1,7dv7,2+f1,7dv7,6f2,1dv1,1f6,1dv1,11CCCCCCCCCCA.Thecharacteristicpolynomialof\FV)]TJ /F5 7.97 Tf 6.59 0 Td[(1isgivenbyp()=j\FV)]TJ /F5 7.97 Tf 6.59 0 Td[(1)]TJ /F7 11.955 Tf 13.51 0 Td[(Ij=[k6,1(f1,4dv4,6+f1,5dv5,6+f1,7dv7,6)+k2,1(f1,4dv4,2+f1,5dv5,2+f1,7dv7,2))]TJ /F7 11.955 Tf 11.04 0 Td[(2],whereki,j:=vi,j=vj,j.SinceVisanM-matrix,weknowthattheinverseofVhasonlynonnegativeelements.Thus,dv5,6,dv7,6,dv5,2,anddv7,2arenonnegative,andthelargestpositiverootofp()is=p k6,1(f1,4dv4,6+f1,5dv5,6+f1,7dv7,6)+k2,1(f1,4dv4,2+f1,5dv5,2+f1,7dv7,2)].Recallthatcvi,jaretheelementsofV)]TJ /F5 7.97 Tf 6.59 0 Td[(1.Thus,cvi,j=()]TJ /F8 11.955 Tf 9.3 0 Td[(1)i+jcj,i=det(V),whereC:=(ci,j)isthecofactormatrixofV.Todeterminedv4,6,dv5,6,dv7,6,dv4,2,dv5,2,anddv7,2,weneedonlycalculatec6,4,c6,5,c6,7,c2,4,c2,5,c2,7,anddet(V).c6,4=v1,1v2,2v3,3v5,5v4,7v7,6c6,5=v1,1v7,6v4,7v3,3(v2,2v5,4+v2,4v5,2)c6,7=v1,1v7,6v5,5[v2,2v3,3v4,4)]TJ /F4 11.955 Tf 11.95 0 Td[(v2,4(v3,2v4,3+v3,3v4,2)]c2,4=v1,1v6,6[v7,5v5,2v3,3v4,7+v7,7v5,5(v3,2v4,3+v3,3v4,2)]c2,5=v1,1v6,6v7,7[v3,2v4,3v5,4+v3,3(v4,2v5,4+v4,4v5,2)]c2,7=v1,1v6,6v7,5[v3,2v4,3v5,4+v3,3(v4,2v5,4+v4,4v5,2)],anddet(V)=v1,1v2,2v7,7[1)]TJ /F4 11.955 Tf 11.96 0 Td[(k2,4(k3,2k4,3+k4,2))]TJ /F4 11.955 Tf 11.96 0 Td[(k7,5k4,7(k5,4+k2,4k5,2)]=1)]TJ /F7 11.955 Tf 60.89 8.08 Td[( vfjf++v+v vfjf++v v++(1)]TJ /F7 11.955 Tf 11.95 0 Td[(v) vfjf++)]TJ /F7 11.955 Tf 21.32 8.08 Td[(vf vf+f v+f+vfjf vfjf++v++ vfjf++v+vfjf vfjf++ 48

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TheinvasionreproductionnumberRfv:=.Makingtheappropriatesubstitutionsinto,wearriveatthefollowingexpressionforRfv: Rfv=f[k6,1(k1,5k7,6k4,7(k5,4+k2,4k5,2)+k1,7k7,6(1)]TJ /F4 11.955 Tf 11.96 0 Td[(k2,4(k3,2k4,3+k4,2)))+k2,1[(k1,5+k1,7k7,5)(k3,2k4,3k5,4+k4,2k5,4+k5,2)]g (2) (1)]TJ /F4 11.955 Tf 11.96 0 Td[(k2,4(k3,2k4,3+k4,2))]TJ /F4 11.955 Tf 11.96 0 Td[(k4,7k7,5(k5,4+k2,4k5,2))g1=2,wherek6,1=fvIf dk1,5=bv(1)]TJ /F4 11.955 Tf 11.96 0 Td[(jf) ^K(vf+)k7,6=vf vf+k4,7=f v+f+k5,4=vfjf vfjf++v+k2,4= vfjf++v+k5,2=vfjf vfjf++k1,7=k1,5k7,6=fv fv+k3,2=v vfjf++k4,3=v v+k4,2=(1)]TJ /F7 11.955 Tf 11.95 0 Td[(v) vfjf++k2,1=vS dk7,5=vf vf+Despiteitscomplicatedform,wecanshowthatthesquareoftheinvasionnumberRfvisthenumberofnewvivax-infectedmosquitoesarisingfromasinglevivax-infectedmosquitoinapopulationatthefalciparumisolatedendemicequilibrium. 49

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Toarriveatthecorrectbiologicalinterpretation,werstobservethatexpandingtheexpressionfor)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(Rfv2revealsthateachtermintheresultingsumrepresentsapathbywhichonevivax-infectedmosquitoleadstoanothermosquitoinfection.Forexample,therstterm(rearranged),k6,1k7,6k4,7k5,4k1,5,representsthenumberofIf-humansinfectedbyavivax-infectedmosquitobeforedying,causingthosehumanstoprogresstotheCfvstage,timesthefractionofpeoplethatsurvivetheCfvstageandprogresstotheIcstage,timesthefractionofindividualsthatsurvivethisstageandaretreatedforfalciparumpriortotreatmentforvivax,enteringtheIvstage,timestheprobabilitythattheseindividualsareinfectedbyafalciparum-infectedmosquitoandprogresstotheCvfstage,andnally,timesthenumberofsusceptiblemosquitoesaCvfhumaninfectspriortoprogressingtotheco-infectiousstageIc.EachterminRfvrepresentssuchapathfromaninfectedmosquitotoanothermosquitoinfection.Thenegativeterms,aswewilldemonstrate,accountforinfectionsthatarisebecauseahumanpassesthroughthesamestagemorethanonce.InthederivationofRfvwearguedthatthedenominatorofRfvispositive.Thus,itmustbethatk2,4(k3,2k4,3+k4,2)+k4,7k7,5(k5,4+k2,4k5,2)islessthanone.UsingthesamereasoningaswedidforRCvandRvf,wecanrewrite1=(1)]TJ /F4 11.955 Tf 12.12 0 Td[(k2,4(k3,2k4,3+k4,2))]TJ /F4 11.955 Tf 12.11 0 Td[(k4,7k7,5(k5,4+k2,4k5,2))asageometricseries,allowingustofullyinterprettheinvasionnumber.Since1=(1)]TJ /F4 11.955 Tf 11.95 0 Td[(x)]TJ /F4 11.955 Tf 11.95 0 Td[(y)=1Pn=0(x+y)nwhenjx+yj<1,wecanrewriteRfvas )]TJ /F4 11.955 Tf 5.48 -9.68 Td[(Rfv2=hk6,1k1,5k7,6k4,7(k5,4+k2,4k5,2)+k2,1(k1,5+k1,7k7,5)(k3,2k4,3k5,4+k4,2k5,4+k5,2)i (2) 1Xn=0[k2,4(k3,2k4,3+k4,2)+k4,7k7,5(k5,4+k2,4k5,2)]n+k6,1k1,7k7,61)]TJ /F4 11.955 Tf 11.96 0 Td[(k2,4(k3,2k4,3+k4,2) 1)]TJ /F4 11.955 Tf 11.96 0 Td[(k2,4(k3,2k4,3+k4,2))]TJ /F4 11.955 Tf 11.96 0 Td[(k4,7k7,5(k5,4+k2,4k5,2)Now,wecanrewritethefractioninthelasttermsothattheexpressionforRfvisfullyinterpretable.Notethatthistermisoftheform(1)]TJ /F4 11.955 Tf 12.99 0 Td[(x)=(1)]TJ /F4 11.955 Tf 12.99 0 Td[(x)]TJ /F4 11.955 Tf 13 0 Td[(y),where 50

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xispreciselyk2,4(k3,2k4,3+k4,2)andyisk4,7k7,5(k5,4+k2,4k5,2).Usingthefactthat1)]TJ /F6 7.97 Tf 6.59 0 Td[(x 1)]TJ /F6 7.97 Tf 6.59 0 Td[(x)]TJ /F6 7.97 Tf 6.59 0 Td[(y=1)]TJ /F6 7.97 Tf 6.59 0 Td[(x)]TJ /F6 7.97 Tf 6.58 0 Td[(y+y 1)]TJ /F6 7.97 Tf 6.59 0 Td[(x)]TJ /F6 7.97 Tf 6.59 0 Td[(y=1+y 1)]TJ /F6 7.97 Tf 6.59 0 Td[(x)]TJ /F6 7.97 Tf 6.59 0 Td[(y,wehavethat 1)]TJ /F4 11.955 Tf 11.96 0 Td[(x 1)]TJ /F4 11.955 Tf 11.96 0 Td[(x)]TJ /F4 11.955 Tf 11.95 0 Td[(y=1+y1Xn=0(x+y)n (2) Hence,wearriveatafullyinterpretableexpressionfortheinvasionnumber )]TJ /F4 11.955 Tf 5.48 -9.68 Td[(Rfv2=hk6,1k1,5k7,6k4,7(k5,4+k2,4k5,2)+k2,1(k1,5+k1,7k7,5)(k3,2k4,3k5,4+k4,2k5,4+k5,2)i (2) 1Xn=0(x+y)n+k6,1k7,6k1,7 1+y1Xn=0(x+y)n!.Thetermsx=k2,4(k3,2k4,3+k4,2)andy=k4,7k7,5(k5,4+k2,4k5,2)representfourdifferenttransmissionpaths.xrepresentstwowaysapersoncanstartin,andreturnto,stageIv.Onepathtravelsthroughthesymptomaticclasswhiletheotherdoesnot.Similarly,yrepresentstwowaysinwhichapersoninIccanarriveatstageCvf.OneofthesetwopathstravelsthroughstageL,whiletheotherpathbypassesstageL.ThenthterminthesummationP1n=0(x+y)nrepresentstheprobabilityoftakinganycombinationofthefourloops,resultinginatotalofexactlynloops.Thenumberoneinparenthesesofequation( 2 )representsthecontributiontosecondaryvivaxcasesbyvivax-infectedindividualswhomakenoloops.Finally,thenthtermintheexpressiony1Pn=0(x+y)nrepresentstheprobabilitythatanindividualrsttakesoneoftheloopsiny,thenmakesatotalofexactlynloopsconsistingofsomecombinationofthefourloopsdescribedbyxandy.Thesecondsummationinequation( 2 )arisesbecausetheonlywayinwhichanindividualcanenterpathx(Iv!L(L!Cv!Iv+L!Iv))frompathk1,7k6,1k7,6(Ic!jv!Cfv!Ic)isbyrstenteringpathy(Ic!Iv(Iv!Cvf+Iv!L!Cvf)).Conversely,pathsxandycanbereachedfromallotherpathsrepresentedinequation( 2 ). 51

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Bycarefullyrewritingtheinvasionnumberstoconsistoftermsthatcanbeinterpretedaseitherprobabilitiesorfractionsofapopulationofindividualsinaparticularstate,wehaveshownthatitispossibletolinkthemathematicalexpressionstoabiologicalinterpretationrelevanttopublichealth.InSection 3.1.3 ,weillustratehowtheseanalyticexpressionscanbeusedtounderstandtheinterplaybetweentheuseofmalariainterventionsandthecompetitionbetweenfalciparumandvivax. 52

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CHAPTER3PARAMETERESTIMATIONANDEVALUATINGUNCERTAINTY 3.1DescriptionofModelParametersandChoiceofParameterValuesfortheYearsfrom1987to1996ToanswerquestionsaboutdiseasedynamicsandtheuseofcontrolmeasuresinIndia,wemustdeterminerealisticestimatestoparameterizeourmathematicalmodel.Todothiswefoundreasonableestimatesfromthemalarialiteratureforallparametersbutthetransmissionparameters(bv,bf,v,andf)andhumanpopulationgrowthparameters(randK).Usingtheseestimatesfromtheliterature,weestimatetheremainingparametersbyttingthemodeltomalariacasedataforIndia.Inthefollowingsectionswerstdiscussthechoiceofestimatesforparametersfoundintheliterature,thenwedescribetheprocedureforestimatingthehumanpopulationintrinsicgrowthrate,carryingcapacity,andthemalariatransmissionparameters. Table3-1. Descriptionofmodelparameterspertainingtomosquitopopulationdynamicsandtheirestimates ParametersDescriptionValueReference dNaturaldeathrate365 14years)]TJ /F5 7.97 Tf 6.59 0 Td[(1[ 13 ] Probabilitythatasusceptiblemosquito670 670+332see 3.1.1.4 thatgetsinfectedbyaco-infectedhumancontractsP.vivax 1)]TJ /F7 11.955 Tf 11.95 0 Td[(Probabilitythatasusceptiblemosquitothatgetsinfectedbyaco-infectedhumancontractsP.falciparum 3.1.1EstimationofParametersfromtheLiterature 3.1.1.1TimetoinfectiousnessFollowingtheonsetofsymptoms,ittakesroughly4daysforP.vivaxinfectionstobecomeinfectiousinahumanhost[ 43 ],andapproximately7daysforP.falciparuminfection[ 65 ].Thus,wetakevandf,therateofprogressionfromsymptomatictoinfectiousforP.vivaxandP.falciparum,respectively,tobe365 4years)]TJ /F5 7.97 Tf 6.58 0 Td[(1forP.vivax,and365 7years)]TJ /F5 7.97 Tf 6.58 0 Td[(1forP.falciparum.Weassumethatbecomingco-infecteddoesnotalterthetimeittakestobecomeinfectious.Thus,weletfv=vandvf=f. 53

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Table3-2. Descriptionofmodelparameterspertainingtohumanpopulationdynamicsandtheirestimates ParametersDescriptionValueReference 1 DurationofP.vivaxliverstage90dayssee 3.1.1.3 1 vTimeuntilinfectiousafter4days[ 43 ]P.vivaxsymptomonset 1 fTimeuntilinfectiousafter7days[ 65 ]P.falciparumsymptomonset 1 vfDurationofCvf1 fsee 3.1.1.1 1 fvDurationofCfv1 vsee 3.1.1.1 Naturaldeathrate1 60.55years)]TJ /F5 7.97 Tf 6.59 0 Td[(1see 3.1.2 vP.vivaxblood-stageparasite1 3days)]TJ /F5 7.97 Tf 6.58 0 Td[(1[ 83 ]clearanceratewithtreatment fP.falciparumtreatmentrecoveryrate1 12days)]TJ /F5 7.97 Tf 6.59 0 Td[(1see 3.1.1.2 rvP.vivaxnaturalblood-stage365/30years)]TJ /F5 7.97 Tf 6.59 0 Td[(1see 3.1.1.3 parasiteclearancerate rfP.falciparumnaturalrecoveryrate365 200years)]TJ /F5 7.97 Tf 6.59 0 Td[(1[ 11 ] vRecoveryratefromIvtoSsee 3.1.1.2 fRecoveryratefromIftoSsee 3.1.1.2 prProbabilityofpost-treatmentrange0.23)]TJ /F8 11.955 Tf 11.96 0 Td[(0.44[ 3 ]P.vivaxrelapsemedian0.2904 P.vivaxrelapserateprvv+(1)]TJ /F7 11.955 Tf 11.96 0 Td[(v)rvsee 3.1.1.3 vProbabilitythataP.vivax0.82[ 89 ]infectedhumanbecomessymptomatic fProbabilitythataP.falciparuminfectedhumanbecomessymptomatic0.90assumed vPv-inducedcross-immunitytoPf1 fPf-inducedcross-immunitytoPv1 Fractionofco-infectedinfectiousindividuals0.75see 3.1.1.5 thatrecoverrstfromP.falciparum vRateofprogressionfromIctoIfvduetoP.vivaxtreatment fRateofprogressionfromIctoIv(1)]TJ /F7 11.955 Tf 11.95 0 Td[()fduetoP.falciparumtreatment 54

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3.1.1.2EstimatingrecoveryratesTherateofrecoveryfromIvandIftothesusceptibleclassSisestimatedbyv=(1)]TJ /F7 11.955 Tf 11.98 0 Td[(v)rv+vvandf=(1)]TJ /F7 11.955 Tf 11.98 0 Td[(f)rf+ff,respectively.Inotherwords,afractionrecoveratthenaturalrecoveryrate,andafractionrecoveratthetreatmentrecoveryrate.SinceChloroquine(CQ)targetsonlytheasexualbloodstagesoftheparasite,theremaystillbegametocytesremainingattheendoftreatment.Ittakesroughly8daysforgametocytestomature,andthelifespanofamaturegametocyteisroughlybetween3.5and4days.Fromthis,weestimatethatindividualstreatedforfalciparumwithdrugsthatdonotkillthegametocytescanremaininfectiousforupto12(8+4)daysaftertreatmentiscompleted.Thus,treatmentofP.falciparuminfectionswithChloroquinereducestheinfectiousperiodfromroughly200days[ 11 ]to12days,andwetakerf=365 200years)]TJ /F5 7.97 Tf 6.59 0 Td[(1,andf=365 12years)]TJ /F5 7.97 Tf 6.58 0 Td[(1.AstudyofP.vivaxgametocytemiafoundthatoutof516patientstreatedwithCQ,only4stillhadnotclearedthegametocytesbythethirddayoftreatment[ 79 ].Usingthisnding,weletv=365 3. 3.1.1.3ParameterizingP.vivaxrelapseJoshietal.[ 61 ]notethatpatternsofP.vivaxrelapsecanbecategorizedintothreegroups.Therstgroupisreferredtoasthetropicaltypewhichischaracterizedbyanearlyprimaryattackwithfrequentrelapses.Thetimeintervalsbetweenrelapsesofthetropicaltypearebetweenoneandthreemonths.GroupIIhasrelapseintervalsofintermediatelength-approximatelybetweenthreeandvemonthslong.Andnally,groupIII,alsoknownasthetemperatetype,ischaracterizedbyalongprimarylatentperiodandrelapsesoccurringeverysixtosevenmonths.Inthismalariamodel,weassumethatP.vivaxinfectedindividualswhorelapsearethosewhoeitherwerenevertreatedorwereunsuccessfullytreated.SinceP.vivaxparasitesinducingshort-termrelapsepatternswerefoundtobelesssusceptibletoanti-relapsedrugs[ 61 ],weassumethatindividualswhowereunsuccessfullytreatedforP.vivaxexhibitgroupIrelapsepatterns.Thus,theyshouldrelapseeveryoneto 55

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threemonths.TherateatwhicharelapsingindividualprogressesfromIvtoListherateatwhichthatindividuallosesinfectiousness(i.e.therateatwhichgametocytesareclearedfromtheblood).Weassumethattreatedindividualsloseinfectiousnessataratev,regardlessofwhethertreatmentwassuccessfulornot,anduntreated(i.e.asymptomatic)individualsloseinfectiousnessataraterv.Adaketal.determinedthat29.04percentofP.vivaxpatientstreatedonlywithChloroquine(CQ)relapsedfollowingtreatment[ 3 ].Thus,therateatwhichindividualsprogressfromIvtotheliverstageclassL,isgivenby=.2904vv+(1)]TJ /F7 11.955 Tf 11.96 0 Td[(v)rv.Ifnooneistreated,then=rv.ThetimebetweenP.vivaxrelapsesisusuallydenedasthetimebetweenclinicalepisodes.However,inthismodelitispossibleforindividualstorelapse,inthesensethattheparasiterepeatsthecycleofinfectionwithinthehumanhost,withoutpassingthroughthesymptomaticstageCv.WewilltakethetimebetweenrelapsestobethetimeittakestoprogressfromIvtoL(1 rvforanuntreatedindividualand1 vforanunsuccessfullytreatedindividual),plusthetimeittakestoprogressfromLtothenextinfectedstage(1 ).Thus,ifwetaketheaveragetimebetweenrelapsestobethreemonthsforanunsuccessfullytreatedindividual,1 +1 v=3months90)]TJ /F8 11.955 Tf 12.54 0 Td[(93days.Since1 visapproximately3dayslong,wetake1 tobe90days.Inotherwords,=365 90years)]TJ /F5 7.97 Tf 6.59 0 Td[(1.AsymptomaticindividualscouldhaverelapsepatternsassociatedwithgroupI,II,orIII-experiencingarelapseanywherefromeveryonetosevenmonths.Thus,wetake1 rv+1 tobetheaverageoffourmonthslong.Inotherwords,1 +1 rv120)]TJ /F8 11.955 Tf 10.17 0 Td[(124days.Fromthisestimateandourestimatefor1=,weassumethatittakesroughly30daysforanuntreatedP.vivaxinfectedindividualtoloseinfectiousness. 3.1.1.4EstimationofAstudyconductedbyPhimpraphietal.[ 84 ]showednosignicantdifferenceingametocyteproductionbyP.vivaxorP.falciparumparasitesinaco-infectedhumanthaninhumanswhowereonlyinfectedwithoneofthetwoparasitespecies.Also,P.vivaxgametocytedensitieswerefoundtobehigherthanP.falciparumdensitiesin 56

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infectedhumans,withroughly670P.vivaxgametocytesperlofbloodand332P.falciparumgametocytesperlofblood.Sincegametocytesaretheinfectiousstageofthemalariaparasitesinhumans,weusethesendingstodeterminearoughestimateoftheparameter,theproportionofmosquitoesinfectedbyahumaninIcthatcontractP.vivax.Weassumethatisthedensityofvivaxgametocytesintheblooddividedbythetotalgametocytedensity.Inotherwords,=670 670+3320.67. 3.1.1.5EstimationofP.vivaxandP.falciparumarealsoendemictoThailandwithroughlyhalfthecasesresultingfromP.vivaxinfectionandhalfduetoP.falciparuminfection.Approximately10percentofcasesinThailandinitiallydiagnosedasP.vivaxcasesand30percentofcasesinitiallydiagnosedasP.falciparumcasesturnedouttobeco-infections[ 109 ].Fromthis,weestimatedthattheproportionofco-infectedcasestreatedrstforP.falciparumis=0.75. 3.1.2EstimationofPopulationGrowthandTransmissionParametersUsingPopulationandMalariaCaseDataforIndia Figure3-1. PlotoftimeseriesdataforIndia'sPopulationSizefrom1950to2009andthebesttofthelogisticcurvetothisdata.Populationdatawasobtainedfrom[ 88 ]. 57

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FromlifeexpectancydataforIndia[ 68 ],weestimatedthattheaveragelifeexpectancybetweentheyears1987and2009isapproximately60.55years,givingus=1=60.55years)]TJ /F5 7.97 Tf 6.59 0 Td[(1.Usingthisestimateandanonlinearleast-squarestofthelogisticequationtoIndia'spopulationdata,estimatesareobtainedfortheparametersrandK(seeTable 3-3 ).ThebesttofthelogisticcurveisillustratedinFigure 3-1 Table3-3. EstimatesofrandK. ParameterDescriptionEstimateCI rGrowthrate0.0398years)]TJ /F5 7.97 Tf 6.58 0 Td[(10.0392.0404KCarryingcapacity7.5616109humans6.29191098.8313109 Assumingthattheuseofcontrolmeasuresremainedfairlysimilarduringtheperiodfrom1987to1996,wecanestimatethetransmissionratesbv,bf,v,fbyimputingtheparametervaluesinTables 3-1 3-2 ,and 3-3 ,andttingthemodeltothemalariacasedata.Moreprecisely,weusedthe`nlint'functioninMATLABtominimizethesumofsquaresofthedifferencebetweenthedataandthesolutionscurvesbycomparingsolutioncurveCvtotheP.vivaxdataandsimilarlycomparingthesolutioncurveCf+Cvf+CfvtotheP.falciparumplusmixed-casedata.Fromthisttingprocedure,weobtainestimatesforthetransmissionparameters,summarizedinTable 3-4 .ThecontrolledreproductionnumbersRCvandRCfcannowbecalculatedusingtheexpressionsinSection 2.3.5 (seeTable 3-4 ). Table3-4. Pre-1997estimatesofthetransmissionparameters. ParameterEstimateCI bv14.540914.2907.7911bf14.04426.2573.8311v191.3306188.3507.3105f51.731223.2618.2006 RCv1.0203RCf1.0052 Theresultingreproductionnumbersarelargerthanone,implyingmathematicallythatatleastoneofthetwomalariaparasiteswillpersist.Yet,inpractice,theextremely 58

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closeproximityofRCvandRCftothepersistencethresholdmakesitdifculttoarriveatanydenitiveconclusionregardingtheoutcomeofmalariainIndia.Asasteptowardsaddressingthisconcern,weuseaparametricbootstrappingproceduretoestimatecondenceintervalsforRCvandRCf.Theprocedure,whichwewillre-iterateherewithslightmodications,isdescribedin[ 29 ]byChowelletal. Table3-5. Pre-1997mean,median,standarddeviations,andcondenceintervalsforRCvandRCf,derivedfromparametricbootstrap. Parametermeanmedianstand.dev.95%CI RCv1.02031.02040.00061.0192.0214RCf1.00521.00520.00011.005.0055 LetusdenotethesolutioncurvesCvandCf+Cfv+CvfthatbesttthedatabySvandSf,respectively.Inoneiterationofthebootstrapprocedure,wesimulatenewvivaxcasedatabydrawingonevalueforeachyear(1987-1996)fromaPoissondistributionwithmeanequaltothevalueofSvatthecorrespondingyear.Inthesameiterationwesimulatenewfalciparum/mixedcasedatainthesamemanner:foreachyear,onenewdatapointisdrawnfromaPoissondistributionwithmeanequaltothevalueofSfatthecorrespondingyear.Inthismanner,wegenerateanewsimulatedsetofP.vivaxandP.falciparum/mixedcasedata.NewestimatesforthetransmissionparametersaredeterminedbyttingCvandCf+Cfv+Cvftothissimulateddata.Thisprocedureofsimulatingnewdataandttingthemodeltothesimulateddataisrepeated1000times.Calculatingtheisolatedcontrolledreproductionnumbersforeachofthe1000runsallowsustoproducehistogramsofthe1000valuesofRCvandRCf.Thesegures(seeFigures 3-2 )revealthatthevaluesofthereproductionnumbersgeneratedbythebootstrappingprocedureappearfairlysymmetric.Consequently,itissimpletodetermineappropriate95%condenceintervalsforRCvandRCf(seeTable 3-5 )bydeterminingthe0.025and0.975quantilesofthe1000estimates.Figure 3-2 illustratesthattheestimatesofRCv,RCf,Rfv,andRvfareconsistentlygreaterthanone.Hence, 59

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wecanconcludethatthetwoPlasmodiumspecieswouldlikelycontinuetocoexistafter1997hadmalariainterventionstrategiesnotimproved. ARCvandRvfbootstrapdata BRCfandRfvbootstrapdataFigure3-2. Histogramsof A RCvandRvfdataand B RCfandRfvdatageneratedbybootstrapfortheperiod1987-1996. 3.1.3EstimationofParametersfortheEnhancedMalariaControlPeriodAround1997,severalprogramsarosethatresultedinanupsurgeinfundingformalariacontrolinIndia.Asaconsequenceofenhancedmalariacontrol,parametersrelatedtodifferentcontrolpoliciesundoubtedlyalsochangedaround1997.Here,weattempttoassessthatchangebyagainttingourmalariamodel,thistimetocasedatafortheperiod1997-2010.Ingeneral,anincreaseintheuseofbednetsdecreasesmosquitobitingrate,increaseduseofinsecticidetreatedbednets(ITNs)bothdecreasesbitingrateandincreasesthemosquitomortalityrate,improvedtreatmentincreasestherecoveryrate,andinsecticidesincreasethemosquitomortalityrate.Acombinationofthesecontrolmeasuresisoftenused.Ourrstgoalhereistounderstandwhichofthesecontrolmeasures,orcombinationofcontrolmeasures,contributedthemosttothedeclineinthenumberofmalariacasesafter1996.Secondly,wewanttounderstandhowtheincreaseinfundingformalariahasaffectedthecompetitionbetweenP.vivaxandP.falciparum.Toaddresstherstquestionwhichparameterscontributedthemosttothepost-1996declineincaseswetthemodeltothe1997-2010dataseveraltimes,each 60

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timeestimatingadifferentcombinationofparametersrelevanttomalariacontrolwhileleavingtheremainingparametersinthemodelxedtotheir1987-1996estimates.Weconsidereachoftheseparameterizationsofthemodeltobeadifferentmodel.Foreachmodel,wecalculatethecorrectedAkaikeInformationCriterion(AICc)-ameasureofthegoodnessoftofamodeltothedata,discountedbythenumberofparametersestimatedrelativetothesizeofthedataset.TheAICcvaluesallowustoorderthemodelsfrombesttoworst:themodelwiththesmallestAICcisthebestmodel,andthemodelwiththelargestAICcistheworstmodel.Tomakethedistinctionbetweenthemodelsclearer,wecalculatethe4AICcforeachmodel:thedifferenceinAICcbetweenthemodelandthemodelwiththesmallestAICc.Thismeansthatthebestmodelhasa4AICcofzero.TheresultsofthismodelcomparisonaresummarizedinTable 3-6 .Theruleofthumbisthatcandidatemodelswith4AICc'sbetween0and2havestrongsupport,modelswith4AICcbetween4and7haveconsiderablylesssupport(butshouldstillbeconsidered),andmodelswith4AICcgreaterthan10shouldbedisregardedaspotentialcandidates[ 20 ].TheresultsofthisanalysisyieldedthatmodelA=fv,f,av,afgcorrespondingtoestimatingtreatmentrecoveryrateandbitingrateparametersbestexplainstheobserveddata.Usingtheruleofthumbfor4AICcvalues,modelBhasstrongsupport,modelsC,D,E,andFhavelesssupportbutshouldstillremaininthepoolofpossiblemodels,andmodelGshouldbediscarded.However,itisimportanttopointoutthatmodelsAandBweresensitivetotheinitialguessfortheparametervaluesinthettingprocedure,whereastheremainingmodelresultswerefairlyrobusttotheinitialguess.ThismeansthattherelationshipbetweenmodelsC,D,E,F,andGremainthesamefordifferentinitialparameterguesseswhileAandBnddifferentpositionsinthelistdependingontheinitialguess.WearrivedattheorderingpresentedinTable 3-6 byrepeatingthettingprocedurefor3differentinitialguessesforeachofthesevenmodels,andchoosingtheestimatescorrespondingtothesmallestcondenceintervals. 61

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Ingeneral,addingtheestimationofd,mosquitodeathrate,toamodelincreasedtheAICcvalue,suggestingthatchangesinmosquitodeathratedonotexplainthedeclineincasesbeginningin1997.Similarly,sincemodelC=fav,afgperformedbetterthanD=fv,fgandlikewiseE=fav,af,dgperformedbetterthanF=fv,f,dg,weconcludethatchangesinmosquitobitingratebetterexplainthedeclineinmalariaprevalencethandochangesintreatmentrecoveryrates.Moreover,asmallerchangeinbitingrate(roughlyhalf)isrequiredtoyieldthesameresultsaschangingthetreatmentrecoveryrate.Someoftheresultsaremoresurprisinganddifculttointerpret.Forexample,theresultsofmodelAsuggestthattreatmentrecoveryratesin1997-2010wereworse,particularlyfortreatmentofvivaxmalaria,thanin1987-1996.Thisoutcomeofthemodelcouldbeaconseqeunceofincreasedparasiteresistancetodrugs. A BFigure3-3. A Besttofmodelto1987-1996casedata; B Besttofmodelto1997-2010data.Datafrom[ 35 ]. ModelsAthroughFcanalsoprovidesomeinsightintohowenhancedcontrolmeasuresaffectthecompetitionbetweenP.falciparumandP.vivax.UsingtheparameterestimatesyieldedbyeachcandidatemodelandtheanalyticexpressionsforthereproductionnumbersRCvandRCfalongwithanalyticexpressionsfortheinvasionnumbersRfvandRvf,wecandetermineinwhichregionofthecompetitiveoutcomegraphthepointfRCv,RCfglies.ThesetofcandidatemodelsfA,Bgyields 62

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A BFigure3-4. A GraphofRfv=1andRvf=1fortheperiod1987-1996(greenandbluelines,respectively)and1997-2010(greylines)asafunctionofRCvandRCf; B Plotofthepoint(RCv,RCf)forIndiabeforeandafter1997.Priorto1997,Indiawasinthecoexistenceregion.Duringperiodofenhancedcontrolmeasures(1997-2010),IndiaisintheregionwhereP.vivaxwilleventuallyoutcompeteP.falciparum. asetofreproductionnumberslyinginaregionwhereP.vivaxoutcompetesP.falci-parum(Figure 3-4 ).Ontheotherhand,thesetofmodelsfC,D,E,FgyieldsasetofreproductionnumberslyingwithincorrespondinginvasionboundarieswhereP.falci-parumoutcompetesP.vivax.AsummaryoftheisolationreproductionnumbersresultingfromeachmodelcandidateisgiveninTable 3-7 .TheresultthatmodelApredictsP.vi-vaxwilloutcompeteP.falciparumissurprisinggiventhatthedatasuggeststheopposite.However,extendingthesolutioncorrespondingtomodelA(Figure 3-5 )totheyear2200conrmsthatthedataispotentiallymisleading.Althoughtheproportionofcasesduetofalciparumhasbeenincreasing,modelArevealsthatthisobservationisinsufcienttodrawconclusionsaboutthelongtermcompetitiveoutcomeofthetwospecies.Whilethistypeofanalysishasthepotentialtounveilinformationregardingthefutureofmalariainaregion,thereproductionnumbersineachcaselieveryclosetotheinvasionboundaries,andconsequentlyitisdifculttodrawdenitiveconclusionsabouttheoutcomeofmalaria.Toaddressthisconcern,weagaincarriedoutaparametric 63

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Table3-6. Post-1996modelsorderedby4AICcvalue(differencefrombestAICcvalue-123.9). ModelParameterPercentchangefromCIAICcpre-1997estimate Aav-43.3-67.3-19.30.0af-1.1-1.2-1.1v-68.1-97.4-38.8f-1.2-1.5-0.9 Bav-45.4-67.9-22.92.8af0.9-7.99.8v-73.3-97.4-49.2f-5.9-22.610.8d9.49.29.5 Cav-3.1-3.3-2.94.1af-0.5-0.5-0.5 Dv6.96.87.05.2f1.00.91.1 Eav-0.6-0.8-0.47.0af2.02.02.1d5.25.15.2 Fv3.73.53.87.1f-1.9-2.4-1.4d2.92.43.4 Gd2.41.83.085.7 Table3-7. 1997-2010estimatesofRCvandRCfforeachcandidatemodel. ABCDEF RCv1.001111.002210.988910.987570.988740.98820RCf0.999900.999971.000181.000141.000151.00022 bootstrapproceduretonotonlyestimatecondenceintervalsforthereproductionnumbers,buttoalsodeterminewhattheprobabilityisthatthereproductionnumberwilllieinanyoneofthefourpossiblecompetitive-outcomeregions.Foreachofthesixcandidatemodels,wealsocalculatedtheAICcforevery1000runsinthebootstraproutinetodeterminewhatthemostfrequentorderingofthesetofcandidatemodelsis.Tomakesurethattheresultsofthebootstrapmethodbetweenmodelsiscomparable,wedraw1000setsofdatafromaPoissondistributionwithmeanequaltothesolution 64

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Figure3-5. ModelAsolutioncurveextendedtotheyear2200. curveassociatedwithmodelAinTable 3-6 thebestttingmodelbasedonAICcvalues.Thebootstrappingprocedureallowedustocompile1000setsofparameterestimatesforeachofthesixmodels(ignoringmodelGbecauseofthepoort),fromwhichwecomputed1000pairsofreproductionnumbers(RCv,RCf).Thenewparametersetsandreproductionnumberpairswereusedtocomputetheinvasionnumbersforthe1000runs,allowingustodeterminewhattheprobabilityisthatamodelwilllandinaparticularcompetitive-outcomeregion.TheresultsarelistedinTable 3-8 .ThecompetitiveoutcomesvarythemostformodelsAandB,whichisconsistentwithourearlierobservationthatthesetwomodelswerethemostsensitivetotheinitialparameterguessusedfortting.Althoughthe1000bootstrappedsamplesresultedin123differentorderings,5orderingsmadeupmorethanhalfofthesamples.TheoriginalorderingfA,B,C,D,E,F,Ggoccurred10.1%ofthetime.29.5%oftherunsledtotheorderingfA,C,D,E,F,B,Gg.9.7%ofthesamplesyieldedtheorderingfC,D,E,F,A,B,Gg.TheorderingfC,D,E,F,B,A,Ggappeared8.4%ofthetime,while4.7%ofthesamplesresultedintheordering 65

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Table3-8. Percentageofbootstraprunsinwhichvivaxandfalciparumwillcoexist(I),vivaxwilloutcompetefalciparum(II),falciparumwilloutcompetevivax(III),andthepercentageofrunsinwhichbothwillbecomeextinct(IV). IIIIIIIV A14.344.019.222.5B411.353.331.4C0.2077.422.4D0074.325.7E0077.922.1F0.4083.4016.2 fB,A,C,D,E,F,Gg.TheAICcvaluesselectedmodelAasthetopmodel54.4%ofthetime.Ofthese544samplesforwhichmodelAwasselectedasthetopmodel,roughly15.1%yieldedtheoutcomethatvivaxandfalciparumwouldcontinuetocoexist,44.9%yieldedthatvivaxwouldoutcompetefalciparum,18.4%yieldedthatfalciparumwouldoutcompetevivax,andnally21.7%yieldedthatbothspecieswouldbecomeextinct.Determiningcondenceintervalsforthereproductionnumbersforeachmodelwasnotasstraightforwardasitwasforthe1987-1996timeperiod.HistogramsofthereproductionnumbersforeachmodelrevealedthatnotallofthesamplesofRCfweresymmetric.Infact,thecollectionofreproductionnumbersRCfformodelBexhibitsabimodaldistribution.Since,modelsAandCwerethemostcommonmodelstakingrstplace,andbecausetheircorrespondingreproductionnumbersexhibitedfairlysymmetricdistributions(seeFigure 3-6 ),wepresentthecondenceintervalsforthesetwomodels.The95%condenceintervalsforRCvandRCf,respectively,correspondingtomodelAare(0.991331.00640)and(0.998901.00075).FormodelC,thecondenceintervalsare(0.986760.99079)and(0.99979.00054).Ultimately,thepercentagesinTable 3-8 providemoremeaningfulinformationthanthecondenceintervalsderivedforthepost-1996reproductionnumbers.Figure 3-6 illustratesthatthespreadoftheRCfdataresultingfromthebootstrapprocedurewasalwayslessthanthespreadofRfv.Conversely,thevarianceinRCvisgreaterthanthatofRvf.This 66

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observation,whichwasconsistentacrossallsixcandidatemodelssuggeststhatRCfislesssensitivethanRfv,andRCvismoresensitivethanRvf,tochangesisparametervalues. AModelARfv,RCfdata BModelARvf,RCvdataFigure3-6. Subgure A presentsahistogramoftheRfvandRCvbootstrapdataformodelA.Subgure B isahistogramoftheRvfandRCfbootstrapdataformodelA. 3.2DiscussionIndia,asistrueformanyothercountries,hasstruggledwiththecontrolofmalaria,experiencingseveralupsanddowns.Whilemorerecenteffortshavebeensuccessfulindramaticallydecreasingthenumberofcases,Indiaisstillfarfromreachingitsgoal.Consequently,knowingwhichofthecontrolstrategiesIndia'ssuccesscanbeattributedtoisvaluabletoIndia'sfuturesuccessandcouldhelpIndiausetheirresourcesmoreefciently.ThepresenceoftwomalariaparasitesinIndiamakesthisachallengingproblem,bothinpracticeandintermsofmathematicalmodeling.Toourknowledge,attemptstomodelbothvivaxandfalciparum([ 86 ],[ 87 ])atthepopulationleveldonotincludethepossibilityofco-infection.Chiyakaetal.[ 27 ]addressco-infectionintheirfalciparum-malariaemalariamodel,howeverthesymmetricnatureofthismodeldoesnotlenditselfwelltotheapplicationtofalciparumandvivax.Ourfalciparum-vivaxmodel 67

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addressestheneedforamodelthatconsidersnotonlythepossibilityofco-infection,butalsothecharacteristicsofvivaxthatdifferentiateitfromfalciparum.Competitionbetweenspeciescanhaveaprofoundeffectonsurvival.Wehaveshownwithourmodel,bystudyingtheinvasionboundaries,thattwospeciescancoexist,eveniftheisolatedreproductionnumberofoneofthespeciesislessthanone.Thishasimportantconsequencesformalariacontrol,sincereducingoneofthereproductionnumbersbelowonemaynotbesufcienttoeradicateeitherpathogenorthedisease.Theemergenceofparasiteresistancetodrugtherapiesisalsoofgreatconcernsincethisforebodingobstacleposesathreattothesuccessofmalariacontrol.Whilewedonotaddressparasiteresistancedirectlyinourmodel,thettingofseveralmodelstotheenhancedmalariacontrolperiodsuggestedthatsufcientuseofbednetsmaybeabletocounteractthenegativeeffectsofincreasedresistancetothetreatmentofmalaria.Infact,themodelselectedasthebestmodelforthemajorityofthebootstrappedsamples(54.4%ofthetime)inSection 3.1.2 ,wasoneinwhichbothbitingrateandtreatmentrecoveryratesdecreasedafter1996.Adecreaseinrecoveryrateincreasestheaveragetimetorecoveryfollowingtheadministrationofanti-malarialdrugs.Asexpected,ourtopmodelindicatesthatdecreasingbitingrateandincreasingthetimetorecoveryfollowingtreatmenthaveopposingeffectsonthereproductionnumber.IncorporatingbothP.falciparumandP.vivaxmalariaintoourmodelprovideduswithawaytodeterminewhatthemostlikelyoutcomesareformalariainIndia.Bootstrappingofthebestpost-1996model(modelA)yieldedthatP.vivaxoutcompetingP.falciparumisthemostlikelyoutcome,whiletheprobabilityofextinctionisonlyslightlymoreprobablethantheprobabilitythatfalciparumwilloutcompetevivaxmalaria(22.5%versus19.2%).TheremainingcandidatemodelspredictedthatP.falciparumoutcompetingvivaxisthemostlikelyoutcome.Aside-by-sidecomparisonofthe 68

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histogramsofthereproductionnumbersandtheinvasionnumbersrevealedthatthevarianceinRCfwasalwayslessthanthevarianceinRfv.Conversely,thevarianceinRCvisgreaterthanthatofRvf.Thismeansthatestimatingthereproductionnumbersalonemaynotbeagoodpredictoroftheoutcomeofthedisease.TheapplicationofourmathematicalmodeltodatasuggestedthatthefutureofmalariainIndiaisuncertain.Althoughweaddressedtheuncertaintyinthemodelpredictions,it'simportantforustonotethatapplyingthesamemethodstodatasetsforsmallerregionsislikelytoproduceverydifferentresults.Inthefuture,wehopetousetheframeworkwehavedevelopedheretomakemorecondentpredictionsabouttheoutcomeofmalariainvariousregionsofIndia. 69

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CHAPTER4ASSESSINGTHEROLEOFSPATIALHETEROGENEITYANDHUMANMOVEMENTINMALARIADYNAMICSANDCONTROLInmanymalariaendemiccountries,resourcesavailabletowardsimplementinginterventionstrategiesareextremelylimited[ 49 ].Interventionstrategiesmustbechosentomaximizetheuseoftheselimitedfundstomostefcientlyreducemalariaburden.Formanydiseases,includingmalaria,humanpopulationmovementcontributesgreatlytothespreadandpersistenceofdisease[ 49 ],andisthereforeanimportantconsiderationwhenimplementinginterventionstrategies[ 110 ].Despitethis,littleisknownabouthumanmovementpatternsandtheirepidemiologicalconsequences[ 104 ].Infact,thefailureoftheGlobalMalariaEradicationProgrammeinthe1950sand1960smaybedue,inpart,tothefailuretotakeintoaccounthumanmovement[ 49 ].Humanmovementoftenlinksareaswithdifferentdegreesofmalariatransmissioncapacity[ 73 ].Localtransmissiondynamicsoftendiffersbetweenareas[ 104 ]duetocharacteristicssuchastopography,mosquitospeciesdensities,pesticideuse,availabilityofmosquitohabitats,ordifferencesincurrentlyimplementedinterventionstrategies[ 47 63 ];forexample,urbanareastypicallyhavemuchlowermalariatransmissionthanruralareas[ 55 72 91 ].Becausehumanmovementcommonlylinksurbanandruralsystemsthatoftenexhibitdramaticallydifferentdegreesofmalariatransmission[ 91 ],urbanizationmaybeanimportantdriverinmalariadynamics.Localtransmissiondynamicsmayinuencetheefcacyofinterventionstrategies.Becausethesetransmissioncharacteristicsmayvarybetweenareasconnectedbyhumanmovement,humanmovementbecomesimportantnotonlyintermsofexpecteddegreeofimportation,butalsointermsofdecidingwheretotargetinterventionstrategiestomostefcientlyuseresources.Inthisstudy,wedevelopamathematicalmodeltoaddresstheimplicationsofmalariamovementbetweenareasofpotentiallyheterogeneoustransmissioncharacteristics,inordertodetermineeffectivetargetedinterventionstrategies. 70

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Mathematicalmodelsareausefultooloftenappliedtobothidentifycontrolmeasuresthataremostimportanttoimplement,aswellasquantifytheeffectivenessofdifferentcontrolstrategiesincontrollingoreliminatingmalariainendemicregions.Forsimplicity,mostmodelsconsidertransmissioninoneregionofinterest,withhomogeneoustransmissionthroughouttheregion.InSection 4.1.1 wedescribeamodicationoftheRoss-Macdonaldmodel,whichistobeusedinthehumanmovementtwopatchmodel.Whilesingle-patchmalariamodelshaveproventobeveryusefulinthestudyofmalariadynamics,weknowfrommanyothersystemsthatspatialstructurecangreatlyinuencethedynamicsofinteractingspecies[ 58 60 85 ],includingpathogensandhosts[ 82 ].Thus,thereisaneedtoexplorehowmalariadynamicsareaffectedbyspatialheterogeneityandtousethisinformationtoinforminterventionstrategies.Toaddressthisneed,inSection 4.1.2 weintroduceatwo-patchmalariametapopulationmodelbasedonthemodiedRoss-Macdonaldmodelthatallowsfordifferentlocaltransmissioncharacteristicsandvariablehumanmigrationratesbetweenpatches.Metapopulationmodelshavebeenusedextensivelyinothersystemstoaddresscommonecologicalissues,suchastheeffectofconnectivitybetweenareas[ 58 ]andtheeffectofmigrationonpredator-preydynamics[ 4 ].Similarmodelshavealsobeenusedtoinvestigatetheimplicationstheseresultsmighthavefordiseasedynamics.Consequently,metapopulationmodelshavebeendevelopedtoexploretheeffectofmigrationondiseasepersistence.Hethcoteetal[ 59 ]foundthatmigrationcouldcauseadiseasetopersistwhereitwouldotherwisedieoutifitwereisolatedusingatwo-patchSIS(Susceptible-Infected-Susceptible)model.Weshowin 4.1.3 thatthisresultisalsopossibleinourtwo-patchmalariamodelforarangeofdifferentimmigrationandemigrationrates.Metapopulationmodelsofvector-bornediseasehavebeenpreviouslystudiedtosomeextentaswell.Cosneretal.consideredtwotypesofmovement,termed 71

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LagrangianandEulerian,intheirvector-bornediseasemetapopulationmodel[ 32 ].IntheLagrangianapproach,individualsareconsideredresidentsofaparticularpatchandspendsomefractionofthetimevisitingotherpatches.IntheEulerianapproach,individualsarenottracked;whilemigrationoccursbetweenpatches,thisapproachdoesnotassignaresidencetoindividualsinthepopulation.Ourtwo-patchmodelincorporatestheEulerianapproachtomodelingmovement.Sinceitisuncommonformosquitoestomovemorethanakilometerthroughouttheirlives[ 33 53 76 78 96 ]whilehumansoftenmovemanykilometersbetweenvillagesorcountries[ 104 ],wemodeledhumanmovementexclusively.Intheirmalariametapopulationmodel,Cosneretal.studiedaspecialcaseofatwo-patchmalariamodelwithnotransmissioninoneofthepatches[ 32 ].Inourstudy,weareinterestedinunderstandinghowhumanmovementaffectsmalariadynamicswhentwopatcheswithdifferent,nonzerotransmissioncharacteristicsareconnectedbyhumanmigration.InSection 4.3.1 ,weparameterizeourmodelusingestimatesfromregionswithvaryinglevelsofmalariaendemicity,toencompassavarietyofpatchandhumanmovementcharacteristicsintheeld.InSection 4.1.3 ,wepresentananalyticexpressionderivedfromthetwo-patchmodelforthebasicreproductivenumber,athresholdquantitydeterminingwhetheradiseasewillpersistorgoextinctinapopulation.Toassesstherelativeefcacyofdifferentcontrolmeasuresandtodeterminewheretotargetthesecontrolmeasures,weperformasensitivityanalysisofthebasicreproductivenumbertodifferentparametersinthemodelinSection 4.2 .InSection 4.4 ,weperformasensitivityanalysisoftheendemicequilibriumandcomparetheseresultstothoseoftheanalysisofthereproductionnumber.Chitnisetal.[ 26 ]performasimilarsensitivityanalysisusingtheirsingle-patchmalariamodel.Theyfoundthatunderbothhighandlowtransmissionsettings,thebasicreproductivenumberwasmostsensitivetothemosquitobitingrate,andtheequilibriumproportionofhumanswasmostsensitivetothehumanrecoveryrate[ 26 ].However,oursensitivityanalysisyieldedadifferentresult,likelyduetodifferent 72

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assumptionsinthemodelformulation:theparameterthebasicreproductivenumberismostsensitivetodependsontherelativedurationoftheextrinsicincubationperiodandmosquitolifespan,andthehumanrecoveryratewasnotthemostimportantfactorintheanalysisoftheendemicequilibrium.Wealsoshowthatourintuitionaboutwherecontrolmeasuresshouldbeimplementedforthegreatestsuccessmaynotalwaysbecorrectandthathavinganideaoftherelativesizesofthemigrationratesbetweenthetwopatchescanprovideinsightintowhichpatchshouldbethetargetofmalariacontrol. 4.1Ross-MacdonaldModelRecallfromSection 1.1 thatintheRoss-Macdonaldmodel,theratesatwhichtheproportionofhumansinfected(x)andtheproportionofmosquitoesinfected(z)changeovertimearegivenbythefollowingsystemofequations: dz dt=acx(1)]TJ /F4 11.955 Tf 11.96 0 Td[(z))]TJ /F4 11.955 Tf 11.95 0 Td[(gzdx dt=mabz(1)]TJ /F4 11.955 Tf 11.96 0 Td[(x))]TJ /F4 11.955 Tf 11.95 0 Td[(rx, (4) where1)]TJ /F4 11.955 Tf 12.33 0 Td[(zand1)]TJ /F4 11.955 Tf 12.33 0 Td[(xaretheproportionofmosquitoesandtheproportionofhumansthataresusceptible,respectively[ 11 ].Wecanrewritetheseequationsintermsofthenumberofhumansinfected,ratherthantheproportioninfected: dz dt=acI N(1)]TJ /F4 11.955 Tf 11.95 0 Td[(z))]TJ /F4 11.955 Tf 11.96 0 Td[(gzdI dt=mabz(N)]TJ /F4 11.955 Tf 11.96 0 Td[(I))]TJ /F4 11.955 Tf 11.96 0 Td[(rI, (4) whereNisthetotalsizeofthehumanpopulation,andIisthenumberofhumansinthatpopulationwhoareinfectedwithmalaria.Susceptiblehumansbecomeinfectedataratemabz,andsusceptiblemosquitoesbecomeinfectedatarateacx.Infectedhumansarelostthroughrecovery,andinfectedmosquitoesarelostthroughdeath.Becausehumanslivemuchlongerthanthedurationofamalariainfectionandthelifespanofamosquito,thehumandeathrateismuchsmallerthananyoftheotherparametersinthismodel,andhenceisnegligible. 73

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Similarly,weignorehumanbirths.However,itispossibletoincludebirthsanddeathsintothemodelinsuchawaythatitreducestothesamesystempresentedhere. 4.1.1ModicationstoRoss-MacdonaldModelInfectedmosquitoesthatdonotsurvivetheextrinsicincubationperiodofmalarianeverhavethechancetotransmitthedisease.Dependingonmosquitodailysurvivalprobabilitiesanddurationoftheextrinsicincubationperiod(whichisdependentuponfactorssuchastemperature),asmanyashalfofinfectedmosquitoesmaynotsurvivetobecomeinfectiousandabletotransmitmalaria[ 47 48 ].Toaccountformosquitosurvival,theRoss-Macdonaldmodelhasbeenmodiedbyreplacing(1)]TJ /F4 11.955 Tf 12.86 0 Td[(z)intheoriginalmodelwithe)]TJ /F6 7.97 Tf 6.58 0 Td[(gn)]TJ /F4 11.955 Tf 13.22 0 Td[(z(seeAppendix1in[ 99 ]).Inotherwords,thepoolofmosquitoesisreducedfromonetotheproportionofindividualsexpectedtosurvivetheextrinsicincubationperiod,whichhaslengthn,iftheirdeathrateisg.Asanalmodication,weeliminatethemosquitoequationbyassumingthattheinfectedmosquitopopulationequilibratesmuchfasterthantheinfectedhumanpopulation.Thisassumptioniscommonlyusedinmalariamodelsbecausethemosquitodynamics(suchastheextrinsicincubationperiodandmosquitodeathrate)operateonamuchfastertimescalethanhumandynamics(suchasthenaturalrecoveryrate)[ 27 62 66 99 ].Thus,byassumingthatthemosquitopopulationdynamicsisatequilibrium,theequationsin( 4 )canbereducedtothesingleequation: dI dt=ma2bcIe)]TJ /F6 7.97 Tf 6.59 0 Td[(gn acI+gN(N)]TJ /F4 11.955 Tf 11.96 0 Td[(I))]TJ /F4 11.955 Tf 11.95 0 Td[(rI. (4) Tosimplifythenotation,welet.=mabe)]TJ /F6 7.97 Tf 6.59 0 Td[(gnand.=acsothatequation( 4 )canbewrittenas dI dt=I I+gN(N)]TJ /F4 11.955 Tf 11.96 0 Td[(I))]TJ /F4 11.955 Tf 11.96 0 Td[(rI. (4) 74

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4.1.2Two-PatchMalariaModelEquation( 4 )isusedinourtwo-patchmodeltodescribethediseasedynamicswithineachpatch.EachpatchcontainsahumanpopulationofsizeNicomposedofSisusceptiblehumansandIiinfectedhumans,withmigrationfrompatchjtopatchioccurringataratekij,regardlessofthehealthstatusofanindividual. Figure4-1. DiseaseDynamicsinaHumanPopulation.Solidboldarrowsindicatetheacquisitionofanewinfection;dashedarrowsindicaterecovery;solidthinarrowsindicatemigrationbetweenpatches. Althoughmosthumanmovementconsistsofshort-termvisits(bestmodeledusingLagrangianframework),ourobjectivewastostudytheeffectoflongtermmigrationondecisionsconcerningmalariacontrol.CombiningtheconceptualizationofthemodelinFigure 4-1 andtheEulerianframeworkformovementwiththedisease-dynamicsgivenbythemodiedRoss-Macdonaldmodel,wearriveatthefollowingsystemofdifferentialequationstodescribethemalariadynamicsofahumanpopulationdistributedbetween 75

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twopatches:dS1 dt=)]TJ /F7 11.955 Tf 23.87 8.09 Td[(11I1 1I1+g1N1S1+r1I1)]TJ /F4 11.955 Tf 11.95 0 Td[(k21S1+k12S2dS2 dt=)]TJ /F7 11.955 Tf 23.87 8.08 Td[(21I2 2I2+g2N2S2+r2I2)]TJ /F4 11.955 Tf 11.95 0 Td[(k12S2+k21S1dI1 dt=11I1 1I1+g1N1S1)]TJ /F4 11.955 Tf 11.96 0 Td[(r1I1)]TJ /F4 11.955 Tf 11.96 0 Td[(k21I1+k12I2dI2 dt=21I2 2I2+g2N2S2)]TJ /F4 11.955 Tf 11.96 0 Td[(r2I2)]TJ /F4 11.955 Tf 11.96 0 Td[(k12I2+k21I1.Inthesystemabove,thepopulationsizeofpatchiisNi=Si+Ii,andthetotalpopulationsizeisN=N1+N2.NotethatdN dt=0,henceNisconstant.AdescriptionofthemodelparameterscanbefoundinTable 4-1 Table4-1. Descriptionofmodelparametersinpatchi ParametersDescription miRatioofthenumberofmosquitoestothenumberofhumans aiHumanbitingrate miaiNumberofmosquitobitesonahumanperunitoftime biTransmissionefciencyfrommosquitotohuman ciTransmissionefciencyfromhumantomosquito giNaturaldeathrateofmosquitoes niLengthofmosquitoincubationperiod imiaibie)]TJ /F6 7.97 Tf 6.59 0 Td[(gin iaici riNaturalhumanrecoveryrate kijMigrationratefrompatchjtopatchi SinceN01=)]TJ /F4 11.955 Tf 9.3 0 Td[(k21N1+k12N2impliesthatN2=k21 k12N1,whereN1andN2denotetheequilibriumvaluesofN1andN2,respectively,andN1+N2=N,wehavethatN1=k12 k12+k21NandN2=k21 k12+k21Natequilibrium.Thus,sinceS1=N1andS2=N2whenthereisnodisease,thedisease-freeequilibrium(DFE)oftheabovesystemis(S1,I1,S2,I2)DFE=k12 k12+k21N,0,k21 k12+k21N,0. 76

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IfdI1 dt=0andmalariaisabsentinpatch1(I1=0),thenI2=0whenk12>0.Likewise,ifdI2 dt=0andmalariaisabsentinpatch2(I2=0),thenI1mustbezerowhenk21>0.Thus,atequilibrium,malariacannotbepresentinonepatchandabsentintheother,providedthatthemigrationratesarenonzero. 4.1.3TheBasicReproductiveNumberR0Recallthatthebasicreproductivenumber,R0,istraditionallydenedtobethenumberofsecondarycasesresultingfromoneinfectiousindividualinanotherwisefullysusceptiblepopulation.Typically,R0providesathresholdcriteriadeterminingwhetheradiseasewillbeabletospreadinapopulation.IfR0<1,thediseasewillbecomeextinct.IfR0>1,thediseasewillpersistinthepopulation.ThebasicreproductivenumberforthemodiedRoss-Macdonaldmodel( 4 )is grwhere=mabe)]TJ /F6 7.97 Tf 6.58 0 Td[(gnand=ac.Thuswithoutmigration,patchiinthetwo-patchmodelwillhaveitsownisolated-patchreproductionnumberR0i=ii giri,wherei=miaibie)]TJ /F6 7.97 Tf 6.59 0 Td[(giniandi=aicifori=1,2.Usingthenext-generationapproach[ 39 106 ],wendthatthebasicreproductivenumber(thedominanteigenvalueofthenextgenerationmatrix)forourtwo-patchmodelwithmigrationisgivenbytheexpression R0=1 2hs1t2+s2t1+p (s1t2+s2t1)2)]TJ /F8 11.955 Tf 11.96 0 Td[(4s1s2i (4) =1 2hs1t2+s2t1+p (s1t2)]TJ /F4 11.955 Tf 11.96 0 Td[(s2t1)2+4s1s2k12k21i (4) where=k12r1+k21r2+r1r2si=ii gi=riR0iti=ri+kjifori=1,2.AlthoughthisexpressionforR0doesnotpossessthebiologicalinterpretationofthetraditionaldenition,itstillprovidesthesameusefulpersistence-extinctionthresholdcriterion. 77

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Observethatwhenthereisnomigrationbetweenpatches,k12=k21=0,thenR0=maxn11 r1g1,22 r2g2o=maxfR01,R02g,thelargerofthetwoisolated-patchreproductivenumbers.Moreover,thevalueoftheglobalreproductivenumberR0forthistwo-patchmodelisalwaysbetweenthetwoisolated-patchreproductivenumbersR01andR02whenmigrationbetweenthetwopatchesispresent.Inoneparameterizationoftheirtwo-patchmalariamodelwithLagrangianmovement,Cosneretal.foundthatitwaspossibletohaveascenarioinwhicheachisolatedpatchreproductionnumber(R01andR02)islessthan1,yettheglobalreproductionnumberR0islargerthanone[ 32 ].Thisndingillustratesthatsomemodelspredictthatitmaybepossibletohaveasystemwherewithoutmigration,thediseasegoesextinctinbothpatches,butonceacertainlevelofmigrationisintroduced,thediseasebecomesendemic.However,asisstatedinthefollowingtheorem,ourtwo-patchmodel,whichassumesEulerianratherthanLagrangianmovement,predictsthatR0willalwaysbeboundedbytheisolatedpatchreproductionnumbers. Theorem4.1. IfR01>R02,thenforallpairsofmigrationrates(k12,k21)2[0,1)[0,1),maxfR01 1+k21 r1,R02gR0R01. Proof. R01>R02impliesthats1 r1>s2 r2.Thus,byassumptionwehavethats1r2>s2r1.WerstevaluateR0atcertainpointsontheboundaryofthedomain[0,1)[0,1).Fromequation( 4 ),wehave R0(k12,0)=1 2r1t2(s1t2+s2r1+js1t2)]TJ /F4 11.955 Tf 11.95 0 Td[(s2r1j). (4) Sincebyassumptions1r2>s2r1,andbecauset2=r2+k12r2,weknowthats1t2)]TJ /F4 11.955 Tf 12.67 0 Td[(s2r1>0,andsojs1t2)]TJ /F4 11.955 Tf 12.66 0 Td[(s2r1j=s1t2)]TJ /F4 11.955 Tf 12.66 0 Td[(s2r1.Thus,equation( 4 )simpliestoR0(k12,0)=R01forallk122[0,1). 78

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Similarly, R0(0,k21)=1 2r2t1(s1r2+s2t1+js1r2)]TJ /F4 11.955 Tf 11.96 0 Td[(s2t1j)=1 r2t1maxfs1r2,s2t1g=maxs1 t1,s2 r2=max(R01 1+k21 r1,R02). (4) Thus,R02R0(0,k21)R01,forallk21intheinterval[0,1).Considerthefunction f(x)=x2)]TJ /F8 11.955 Tf 11.95 0 Td[((s1t2+s2t1)x+s1s2.(4)fispreciselythecharacteristicpolynomialofthenext-generationmatrixusedtoderiveR0inequation( 4 ).R0isthelargerofthetworootsoftheconcave-upparabolaf(x).Consequently,f(R0)=0andf0(R0)>0.Fromthisweknowthatforanyrealnumberxsatisfyingtheinequalityf(x)<0,thenxmustbelessthanR0.Ontheotherhand,ifxissuchthatf(x)>0andf0(x)>0,thenxisgreaterthanR0.Supposek12andk21arepositive.Then,f(R01)=fs1 r1=s1 r12)]TJ /F8 11.955 Tf 11.96 0 Td[((s1t2+s2t1)s1 r1+s1s2=s1 r1(k12r1+k21r2+r1r2)s1 r1)]TJ /F8 11.955 Tf 11.95 0 Td[((s1t2+s2t1)+s2r1=s1 r1s1k12+s1k21r2 r1+s1r2)]TJ /F8 11.955 Tf 11.96 0 Td[((s1t2+s2t1)+s2r1=s1 r1s1t2+s1k21r2 r1)]TJ /F8 11.955 Tf 11.96 0 Td[((s1t2+s2t1)+s2r1=s1 r1s1r2 r1k21)]TJ /F4 11.955 Tf 11.96 0 Td[(s2k21=s1k21 r21(s1r2)]TJ /F4 11.955 Tf 11.96 0 Td[(s2r1). 79

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Byassumption,s1r2)]TJ /F4 11.955 Tf 11.95 0 Td[(s2r1>0,hencef(R01)>0.Similarly,wecanshowthat f(R02)=fs2 r2=s2k12 r22(s2r1)]TJ /F4 11.955 Tf 11.95 0 Td[(s1r2) (4) f R01 1+k21 r1!=)]TJ /F13 11.955 Tf 11.29 16.85 Td[(s1 t12k12k21. (4) ClearlyfR01 1+k21 r1<0andsinces1r2)]TJ /F4 11.955 Tf 11.96 0 Td[(s2r1>0,wealsohavethatf(R02)<0.Now,f0(x)=2x)]TJ /F8 11.955 Tf 11.96 0 Td[((s1t2+s2t1).So, f0(R01)=f0s1 r1=2s1 r1)]TJ /F8 11.955 Tf 11.96 0 Td[((s1t2+s2t1) (4) =2(r1r2+r1k12+r2k21)s1 r1)]TJ /F4 11.955 Tf 11.96 0 Td[(s1r2)]TJ /F4 11.955 Tf 11.95 0 Td[(s1k12)]TJ /F4 11.955 Tf 11.96 0 Td[(s2r1)]TJ /F4 11.955 Tf 11.95 0 Td[(s2k21 (4) =(s1r2)]TJ /F4 11.955 Tf 11.95 0 Td[(s2r1)+s1k12+(2s1r2)]TJ /F4 11.955 Tf 11.95 0 Td[(s2r1)k21 r1. (4) Again,sinces1r2)]TJ /F4 11.955 Tf 11.95 0 Td[(s2r1>0,f0(R01)>0.Thus,fork12andk21positive,f(R02)<0andfR01 1+k21 r1<0impliesthatR0>maxR02,R01 1+k21 r1.Also,f(R01)>0andf0(R01)>0impliesthatR0R02.ConsiderR0(k12,k21)tobeafunctionofbothmigrationratesk12andk21,wherek12,k212[0,1).Foraxedintheinterval[0,1),R0(k12,)isanincreasingfunctionofk12andR0(,k21)isadecreasingfunctionofk21. Proof. WehaveshownintheproofofTheorem 4.1 thatR0(0,)=maxR01 1+ r1,R02andR0(k12,)>maxR01 1+ r1,R02fork12>0.ThusR0(k12,)R0(0,)forallk120.So,weneedonlyshowthatR0(k12,)ismonotonicink12toshowthatitisanincreasingfunctionink12.Similarly,fromTheorem 4.1 wealsoknowthatR0(,0)= 80

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R01R0(,k21)forallnon-negativek21.Soagain,weneedonlyshowthatR0(,k21)ismonotonicink21toshowthatitisadecreasingfunctionink21.First,weshowthatR0(k12,)ismonotonicink12.SinceR0(k12,)iscontinuousink12,itismonotonicwithrespecttok12ifforeveryC2(0,1)suchthatR0(k12,)=Chasanon-negativesolutionk122[0,1),thenthissolutionisunique.SupposeR0(k12,)=C.Then,bythedenitionofR0(eqn 4 ),wehavethat 1 2q+p q2)]TJ /F8 11.955 Tf 11.95 0 Td[(4s1s2=C, (4) whereq=s1t2+s2t1=s1(r2+k12)+s2(r1+)and=r1r2+r1k12+r2.Equation( 4 )impliesthat C2)]TJ /F4 11.955 Tf 11.95 0 Td[(qC+s1s2=0. (4) Observethatbothandqarelinearink12.Thus,equation( 4 )islinearink12,implyingthatifthereexistsak122[0,1)thatisasolutiontoequation( 4 ),thenthissolutionisunique.Hence,R0(k12,)ismonotonicforeach2[0,1).Bythesameargument,R0(,k21)ismonotonicforeach2[0,1).SinceR0(k12,)ismonotonicfornon-negativek12andR0(0,)R0(k12,),foreachxedk21=2[0,1),R0isanincreasingfunctionofk12.Likewise,sinceR0(,0)R0(,k21)fornon-negativek21,foreachxedk12=2[0,1),R0isadecreasingfunctionofk21. TheproofofTheorem 4.1 ,assumingthatR01>R02,alsoshowsthattheminimumvalueofR0(k12,k21)onthedomain[0,1)[0,)ismaxR01 1+ r1,R02andthemaximumvalueisR01.Thus,ifR02<1andR01 1+ r1>1forsome>0(andhenceR01>1),thenR0(k12,k21)>1forallmigrationpairs(k12,k21)in[0,1)[0,).Thisindicatesthatitispossibletohaveasituationinwhichwithoutmigration,thediseasediesoutinonepatch 81

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butnottheother,yetwithmigrationthediseasepersistsinbothpatchesforallk120andfor0k21.IfR02andR01 1+ r1arelessthanonebutR01>1,thenforsomemigrationratepairs(k12,k21),R0willbelargerthanone,andforotherpairs,R0willbelessthanone.Furthermore,thereexistsavalue1forall(k12,k21)in[0,1)[0,).Finally,ifR02andR01arebothlessthanone,thenR0willalwaysbelessthanone,regardlessofthemigrationratesbetweenpatches. 4.2SensitivityAnalysisPopulationbiologistsmakeuseofsensitivityandelasticityanalysestoevaluatetheeffectofperturbationsinpopulationfecundity,growth,andsurvivalontheoverallgrowthofthepopulation,andtodeterminewhichlifestageapopulation'sgrowthismostsensitiveto[ 57 107 ].Thesensitivityofaquantitytoaparameterpiscalculatedass=@ @p,andisusedtodeterminetheamountofchangethatoccursininresponsetochangesinelementsp;sensitivitycanthenbeusedtocomparehowabsolutechangesinvariousparametersaffect[ 36 ].However,wecannoteasilycomparesensitivitieswithrespecttoparametersofdifferentscales.Elasticity,ontheotherhand,expressestheproportionalchangeinresultingfromaproportionalchangeinp.Thus,computingtheelasticityofR0toaparameterp,ratherthanthesensitivity,allowedustocompareparametersofdifferentordersofmagnitudeanddifferentunits: "p=@R0 @pp R0. (4) Thevalue"pdescribeshowmuch,andinwhatway(positivelyornegatively),thereproductionnumberwillbeaffectedbyasmallchangeinaparametervaluep.Moreprecisely,wecaninterprettheelasticityasfollows:iftheelasticityofaquantitywithrespecttoaparameterpis"p,thena1%changeinpwillresultinan"p%changein.Wecomputetheanalyticexpressionsoftheelasticitiesforeachparameterinboth 82

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thesinglepatchmodelwithoutmigrationandthetwo-patchmodelsothatwemaydrawsomegeneralconclusionsabouttherelativeimportanceofeachparameterinthesetwoscenarios. 4.2.1ElasticitiesforaSinglePatchwithoutMigrationFromequation( 4 )andthesingle-patchexpressionforR0,ma2bce)]TJ /F14 5.978 Tf 5.76 0 Td[(gn rg,wendthat"m="b="c=1"r=)]TJ /F8 11.955 Tf 9.3 0 Td[(1"a=2"g=)]TJ /F8 11.955 Tf 9.3 0 Td[((gn+1)"n=)]TJ /F4 11.955 Tf 9.3 0 Td[(gn.Thus,wehave(forp=m,b,c)thatj"nj<1=j"rj="p2 g,then1=j"rj="p<"a1=g;i.e.whentheincubationperiodislongerthantheaveragemosquitolifespan.Unliketheelasticitiesrelatedtotheparametersm,a,b,c,andr,theelasticitiesofR0withrespecttothemosquitodeathrategandincubationperiodnare 83

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linearfunctionsofbothgandn.Thus,increasingmosquitodeathrateorlengtheningtheextrinsicincubationperiodenhancestheeffectofsuchcontrolmeasuresonR0. 4.2.2ElasticitiesfortheTwo-PatchMetapopulationModelTheanalyticexpressionsfortheelasticitiesinthetwo-patchmodelwithmigration,whilemorecomplicatedinformthanthoseofthesingle-patchmodel,providesomeinsightintotherelativeimportanceofthemodelparameters.Forpi=mi,ai,bi,ci,gi,ni,wehavethat @R0 @p1=@s1 @p11 2t21+s1t2)]TJ /F4 11.955 Tf 11.96 0 Td[(s2t1 p +2s2k12k21 p (4) @R0 @p2=@s2 @p21 2t11)]TJ /F4 11.955 Tf 13.15 8.09 Td[(s1t2)]TJ /F4 11.955 Tf 11.96 0 Td[(s2t1 p +2s1k12k21 p (4) where=(s1t2)]TJ /F4 11.955 Tf 11.97 0 Td[(s2t1)2+4s1s2k12k21,@si @pi=si piforpi=mi,bi,ci,@si @ai=2si ai,@si @gi=)]TJ /F4 11.955 Tf 9.3 0 Td[(sigini+1 gi,and@si @ni=)]TJ /F4 11.955 Tf 9.29 0 Td[(gisi.Thus,theelasticitiesforpi=mi,bi,ciaregivenbytheexpressions "p1=s1 2R0t21+s1t2)]TJ /F4 11.955 Tf 11.96 0 Td[(s2t1 p +2s2k12k21 p (4) "p2=s2 2R0t11)]TJ /F4 11.955 Tf 13.15 8.09 Td[(s1t2)]TJ /F4 11.955 Tf 11.96 0 Td[(s2t1 p +2s1k12k21 p (4) and "ai=2"pi (4) "gi=)]TJ /F7 11.955 Tf 9.3 0 Td[("pi(gini+1) (4) "ni=)]TJ /F4 11.955 Tf 9.3 0 Td[(gini"pi (4) Fork12,k216=0,becausep js1t2)]TJ /F4 11.955 Tf 12.73 0 Td[(s2t1j,1+s1t2)]TJ /F6 7.97 Tf 6.59 0 Td[(s2t1 p and1)]TJ /F6 7.97 Tf 13.15 4.88 Td[(s1t2)]TJ /F6 7.97 Tf 6.59 0 Td[(s2t1 p ,whichappearintheexpressionsfor"p1and"p2,respectively,lieintheinterval(0,2).Thus,"piispositiveforpi=mi,ai,bi,ciandnegativeforpi=gi,ni.Sincegini+1>1,"pi<"ai1 gi,and"pi
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Theelasticityfortheremainingmodelparametersri,kij,andnaregivenby"r1=r1 2R0s21)]TJ /F4 11.955 Tf 13.15 8.09 Td[(s1t2)]TJ /F4 11.955 Tf 11.95 0 Td[(s2t1 p )]TJ /F4 11.955 Tf 13.15 8.09 Td[(r1t2 "r2=r2 2R0s11+s1t2)]TJ /F4 11.955 Tf 11.95 0 Td[(s2t1 p )]TJ /F4 11.955 Tf 13.15 8.09 Td[(r2t1 "k21=k21 2R0s21)]TJ /F4 11.955 Tf 13.15 8.08 Td[(s1t2)]TJ /F4 11.955 Tf 11.95 0 Td[(s2t1 p +2s1s2k12 )]TJ /F4 11.955 Tf 13.15 8.08 Td[(r2k21 "k12=k12 2R0s21+s1t2)]TJ /F4 11.955 Tf 11.95 0 Td[(s2t1 p +2s1s2k21 )]TJ /F4 11.955 Tf 13.15 8.09 Td[(r1k12 "n1=)]TJ /F4 11.955 Tf 10.49 8.09 Td[(s1g1n1 2R0t21+s1t2)]TJ /F4 11.955 Tf 11.95 0 Td[(s2t1 p +2s2k12k21"n2=)]TJ /F4 11.955 Tf 10.49 8.09 Td[(s2g2n2 2R0t11)]TJ /F4 11.955 Tf 13.15 8.09 Td[(s1t2)]TJ /F4 11.955 Tf 11.95 0 Td[(s2t1 p +2s1k12k21Ifri
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Whiletherelationshipbetweentheelasticitiesforparametersmi,ai,bi,ciandgiinpatchiareclear,therelationshipbetweentheelasticitieswithrespecttotheremainingparameters(riandkji)andtherelationshipbetweentheelasticitiesofparametersofdifferentpatches,arenotobviousfromtheanalyticexpressionspresentedinthissection.Inthefollowingsectionswewillestimateparametervaluesandusetheseestimatestoderiveelasticitiesforallmodelparametervaluesundervariousscenarios,whichwewilldeneusingcombinationsofdifferentparametersetscorrespondingtohightransmission,lowtransmission,fastmigration,andslowmigration. 4.3NumericalResults 4.3.1ParameterEstimatesRealisticparametervalueswereneededtogainanunderstandingofhowspatialheterogeneityinmalariatransmissionaffectstheprevalenceofmalaria,malariatransmission,andmalariacontrolinourtwo-patchmodel.Wecompiledbaselineparametervaluesforfourdifferentsituations,estimatedfrompublishedstudies.First,wecompiledvaluesforhightransmissionareas,andlowtransmissionareas.Hightransmissionparametersweregatheredfromstudiesinsub-SaharanAfrica.LowtransmissionparametersweretakenfromstudiesintheAmericas,especiallySouthAmerica,wherethenumberofcasesisgenerallylow.Amongregionsthatareconsideredhightransmission,thereisstillalotofvariabilityintheirlevelsofmalariatransmission.Thus,toencompasssomeofthisvariability,wecompiledparametersassociatedwiththedryseasoninahightransmissionregionandparametersassociatedwiththewetseasoninahightransmissionregion.Similarly,notalllowtransmissionregionscanbedescribedbythesamesetoftransmissionparameters.So,weagaincompileddryseasonandwetseasonparametersforalowtransmissionregion.Usingtheseestimatesfromdifferentseasonsinbothhighandlowtransmissionareas,weobtainedfourparametersetsrepresentativeofahigh-transmission/wet-conditionspatch,high-transmission/dry-conditionspatch, 86

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low-transmission/wet-conditionspatch,andlow-transmission/dry-conditionspatch(Table 4-2 ). Table4-2. Wetanddryconditionestimatesofmodelparametersforlowandhightransmissionsettings.Note:mwasnotdirectlydeterminedfromeldstudies.mafordryconditionsisknownfrom[ 45 ].Weassumeforwetconditionsthevaluetobe10timesgreateraccordingto[ 102 ]. LowHigh ParametersWetDryReferenceWetDryReference mi176.1917.619395.4560.81[ 77 ] ai0.1050.105[ 71 ]0.410.265[ 70 77 ] miai18.51.85[ 45 ]161.147716.114[ 70 ] bi0.10.1[ 14 ]0.0970.097[ 77 ] ci0.2140.2140.2140.214[ 19 31 ] gi0.1670.167[ 48 93 ]0.1810.26[ 70 ] ni10days10days10days10days[ 77 ] ri1=150days1=150days[ 15 30 ]1=150days1=150days Theratioofmosquitoestohumansmwasnotdirectlycalculatedforthewetanddryconditions.Instead,thevaluemawasmeasuredinvariouseldstudiesbyestimatingtheaveragenumberofbitesonahumanpernight.Theproportionofbitesonhumansoutofallbitesfromthevectorspecieswasdividedbytheaveragetimebetweenbloodmealstocalculatea,thehumanbitingrate.Byknowingmaanda,wecalculatedm.Thedifferenceinmabetweenwetanddryconditionswasnotdirectlyfoundforthehightransmissionscenario;however,eldstudieshaveshownthatthebitingrateonhumansinthewetseasonistenfoldthatofthedryseason[ 102 ];weassumedmaforthedryseasonwas1=10thatofthewetseason,anddeterminedmfromtheresultantmavalue.Foreachofourfourbaselineparametersets,wecalculatedthebasicreproductivenumberforanisolatedpatchwiththoseparametervalues(seeTable 4-3 ).Thesefourparametersetswereusedtodescribethewithin-patchmalariatransmissionparametersinourtwo-patchmodel.UsingthefourparametersetsandtheanalyticexpressionsderivedforR01andR02,wecalculatedandusedthese 87

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Table4-3. R0forthefourscenarios HightransmissionLowtransmission Wet187.157.03 Dry3.800.70 isolated-patchreproductivenumberestimatesasabaselinetocomparetheglobalR0valuetoindifferentparameterizationsofthetwo-patchmodel.Wepresentresultsforthreescenarios:inthersttwoscenariospatchoneishightransmissionandpatchtwoislowtransmission.Intherstscenario,bothpatcheshavedryconditions.Inthesecondscenario,bothpatchesexperiencewetconditions.Finally,bothpatchesareidenticallowtransmission,dryconditionpatchesinthethirdscenario. 4.3.2EffectofMigrationonR0WerstcalculatedtheglobalR0accordingtoEquation( 4 ),witheachpatchusingparametersfromoneoftheparametersetsinTable 4-2 .Wevariedk12andk21toexaminethepatternsofglobalR0withvaryingmigrationrates.AssumingforsimplicitythatR01>R02,fromSection 4.1.3 ,weknowthatmin(R0)=maxR01 1+ r1,R02for(k12,k21)2[0,1)[0,),andmax(R0)=R01.UsingthisfactaboutthemaximumandminimumvaluesofR0alongwithourestimatesoftheisolated-patchreproductivenumbersunderthefourpatchcharacteristicsidentiedinTable 4-3 (High-Wet,High-Dry,Low-Wet,Low-Dry),wecandeterminewhattherangeofR0willbewithmigrationundereachscenario.NumericalsimulationoftheglobalR0(Figure 4-2 )asafunctionofthemigrationratessuggeststhatifthemigrationtermsk12,k21arezero,theglobalR0isequaltoR01(assumingR01>R02).Fork21>0,ask12increases,theglobalR0getsclosertothevalueofR01.Thisislikelybecauseask12increases,ahigherproportionofindividualsexposethemselvestothetransmissioncharacteristicsofpatch1,causingthatpatchtocontributemoretotheglobalR0.Similarly,ask21increases,theglobalR0becomesclosertotheminimumR0value,R02.Thesenumericalndingsareinagreementwith 88

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theproofofTheorem 4.1 .So,ifpatch2hasR02<1,butR01>1andk12>>k21,morepeoplewillbeexposedtothetransmissioncharacteristicsofpatch1thanofpatch2,makingitmorelikelythatthediseasewillpersistinbothpatches.Theoppositeisalsotrue;ifmanymoreindividualsaremovingintopatch2thanpatch1(k21>>k12),thediseaseislikelytogoextinct,sincemorepeopleareexposedtothetransmissioncharacteristicsofthelowtransmissionpatchthantothoseofthehightransmissionpatch. Figure4-2. R0plottedasafunctionofthemigrationratesk21andk12.ThetopandbottomplanesrepresentR01andR02,respectively.Inthisgraph,patch1isahightransmissionpatch,patchtwoisalowtransmissionpatch,andconditionsaredryinbothpatches(R01=3.80,R02=.70). AsshowninSection 4.1.3 ,ifthetwopatcheshaveR0i>1,thenthediseasealwayspersists.Forexample,supposebothpatchesareintherainyseasonbutpatchoneisahigh-transmissionpatchandpatchtwoisalowtransmissionpatchsothatR01=187.15andR02=7.03.Then,weknowthat7.03R0187.15.Thus,itisclearinthisscenariothatnomatterwhattherateofmigrationisbetweenthetwopatches,thediseasewillpersist.Ournumericalsimulationsofscenarioswherebothisolatedpatchreproductionnumbersaregreaterthanone(Figure 4-3 )suggestthatmigrationcausesthesystem 89

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toreachequilibriumsoonerthanifthetwopatcheswereisolated.Notethatineachmodelsimulation,wesettheinitialpatchpopulationsizesequaltotheequilibriumpatchpopulationsizessothateachpatch'spopulationremainsconstantovertime;thatisNi(t)=Niforalltimet>0. A BFigure4-3. Numberofinfectedindividualsintwowetconditionpatches.Patch1ishightransmission(R01=187.15),patch2islowtransmission(R02=7.03). A Patchesareisolated, B patchesareconnectedviahumanmovementwithmigrationratesk12=.001,k21=.005,resultinginR0=113.66. IfthetwopatcheshaveR0i<1,thenthediseasealwaysgoestoextinction.ThisscenarioisillustratedinFigure 4-4 withtwolow-transmission,dry-conditionpatcheswiththesameisolatedpatchreproductionnumbers,R01=R02=0.70.Withoutmigration,thedynamicswithineachpatchareidentical.Undertheinuenceofhumanmovementthediseasestillgoesextinctinbothpatches,however,thenumberofcasesdecreasesmoresharplyinpatch1,sincemigrationintothispatchisvetimesfasterthanmigrationintopatch2.Ifbothpatchesaredrywithpatch1beingthehigh-transmissionpatchandpatch2thelow-transmissionpatch,thenR01=3.80andR02=0.7021.ThecontinuityofR0withrespecttothemigrationparametersindicatesthatthereexistsaratesuchthatmin(R0)>1forall(k12,k21)2[0,1)[0,).Infact,if1forall(k12,k21)2[0,1)[0,).Sincer1=1=150,R0>1if(k12,k21)2[0,1)[0,0.0187).Thisexampleillustratesthatalthoughthediseasewouldbecomeextinctinpatch2ifthe 90

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A BFigure4-4. Numberofinfectedindividualsintwolowtransmission,dryconditionpatches(R01=R02=0.70). A Patchesareisolated, B patchesareconnectedviahumanmovementwithmigrationratesk12=.001,k21=.005,resultinginR0=0.70. twopatcheswereisolated,thepresenceofsufcientlyslowmigrationfrompatch1topatch2inourtwo-patchmodelallowsthediseasetopersistinbothpatches.ThisresultisalsodemonstratedinFigure 4-5 ,whichplotsthenumberofinfectedhumansovertimeforthisHigh-Dry/Low-Dryscenario.Subgure 4-5A showsthatifthetwopatchesareisolated,thediseasediesoutinpatch2butpersistsinpatch1.However,ifthetwopatchesareconnectedbyhumanmovement(asinsubgure 4-5B ),althoughtheprevalenceofmalariainpatch1atsteadystatedecreases,thereisnowpersistenceofthediseaseinbothpatchesandthetotalprevalenceishigherthanwhenthepatchesareisolated.Conversely,Figure 4-6 illustrateshowsimplychangingthemigrationratessothatk21>>k12canbringthereproductionnumberbelowone,resultingineventualextinctionofthediseaseinbothpatches. 4.3.3EffectofMigrationonElasticityTheelasticityoftheglobalR0tomostofaparticularpatch'sparametersisentirelydependentupontwofactors:thedifferenceintheparametervaluesbetweenthetwopatches,andmigration.SimilartotheeffectofmigrationonR0,thefasterthemovementfrompatchjtopatchiis(inotherwords,thelargerkijis),themoreinuencetheparametersinpatchihaveonthesystem.Forexample,ifaparameter,suchasbiting 91

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A BFigure4-5. Numberofinfectedindividualsintwodryconditionpatches.Patch1ishightransmission(R01=3.80),patch2islowtransmission(R02=0.70). A Patchesareisolated, B patchesareconnectedviahumanmovementwithmigrationratesk12=.001,k21=.005,resultinginR0=2.35. Figure4-6. Numberofinfectedindividualsintwodryconditionpatches.PatchconditionsareidenticaltothoseinFigure 4-5 withtheexceptionofthemigrationrates.Patch1ishightransmission(R01=3.80),patch2islowtransmission(R02=0.70).Patchesareconnectedviahumanmovementwithmigrationratesk12=.0001,k21=.02,resultinginR0=0.99. rate,hasthesamevalueinbothpatches,thentherelativesizeofthemigrationratesandtherelativevaluesofR01andR02determinewhetherthebasicreproductivenumberismoreelastictotheparameterassociatedwithpatch1ortheparameterassociatedwithpatch2.Ontheotherhand,ifbitingratea1>a2,(k12=k21),and(R01=R02),thentheelasticityofR0toa1islargerthantheelasticityofR0toa2. 92

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Ifa1>a2,thenifk21ischosenappropriatelyandsatisestheinequalityk21>k12andR01andR02arenotverydifferentinvalue,thentheelasticitiesrelatedtothesetwoparametersmaybecomeequal,ortheelasticityofR0toa1maybecomesmallerthantheelasticityofR0toa2.Inotherwords,whethercontrolmeasuresshouldtargetpatch1orpatch2dependsontherateofhumanmovementbetweenthepatches.InSection 4.2.2 ,weprovedthat"p1+"p2isconstantandequaltotheelasticitiesintheisolated-patchcaseforpi=mi,bi,ci,andai.Thisresult,whichwedemonstratevianumericalsimulationinFigure 4-7B ,suggeststhatitmaybepossibletodivideresources,suchasinsecticidetreatedbednets,betweentwoconnectedpatchesinsuchawaythatthecontrolmeasuresareasefcientasifthetwopatcheswereasinglehomogeneouspatch.Still,numericalsimulationsareneededtoidentifywhichpatchshouldbetheprimarytargetformalariacontrol. A BFigure4-7. A ElasticityofR0toallparameters,withk12=0.001,plottedasfunctionsofk21.Patch1isahightransmission,dryconditionspatch(R01=3.80);patch2isalowtransmission,dryconditionspatch(R02=0.70).Bluelinesareelasticitieswithrespecttopatch1parameters;redlinesareelasticitieswithrespecttopatch2parameters. B Sumoftheelasticities"p1and"p2foreachparameterp. Intuitively,wemightchoosetoalwaystargetthehighertransmissionpatch,howeverournumericalsimulationsindicatethatthisisnotalwaysthebeststrategyforreducingmalariatransmission.Figure 4-7 illustrates,usingthedryseasonparameterizations, 93

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howtheelasticityofR0withrespecttothemodelparameterschangesfordifferentratesofmigrationbetweenahightransmissionpatchandalowtransmissionpatch.Inthisexample,k12=.001andtheelasticitiesareplottedasafunctionofk21.Forthishigh-dry/low-dryscenario,ifk21isclosetozero(i.e.nomigrationintopatch2),thenanyresourcesusedtocontrolmalariainthesecondpatchwillessentiallybewasted.Inparticular,whenk21issmall,R0ismostelastictomosquitodeathrateinpatch1(g1).Ask21isincreased,R0becomeslesssensitivetog1andeventuallyreachesapointwhereR0isequallyelastictog1andtog2.Hence,atthisintermediatek21(approximatelyk21=.04),resourcestargetingmosquitodeathrateshouldbeevenlydividedbetweenthetwopatches.Ifk21increasesfurther,R0ismostsensitivetomosquitodeathrateinpatch2(g2).Theelasticitieswithrespecttoparametersinpatch1approachzeroask21increases,suggestingthatifmigrationintopatch2isveryfast,resourcesusedforcontrolinpatch1willbewasted.BecauseR0willbelessthanoneforlargeenoughk21inthehigh-dry/low-dryscenario,theresultthatweshouldtargetthelowertransmissionpatchwhenk21exceedsacertainratemaynotbesurprisinginasystemwhereR01>1andR02<1.However,asimilarresultholdsevenwhenbothisolated-patchreproductivenumbersaregreaterthanone.Suppose,forexample,thatpatch1haslow-wettransmissioncharacteristicsandpatch2hashigh-drytransmissioncharacteristicssothatR01>R02>1.TheelasticitiesofR0tothemodelparametersareplottedinFigure 4-8 .AswiththepreviousexamplewhereR01>1>R02,ifmigrationintopatch2isslow,controlmeasuresshouldtargetmosquitodeathrateinpatch1.However,ifthemigrationratek21exceedsacertainvalue(approximatelyk21=.007),controleffortsshouldtargetthepatchwiththelowerreproductivenumber(patch2).Migrationratesareoftendifculttoestimate.However,havinganideaoftherelativesizesofthemigrationparametersk12andk21mightbesufcienttoprovideinsightintowherecontrolmeasuresshouldbeimplemented.Ingeneral,wefoundthatinscenarios 94

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Figure4-8. ElasticityofR0toallparameters,withk12=0.001,plottedasfunctionsofk21.Patch1isalowtransmission,wetconditionspatch(R01=7.03);patch2isahightransmission,dryconditionspatch(R02=3.80).Bluelinesareelasticitieswithrespecttopatch1parameters;redlinesareelasticitieswithrespecttopatch2parameters. whereR01>R02>1,orwhenR01>1>R02,controlmeasuresshouldtargetpatch2ifk21>>k12,andpatch1shouldbetargetedotherwise.Moreover,thelargerk12is,thesmallertheratioofk21tok12needstobeinorderforpatch2tobethemoreappropriatetargetformalariacontrol.Forexample,ifpatch1haslowtransmission,wetconditionsandpatch2hashightransmission,dryconditions,sothatR01>R02>1,thenfork12=0.0001,k21mustbemorethan50timesgreaterthank12inorderforpatch2tobethemoreimportanttargetformalariacontrol.Ifk12=0.001,k21needonlyberoughly6.5timesgreaterthank12,andifk12=0.01,k21mustonlybetwiceasbigask12towarranttargetingpatch2.Consequently,beingabletoclassifymovementintothehightransmissionpatchaseitherfastorslowprovidesadditionalinsightintohowresourcesformalariacontrolcanbebestallocated.Infact,ifbothmigrationratesarelargeandequal,thenplacingcontrolmeasuresinpatch1willonlybeslightlymoreeffectiveatreducingtransmissionthantargetingpatch2intheR01>R02>1case,whereasifbothmigrationratesaresmallandequal,targetingpatch1shouldyieldsubstantiallybetterresultsthantargetingpatch2.Whiledecisionsregardingwhere 95

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toallocateresourcesintheR01>1>R02casefollowthesamegeneralrulesasintheR01>R02>1case,themostsignicantqualitativedifferencebetweenthetwotransmissionscenariosoccurswhenbothmigrationratesarelargeandequal:unlikewiththeR01>R02>1case,patch1isamuchbettertargetforcontrolthanpatch2.WesummarizetheserulesofthumbforwheretotargetcontrolmeasuresinTables 4-4 and 4-5 Table4-4. TargetpatchformalariacontrolintheR01>R02>1case ratiok12slowk12fast k21 k12>>1patch2patch2 k21 k12=1patch1patch1slightlybettertarget k21 k12<<1patch1patch1 Table4-5. TargetpatchformalariacontrolintheR01>1>R02case ratiok12slowk12fast k21 k12>>1patch2patch2 k21 k12=1patch1patch1 k21 k12<<1patch1patch1 4.4ElasticityAnalysisofEndemicEquilibriumTheelasticityanalysisofthebasicreproductionnumberprovidedinsightintoappropriatemalariainterventionsforreducingtransmissionintensity.Wenowturntothestudyoftheendemicequilibriumtodeterminewhetherthegoalofreducingmalariaprevalencerequiresaqualitativelydifferentapproachtomalariacontrol.ConsiderourequilibriumequationdIi dt=iiIi iIi+giNi(Ni)]TJ /F4 11.955 Tf 12.6 0 Td[(Ii))]TJ /F8 11.955 Tf 12.6 0 Td[((ri+kji)Ii+kijIj=0.MultiplyingthroughoutbyiIi+giNiyieldsfi(I1,I2):=iiIi(Ni)]TJ /F4 11.955 Tf 11.96 0 Td[(Ii))]TJ /F8 11.955 Tf 11.96 0 Td[((ri+kji)Ii(iIi+giNi)+kijIj(iIi+giNi)=0,i=1,2. (4)Wendtheelasticityoftheendemicequilibriumbydifferentiatingsystem( 4 )implicitly,rstwithrespecttotheparametersi,i,ri,gi.Fromtheseelasticities,wecanderivetheelasticitiesformiandaisincei=miaibie)]TJ /F6 7.97 Tf 6.59 0 Td[(giniandi=aici.Implicit 96

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differentiationoff1(I1,I2)=0andf2(I1,I2)=0withrespecttoaparameterp1ofpatch1leadstothefollowingsystemoftwoequations:A1@I1 @p1+B1@I2 @p1+Cp1=0 (4)B2@I1 @p1+A2@I2 @p1=0whereAi=ii(Ni)]TJ /F8 11.955 Tf 11.96 0 Td[(2Ii))]TJ /F8 11.955 Tf 11.96 0 Td[((ri+kji)(2iIi+giNi)+kijiIjBi=kij(iIi+giNi)andC1=1I1(N1)]TJ /F4 11.955 Tf 11.95 0 Td[(I1)C1=1I1(N1)]TJ /F4 11.955 Tf 11.96 0 Td[(I1)+k12I1I2)]TJ /F4 11.955 Tf 11.96 0 Td[(I21(r1+k21)Cr1=)]TJ /F4 11.955 Tf 9.3 0 Td[(I1(1I1+g1N1)Cg1=)]TJ /F4 11.955 Tf 9.3 0 Td[(n111I1(N1)]TJ /F4 11.955 Tf 11.95 0 Td[(I1))]TJ /F8 11.955 Tf 11.96 0 Td[((r1+k21)I1N1+k12I2N1 Proposition4.1. A1A2)]TJ /F4 11.955 Tf 11.95 0 Td[(B1B26=0.Furthermore,A1A2)]TJ /F4 11.955 Tf 11.96 0 Td[(B1B2>0. Proof. Fromsystem( 4 ),wehavethat(A1A2)]TJ /F4 11.955 Tf 11.96 0 Td[(B1B2)@I1 @p1+Cp1A2=0. (4)Observethatfi(I1,I2)=AiIi+hi,wherehi:=iiI2i+(ri+kji)iI2i+kijIjgiNiisstrictlypositive.Thus,f(I1,I2)=0andhi,Ii>0implythatAi<0.Hence,equation( 4 )impliesthatA1A2)]TJ /F4 11.955 Tf 12.47 0 Td[(B1B2and@I1 @p1arenonzeroaslongasCp16=0.ClearlyC1>0andCr1<0.WemustverifythatC1andCg1arealsononzero.Notethatfi(I1,I2)=0implies(ri+kji)Ii)]TJ /F4 11.955 Tf 11.96 0 Td[(kijIj=iiIi iIi+giNi(Ni)]TJ /F4 11.955 Tf 11.96 0 Td[(Ii)>0. (4) 97

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Usingequation( 4 ),wendthatC1=1I1(N1)]TJ /F4 11.955 Tf 11.96 0 Td[(I1)1)]TJ /F7 11.955 Tf 33.08 8.09 Td[(1I1 1I1+g1N1>0Cg1=)]TJ /F4 11.955 Tf 9.3 0 Td[(n111I1(N1)]TJ /F4 11.955 Tf 11.95 0 Td[(I1))]TJ /F8 11.955 Tf 11.96 0 Td[(((r1+k21)I1)]TJ /F4 11.955 Tf 11.95 0 Td[(k12I2)N1<0.So,Cp1A26=0impliesA1A2)]TJ /F4 11.955 Tf 11.96 0 Td[(B1B2and@I1 @p1arenonzero.Now,A1A2)]TJ /F4 11.955 Tf 12.59 0 Td[(B1B2nonzeroandcontinuousinallparametersimpliesthatitmusthaveadenitesign:eitherstrictlypositiveorstrictlynegative.Considerk12=k21=0.ThenA1A2)]TJ /F4 11.955 Tf 12.03 0 Td[(B1B2=A1A2>0impliesthatA1A2)]TJ /F4 11.955 Tf 12.04 0 Td[(B1B2isstrictlypositiveforallpositivedisease-relatedparametersandforallnonnegativemigrationratesk12,k21. Solvingsystem( 4 )forthesensitivities@I1=@p1and@I2=@p1withp1=1,1,r1,g1,wehave@I1 @p1=)]TJ /F4 11.955 Tf 29.1 8.08 Td[(Cp1A2 A1A2)]TJ /F4 11.955 Tf 11.95 0 Td[(B1B2 (4)@I2 @p1=Cp1B2 A1A2)]TJ /F4 11.955 Tf 11.95 0 Td[(B1B2 (4)Weobtainedanalogousequationsforthesensitivitieswithrespecttopatch2parameters.TheelasticityofIiwithrespecttoaparameterpjisEipj=pj Ii@Ii @pjfori,j2f1,2g.Moreover,theelasticityofthetotalmalariaprevalence(I=I1+I2)withrespecttoparameterpiisgivenbyEpi=I1 I1+I2E1pi+I2 I1+I2E2pi.Fromouranalyticexpressions,wecandeterminethesignofeachelasticity.Aswewouldexpect,elasticitieswithrespecttoparametersm,a,b,carepositiveandelasticitieswithrespecttoparametersr,g,andnarenegative.Tovisualizetheelasticities,werstsolvefortheendemicequilibrium(I1,I2)numericallyforarangeofmigrationrates.Next,wesubstitutethesevaluesintoouranalyticexpressionsfortheelasticitiesand,aswedidfortheelasticitiesofR0,we 98

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plottheendemicequilibriumelasticities(inabsolutevalue)asafunctionofk21,withk12=0.001xed(seeFigure 4-9 ). AR01>1>R02 BR01>R02>1Figure4-9. Elasticityofendemicequilibriumtoallparameters,withk12=0.001,plottedasfunctionsofk21.Bluelinesareelasticitieswithrespecttopatch1parameters;redlinesareelasticitieswithrespecttopatch2parameters.( A )Patch1isahightransmission,dryconditionspatch(R01=3.80);patch2isalowtransmission,dryconditionspatch(R02=0.70).Notethatfork210.028,R0<1.( B )Patch1isalowtransmission,wetconditionspatch(R01=7.03);patch2isahightransmission,dryconditionspatch(R02=3.80). 99

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Figure 4-9A illustratestheelasticitiesoftheendemicequilibriuminthehigh-dry/low-drysetting,plottedfortherangeofmigrationratesforwhichtheendemicequilibriumexists.Fork210.028,R0<1,andhencethedisease-freeequilibriumistheonlyequilibrium.Figure 4-9 revealsthatstudyingtheendemicequilibriumyieldsresultsthatarequalitativelysimilartothoseobtainedinourstudyofR0.Inparticular,thewithinpatchrankingsofelasticitiesforagiventransmissionsettingareessentiallythesameintheanalysisofR0andtheanalysisoftheendemicequilibrium(EE).Inthehigh-dry/low-drysetting(R01>1>R02),theorderingofboththeR0elasticities(Figure 4-7A )andtheEEelasticities(Figure 4-9A )isjEg1j>jEn1j>Ea1>Ep1forpatch1parametersandjEg2j>Ea2>jEn2j>Ep2forpatch2parameters.WedonotincludejEr2jinthisorderingbecausethepositionthiselasticitytakesintherankingchangesasthemigrationratek21changes.Inthelow-wet/high-drysetting(R01>R02>1),theelasticitiesoftheEE(Figure 4-9B )alsoretainthesameorderingastheelasticitiesofR0(Figure 4.3.3 ):jEg1j>Ea1>jEn1j>Ep1>jEr1jandjEg2j>jEn2j>Ea2>jEr2j>Ep2.TheelasticitiesoftheendemicequilibriumalsoretainsomeofthequalitativebehaviorastheelasticitiesofR0withrespecttowhichpatchisthebettertargetforcontrol.Inthelow-wet/high-drysetting,forverysmallvaluesofk21,analysisoftheEErevealspatch1isthebettertargetforcontrol;beyondacertainmigrationrate,patch2becomesthebettertarget.Similarly,patch1isthebesttargetforcontrolinthehigh-dry/low-drysetting.However,sincetheendemicequilibriumonlyexistsforaconstrainedsetofmigrationratesk21,thereisnomigrationrateforwhichpatch2becomesthebettertargetforcontrolunderthisparameterization.PerhapsifR02werelarger,butstilllessthanone,wemightseethatpatch2doesbecomeabettertargetforcontrolbeforetheglobalreproductionnumberfallsbelowone.Thedifferencesbetweentheanalysisoftheendemicequilibriumandthereproductionnumberarequantitativeinnature.Themoststrikingdifferenceisthevalueofk21forwhichweshouldswitchfromtargetingpatch1totargetingpatch2.Thisswitchin 100

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strategyoccursformuchsmallervaluesofthemigrationratek21inourstudyoftheEEthaninourstudyofR0.Thisindicatesthatdetermininganappropriateinterventionstrategywilldepend,inpart,onwhetherthegoalistoreducetransmissionpotentialorwhetherthegoalistoreducetheoverallprevalenceofmalaria.Thequalitativesimilaritiesofwithin-patchelasticities,ontheotherhand,suggestthatthebesttypeofintervention(bednets,treatment,insecticides,etc.)willbethesamewitheithergoalinmind. 4.5ConclusionSimplemalariamodels,suchastheRoss-Macdonaldmodel,thatassumepopulationsareisolatedandhomogeneoushavemadealargecontributiontotheareaofmalariaresearchoverthelastseveraldecades.However,thediversityinherenttothisdiseaserequiresmodelsthatincorporateheterogeneitysothattheymayprovidegreaterinsightintohowweshouldapproachmalariacontrol.Ourstudyofatwo-patchmalariamodelsuggeststhatusingintuitiontoguidedecisionmakinginmalariacontrolmaynotbesufcient.Forexample,targetingregionswiththehighesttransmissionratesmaynotbethemosteffectiveuseofresourcesiftheyarestronglyconnectedtolowertransmissionregionsviaemigration.Furthermore,usingsingle-patchmodelstoestimateparametersrelevanttomalariadynamicsandmalariacontrol,suchasthebasicreproductionnumber,mayprovideaninaccurateassessmentoftransmissionpotentialinaregion.ThisdiscrepancybecameclearinourexplorationofthescenariowhereR01>1>R02.Thetwo-patchmalariamodelindicatesthathumanmovementcanresultinthepersistenceofmalariainregionswheremalariawoulddieoutifisolated,whereasasingle-patchmalariamodelwouldinaccuratelypredictextinctionofmalariainsuchsituations.OurresultsaresimilartothosefoundbyCosner,etal[ 32 ],asourresultsalsoshowthathumanmovementcancausemalariatobeendemicinanareawithareproductionnumberbelow1.However,unlikeCosner'stwo-patchmalariamodelwithLagrangian 101

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movement,itisnotpossibleformalariatopersistinourmodelifbothregionshaveanisolatedreproductionnumberbelowone.WhereasCosner'sexplorationofatwo-patchsystemassumeszerotransmissioninonepatch,ourresultsallowforanadditionallevelofresolution.Becauseweassumeeachpatchiscapableofsupportingmalariatransmission,weareabletocomparetheelasticityoftheglobalR0toreductionsintransmissioninboththehighandlowtransmissionpatches.Thisallowedustoinvestigatehowimplementingcontrolmeasuresindifferent,butconnected,regionsimpactstheoverallleveloftransmission.Theresultsfromourelasticityanalysisofthetwo-patchmodelR0arealsofundamentallydifferentfromelasticityanalysesperformedonsingle-patchmalariamodels.Thisisespeciallytruefortheparameters1=gandn,whichrepresentthemosquitolifespanandtheextrinsicincubationperiod;manyothermodelsthathaveassessedelasticitiesdidnotincludetheextrinsicincubationperiod,andconsequentlyreportedbitingrateatobethemostimportantparametertotarget[ 26 ].Inallfourofourparametersets,theaveragelifespanofamosquitowasshorterthantheextrinsicincubationperiod.Fromourelasticityanalysis,thisimpliedthatR0wasmoresensitivetomosquitodeathratethantobitingrateinallscenarios.Theseresultssuggestthatusingmulti-patchmalariamodelscanhelpinforminterventionstrategyusageinareasofheterogeneousmalariatransmissionconnectedbyhumanmovement.Forexample,inHispaniola,whileconventionalwisdommaysuggestthatresourcesshouldbefocusedonHaiti,whichhashighermalariatransmission,ourresultssuggestthisisnotnecessarilythecase;efcientapplicationofinterventionstrategieswilldependalsoonlong-termhumanmovementpatternsbetweenthetwocountries.ComparingtheendemicequilibriumelasticityresultstotheR0elasticityresultssuggeststhatitmaybenecessarytodevelopdifferentcontrolstrategiesdependingonwhetherthegoalistoreducethetransmissionpotential,orwhetherthegoalistoreduce 102

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diseaseprevalenceinamalariaendemicsetting.Onceagoalisestablished,knowledgeabouthumanmigrationrateswillbeessentialtoidentifyinganeffectivecontrolstrategythatmakesefcientuseofavailableresources.Althoughthisexplorationofmalariadynamicsandmalariacontrolinthecontextofatwo-patchmodelisstillanoversimplicationofreality,ithighlightstheneedformorecomplexmathematicalmodelsincorporatingbothspatialheterogeneityandhumanmovementtoguidepublichealthofcialsintheprocessofmakingdecisionsthatwillmakethebestuseofthelimitedresourcestheyhave.Ourstudyalsostressestheimportanceofcollectingmalariaprevalencedataandhumanmovementdatainmalariaendemicregionssothatthesemoresophisticatedmodelscanprovidereasonable,region-specicanswersabouthowtobestallocateresources. 103

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CHAPTER5OPTIMALTEMPORALDISTRIBUTIONOFMALARIAVACCINESINAHETEROGENEOUSTRANSMISSIONLANDSCAPEThesuccessofvaccinesforinuenza,measles,rubella,andsmallpox,tonameafew,makestheavailabilityofamalariavaccineextremelydesirable.However,developingavaccineformalariahasproventobeachallengebecauseofthecomplexbiologyofPlasmodiumparasitesandtheirabilitytoevadethehostimmunesystem[ 90 ].OneofthemostpromisingvaccinecandidatesistheRTS,Svaccine,whosepurposeistoreducetheinfectivityofthesporozoitestageinhumans[ 2 ].Thisvaccine,whichwasdevelopedroughly25yearsago,iscurrentlyinthethirdphaseofvaccinetrials.ThephaseIItrialsestimatedbetween20and50%efcacyofthevaccineagainstclinicalmalaria[ 2 ].Thepotentialavailabilityofmalariavaccinesinthenearfuture,particularlyvaccinesthatarefarfrom100%effective,bringstolighttheneedtostudythepotentialconsequencesofamalariavaccinationprogram.Moreover,sincevaccinesarenotanunlimitedresource,andtheinfrastructuretodeliverthemalariavaccinesmaybelimited,itisimportanttounderstandapriorihowthesevaccinescanbestbedistributedovertime.Tothisendweintroduceamulti-patchmalariamodelwithvaccinationandshort-termhumanmovement.Applyingideasfromoptimalcontroltheorytothismodel,wedeterminednecessaryandsufcientconditionsforanoptimalvaccinationstrategyinaheterogeneoustransmissionlandscape.Wethenexplorehowthisstrategychangesaswevaryparametersassociatedwithvaccineefcacyandhumanmovement. 5.1MalariaModelswithVaccinationSomemalariamodelsincorporatemalariavaccinationbyassumingarandomfractionpofnewbornindividualsarevaccinated[ 8 28 ].Incorporatingvaccinationinthiswayreducesthebasicreproductionnumberbytheproportion(1)]TJ /F4 11.955 Tf 12.45 0 Td[(p).Inadditiontovaccinatingaproportionofnewbornindividuals,Chiyakaetal.[ 28 ]consideredtheeffectofvaccinatingsusceptibleindividualsataratev.Comparingthetwovaccinationstrategiesrevealedthatvaccinatingsusceptibleindividualsinadditiontonewborns 104

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resultsinlowermalariaprevalencethanvaccinatingnewbornsalone.Teboh-Ewungkemetal.[ 105 ]developedapopulation-levelmalariamodeltostudytherolegametocytedensity(theinfectiousstageoftheparasitelife-cycleinhumans)playsinmalariatransmissioninthepresenceofanimperfectvaccine.Thismodeldividesthepopulationofinfectioushumansintotwoclasses:individualswithlowgametocyte-densityandindividualswithhighgametocyte-density.Infectioushumanscanpassback-and-forthbetweenthesetwostagesbeforeenteringatemporarilyimmunestage;susceptiblehumansarevaccinatedatsomerateh.Theauthorsdeterminedathresholdquantity5determiningwhethervaccinationwillhaveapositiveornegativeimpactondiseaseprevalence.Inparticular,if5<1,vaccinationhasapositiveimpact,if5=1,vaccinationhasnoimpact,andif5>1,vaccinationhasanegativeimpactonmalariaprevalence[ 105 ].Thisresultstressestheimportanceofstudyingthepotentialconsequencesofstartingamalariavaccinationprogramandtheneedtodesignvaccinationstrategiesthatwillavoidnegativeoutcomes. 5.2OptimalControlTheoryAppliedtoMalariaModelsApplicationsofoptimalcontroltheorytovector-bornediseasemodelsappearin[ 17 18 21 92 ].In[ 21 ]Caetanoetal.studiedtheoptimalcontrolofDengueusinginsecticidesandeducationalcampaignsaimedatmotivatingindividualstominimizestandingwater.ReducingtheamountofstandingwaterwouldinturnreducetheavailablebreedinghabitatfortheDenguevector.Bytryingtojointlyminimizethecostofinsecticidespraying,educationalcampaigns,andthecostofinfectedindividuals,theauthorsnumericallydeterminedanoptimalcontrolpolicy.However,believingthispolicywouldbedifculttoimplement,Caetanoetal.determinedasub-optimalpolicybychoosingthebeststrategyoutofthepossiblesetofpiecewiseconstantcontrolpolicies.Thissub-optimalpolicyyieldedresultsverysimilartotheoptimalpolicy,atasimilarcost[ 21 ].Rodriguesetal.[ 92 ]attemptedtosolvethecontrolproblemproposedbyCaetanoetal.intwoways:byusingoptimalcontroltheoryandbydiscretizingtheproblemto 105

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uselinearprogrammingtechniques.Usingoutoftheboxsolversappropriateforoptimalcontrolproblems,whichtheauthorsargueddoabetterjobofndingglobaloptimathanothernumericalmethods,theywereabletoobtainacontrolpolicythatperformsbetterthanthepolicyderivedin[ 21 ].Blaynehetal.[ 18 ]studiedtheproblemofcontrollingWestNileVirus.Theirgoalwastojointlyminimizethenumberofexposedandinfectedhumans,thevectorpopulation,andthecostofimplementingcontrolmeasures.Thecontrolmeasuresconsideredwerevectorcontrolandthepreventionofvector-to-humancontacts.NumericalresultsoftheoptimalcontrolproblemappliedtoWestNilesuggestedthatcontroleffortsshouldfocusmoreonvectorcontrolthanonpreventionofcontacts,withtheimportantcaveatthatmoreknowledgeisneededregardingcostsandparametervaluesfortheregionofinterest[ 18 ].Tothebestofourknowledge,optimalcontrolhasnotbeenappliedtovector-bornediseasesinametapopulationcontext.However,Rowthornetal.[ 94 ]studiedthecontrolofadirectlytransmitteddiseaseunderabudgetconstraintinatwo-patchframework.Theauthorsfoundthatundercertaininitialconditions,equalizingthenumberofinfectedindividualsineachpatchwastheworststrategy.Furthermore,treatmentisbestallocatedinregionswithalargesusceptiblepopulation,ratherthaninpopulationswithahighnumberofinfectedindividuals[ 94 ].Thesenon-intuitiveresultsemphasizethepotentialutilityofoptimalcontroltheoryindeterminingeffectivecontrolstrategiesinheterogeneousenvironments. 5.3MalariaModelwithShort-TermMovementandVaccinationFirst,weconsiderahumanpopulationofconstantsizeNsegmentedintonresidentpatchesofpopulationsizeNi,i=1,...,n.Eachindividualisaresidentofoneofthesenpatchesandabletotemporarilyvisitanyoftheremainingn)]TJ /F8 11.955 Tf 13.15 0 Td[(1patches,butneverchangesresidence.Weassumethatthesemovementsareshortenoughthattheyhavenegligibleeffectonpatchpopulationsize,andconsequentlywetakeNi(t)=Ni(0)foralltimet.Mosquitoesarealsoassignedaresidencepatch;however, 106

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becausemosquitoestravelrelativelyshortdistances,weassumemosquitoesneverleavetheirpatchofresidence.ThesystemofordinarydifferentialequationsinSystem( 5 )describesthediseasedynamicsinhumansandmosquitoesthatareresidentsofpatchi.Parametersaredenotedwithsubscriptssothatthemodelisexibleenoughtoincorporatedifferentpopulationandtransmissiondynamicsforeachofthenpatches.Initially,humansarebornsusceptibleintooneofthepatches;thisstatevariableisdenotedbySN,i.Thesesusceptibleindividualscaneitherbecomevaccinatedatarateviandprogresstothesusceptible/vaccinatedstageSV,i,ortheycanbecomeinfectedatsomeratehN,i,progressingtoaninfectiousstage,IN,i,inwhichtheyare(atleastpartofthetime)clinicallyill.IndividualsleavetheIN,istageoftheinfectionandenterRN,iataratei,atwhichpointtheyaretemporarilyimmune,nolongerclinicallyill,andlessinfectioustomosquitoes.Theseasymptomatic,temporarilyimmune,partiallyinfectiousindividualshavebeenpreviouslyincorporatedintomathematicalmodelsofmalaria[ 8 24 ].TheseRN,iindividualswilleventuallycleartheinfectionataraterN,iandreturntothesusceptible,non-immunestage,SN,i.Vaccinated,susceptibleindividualsarepartiallyimmuneandwilleventuallyeitherlosetheirimmunityatarate!iandreturntothesusceptible/non-immuneclass,orcanbecomeinfectedataratehV,i.Sincethevaccineofferspartialprotectionagainstmalaria,hV,i
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havedifferentinfectivityandsusceptibility.Consequently,wedenotethetransmissionratefromapatchjmosquitotoapatchinon-immunesusceptiblehumanbyN,ji,andthetransmissionratefromapatchjmosquitotoapatchivaccinatedsusceptiblehumanbyV,ji.Similarly,ij,N,ij,andV,ijdenotethetransmissionratesfromahumaninstageIN,j,RN,j,andRV,j,respectively,toapatchimosquito.WeassumethatindividualsinRV,iandRN,iarelessinfectioustomosquitoesthanindividualsinIN,i,andconsequently,N,ij,V,ij
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ThedescriptionofthemodelandthecorrespondingdiagraminFigure 5-1 canbedescribedmathematicallyasfollows:S0N,i=iNi)]TJ /F4 11.955 Tf 11.96 0 Td[(hN,iSN,i)]TJ /F4 11.955 Tf 11.95 0 Td[(viSN,i+!iSV,i+rN,iRN,i)]TJ /F7 11.955 Tf 11.95 0 Td[(iSN,iI0N,i=hN,iSN,i)]TJ /F8 11.955 Tf 11.96 0 Td[((i+i)IN,iR0N,i=iIN,i)]TJ /F8 11.955 Tf 11.95 0 Td[((rN,i+i)RN,i (5)S0V,i=rV,iRV,i+viSN,i)]TJ /F8 11.955 Tf 11.96 0 Td[((hV,i+!i+i)SV,iR0V,i=hV,iSV,i)]TJ /F8 11.955 Tf 11.96 0 Td[((rV,i+i)RV,iz0i=ki(1)]TJ /F4 11.955 Tf 11.95 0 Td[(zi))]TJ /F4 11.955 Tf 11.95 0 Td[(gizi,wherehN,i:=nXj=1pjiN,jizj (5)hV,i:=nXj=1pjiV,jizj (5)ki:=nXj=1pij Nj(ijIN,j+N,ijRN,j+V,ijRV,j) (5) Table5-2. Descriptionofvaccinationmodelpatchiparameters ParameterDescription N,jiPatchjmosquito-to-patchisusceptible/non-immunehumantransmissionrateV,jiPatchjmosquito-to-patchisusceptible/vaccinatedhumantransmissionrateijIN,j-to-patchimosquitotransmissionefciencyX,ijRX,j-to-patchimosquitotransmissionefciency,X=N,V1=iDurationofclinicalepisodeinpatchiiHumannaturalmortalityrateinpatchirX,iRX,iHuman'sparasiteclearancerateinpatchi,X=N,V!iRateoflossofvaccine-acquiredimmunityinpatchiviVaccinationrateinpatchipjiProportionoftimearesidentofpatchispendsinpatchj 109

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. SV,i RV,i SN,i IN,i RN,i hV,i rV,i vi !i hN,i i rN,i iNi Figure5-1. Diagramofdiseasedynamicsinhumans. Fromthissystemofdifferentialequations,wecaneasilydeterminethediseasefreeequilibriumE0:=(~SN,~IN,~RN,~SV,~RV,~z)=(~SN,~0,~0,~SV,~0,~0),whereeachvectorisarowvectoroflengthn,andtheithcomponentsof~SNand~SVareSN,i=!i+i !i+i+viNiandSV,i=vi !i+i+viNi,respectively.Linearizingthesystemaroundthisdisease-freeequilibriumandndingthedominanteigenvalueofthenextgenerationmatrix(asdescribedinpreviouschapters),wedeterminedthebasicreproductionnumberforasingleisolatedpatch:R0i=s N,ii(!i+i) gi(!i+i+vi)ii i+i+N,iii (i+i)(rN,i+i)+V,iivi gi(!i+i+vi)V,ii rV,i+i. (5)TherstterminthesuminR20iisthecontributiontothereproductionnumberbyinfectedindividualswhoarenotvaccinated;thesecondtermrepresentsthecontributionbyinfected,vaccinatedindividuals.WewilllaterassumethatV,ji=(1)]TJ /F4 11.955 Tf 12.57 0 Td[(e)N,ji,whereedenotesthevaccineefcacy.DifferentiatingthereproductionnumberwithrespecttothevaccinationratevirevealsthatR0iisnotalwaysadecreasingfunctionofvi.Forexample,ifthevaccineispoorandindividualsinpatchiundergotreatment,thereby 110

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increasingparameteri,thenR0icaninfactbeanincreasingfunctionofvi.Thisobservationemphasizestheneedtounderstandtheconsequencesofadministeringimperfectvaccinesandtheneedtodeterminehowtodistributethemeffectively.Inparticular,poorvaccinesmayjeopardizecontroleffortsinregionswheretreatmentremainsaneffectivecontrolstrategy. 5.4OptimalControlProblemThegoalofsolvinganoptimalcontrolproblemistodetermineavaccinationstrategythatjointlyminimizestheproportionofthepopulationinfectedoveragiventimeinterval,thetotalnumberofmalariavaccinesdistributedoverthatintervalperperson(i.e.thenumberofvaccinesused,scaledbytheinverseofthetotalpopulationsize),andthevaccinationrate.Moreformally,theoptimalcontrolproblemistodetermineavectorfunction~v(t)thatminimizestheobjectivefunctionalJ=ZT0nXi=1IN,i+RN,i+RV,i+viSN,i N+Wiv2idtsubjecttosystem( 5 )withinitialconditionvector~x0=~SN0,~IN0,~RN0,~SV0,~RV0,~z0i2R6n,whereeachelementof~x0isrow-vectoroflengthn,andWiisavaccinationweight.Wealsorequire~v2U,whereU:=[0,vmax,1][0,vmax,2][0,vmax,n],andvmax,idenotesthemaximumvaccinationratefeasibleforpopulationi.Restrictingthevaccinationrateisessentialsinceacontrolpolicyshouldnotoverburdenhealthfacilitiesbymakingunrealisticdemands.Forsimplicityofnotation,wewilldenotethevectorofstatevariables(~SN,~IN,~RN,~SV,~RV,~z)by~x,therighthandsideofsystem( 5 )by~f,andtheintegrandoftheobjectivefunctionJbyF.Moreprecisely,wedenetheelementsof~xand~fasfollows: xi,1:=SN,ixi,4:=SV,ixi,2:=IN,ixi,5:=RV,ixi,3:=RN,ixi,6:=zifi,k:=dxi,k dt 111

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Wedenetherowvectors~fk:=(f1,k,f2,k,...,fn,k)sothatwearriveatthenotation~f:=(~f1,~f2,...,~f6)T.Thus,theoptimalcontrolproblemcanbestatedasfollows:MinimizeJ=ZT0F(t,~x(t),~v(t))dt (5)subjecttod~x(t) dt=~f(t,~x(t),~v(t)) (5)withinitialconditions~x(0)=~x0, (5)naltimeconditions~x(T)free, (5)and~v(t)=(v1(t),v2(t),...,vn(t))2URn (5)Usingthefollowingstandarddenitionandtheoremsfromoptimalcontroltheory[ 97 ],weprovedtheexistenceofanoptimalcontrolpair(~x,~v)andderivednecessaryandsufcientconditionsfortheoptimalcontrol. Denition1. SupposeFandthecomponentsofthevector~fhavepartialderivativeswithrespecttothestatevariablesthatarecontinuousintime,inallstatevariables,andinallcontrolvariables.(~x(t),~v(t))isanadmissiblepairifeachcomponentof~v(t)ispiecewisecontinuous,~v(t)2U,and~x(t)isthecorrespondingcontinuousandpiecewisedifferentiablevectorfunctionsatisfying( 5 )-( 5 ). 112

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Theorem5.1(Filippov-CesariExistenceTheoremadaptedfrom[ 97 ]). Forall(t,~x)2Rl+1,wherelisthedimensionof~x(t),denethesetN(t,~x):=nF(t,~x,~v)+,~f(t,~x,~v):0,~v2Uo.Supposethat (a) N(t,~x)isconvexforevery(t,~x), (b) Uiscompact, (c) thereexistsanumberb>0suchthatk~x(t)kbforallt2[0,T]andalladmissiblepairs(~x(t),~v(t)).Then,thereexistsanoptimalpair(~x(t),~v(t)),wherethecontrol~v(t)ismeasur-able. Theorem5.2(TheMinimumPrinciple(NecessaryConditions)adaptedfrom[ 97 ]). Suppose(~x(t),~v(t))isanoptimalpairforproblem( 5 )-( 5 ).Then,thereexistsaconstant0,with0=0or1,andacontinuous,piecewisedifferentiablevectorfunction~(t)=(1(t),...,l(t)),calledadjointfunctions,suchthatforallt2[0,T],(0,~(t))6=(0,~0),and (a) Thecontrolfunctionv(t)minimizestheHamiltonianH(t,~x(t),~v(t),~(t))for~v2U.Inotherwords,H(t,~x(t),~v,~(t))H(t,~x(t),~v(t),~(t))8~v2U (b) Theadjointfunctionssatisfydi dt=)]TJ /F7 11.955 Tf 10.49 8.09 Td[(@H(t,~x(t),~v(t),~(t)) @xi,fori=1,...,r,wherever~v(t)iscontinuous (c) ~(T)=~0. Theorem5.3(Mangasarian(SufcientConditions)adaptedfrom[ 97 ]). SupposeUisconvexinproblem( 5 )-( 5 )andthatthepartialderivatives@F=@vjand@fi=@vj 113

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allexistandarecontinuous.Ifthepair(~x(t),~v(t)),togetherwithacontinuousandpiecewisedifferentiableadjointfunction~(t),satisesalltheconditionsintheMaximumPrinciple(Theorem 5.2 )with0=1,andifH(t,~x,~v,~)isconcavein(~x,~v)forallt2[0,T],then(~x(t),~v(t))solvestheproblem.IfH(t,~x,~v,~)isstrictlyconcavein(~x,~v),then(~x(t),~v(t))istheuniquesolutiontotheproblem. Theorem5.4. Theoptimalcontrolproblemgivenby( 5 )-( 5 )hasanoptimalsolution. Proof. (i) LetN(t,~x)bedenedasinTheorem 5.1 .Lety1,y22N(t,~x).ToshowthatN(t,~x)isconvexforeach(t,~x),wewillshowthatthelineconnectingy1andy2iscontainedintheset.Inotherwords,ourgoalistoshowthaty3:=y1+(1)]TJ /F7 11.955 Tf 11.95 0 Td[()y22N(t,~x)82[0,1].yi2N(t,~x)impliesthatthereexist1,2andcontrolvectors~u1,~u22U(~ui:=)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(ui1,ui2,...,uin)suchthatyi=fF(t,~x,~ui)+i,~f1(t,~x,~ui),...,~f6(t,~x,~ui)gfori=1,2.Let~w:=(w1,w2,...,w6n+1)besuchthaty3=~w.Then,w1= nXi=1IN,i+RN,i+RV,i+u1iSN,i N+Wi(u1i)2+i!+(1)]TJ /F7 11.955 Tf 11.95 0 Td[() nXi=1IN,i+RN,i+RV,i+u1iSN,i N+Wi(u2i)2+2!=nXi=1IN,i+RN,i+RV,i+u1iSN,i N+Wi(u1i)2+(1)]TJ /F7 11.955 Tf 11.96 0 Td[()Wi(u2i)2+1+(1)]TJ /F7 11.955 Tf 11.96 0 Td[()2.Lettingu3i:=p (u1i)2+(1)]TJ /F7 11.955 Tf 11.96 0 Td[()(u2i)2and3i:=i+(1)]TJ /F7 11.955 Tf 12.02 0 Td[()2,wehaveshownthatw1=F(t,~x,~u3)+3.Because2[0,1]impliesthat30and0u3ivmax,i(and 114

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consequently~u32U),thisprovesthaty32N(t,~x).Notethatfork=2,3,5,6,~fkareindependentofthecontrolfunction.Thus,itisclearthatforj6=i+1,4n+i+1,wjisconstantforall2[0,T],i=1,...,n.So,weneedonlycheckthatwi+1andw4n+i+1satisfytherightbounds.Fori=1,...,n,wendthatwi+1=fi,1(t,~x,u1i)+(1)]TJ /F7 11.955 Tf 11.96 0 Td[()fi,1(t,~x,u2i)=iNi)]TJ /F4 11.955 Tf 11.95 0 Td[(hN,iSN,i)]TJ /F8 11.955 Tf 11.96 0 Td[((u1i+(1)]TJ /F7 11.955 Tf 11.95 0 Td[()u2i)SN,i+!iSV,i+rN,iRN,i)]TJ /F7 11.955 Tf 11.95 0 Td[(iSN,i=fi,1)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(t,~x,u1i+(1)]TJ /F7 11.955 Tf 11.96 0 Td[()u2iSimilarly,wendthatw4n+i+1=fi,4(t,~x,u1i)+(1)]TJ /F7 11.955 Tf 11.96 0 Td[()fi,4(t,~x,u2i)=fi,4(t,~x,u1i+(1)]TJ /F7 11.955 Tf 11.95 0 Td[()u2i).Letting^u3i:=u1i+(1)]TJ /F7 11.955 Tf 11.96 0 Td[()u2i,weeasilydeterminethat^u3i2[0,vmax,i]82[0,1].Wehaveshownthat~w2N(t,~x).Therefore,N(t,~x)isconvexforeach(t,~x). (ii) U=[0,vmax,1][0,vmax,2][0,vmax,n]isclosedandboundedinRn,andhenceiscompact. (iii) k~x(t)k= nXi=1)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(S2N,i(t)+I2N,i(t)+R2N,i(t)+S2V,i(t)+R2V,i(t)+z2i(t)!1=2 nXi=1(6N2i+1)!1=2= n+nXi=16N2i!1=2q n(1+6maxfN2igni=1),8t2[0,T].FromtheFilippov-CesariTheorem( 5.1 ),itfollowsthatthereexistsanoptimalpair(~x(t),~v(t)),where~v(t)ismeasurable,thatsolvesouroptimalcontrolproblem. 115

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Itfollowsfromthetheoremsforderivingnecessaryandsufcientconditions(Theorems 5.2 and 5.3 )thatouroptimalcontrolproblem( 5 )-( 5 )canbereformulatedasfollows:TheproblemofminimizingtheobjectivefunctionalisequivalenttominimizingtheHamiltonianwith0=1.Thus,wewanttondvi(t)=vi(t)suchthat@H @vi=0and@2H @v2i>0fori=1,...,n.First,weintroducesomenotationfortheadjointsystem:q0i,k(t):=)]TJ /F7 11.955 Tf 13.76 8.08 Td[(@H @xi,k~qk:=(q1,k,q2,k,...,qn,k)~:=(~q1,~q2,...,~q6)fori=1,...,nandk=1,...,6.TheHamiltonianforoursystemisdescribedbyH(t,~x(t),~v(t),~(t))=F(t,~x,~v)+nXi=16Xk=1qi,k(t)fi,k(t,~x,~v).Wendthat@H=@vi=2Wivi+(1=N)]TJ /F4 11.955 Tf 12.86 0 Td[(qi,1+qi,4)SN,i=0,whichimpliesvi(t)=)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(qi,1(t))]TJ /F4 11.955 Tf 11.96 0 Td[(qi,4(t))]TJ /F5 7.97 Tf 14.23 4.71 Td[(1 NSN,i(t) 2Wi,and@Hi=@v2i>0,whereqi,1andqi,4aresolutionstothesystem:q0i,1=)]TJ /F4 11.955 Tf 11.98 8.09 Td[(vi Ni+qi,1(hN,i+vi+i))]TJ /F4 11.955 Tf 11.95 0 Td[(qi,2hN,i)]TJ /F4 11.955 Tf 11.96 0 Td[(qi,4viq0i,2=)]TJ /F8 11.955 Tf 13.17 8.09 Td[(1 Ni+qi,2(i+i))]TJ /F4 11.955 Tf 11.96 0 Td[(qi,3i)]TJ /F6 7.97 Tf 18.31 14.95 Td[(nXj=1qj,6pji Niji(1)]TJ /F4 11.955 Tf 11.96 0 Td[(zj)q0i,3=)]TJ /F8 11.955 Tf 13.17 8.09 Td[(1 Ni)]TJ /F4 11.955 Tf 11.95 0 Td[(qi,1rN,i+qi,3(rN,i+i))]TJ /F6 7.97 Tf 18.31 14.94 Td[(nXj=1qj,6pj,i NiN,ji(1)]TJ /F4 11.955 Tf 11.96 0 Td[(zj)q0i,4=)]TJ /F4 11.955 Tf 9.29 0 Td[(qi,1!i+qi,4(hV,i+!i+i))]TJ /F4 11.955 Tf 11.95 0 Td[(qi,5hV,iq0i,5=)]TJ /F8 11.955 Tf 13.17 8.09 Td[(1 Ni)]TJ /F4 11.955 Tf 11.95 0 Td[(qi,4rV,i+qi,5(rV,i+i))]TJ /F6 7.97 Tf 18.31 14.94 Td[(nXj=1qj,6pji NiV,ji(1)]TJ /F4 11.955 Tf 11.95 0 Td[(zj)q0i,6=nXj=1pij(qj,iijSN,j)]TJ /F4 11.955 Tf 11.95 0 Td[(qj,2ijSN,j+qj,4V,ijSV,j)]TJ /F4 11.955 Tf 11.95 0 Td[(qj,5V,ijSV,j)+qi,6(ki+gi) 116

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fori=1,...,n,andwithnaltimeconditionsqi,k(T)=0foralli=1,...,nandk=1,...,6.Asanexample,wecarefullyderivetheequationforq0i,2;thederivationsoftheremainingadjointequationsfollowsimilarly.q0i,2=)]TJ /F7 11.955 Tf 13.57 8.09 Td[(@H @xi,2=)]TJ /F7 11.955 Tf 13.5 8.09 Td[(@H @IN,i=)]TJ /F7 11.955 Tf 13.84 8.09 Td[(@F @IN,i)]TJ /F4 11.955 Tf 11.95 0 Td[(qi,2@fi,2 IN,i)]TJ /F4 11.955 Tf 11.95 0 Td[(qi,3@fi,3 IN,i)]TJ /F6 7.97 Tf 18.3 14.94 Td[(nXj=1qj,6@fj,6 @IN,i=)]TJ /F8 11.955 Tf 13.18 8.08 Td[(1 Ni+qi,2(i+i))]TJ /F4 11.955 Tf 11.95 0 Td[(qi,3i)]TJ /F6 7.97 Tf 18.31 14.94 Td[(nXj=1qj,6(1)]TJ /F4 11.955 Tf 11.95 0 Td[(zj)@kj @IN,iSince@kj @IN,i=pji Niji,wearriveattheequationforq0i,2:q0i,2=)]TJ /F8 11.955 Tf 13.18 8.09 Td[(1 Ni+qi,2(i+i))]TJ /F4 11.955 Tf 11.95 0 Td[(qi,3i)]TJ /F6 7.97 Tf 18.31 14.95 Td[(nXj=1qj,6pji Niji(1)]TJ /F4 11.955 Tf 11.95 0 Td[(zj).Theexpressionfortheoptimalcontrolisgivenmorepreciselybyvi(t)=8>>>><>>>>:0if)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(qi,1(t))]TJ /F4 11.955 Tf 11.95 0 Td[(qi,4(t))]TJ /F5 7.97 Tf 14.22 4.71 Td[(1 N<0vmax,iif)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(qi,1(t))]TJ /F4 11.955 Tf 11.95 0 Td[(qi,4(t))]TJ /F5 7.97 Tf 14.22 4.71 Td[(1 NSN,i(t) 2Wi>vmax,i)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(qi,1(t))]TJ /F4 11.955 Tf 11.95 0 Td[(qi,4(t))]TJ /F5 7.97 Tf 14.22 4.7 Td[(1 NSN,i(t) 2Wiotherwise (5) 5.5NumericalResultsToinvestigatetheroleofshort-termmovementinaspatiallyheterogeneoustransmissionlandscapeontheoptimalvaccinationstrategy,webeginbynumericallysolvingtheoptimalcontrolproblemfortwopatches. 5.5.1VaccinationModelParametersForthenumericalexplorationofthevaccinationmodelwithtwopatches,weassumeforsimplicitythatX,21=X,22andX,12=X,11,forX=N,V.Similarly,weassume12=11,21=22,X,12=X,11,andX,21=X,22forX=N,V.Inotherwords,the 117

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transmissionratesaredeterminedcompletelybywhereanindividualislocated,andnotwheretheyarefrom,whentheyareexposedtomalaria.Thereciprocalofviistheaveragetimeapatch-iresidentremainsinthesusceptible,non-immunestagebeforereceivingavaccine.StagetwotrialsoftheRTS,Svaccinesuggestedthatthevaccineprovidedimmunityforatleast18months[ 5 ].Thus,weassumethattherateatwhichvaccine-inducedimmunitywanesisgivenby!i=1=(1830)days)]TJ /F5 7.97 Tf 6.59 0 Td[(1.Furthermore,wethinkitunlikelythatindividualswillreceiveavaccinemorefrequentlythaneverytwoyearsgiventhatthevaccineshouldremaineffectiveforatleastoneandahalfyears.Consequently,wechosethemaximumvaccinationrateineachpatch(vmax,1andvmax,2)tobe1=(2365)days)]TJ /F5 7.97 Tf 6.59 0 Td[(1.Estimatesfortherateatwhichnaturallyacquiredimmunitywanesvaryintheliterature.WetaketherateatwhichthistemporaryimmunityislosttoberN,i=1=365days)]TJ /F5 7.97 Tf 6.58 0 Td[(1asin[ 26 ].Whilethemalariavaccinemayreducethedurationoftheinfectiousperiod(RV,i),weassumethatrV,i=rN,i.TransmissionrateestimatesforthevaccinationmodelwerederivedusingtheLow-WetparametersetforpatchoneandtheHigh-Dryparametersetforpatchtwo,presentedinTable 4-2 .Moreprecisely,wesetN,ii=miaibie)]TJ /F6 7.97 Tf 6.59 0 Td[(giniandii=aici,fori=1,2.Theseestimatesofthemosquito-to-humantransmissionratesyieldedprevalencescloseto100%inthevaccinationmodel;wesubsequentlyreducedthemosquitodensities10foldtoyieldmorereasonableestimatesofprevalence.Ourrstgoalofthisoptimalcontrolstudywastocomparetheeffectofdifferentvaccineefcacies,denotedbye,ontheoptimalvaccinationstrategy.Tothatend,weassumedV,ii=(1)]TJ /F4 11.955 Tf 12.86 0 Td[(e)N,ii,whereedenotesthevaccineefcacy.Thesporozoite-blockingvaccinemayreduceinfectivityinRV,iindividualstomosquitoes.Similarly,individualsinRN,iarelessinfectioustomosquitoes.WeletsdenotethereducedinfectivityoftheseindividualstomosquitoessothatV,i=N,i=sii.Forthepurposeofournumericalexploration,weassumedthatthesetemporarilyimmuneindividualshaveaninfectivity 118

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reducedbyhalfcomparedtonon-immune,infectedindividuals.Inotherwords,s=.5.Finally,wechoseiequaltoourestimateforriinTable 4-2 ,andthehumannaturalmortalityratetobe1=(65365)days)]TJ /F5 7.97 Tf 6.59 0 Td[(1.Recallthatourgoalistojointlyminimizetheproportionofthehumanpopulationinfected,thenumberofvaccinesdistributedoverTdays,andthesquareofthevaccinationrateineachpatch.Becausethesquareofthemaximumvaccinationrateisontheorderof10)]TJ /F5 7.97 Tf 6.59 0 Td[(6,wewillneedtochoosethevaccinationweightsWisothatprevalenceofmalariaandthesquareofthevaccinationrateareofthesameorderofmagnitude.Theparameterestimatesaboveyieldedprevalencesontheorderof10)]TJ /F5 7.97 Tf 6.59 0 Td[(2to10)]TJ /F5 7.97 Tf 6.59 0 Td[(1.Thus,wechooseWi=105,fori=1,2.ParametervaluesusedinournumericalsimulationsaresummarizedinTable 5-3 .Usingtheexpressionderivedfortheisolatedreproductionnumbers(Equation( 5 ))andourparameterestimatesyieldsR01=1.22andR02=0.92intheabsenceofvaccination.Ifweassumeaconstantvaccinationratevi(t)=vmax,i=1=(2365),thecontrolledisolatedreproductionnumbershavevaluesRC1=1.02andRC2=0.77.Theseestimatesimplythatahighervaccinationratemustbesustainedinordertoguaranteeeliminationofmalariaunderallmovementregimes. Table5-3. Descriptionofvaccinationmodelpatchiparametervaluesindays)]TJ /F5 7.97 Tf 6.58 0 Td[(1 ParameterEstimateParameterEstimateParameterEstimate N,110.0338110.02251=i1=150N,12N,11N,11s11i1=(65365)N,220.0120V,11s11rN,i1=365N,21N,221211rV,i1=365V,11(1)]TJ /F4 11.955 Tf 11.95 0 Td[(e)N,11N,12s12!i1=(1830)V,21(1)]TJ /F4 11.955 Tf 11.95 0 Td[(e)N,21V,12s12vi1=2365V,22(1)]TJ /F4 11.955 Tf 11.95 0 Td[(e)N,22220.0567V,12(1)]TJ /F4 11.955 Tf 11.95 0 Td[(e)N,12V,22s222122N,21s21V,21s21 119

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5.5.2NumericallySolvingtheOptimalControlProblemTonumericallysolvetheoptimalcontrolproblem,weusedtheForward-BackwardSweepmethoddescribedin[ 67 ].Thismethodinvolvesmakinganinitialguessforthecontrolfunction,usingtheinitialconditionvector~x0tosolvethestatevariablesysteminforwardtime,thenusingthesolutionstothestatevariablesystemandterminaltimeconditions~(T)=~0tosolvetheadjointsysteminbackwardtime.WesolvethesesystemsnumericallyusingtheFourthOrderRunga-Kuttamethod[ 10 ].Oncethestateandadjointsystemsaresolved,weinputthecorrespondingsolutionsintotheexpressionfortheoptimalcontrol~v(t)(equation( 5 ))derivedintheprevioussection.Weupdatetheestimateoftheoptimalcontrolbysettingthecontrolfunctionequaltoaconvexcombinationoftheoldcontrolestimateandtheestimateobtainedfromtheexpressionfor~v(t).Wecalculatetheerrorbetweenthenewandoldestimatesofthecontrol,aswellasforthenewandoldsolutionstothestateandadjointsystems.Thisprocedureisrepeateduntilthealgorithmconverges;theconvergencecriterionfortheforward-backwardsweepalgorithmis:Xjvi(t)j)]TJ /F13 11.955 Tf 17.94 11.35 Td[(Xjvi(t))]TJ /F4 11.955 Tf 11.95 0 Td[(vi,old(t)j0,wherewetake=0.001. 5.5.2.1OptimalstrategyinamalariaendemicsettingTheinitialconditionvector~x0issetequaltotheequilibriumvaluesof~x(t)whenthereisnovaccinationprogram,i.e.~x0=limt!1~x(t),sothatwemaystudythequalitativeeffectoftheoptimalvaccinationstrategyontheprevalenceofmalariainthistwo-patchsystemwithdifferentvaccineefcaciesanddifferentshort-termmovementpatterns.Wenumericallysolvedtheoptimalcontrolproblemforvaccineefcaciese=0.3,0.5,0.8,1andfortheshort-termmovementparameterpairs(p11,p22)=(1,1),(1,0.5),(0.5,1),(0.5,0.5),(0.8,0.3),(0.3,0.8),overaspanofT=10years.Notethatchoosingvaluesforp11andp22automaticallydeterminesthevaluesofthe 120

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remainingpji'sinthetwo-patchcase:p21=1)]TJ /F4 11.955 Tf 12.08 0 Td[(p11andp12=1)]TJ /F4 11.955 Tf 12.08 0 Td[(p22.Also,becausewehavechosentransmissionparameterstodependsolelyonthelocationofanindividualandnottheirpatchresidence,whenp11+p22=1,thetwo-patchsystemessentiallybecomesahomogeneoussystem,withidenticaldynamicsineachpatch.Thus,inthe(p11,p22)=(0.5,0.5)case,weexpecttheoptimalstrategytobeidenticalinbothpatches(v1(t)=v2(t)8t2[0,T]).Theresultsforvaccineefcacye=0.5underthemovementpatterns(p11,p22)=(1,0.5),(0.5,1)areillustratedinFigure 5-2 ;theresultsformovementpatterns(p11,p22)=(0.5,0.5),(0.8,0.3),(0.3,0.8)arepresentedinFigure 5-3 .Inthegures,eachrowcorrespondstoadifferentmovementpattern.Theoptimalcontrolstrategyforthatmovementpatternisshownontheleft,andthecorrespondingnumberofinfectedindividualsineachpatchisshownontheright.Theshadedregionillustratesthediseasedynamicsafterthe10-yearvaccinationpolicyhasbeenlifted,allowingustoassessthelongtermbenetsoftheoptimalstrategy.Inreality,itisunlikelythatmalariavaccination,onceimplemented,willbelifted.However,asconditionsimproveinapopulation,itispossiblethatindividualswillbecomelesslikelytoseekvaccination.Applyinganoptimal10-yearvaccinationpolicyandstudyingthelongtermdynamicsrepresentsaworst-casescenario.When(p11,p22)=(1,1),inotherwordswhenindividualsneverleavetheirresidencepatch,malariaisendemicinpatch1,butextinctinpatch2.However,allowingshorttermmovementinatleastoneofthepatchesledtopersistenceofthediseaseinbothpatches.Theoptimalstrategyinallscenarioswasoneinwhichweshouldvaccinateatthemaximumrateforatleastveyears,followedbyasharpdeclineinthevaccinationrate.Thenumericalresultsofouroptimalcontrolproblemcoincide,atleastqualitatively,withourintuitionforhowvaccinesshouldbedistributedovertimebetweentworegions.Theoptimalpolicysuggestssustainingthemaximumvaccinationratelongerinthepatchwithhighermalariatransmission.Figures 5-2C and 5-2D illustratethatevenwhenashort-termmovementregimeleadstohigherprevalenceinthelowtransmissionpatch 121

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(patch2),theoptimalstrategyisoneinwhichwesustainthemaximumvaccinationratelongerinthepatchwithintrinsicallyhighertransmissioncharacteristics(patch1).Consequently,thisresultsuggestswecannotrelysolelyoncurrentpatchprevalencestoguidedecisionsregardingwheretoplacemoreresources.However,Figures 5-2 and 5-3 dosuggestthatweshouldsustainvaccinationeffortslongerundermovementregimesthatleadtohigheroverallprevalence.Vaccinationeffortsshouldbesustainedlongestunderthemovementregimes(p11,p22)=(1,0.5)and(0.8,0.3),whichleadtothehighestoverallprevalences.Furthermore,whetherweshouldsustainthemaximumvaccinationratemuchlongerinthehightransmissionpatchthaninthelowtransmissionpatchappearstobedeterminedbytherelativedifferencebetweenthetwopatchprevalences.Inotherwords,ifamovementregimeleadstoamuchhigherprevalenceinpatch1thaninpatch2intheabsenceofavaccinationprogram,thedifferenceinthetimeatwhichweshouldstopvaccinatingatthemaximumrateineachpatchisgreaterthanifamovementregimeleadstomoresimilarprevalencesineachpatch.Whenshort-termmovementleadstoverysimilarprevalencesineachpatch,theoptimalstrategiesineachpatcharenearlyidentical,requiringthemaximumvaccinationratetobesustainedonlyslightlylongerinthepatchwithhighertransmission.Comparingtheoptimalsolutionsfordifferentvaccineefcaciesrevealedthathighervaccineefcaciesyieldedanoptimalvaccinationstrategyinwhichthevaccinationratedeclinesearlierduringthe10yearperiod,whilestillleadingtoasmallernumberofinfectedindividualsattheendofthevaccinationperiod,comparedwithlowervaccineefcacies.IncreasingthevaccinationweightsWidecreasedtheamountoftimeweshouldsustainthemaximumvaccinationrate.NoneoftheoptimalstrategiespresentedinFigures 5-2 and 5-3 successfullyeliminatemalaria.Vaccinationappearstohaveonlyshort-termbenetsundermovementregimesresultinginhighprevalencesintheabsenceofvaccination.Althoughvaccinatingatthemaximumratecansubstantiallydecreasethenumberofcasesinmidtohighendemicityscenarios,oncethiscontrol 122

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policyisrelaxed,thenumberofcasesbeginsincreasing,rapidlyreturningtothepre-vaccinationprogramprevalence(seeFigures 5-2B and 5-3D ). Ap11=1,p22=.5 Bp11=1,p22=.5 Cp11=.5,p22=1 Dp11=.5,p22=1Figure5-2. Eachrowillustratestheoptimalstrategyinamalariaendemicregion(leftcolumn),andcorrespondingnumberofinfectedindividuals(rightcolumn)fordifferentshort-termmovementpatterns.Vaccineefcacyise=0.5.Thegreyshadedareaillustratesthediseasedynamicsfollowingthe10-yearcontrolperiod. 5.5.2.2Optimalstrategyinanon-endemicsettingThenumericalexplorationdescribedintheprevioussectionallowedustogainsomeunderstandingaboutwhatvaccinationstrategytoemployinaheterogeneous,malariaendemicenvironment,andtheoutcomeofthosestrategies.Whilereducingtheburdenofmalariainendemicregionsisanimportantgoal,theriskofepidemicsinnon-endemicsettingsisalsoasubstantialpublichealthconcern.Repeatingour 123

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Ap11=.5,p22=.5 Bp11=.5,p22=.5 Cp11=.8,p22=.3 Dp11=.8,p22=.2 Ep11=.3,p22=.8 Fp11=.3,p22=.8Figure5-3. Optimalvaccinationstrategiesforvaccineefcacye=0.5areshowninleftcolumn.Intherightcolumn,thenumberofinfectedindividualsinpatch1areshowninblue,thenumberofinfectedindividualsinpatch2areshowningreen.Dashedlinesrepresentthepatchprevalencewithnovaccinationprogram;solidlinesrepresentthenumberofinfectedindividualsundertheoptimalstrategy. 124

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numericalanalysesatthebeginningofamalariaepidemic,ratherthanwhenthediseasehasreachedequilibrium,revealedthatanoptimalvaccinationstrategycandelaytheepidemic,orpreventtheepidemicfromtakingoff(seeFigures 5-4 and 5-5 ).Inthemovementregime(p11,p22)=(1,0.5)(Figure 5-5F ),theoptimalvaccinationstrategyslowsthegrowthofinfectedindividualsineachpatchandthediseasebeginstoapproachanew,smaller,equilibriumvalue.However,oncethecontrolmeasuresarerelaxed,thenumberofcasesincreasesagain,eventuallyapproachingthesameequilibriumnumberofcasesobservedwhennovaccinationpolicyisimplemented.Undertheshort-termmovementregime(p11,p22)=(0.8,0.3)(Figure 5-5D ),theoptimalvaccinationstrategymaintainsthenumberofmalariainfectionsatafairlylowlevelthroughoutthecontrolperiod,butfailstoeliminatemalariaduringthat10-yearinterval.Thisfailuretoeliminatemalariaduringthecontrolperiodpermitsthenumberofmalariainfectionstogrowoncethecontrolpolicyislifted.Thus,inthiscase,theoptimalpolicysucceedsonlyindelayingthemalariaepidemic.Outofthevemovementpatternspresented,wearesuccessfulinpreventingamalariaepidemicwitha50%effectivevaccineunderthe(p11,p22)=(0.5,1)(Figure 5-4D ),(p11,p22)=(0.5,0.5)(Figure 5-5B ),and(p11,p22)=(0.3,0.8)(Figure 5-5F )movementregimes.Thesemovementregimesnaturallyyieldedthelowestprevalenceswithoutanyvaccinationprogram.Theoptimalstrategiesderivednumericallyforthenon-endemicsettingsarequalitativelyverysimilartothoseintheendemicsettingspresentedinSection 5.5.2.1 .Weagainobservedthatvaccinationshouldbemaintainedatthemaximumrateforseveralyears,withvaccinationterminatingearlierinthepatchwithlowermalariatransmissionintensity.Theoptimalstrategiesineachpatchbecamemoresimilarundermovementregimesthatledtosimilarprevalencesintheabsenceofcontrol.However,itisimportanttonotethatinarealnon-endemicsetting,wewouldnotknowthewithout-vaccinationprevalenceapriori.Consequently,knowledgeaboutshort-termmovementbetweenpatchesandestimatesofwithin-patchtransmissionintensities 125

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isnecessaryinthenon-endemicsettingtodeterminewhethervaccinationstrategiesshouldbeessentiallythesameineachpatch,orwhethereffortsshouldbemaintainedlongerinthepatchwithhighertransmissionintensity. Ap11=1,p22=.5 Bp11=1,p22=.5 Cp11=.5,p22=1 Dp11=.5,p22=1Figure5-4. Eachrowillustratestheoptimalstrategyinanon-endemicregion(leftcolumn),andcorrespondingnumberofinfectedindividuals(rightcolumn)fordifferentshort-termmovementpatterns.Vaccineefcacyise=0.5.Thegreyshadedareaillustratesthediseasedynamicsfollowingthe10-yearcontrolperiod. 5.6FutureDirectionsResidentsofregionswithhighmalariaendemicitytendtobefrequentlyre-exposedtoinfectiousbites.Whilemalariaconfersonlytemporaryimmunity,thisrepeatedexposuretomalariaboostsanindividual'simmunitytothedisease.Themalariamodelwithvaccinationandshort-termhumanmovementpresentedinthischapterdoes 126

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Ap11=.5,p22=.5 Bp11=.5,p22=.5 Cp11=.8,p22=.3 Dp11=.8,p22=.3 Ep11=.3,p22=.8 Fp11=.3,p22=.8Figure5-5. Optimalvaccinationstrategiesforvaccineefcacye=0.5areshowninleftcolumn.Intherightcolumn,thenumberofinfectedindividualsinpatch1areshowninblue,thenumberofinfectedindividualsinpatch2areshowningreen.Dashedlinesrepresentthepatchprevalencewithnovaccinationprogram;solidlinesrepresentthenumberofinfectedindividualsundertheoptimalstrategy.Resultsareshownforveshort-termmovementpatterns. 127

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nottakeintoconsiderationthispossiblebuildupofnaturallyacquiredimmunityinthepopulation.Thenextstepinthestudyofoptimalvaccinationpoliciesinaheterogeneoustransmissionlandscapeistoadaptthemodelpresentedheretoincludethebuildupofnaturallyacquiredimmunityinthepopulation.Whiletheresultsofouroptimalcontrolproblemwiththecurrentmodelappearfairlyintuitive,theadditionofnaturallyacquiredimmunityintothemalariamodelwillforceustoweighthebenetsofnaturalimmunityagainstthebenetsofvaccine-acquiredimmunity.Infact,Aron[ 8 ]discoveredthatincorporatingboostedimmunitycanleadtomuchmorecomplicateddiseasedynamics,includinganon-monotonerelationshipbetweenprevalenceandtransmission.Withtheexcitingpossibilityofamalariavaccineavailableinthenearfuture,analyzingamodeladdressingbothvaccinationandnaturallyacquiredimmunitywillallowustostudythepotentiallyunexpectedconsequencesofdistributingimperfectvaccinesinregionswithnaturallyhighmalariaendemicity.Optimalcontroltheoryisoneapproachtounderstandinghowwemightavoidunexpectednegativeconsequencesofavaccinationpolicy,suchasreducingthelevelofnaturalimmunityinapopulationbelowalevelwhichmakesthepopulationathighriskforlargeoutbreaksofmalaria.Onceavailabletothepublic,thestockpileofmalariavaccinesforagivenregionwillbelimited.Inadditiontoincorporatingnaturallyacquiredimmunityintoourvaccinationmodel,wewilldevelopthetoolsnecessarytoconsidertheoptimalcontrolproblemunderthisresourceconstraint.Preliminaryresultsusingtheoremspresentedin[ 97 ]aboutconstrainedoptimizationproblemsindicatethatanoptimalsolutiontotheproblemundervaccineresourcelimitationsexists.Formulatingthenecessaryandsufcientconditionsfortheoptimalsolutionisamorechallengingproblemthanintheunconstrainedcasepresentedhere,butwilladdressaveryimportantissueformalariacontrol:howshouldalimitednumberofmalariavaccinesbedistributedinaspatiallyheterogeneousenvironment? 128

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BIOGRAPHICALSKETCH OliviaF.ProsperwasborninGrenoble,Francein1984toherparentsDr.HarrisonandDr.Marie-FranceProsper.ShemovedtoIllinoiswithherfamilyatagetwo.HeryoungersisterLeilaarrivedafewyearslater.In1993,theymadethemovetoTallahassee,Florida,whereOliviaeventuallyearnedherhighschooldiplomafromLincolnHighSchoolin2002.Tornbetweenstudyingpiano,writing,andscience,herloveofpuzzleseventuallymaterializedintoapassionformathematics.Thispassion,downtheroad,evolvedintoapassionforresearchintheareaofmathematicalbiology.OliviaearnedherBachelorofSciencedegreein2006,herMasterofSciencedegreein2008,bothinmathematics,andaminorinFrenchattheUniversityofFlorida.OliviareceivedherPh.D.inmathematicsfromtheUniversityofFloridaandlooksforwardtotakingthenextstepinhercareeratDartmouthCollege. 138