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Population Thresholds and Disease Ecology

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Title:
Population Thresholds and Disease Ecology
Creator:
Borchering, Rebecca K
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[Gainesville, Fla.]
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University of Florida
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Doctorate ( Ph.D.)
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University of Florida
Degree Disciplines:
Mathematics
Committee Chair:
MCKINLEY,SCOTT
Committee Co-Chair:
PULLIAM,JULIET
Committee Members:
PILYUGIN,SERGEI S
JURY,MICHAEL THOMAS
PONCIANO CASTELLANOS,JOSE MIGUEL

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Subjects / Keywords:
diffusion -- ecology -- elimination -- encounters -- extinction -- invasion -- stochastic
Mathematics -- Dissertations, Academic -- UF
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Mathematics thesis, Ph.D.

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Abstract:
Population dynamics begin with invasion and end with extinction. Understanding invasion and extinction events are thus key to understanding how populations function. These phenomena are challenging to study because they involve dynamics at two scales. Large population scale dynamics tend to look deterministic, while small population scale dynamics are inherently stochastic. In this dissertation, we draw from applications in disease ecology to understand how population size and the relationship between birth (infection) and death rates affect model suitability for a given system. It is often necessary to use approximation methods to gain insight in stochastic settings, but as we will show, these approximations can vary substantially in their ability to accurately represent the original stochastic process. Invasion dynamics first arise in Chapter 2, where we consider whether changes in carcass density can indirectly facilitate the spread of rabies in a jackal population. In Etosha National Park, rabies is introduced to the local jackal population from spillover infections originating from individuals outside of the park. We propose and analyze a consumer encounter model at resource sites to study how jackal-jackal encounter rates might respond to increases in carcass abundance during anthrax outbreaks and ask whether these changes would potentially allow rabies to spread in the jackal population. In Chapter 3, we further investigate invasion dynamics by calculating the probability of invasion for a class of birth and death processes. We calculate an exact solution and then compare this value to results from several popular approximation methods. We find that population size and the relationship between the birth and death rate can both substantially influence which method best approximates the exact solution. In Chapter 4, we study pathogen elimination dynamics when vaccination is used as a control. The duration of the vaccination campaign is based on an imperfect case detection process. We investigate the effect that this could have on the success of a vaccination campaign and thus on an elimination effort. ( en )
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In the series University of Florida Digital Collections.
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Includes vita.
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Includes bibliographical references.
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Description based on online resource; title from PDF title page.
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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.
Thesis:
Thesis (Ph.D.)--University of Florida, 2017.
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Adviser: MCKINLEY,SCOTT.
Local:
Co-adviser: PULLIAM,JULIET.
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RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2017-11-30
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by Rebecca K Borchering.

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11/30/2017
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POPULATIONTHRESHOLDSANDDISEASEECOLOGY By REBECCAK.BORCHERING ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2017

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c 2017RebeccaK.Borchering

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ACKNOWLEDGMENTS First,IwouldliketothankmyadvisorsDr.ScottA.McKinleyandDr.JulietR.C.Pulliam. IcertainlywouldnotbetheresearcherIamtodaywithoutthem.Specically,Iwanttothank Scottforagreeingtobemyadvisor,despitehisinitialreservations.Iwouldalsoliketothank himforthetimeheinvestedinmyacademicdevelopmentandforhispersistententhusiasm. IwanttothankJulietforhermentorshipandforprovidinganabundanceofinvaluable opportunities.Ialsowanttothankherforintroducingmetotheprocessofworkingwithdata. IthankMargaretforprovidinglocalsupportwhenIneededitmost. Iwouldliketoacknowledgetheadministration,faculty,andsta!intheUniversityof FloridaMathematicsandBiologyDepartments.Iwouldliketothankmypastandpresent o "cematesandmyacademicsiblingsforpartakingingoodtimesandforprovidingperspective duringdi" culttimes.Ithankmyfamiliesfortheirendlessloveandencouragement.Lastly, IthankFrancisforthemuchneededsupportheprovidedthroughoutgraduateschool:from rst-yearalgebra,throughthenalstagesofdissertationsubmission,andeverywherein between. IamimmenselygratefulforfundingfromtheMathematicsDepartment,theNational ScienceFoundationunderGrant0801544intheQuantitativeSpatialEcology,Evolutionand EnvironmentProgramattheUniversityofFlorida,andtheCLASDissertationFellowship, fundedbytheKennethandJanetKeeneEndowedFellowship. Chapter 2 : IwouldliketothankWayneM.Getz,BenjaminM.Bolker,CraigW.Osenberg, andAndrewM.Heinfortheirsupportandthoughtfulconversationsinthedevelopmentofthis work.TheworkinthischapterwassupportedbytheInternationalClinicsonInfectious DiseaseDynamicsandData(ICI3D)program,withfundingfromtheSouthAfricanCentrefor EpidemiologicalModellingandAnalysis(SACEMA),theAfricanInstituteforMathematical Sciences(AIMS),andtheUSNationalInstitutesofHealth(NIGMSawardR25GM102149 toJRCPandAWelte).DatacollectionforthisworkwasconductedbytheBerkeleyEtosha AnthraxProjectwithsupportfromanNSF-NIHEcologyofInfectiousDiseaseGrant(award 3

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GM83863toWMGetz)andwithassistancefromtheEtoshaEcologicalInstitute,(Namibian MinistryofEnvironmentandTourism).AdditionalfundingwasprovidedfromtheCenterfor InferenceandDynamicsofInfectiousDiseases(MIDAS-NIGMS,awardU54GM111274toME Halloran).JRCPwassupportedbytheResearchandPolicyonInfectiousDiseaseDynamics (RAPIDD)ProgramoftheFogartyInternationalCenter,NationalInstitutesofHealthand ScienceandTechnologyDirectorate,DepartmentofHomelandSecurity.SEBwassupported byNSF-NIH(GM83863toWMG),MIDAS-NIGMS(awardU01GM087719toLAMeyersand APGalvani)andNIAID(K01AI125830toSEB).SAMandJMFwerepartiallysupportedby theArmyResearchO"ce(ARO64430-MA).RKBwaspartiallysupportedbytheNational ScienceFoundationunderGrant0801544intheQuantitativeSpatialEcology,Evolutionand EnvironmentProgramattheUniversityofFlorida,andtheCLASDissertationFellowship, fundedbytheKennethandJanetKeeneEndowedFellowship. Chapter 3 : IwouldliketothankJayM.Newbyforthoughtfulconversationsinthe developmentofthiswork.FundingStatement:SAMwaspartiallysupportedbytheArmy ResearchO"ce(ARO64430-MA).RKBwaspartiallysupportedbytheNationalScience FoundationunderGrant0801544intheQuantitativeSpatialEcology,Evolutionand EnvironmentProgramattheUniversityofFlorida,andtheCLASDissertationFellowship, fundedbytheKennethandJanetKeeneEndowedFellowship.Additionalfundingwasprovided fromtheCenterforInferenceandDynamicsofInfectiousDiseases(MIDAS-NIGMS,award U54GM111274toMEHalloran). 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ................................... 3 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 10 CHAPTER 1INTRODUCTION ................................... 12 2ENCOUNTERRATESANDDISEASEECOLOGY .................. 16 2.1Introduction ................................... 16 2.2Modeldevelopmentandpreliminaryanalysis .................. 19 2.2.1Resource-drivenencounters ....................... 19 2.2.2Introductionofpathogenandoutbreakofdisease. ........... 21 2.2.3Datacollectionandanalysis ....................... 24 2.3Results ...................................... 25 2.3.1Analysisoftheconsumer-resourcemodel ................ 26 2.3.2Therelationshipbetweenresourcedensityandsitevisitation ...... 28 2.3.3Therelationshipbetweendefendableterritorysizeandthedistanceof detectionandresponse .......................... 29 2.3.4PlacingmodelresultsinthecontextofDiseaseEcology ........ 31 2.4Discussion .................................... 33 2.4.1Opportunitiesforintegratingmoredetailedanimalbehavior ...... 34 2.4.2Usingseasonalityasatoolforinvestigation ............... 37 2.5MathematicalAnalysis .............................. 38 2.5.1Smallresourcedensityand/orsmalldetectiondistance ......... 40 2.5.2Analysisinthehighresourceandlargedistanceofdetectionregimes .. 42 2.5.3Convertingresultsfornon-unitconsumerdensity ............ 44 2.5.4Branchingprocessapproximation .................... 45 3INVASIONPROBABILITIES ............................. 53 3.1Introduction ................................... 53 3.1.1MathematicalFramework ........................ 56 3.1.2ApproximationsforInvasionProbabilities ................ 59 3.2Exactsolutionforinvasionprobabilities ..................... 62 3.3Di! usionapproximation ............................. 64 3.3.1MotivationsfortheDi!usionApproximation ............... 64 3.3.2Analysisof p di! usion ( k ) .......................... 67 3.4Exponentialapproximation ............................ 76 3.4.1MotivationfortheExponentialApproximation .............. 76 5

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3.4.2Analysisof p exp ( k ) ............................ 79 3.5Numericalobservationsforinvasionprobabilityapproximations ......... 82 3.5.1Di! usionapproximationmethodsfailforlargepopulationswhendynamics aresupercritical .............................. 82 3.5.2Di! usionapproximationmethodscanworkwellforsmallpopulations thatexhibitnearcriticaldynamics .................... 83 3.5.3Whenleadingordertermsmatch,higherordertermsmatter:forsmall, butnotlargepopulationsizes ...................... 84 3.5.4Approximationsuccessdependsontheinitialnumberofindividualsintroduced inthepopulation ............................. 85 3.6Discussion .................................... 86 3.7SeriesFormulas ................................. 88 4ELIMINATIONPROBABILITIES ........................... 96 4.1Context ...................................... 96 4.1.1RabiesvaccinationofwildlifeintheeasternUnitedStates ....... 97 4.1.2Pathogeneliminationsettingsandmodeling ............... 98 4.1.3QuestionsandAims ........................... 99 4.2Stochasticmodeldevelopment .......................... 100 4.2.1Traditionalobservationmodel:timesinceobservation ......... 101 4.2.2Unconventionalobservationmodel:Markovmodelfor"forgetting"observed cases ................................... 101 4.2.3Fundamentalmathematicalquestion ................... 102 4.3Deterministicframeworkformodelingeliminationandre-invasionphenomena. 104 4.3.1ODEmodelforSISdiseasedynamics .................. 105 4.3.22DODEmodelanalogueto2DCTMCmodel .............. 105 4.3.3InterpretingtheODEmodelforre-invasionprobabilities. ........ 111 4.4Stochasticanalysisoftheapparenteliminationprobability. ........... 112 4.5Numericalinvestigationoftheprobabilityofre-invasion. ............ 121 4.5.1Thetwo-dimensionalmodelthat"forgets"observedindividualsbehaves wellcomparedtothemodelbasedonobservationtimes. ........ 121 4.5.2Thedeterministicpredictionforthere-invasionprobabilitydoesnot performwell. ............................... 123 4.5.3Usingourapproximationfortheapparenteliminationprobability. .... 124 LISTOFREFERENCES ................................... 133 BIOGRAPHICALSKETCH ................................. 139 6

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LISTOFTABLES Table page 2-1Parametersusedinthediseaseecologyanalysis .................... 38 3-1Summaryoftheasymptoticinvasionprobabilitiesforlargepopulations ....... 61 4-1Parametersusedintheeliminationprobabilityanalysis ................ 109 7

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LISTOFFIGURES Figure page 2-1CollaredjackalmovementandknowncarcasslocationsinEtoshaNationalPark ... 47 2-2Voronoidiagramsdisplayingtheregionsof"attraction"foreachresource ...... 48 2-3Invasioneventsresultingfromspilloverinfections ................... 48 2-4Simulationsandasymptoticpredictionsofresource-drivenencounterratesbasedon resourcedensityanddetectiondistance ........................ 49 2-5Detectiondistancee! ectandpeaknumbersofresource-drivenencounters ..... 49 2-6Monthlycarcassavailabilityandjackalvisitationtocarcasssites ........... 50 2-7Jackalvisitationbasedonaveragedistancefromcarcasssite ............. 51 2-8Simulatednumberofresource-drivenencountersfortwochoicesofthethedetection distanceparameteroverarangeofresourceintensities ................ 51 2-9Time-dependentreproductiveratiobasedonthecorrespondingnumberofresource-driven encountersforeachmonth .............................. 52 3-1Numberofinfectiousindividualsresultingfromtheintroductionofoneinfectious individual ....................................... 91 3-2Comparisonoftheinvasionprobabilitycontinuumapproximationresultstotheexact solutionforExamples1-3 ............................... 92 3-3Comparisonoftheinvasionprobabilitycontinuumapproximationstotheexactsolution forrateequationswithmismatchedleadingorders .................. 93 3-4Comparisonoftheinvasionprobabilitycontinuumapproximationstotheexactsolution forrateequationswithidenticalleadingorderterms ................. 94 3-5Comparisonoftheinvasionprobabilitycontinuumapproximationstotheexactsolution fordi! erentnumbersofinitiallyintroducedindividuals ................ 95 4-1Diagramoftheframeworkunderlyingthetrueeliminationvs.re-invasionproblem 125 4-2Solutionstothedeterministicsystemintheeliminationframework ......... 126 4-3Theproportionofthepopulationremainingintheunobservedinfectiousclassat theendofthevaccinationcampaignfortheODEmodel ............... 127 4-4Phaseplotsofstochasticsimulationsfortheeliminationprobabilityhittingboundary problem ........................................ 128 4-5Comparisonofthedistributionofthenumberofremaininginfectiousindividualsat theendofthevaccinationcampaignforthestochasticmodels( N =1000 ) ..... 129 8

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4-6Comparisonofthedistributionofthenumberofremaininginfectiousindividualsat theendofthevaccinationcampaignforthestochasticmodels( N =100 ) ..... 129 4-7Comparisonoftheprobabilityofre-invasionforthetwostochasticmodelsandthe deterministicapproximation .............................. 130 4-8MeansquareerrorbetweentheODEapproximationforthere-invasionprobability andresultsfromthetwostochasticmodels ...................... 131 4-9CharacteristiccurvesforthesolutiontothePDEcharacterizingtheprobabilityof apparentelimination .................................. 132 4-10Durationofmemoryvalueswhosecharacteristiccurvespassthroughthecorresponding initialcondition .................................... 132 9

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy POPULATIONTHRESHOLDSANDDISEASEECOLOGY By RebeccaK.Borchering May2017 Chair:ScottA.McKinley Cochair:JulietR.C.Pulliam Major:Mathematics Populationdynamicsbeginwithinvasionandendwithextinction.Understanding invasionandextinctioneventsarethuskeytounderstandinghowpopulationsfunction. Thesephenomenaarechallengingtostudybecausetheyinvolvedynamicsattwoscales.Large populationscaledynamicstendtolookdeterministic,whilesmallpopulationscaledynamics areinherentlystochastic.Inthisdissertation,wedrawfromapplicationsindiseaseecology tounderstandhowpopulationsizeandtherelationshipbetweenbirth(infection)anddeath ratesa!ectmodelsuitabilityforagivensystem.Itisoftennecessarytouseapproximation methodstogaininsightinstochasticsettings,butaswewillshow,theseapproximationscan varysubstantiallyintheirabilitytoaccuratelyrepresenttheoriginalstochasticprocess. InvasiondynamicsrstariseinChapter2,whereweconsiderwhetherchangesincarcass densitycanindirectlyfacilitatethespreadofrabiesinajackalpopulation.InEtoshaNational Park,rabiesisintroducedtothelocaljackalpopulationfromspilloverinfectionsoriginating fromindividualsoutsideofthepark.Weproposeandanalyzeaconsumerencountermodel atresourcesitestostudyhowjackal-jackalencounterratesmightrespondtoincreasesin carcassabundanceduringanthraxoutbreaksandaskwhetherthesechangeswouldpotentially allowrabiestospreadinthejackalpopulation.InChapter3,wefurtherinvestigateinvasion dynamicsbycalculatingtheprobabilityofinvasionforaclassofbirthanddeathprocesses. Wecalculateanexactsolutionandthencomparethisvaluetoresultsfromseveralpopular approximationmethods.Wendthatpopulationsizeandtherelationshipbetweenthebirth 10

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anddeathratecanbothsubstantiallyinuencewhichmethodbestapproximatestheexact solution.InChapter4,westudypathogeneliminationdynamicswhenvaccinationisusedas acontrol.Thedurationofthevaccinationcampaignisbasedonanimperfectcasedetection process.Weinvestigatethee!ectthatthiscouldhaveonthesuccessofavaccination campaignandthusonaneliminatione! ort. 11

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CHAPTER1 INTRODUCTION Inthisdissertation,weinvestigatepopulationmodelsandthethresholdsthatcharacterize theirdynamics.Ourprimaryfocusisonapplicationsfromdiseaseecology.Weseektonda balancebetweenusingsimpletheoreticalmodelsthatareconducivetomathematicalanalysis andmodelsthataredetailedenoughtocharacterizeparticularbiologicalsystems.Simulations areusedtoverifyanalyticalresultsandenhancetheutilityofthetheoreticalmodels. Modelsforsmallandlargepopulationdynamics: Oneoftherstconsiderationsin choosinganappropriatemathematicalmodelforstudyingapopulationshouldbethesizeof thepopulationofinterest.Ordinarydi! erentialequation(ODE)modelscanbeusedtostudy populationdynamicswhenthepopulationsizeislarge.Ingeneral,stochasticmodelsshould beusedtomodelsmallpopulationssincethee!ectsofrandomuctuationsarenotaveraged outwhenthereisonlyasmallnumberofindividuals.Stochasticmodelscancaptureextinction eventsthatoccurinnitetime,whileODEmodelscanonlyapproachzerowhenevertheir initialstateispositive. WeconsidercontinuoustimeMarkovchain(CTMC)modelstobethe"true"models forpopulationdynamics.Thephenomenaofinvasionandeliminationthatweinvestigate involvesmallpopulations.FrequentlywewilluseaSusceptible-Infectious-Susceptible(SIS) modelforinfectiousdiseasedynamics.Inthismodel,thepopulationsizeisxedandthusthe systemisfullycharacterizedbythenumberofinfectiousindividualsattime t .Forthepurpose ofillustration,wewillassumethattransmissionratesdonotdependonthedensityofthe population.AnalogousCTMCandODEmodelsforSISdynamicsarepresentedbelow.Both typesofmodelsassumethatthetimesbetweeneventsareexponentiallydistributed. Forapopulationsize N ,andconstants b > 0 and d > 0 ,let X N ( t ) bethesizeofthe infectiouspopulationinaCTMCmodel.Fromthestate X N ( t )= n ,theSISsystemwithout 12

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densitydependenceisdenedbythefollowingtransitioneventsandrates infections: n n +1 atrate bn ( N # n ) N deaths: n n # 1 atrate dn Forasu"cientlylargepopulation,theanalogousdeterministicmodelfortheproportion ofthepopulationthatisinfectious, x ( t ) ,withinfectionanddeathrateparameters b > 0 and d > 0 respectivelysatises x = bx (1 # x ) # dx When b > d ,thisODEadmitsanendemicequilibrium(i.e.apositivexedpointsatisfying x =0 )at x =1 # d / b .Thereisalwaysatrivialequilibriumat x =0 Thresholddynamics: Whenaparametervariesinasystem,weoftenaskifthereapoint atwhichchangingtheparametervalueresultsinaregimeshift.Theparameteratwhichsuch asshiftoccursisconsideredathreshold.Ininfectiousdiseasemodeling,thecommonthreshold parameterofstudyisthebasicreproductionnumber.Thebasicreproductionnumber,often denoted R 0 ,isdenedtobetheaveragenumberofsecondaryinfectionsgeneratedbya singleinfectiousindividualduringitslifetimeinanotherwisefullysusceptiblepopulation.When R 0 < 1 ,infectiousindividualsareonaveragenotbeingreplacedwithnewinfectiousindividuals beforetheydie.Inthiscase,thesystemisconsideredsubcriticalanditisunlikelythatlarge outbreakswilloccur.Alternatively,when R 0 > 1 ,infectiousindividualsareonaveragebeing replacedwithatleastasmanynewinfectiousindividuals.Inthiscase,thesystemisconsidered supercriticalandlargeoutbreaksarepossible.Forpositiveinitialconditions,whenanendemic equilibriumexists,ODEmodelsofsupercriticalsystemsalwayspredictthatthestateofthe systemwillapproachtheendemicequilibrium.CTMCmodels,ontheotherhand,alwaysallow forthechancethatthepopulationwillgoextinctbeforeestablishmentoccurs. InChapter 2 ,weinvestigateanapplicationfromdiseaseecologythatconsidershow conspecicjackal-jackalencounterratesrespondtochangesincarcassabundanceandwhether thesee! ectscouldmakethejackalpopulationvulnerabletorabiesoutbreaks.APoisson 13

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arrivalprocessisusedtomodelintroductionsofrabiesfromindividualslivingoutsideofEtosha NationalPark(ENP).Abranchingprocessisusedtocharacterizewhetherrabiesisthenable tospreadintheENPjackalpopulation.AspatialPoissonprocessisusedtodeterminethe expectednumberofencountersthatjackalswouldexperienceatcarcasssitesbasedonthe carcassdensity.Theimportantthresholdsthatcharacterizethedynamicsofthissystemarethe basicreproductionnumberthatdetermineswhetherrabiescaninvadethejackalpopulationand theresourcedensitythatseparatestheregimeswhereincreasingcarcassabundanceincreases versusdecreasestheexpectednumberofjackalencountersatagivencarcasssite. InChapter 3 ,wedelvedeeperintocharacterizingtheconditionsforpathogeninvasion thataroseinChapter 2 .Weconsidertheprobabilityofinvasioninthebroadercontextofbirth anddeathprocesses.Thisprimarilytheoreticalchapteraddressesthequestionofhowshould populationsizeandstateofcriticality(i.e.whethertheexhibiteddynamicsaresubcritical orsupercritical)a!ectapractitioner'schoiceofapproximationmethod.Wendanexact solutionfortheprobabilityofinvasionforaCTMCmodelandthencomparetheresultsof severalapproximationstothissolution:includingapproximationbyabranchingprocessand byastochasticdi!erentialequation(SDE).Denitionsandmotivationfortheapproximation methodsarepresentedinChapter 3 .Westudythedynamicsoflargepopulations(astheir sizetendstoinnity)analyticallyandstudythedynamicsofnitepopulationsnumerically.In thischapter,thekeythresholdisbetweenapopulationexhibitingsubcriticalorsupercritical dynamicsandisdeterminedbytherelationshipbetweenthebirthanddeathratesofthe population. InChapter 4 ,weinvestigateextinctionphenomenainthecontextofpathogenelimination whenashort-termvaccinationcampaignisimplementedasacontrolmeasure.Weaskthe questionofwhetherornotitislikelythateliminationwillbeachievedwhenthedecisionto endavaccinationcampaignisbasedonthenumberofobservedinfectiousindividualsrather thanthe(inpractice)unknowabletruenumberofinfectiousindividuals.Thisquestionis addressedbysolvingaboundaryhittingproblemforanassociatedSDE.Weapplytheresults 14

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ofChapter 3 toapproximatetheprobabilityofre-invasionwhenthereareremainingunobserved infectiousindividualsattheendofavaccinationcampaign.Thischapterisanattemptat movingtowardsaplacewhereanalyticalmathematicalresultsforsimplemodelscanbeusedto informpolicydecisionsandultimatelydecreaseourdependenceonmassivesimulationstudies topredictthee! ectsofcontrolmeasures.Wealsodevelopadeterministicframeworkfor approximatingtheprobabilityofre-invasionandevaluateitsperformanceagainstsimulations fromtwostochasticmodels.Inthischapter,thethresholdcomponentofourinvestigation centersaroundthebasicreproductionnumber.Oneofthedeningcharacteristicsofthe proposedmodelisthattheimplementationofvaccinationtemporarilyaltersthestateofthe diseasesystemfromsupercritical( R 0 > 1 )tosubcritical( R 0 < 1 ).Whenvaccinationends, thepathogenhastheopportunitytoreboundifthereareremaininginfectiousindividualssince thesystemisrestoredtoitssupercriticalstate. 15

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CHAPTER2 ENCOUNTERRATESANDDISEASEECOLOGY AbstractforSubmittedManuscript2016. 1 Territorialanimalsshareavarietyofcommon resources,whichcanbeamajordriverofconspecicencounterrates.Weexaminehow changesinresourceavailabilityinuencetherateofencountersamongindividualsina consumerpopulationbyimplementingaspatiallyexplicitmodelforresourcevisitationbehavior byconsumers.Usingdatafrom2009and2010inEtoshaNationalPark,weverifyourmodel's predictionthatthereisasaturatione!ectintheexpectednumberofjackalsthatvisita givencarcasssiteascarcassesbecomeabundant.However,thisdoesnotdirectlyimplythat theoverallresource-drivenencounterrateamongjackalsdecreases.Thisisbecausethe increaseinavailablecarcassesisaccompaniedbyanincreaseinthenumberofjackalsthat detectandpotentiallyvisitcarcasses.Usingsimulationsandmathematicalanalysisofour consumer-resourceinteractionmodel,wecharacterizekeyfeaturesoftherelationshipbetween resource-drivenencounterrateandmodelparameters.Theseresultsareusedtoinvestigatea standinghypothesisthattheoutbreakofafataldiseaseamongzebrascanpotentiallyleadto anoutbreakofanentirelydi! erentdiseaseinthejackalpopulation,aprocesswerefertoas indirectinductionofdisease. 2.1Introduction Duetotherapidgrowthinhigh-resolutionanimalmovementdata,thereisagrowing recognitionthatclassicalmodelsforencounterratesamonganimalsshouldberevisited [ 32 41 ].Somerecentprogressonthispointhasbeenmadeinforagingtheory.Theoreticians haveshownthatpredictionsforsearche" ciency,anditsdependenceonpreydensity,canbe substantiallydi! erentwhenmovementmodelsincludeintermittentlong-rangerelocationevents [ 74 ]orlocalsensinganddecision-making[ 37 ].Theresultingnonlineardependenceonprey 1 Authorlist:RebeccaK.Borchering,SteveE.Bellan,JasonM.Flynn,JulietR.C.Pulliam, ScottA.McKinley. 16

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densityyieldsnovelformsforfunctionalresponseinpredator-preysystems,andraisesquestions aboutwhatothertheoreticalframeworkswouldbenetfromtheinclusionofmorerealistic animalbehavior. Understandingtheimpactofsuddenenvironmentalchangesonanimalbehaviorisa particularlycompellingapplication.Intentionallyorunintentionally,humansfrequentlyalterthe availabilityofresourcesforconsumerspecies,invariablyleadingtounintendedconsequences. AsreviewedbyOroetal.[ 56 ],considerableworkhasbeendevotedtoidentifyinginstances ofanthropogenicresourceprovisioning.Otherworkhasexaminedtheimpactofnaturally occurringresourcesubsidiesthatoccuraspulsesineitherspaceortimeonconsumer populationdynamics,behaviorandcommunitystructure(Rose&Polis[ 61 ],Andersonet al.[ 5 ],Clotfelteretal.[ 18 ],Holt[ 39 ],Ostfeld&Keesing[ 57 ],Yangetal.[ 73 ]).Asmallerbut substantialbodyofwork(reviewedbyBeckeretal.[ 9 ]andSorensenetal.[ 64 ])specically considersthee! ectsthatsuchprovisioningcanhaveforinfectiousdiseasesofconsumers. Inordertodevelopapredictiveframeworkfortheimpactofresourcesupplementationon thespreadofdiseaseamongconsumers,onemustestablisharelationshipbetweenresource densityandarateofconspecicconsumerencounters.Thisisespeciallytruewhenconsidering directlytransmittedpathogenssuchasrabiesvirusorcaninedistempervirus.Beckeretal.[ 9 ] recentlyconsideredthisrelationshipexplicitlyintheirreviewonthelinkbetweenanthropogenic resourcesandwildlife-pathogendynamics.Motivatedbytheprevailingtrendrevealedbytheir meta-analysis,theyintroducedtheassumptionthatincreasedresourceavailabilitywillleadto increasedconsumeraggregation,whichinturnimpliesincreasedinfectionriskwhenapathogen ispresent.Incontrast,wehighlightthattherelationshipbetweenresourceavailabilityand consumerencountersneednotbestrictlyincreasing.Atsomelevelofresourceavailability, consumersmaynolongerneedtoshareresourcesites,meaningthatconsumerencounters mightactuallydecrease.Thequestionthenconcernswhetherthereisacriticalresourcedensity abovewhichconsumeraggregationnolongerincreases,andhowthatdensitywoulddependon parametersthatcanbeinferredfromdata. 17

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Withtheforegoingdiscussionasmotivation,inthisworkweinvestigatetheroleresources playinhelpingtomaintainpathogentransmissionorfacilitatediseaseemergence.Specically, wequantifytherelationshipbetweenincreasesinresourceavailabilityandtheconsequent changesinconspecicencounterratesamongconsumers.Furthermore,weprovidecontext fromdiseaseecologytoaddressthequestion"Howbigisbig?"whenitcomestoencounter ratechanges.Tothisend,weconsidertherelationshipbetweenthepotentialfordisease maintenanceamongapopulationofjackalsandtheannualoccurrenceofanthraxoutbreaks amonglocalherbivoresinEtoshaNationalPark(ENP)inNamibia[ 10 49 69 ].Thissurgein availablecarcassesservesasasupplementalresourceforthelocaljackalpopulation,anddue totwoyearsworthoftrackingdata,wehavenewunderstandingabouthowjackalsrespondto temporarilyavailableresources[ 10 ].Thejackalsliveinterritorialfamilygroups.Theyregularly huntandforagewithinandnearbytheirdefendableterritory,andopportunisticallyscavengeon carcasseswhentheyareavailable.Instudyingthedata,wealsoobservethatjackalssometimes makelongtrekstovisitresources(seeFigure 2-1 ),possiblycrossingthroughtheterritoriesof neighboringfamilygroups. Thesemovementpatternshaveinterestingimplicationsforthepotentialspreadofdisease. Itispossiblethatduringresourcepulses,jackalswillhaveincreasedcontactwithindividuals outsidetheirfamilygroup.Asaconsequence,thoughanthraxbacteriararelycausedisease incarnivores,anintenseuptickinjackal-to-jackalencounterscouldleadtoanoutbreakof a di erent diseaseinthejackalpopulation[ 10 ].Inthisparticularsense,wemightsaythat anthraxcan"cause"arabiesepidemic.Werefertothisprocessas indirectinductionofdisease becausechangeinresourceavailabilitydoesnotintroduceapathogen,itsimplychangesa population'scontactnetworkstructureinsuchawaythatthepopulationbecomessusceptible toinvasionbyapathogenthatwouldnototherwisebeabletotakehold.Infact,asweargue later,inductionofdiseasecanresultfroma decrease inresourceavailabilityaswell. 18

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2.2Modeldevelopmentandpreliminaryanalysis 2.2.1Resource-drivenencounters WiththejackalpopulationfromENPinmind,weintroducethefollowinggeneral assumptions: thelocationsofbothconsumersandresourcesarerandomlydistributedthroughouta spatialregionofinterest; theresourcesareonlyavailableforagivenintervaloftime 1 andnewresourcesare locatedindependentlyofpreviousones; consumersareterritorial,spendingmostoftheirtimenearahomelocation,andhavea limitedrangeofdetection,characterizedbyalengthscale ; consumersprefertovisitthenearestresourcetheydetect; theyrespondtoresourcesindependentlyofotherconsumers;and, theyaresatiatedaftervisitingaresource,andthereforevisitatmostoneresourceper unitoftime 2 Weareinterestedinthenumberofconspecicencountersatypicalconsumerwillhaveasa resultoftemporarilyavailableresources. Forthesakeofsimplicity,andbecausewebelievedthechoicewasreasonableforthe jackalpopulationinENP,wechoosethetimeparameterstobethesame, 1 = 2 = = oneweek.Weuse O todenotethespatialregionwearestudying.Foreachweek,resources aredistributedthroughout O accordingtoaPoissonspatialprocess,withintensityparameter # .Thismeansthatforanyregionofarea A containedin O ,thenumberofresourcesin thatregionisPoissondistributedwithmean # A .Moreover,iftworegionsaredisjoint,their respectivenumbersofresourcesareindependent.Weassumethereisaconsumerlocated attheorigin,referredtoasthe focalconsumer .Theremainingconsumersaredistributed throughout O accordingtoaPoissonspatialpointprocesswithintensity $ .Theseintensity parameterscorrespondtotheexpectedpopulationdensityproducedbythemodelforthe consumersandresources,respectively.Inoursimulationandmathematicalanalysis,thesizeof thelandscapeistakentobesu"cientlylargethatthepresenceofaboundarydoesnothavean e ectonquantitiesofinterest. Tomodeltheconsumer'slimitedabilitytodetectresourcesand/ortraveltoresourcesthat havebeendetected,weassumethereisamaximumdistance withinwhichagivenconsumer 19

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willdetectresources.Moreover,weassumethatconsumerswilldetectallresourceswithina surroundingcircleofradius andwillchoosetovisitthenearestofthesedetectedresources. Tounderstandtheconsumer-resourcelandscape,itishelpfultoconstructVoronoitessellations oftheregion O generatedusingthesetoftheresourcelocations[ 55 ].Usingthe R -package deldir [ 68 ],wedisplaythreesuchtessellationsinFigure 2-2 .Consumerlocationsaredisplayed assquareswhileresourcelocationsaretriangles.Thefocalconsumerappearsinblack.Each subregionofthetessellation,referredtoasa Voronoicell ,containsexactlyoneresourceandis comprisedofallpointsthatareclosertothislocalresourcethananyother.Wealsorefertoa resource'sVoronoicellasits basinofattraction .Westressthatwhenresourcesarerare,the basinofattractionwillusuallycontainmanypointsthatareadistancegreaterthan fromthe resource.Ifaconsumerislocatedatsuchapoint,itwillnotvisitanyresourcesduringthat unitintervaloftime. Thefundamentalgoalintheanalysisofourmodelistounderstandthenumberof encountersthatoccurduetothepresenceofaparticulartypeofresource.Wedenethe resource-drivenencounterrate E tobetheexpectednumberofconsumersthatchoosethe sameresourceasthefocalconsumerperunitintervaloftime.InFigure 2-2 A, 2-2 B,and 2-2 C, thefocalconsumerhas0,2and1encountersrespectively.Thisrevealsafundamentaldynamic inthemodel:thatintermediateresourceavailabilitycanproducethehighestencounterrates. Whenresourcesarescarce,resource-drivenencountersarerarebecauseitisunlikelythatthe focalconsumerisnearenoughtoaresourcetodetectit.Ontheotherhand,whenresources arecommon,encountersarerarebecausenearbyconsumershavelocalresourcesoftheirown tovisit. Toestimate E foragivenparametertriplet ( $ # ) ,wesimulated1000independent landscapes,calculatedtheresultingnumberofencountersforthefocalconsumerineach, andthentooktheaverageoftheseobservations.Formostofoursimulations,weusedthe parameterranges # $ (0,10) and $ (1,14) .AsdescribedinSection 2.5.3 ,foreverytriplet ( $ # ) thereisanassociatedtriplet (1, # ) forwhich E isthesame.Wethereforealwaysset 20

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$ =1 inoursimulationsandusethetransformation # = # / $ and = % $" tocompute E when $ & =1 2.2.2Introductionofpathogenandoutbreakofdisease. Toplaceourencounterrateresultsinthecontextofdiseaseecology,weemployasimple stochasticmodelofpathogenspilloverbetweentwo"adjacent"populations.Becausewe assumethatthediseaseisinitiallynotpresentinthetargetpopulation,weincorporatearate % spillover ofpathogenintroductioneventsfromamaintenancepopulationinwhichthediseaseis endemic.Duetoourinterestintransientseasonale! ects,theresultsareexpressedintermsof theduration T oftheresourceincrease. Wemakethreecentralassumptions: thetimescaleofanoutbreakissmallrelativetothetimeittakesforsignicantchanges inpopulationsizetooccur; eachintroductionofapathogeninvolvesjustoneinitialinfectiousindividual;and, thearrivaltimesofpathogenspillovereventsareindependent. Undertheseassumptions,theinitialpathogeninvasionprocessisintrinsicallystochastic. WemodeltheintroductionsasaPoissonarrivalprocesswithrateparameter % spillover ,an assumptionsimilartotheinvasionmodelproposedbyDruryetal.[ 22 ].Forthetransmission eventsamongindividualsinthetargetpopulation,weuseastochasticSusceptible-Infectious-Susceptible (SIS)model.Becausethetotalpopulationsizeisxedinthismodel,itisonlynecessaryto trackthestatetransitionsfortheinfectiousgroup,whosepopulationsizeattime t isdenoted I ( t ) .Thetransitionratesforourcontinuous-timeMarkovchainaregivenby I I +1 atrate & ( I )= ( % spillover + bI ) 1 # I N I I # 1 atrate ( I )= I where N isthetargetpopulationsize, istheclearance(ordisease-relatedmortality)rate,and b (1 # I / N ) istheaveragenumberoftransmissionsperunitintervaloftimebyaninfectious individualwhentheinfectiouspopulationhassizeI. 21

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Whilethissimplemodeloftransmissionignoresotherpotentiallyrelevantcharacteristics (e.g.latentperiods,populationturnover,andacquiredimmunity),ourpresentfocusisonhow consumer-resourceinteractionsmodulatetransmissiondynamicsintheearlyintroductionphase. Wearespecicallyinterestedintheprobabilitythatthelevelofinfectioncanreachanendemic stateinthetargetpopulationbeforetheperiodofresourceincreasedissipates.Givenour contextthatthediseasedynamicstakeplaceoveralargeareaandthepathogenintroductions arerelativelyrare,weintroduceafourthassumption:eachpathogenintroductionresolves itselfindependentlyinthetargetpopulation(eithertoextinctionorinvasion).Mathematically, thisistantamounttoomittingthe % spillover terminthetransitionrateformulasandtreating eachpathogenintroductioneventindependently.The"endemicequilibrium"istheminimum sizefortheinfectiouspopulationsuchthattherateofincreaseequalstherateofdecrease. Weconsiderapathogenintroductiontobe"successful"ifthesizeoftheresultinginfectious populationeventuallyexceedstheendemicequilibriumvalue: I =min # i $ { 1,..., N } : & ( i ) ( i ) $ (21) Wethenstudythecontinuous-timeMarkovchain { I ( t ) } t 0 withthetransitionrates I I +1 atrate & ( I )= bI % 1 # I N & I I # 1 atrate ( I )= I andcomputetheprobabilityofsuccessfulinvasionassumingthatapathogenhasbeen introducedattimezero: p invasion = P { I ( t ) hits I before 0 | I (0)=1 } InasensemaderigorousbyKurtz[ 47 ],when N islargethisstochasticsystembehaves moreandmorelikeanassociatedODE, y = by (1 # y ) # y (22) 22

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whereweinterpret y ( t ) astheproportionofthepopulationthatisinfectiousattime t .If b > and y (0) > 0 ,then y ( t ) convergestotheequilibriumvalue y =1 # / b .Otherwise y ( t ) 0 as t "( .FollowingtheterminologyusedbyBall[ 7 ](seealsoHe!ernanet al.[ 36 ]),wereferto R 0 = b / asthe reproductiveratio IncontrasttotheODEmodel,nomatterhowlarge N is,inthestochasticmodelthereis alwaysachancethataninfectiouslineagewillgoextinctbeforeitreachesanendemicstate. InFigure 2-3 wedisplaytenstochasticSISpathswithapopulationsizeof50with b =2 and =1 .Someofthesepathsquicklygoextinct,whileothersreachtheendemicstate.Overlaid onthestochasticpathsis Ny ( t ) ,therescaledsolutiontotheassociatedODE( 22 ),with initialcondition Ny (0)=1 JustasitisfortheODEmodel,thereproductiveratioisacriticaldimensionless parameterinthestochasticmodel.When R 0 1 ,thenas N "( p invasion 0 [ 38 ]. Ontheotherhand,when R 0 > 1 ,thenas N "( ,theprobabilityofinvasionisstrictly greaterthanzero.AsdescribedinAppendix 2.5.4 p invasion iscommonlyapproximatedby computingthecomplementofanextinctionprobabilityforanassociatedbranchingprocess[ 8 ]. Thisgivestheapproximation p invasion ) ( ) ( 1 # b b > 0, b ' (23) Withthisinhand,wecanestimatetheprobabilitythatthereisasuccessfulpathogen invasioninthetargetpopulationduringtheperiodofincreasedresourceavailability, t $ [0, T ] Foreachpathogenintroduction,welabelit"successful"withprobability p invasion andthen notethat,fromMarkovchaintheory[ 23 ]thearrivalof successful introductionsisalsoa Poissonprocess,butwitha"thinned"intensity % spillover p invasion .Thisimpliesthatthetimeof thearrivaloftherstsuccessfulspilloverhasanexponentialdistributionwithrateparameter % spillover p invasion .Therefore,theprobabilityofasuccessfulinvasionoccurringduringtheresource 23

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pulsehastheform 1 # e # T spillover p invasion 2.2.3Datacollectionandanalysis JackalswerecapturedfromJanuary2009toJuly2010incentralENPaspartofalarger studyonjackalmovementandanthraxecology(Bellanetal[ 10 ]).Twenty-twoadultjackals werettedwithGPS(globalpositioningsystem;AfricanWildlifeTracking,Pretoria,Republic ofSouthAfrica)collarsbasedontherequirementthattheywerelargeenoughtolimitthe collartolessthan6%ofbodyweight.MovementdatawasacquiredfromcollarsbyVHF radio-trackinganimalsanddownloadingrecordedhourlyGPSxeswithUHFdownload.Due tochallengesassociatedwithacquiringdownloads,thereissomevariationinthetimeintervals betweenrecordedlocations.Insomecasestherearemissingdatapoints;andinafewcases, observationsweremademorefrequentlythanonceperhour.Thedurationoftimeeach collaredanimalwasobservedalsovariedgreatly,fromafewweeksto2years,foratotalof 13.5jackal-yearsof(roughly)hourlylocationdata. Inadditiontojackalpositiondata,carcasssurveillancedatawasrecordedfromJanuary 2009toNovember2010(see[ 10 ]and[ 11 ]foradditionalinformation).Multiplecharacteristics ofacarcasswererecorded,suchas:species,dateofobservation,levelofdegradation,and causeofdeath.Particularlyrelevanttoourinvestigation,therearejackalcountsrecordedfor 299outof411carcasssites(178outof244zebracarcasssites).Thesedataaredisplayedin Figure 2-6 Resourcevisits. Foreachrecordedinstanceofacarcass,weassigneda"carcassactive interval"basedonitsestimatedtimeofdeathandthelevelofdegradationatthetimeof discovery(ifrecorded).Thiswindowlasteduptosixdays.Sixdaysalsoservedasthebaseline durationofavailability,usedwhenlowornolevelofdegradationwasrecorded.Foreach jackalthatwastrackedintheparkcontemporaneouslywithaknowncarcass,wecomputeda "time-localaverageposition,"i.e.themeanofallrecordedpositionsofthejackalduringthe carcassactiveinterval.Thedistancebetweenthisaveragepositionandthelocationofthe 24

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activecarcasswasassignedtobethedistanceofaresourcevisitornon-visit.Ifthejackal's minimumdistancefromthecarcassofinterestduringthisperiodwaslessthan100m,we classiedtheeventasaresourcevisit. Ouruseofthelocationdatasetsforcollaredjackalstoidentify"resourcevisits"assumes thatvisitstocarcasssitesaremeaningfullycapturedwithinthemovementdata.Ifjackal movementwerenotinuencedbythedistributionofcarcassesonthelandscape,wewould expecttondnoassociationbetweenjackallocationsandidentiedcarcasssites.We performedarandomizationtesttoassessthenullhypothesisthatthereisnoassociation betweenjackallocationsandidentiedcarcasssites.Foreachsite,weheldthetimethatthe sitewasavailableconstantandreassignedthelocationbysampling(withoutreplacement)from therecordedcarcasslocations.Foreachpermuteddataset,wethencalculatedthenumberof "resourcevisits"inthesamewayasdescribedabovefortheobserveddata.Usingthetrue carcasslocationdatatherewere10and44visitsfromjackalswithatime-localaverageof10+ kmand5+kmrespectivelyfromthecarcasslocations.Outof1000setsofrandomizedcarcass locations,themaximumnumberof10+kmand5+kmvisitswas6and15,corresponding toa p -valueof0.001forbothdistances,andindicatingthatthereisahighlysignicant associationbetweenjackallocationsandidentiedcarcasssites. 2.3Results Bywayofsimulationandanalysis,weareabletocharacterizethemostprominent qualitativefeaturesoftheexpectednumberofencountersexperiencedbyafocalconsumer,as itdependsontheexpectedresourcedensity # andthemaximumdistanceofdetection .While wereporttheresultsofthespecicmodeldescribedintheprevioussection,aslongasamodel isconsistentwiththelistedassumptions,thenourfundamentalconclusionsarethesame:there isanon-monotonicrelationshipbetweentheexpectedresource-drivenencounterrateandthe resourcedensity;themaximumpotentialencounterratecanbelargeintermsofitsimpacton thecriticaldiseaseecologyparameter R 0 ;and,somewhatsurprisingly,lowresourcedensities areassociatedwiththelargestincreasesinencounterrates. 25

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2.3.1Analysisoftheconsumer-resourcemodel Wesummarizethepredictionsofthemathematicalmodelasfollows.Forxedvaluesof $ and : Fromthepointofviewofanavailableresource(carcass),thenumberofvisitors decreaseswith # Thepresenceofadditionalresourcesincreasesthenumberofoptions forconsumersandso,as # increases,theexpectednumberofvisitorsatagivenresource sitedecreases. Fromthepointofviewofanindividualconsumer,thenumberofencountersincreases, thendecreaseswith # Whenresourcesarescarce,mostconsumerswillnotbenear enoughtodetectthem.Increasing # meansthatmoreandmoreconsumersvisit resources,leadingtoincreasedconsumer-consumerinteractions.Thee!ectisnot monotonicthough.Whenresourcesareabundant,consumerswillgenerallydetectmore ofthem.Duetothisincreaseinavailableoptions,itbecomeslesslikelythatmultiple consumerswillvisitthesameresource. Fromthepointofviewofagivenresourcesite,therearetwolimitingfactorsonthe numberofvisitors:1)thesizeoftheresource'sbasinofattraction,asdenedbytheVoronoi tesselationdescribedinSection 2.2.1 andpresentedinFigure 2-2 ;and2)theconsumers' limiteddistanceofdetection.As # increases,theresource'sbasinofattractiondecreasesin size,thereforelimitingthepoolofconsumersthatwouldchooseit.InSection3.2wepresent ananalysisoftheENPdataset,whereinwendsomeevidencethatthenumberofjackals expectedataparticularcarcassdecreaseswiththenumberofcarcassesavailableatthetime. Fromtheconsumerpointofview,afocalconsumerisalwaysinthebasinofattraction ofsomeresource;however,whenresourcesarescarceitisunlikelythatitwillbecloseenough todetectthenearestresource.Ontheotherhand,whenresourcesareabundant,theareaof thebasinofattractioncanbeverysmallcomparedtothefocalconsumer'sdetectionarea, limitingthepoolofpotentialconsumersthatmightsharetheresource.InFigures 2-4 and 2-5 weprovideacomprehensiveviewofthedependenceofafocalconsumer'sencounterrate E onresourcedensity.InAppendix 2.5 ,weprovidethedetailsofamathematicalanalysisofthe modelandrigorouslydemonstratecertainprominentfeaturesoftherelationship:namely,the asymptoticpowerlawinboththescarceandabundantresourceregimes,aswellasinthesmall andlargedistanceofdetectionextremes.Furthermore,weprovideanapproximateformulafor 26

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theresourcedensity # thatleadstothemaximumnumberofencountersforagivendistance ofdetectionandconsumerdensity. Asymptoticresults. InFigure 2-4 weseethat E ,theexpectednumberofencounters forthefocalconsumer,hasaveryregularpowerlawbehaviorinboththesmall-and large# regimes.Regardlessofthevalueof ,alloftheencounterratecurvesoverlapin thelarge# regime.Forsmall # ,thelog-logslopeisthesameforall ,buttheleading coe" cientdi! ers.InTheorem 2.5.1 wedemonstratethatwhenresourcesarescarce(small # ),theresource-encounterfunctionisasymptoticallylinearin # .Furthermoreweareableto establishtheleadingcoe" cient,yieldingthe small# approximation E ) $#( 2 4 ,whichisalso validatedbysimulation.Forexample,inFigure 2-4 ,thelowerblackdottedlineisthesmall# approximationwhen =1 andweseegoodagreementfor # < 0.1 .Whenresourcesare abundanttheanalysisleadstoanunsolvedprobleminspatialpointprocesstheoryconcerning thedistributionofVoronoicell(basinofattraction)sizesintessellationsgeneratedbyPoisson spatialprocesses.Nevertheless,wearguethat E scaleswith # # 1 intheabundantresource limit.FollowingthediscussioninSection 2.5.3 ,wepresentthe large# approximation E ) $ /# (Figure4,blackdashedline).Thecorrectleadingcoe"cientappearstobelargerthan $ ,but wewereunabletoobtaintheexactvaluebymathematicalanalysis. Characterizingtheencounterratepeak. ForreasonsdiscussedinSection 2.3.4 perhapsthemostimportant"landmark"oftheresource-encounterfunctionisitspeak. Unfortunatelyitisdi"culttodirectlyanalyzethemagnitudeofthepeakandthecorresponding criticalresourcedensity.However,thereisanaturalrst-orderestimatethatinvolvesthe small-andlarge# approximations.Solvingfortheirintersectionyieldstheestimate # ) (1 / ( ) # 2 and E ( # ) ) $(" 2 where # istheresourceintensitythatleadstothemaximum resource-drivenencounterrate.FromFigure 2-4 ,itisclearthatthisisanoverestimate,butnot dramaticallyso.Using1000simulationsatanarrayof # and values,wefoundthefollowing estimatesusingalinearregression: # ) 0.536 # 2 and E ( # ) ) 1.48 $" 2 (24) 27

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Whiletheexponentsalignwellwiththeintersectionofthesmall-andlarge# approximations, wewereunabletoobtainasatisfactoryexplanationoftheleadingcoe"cientsthroughdirect mathematicalanalysis. Dependenceondistanceofdetection. Inadditiontocharacterizingtheencounter rate'sdependenceon # ,wearealsoabletoobtainanasymptoticunderstandingofthe dependenceoftheresource-drivenencounterrateonthemaximumdistanceofdetection parameter .As " 0 ,theencounterratefunctionbehaveslike 4 (Theorem 2.5.1 ).Asmight beexpected,thisfunctionismonotonicallyincreasingin andsaturatestoalimitingvaluefor large (Theorem 2.5.4 ).ThecorrespondingsimulationresultsaredisplayedinFigure 2-5 A. Thelimitingvaluecorrespondstotheexpectedareaofthebasinofattractioninwhichthe focalconsumerresides.Asexplainedbefore,thisexactvalueisnotknown,butitscaleslike # # 1 ,whichiswhythelimitingvaluesinFigure 2-4 arelargestforthesmallestvaluesof # 2.3.2Therelationshipbetweenresourcedensityandsitevisitation Themathematicalmodelmakespredictionsaboutbothfull-populationscaleencounter ratesandlocalsingle-resourcesiteencounters.Forthelatter,fromtheperspectiveofagiven carcasssite,themodelpredictsthatthemaximumnumberofvisitorsshouldbeobserved whentheresourcedensityisthelowest.Thisisbecauseinthesparseresource-densityregime thereislittletonocompetitionforconsumers.Astheresource-densityincreases,theexpected numberofvisitorsshoulddecrease.WeconsultedtheENPdatasettoinvestigatewhetherthis e ectcanbeobservedforjackalsandtheirtendencytovisitcarcassesthatseasonallyvaryin abundance. Inthestudyarea[ 10 ],thenumberofcarcassesavailableforjackalscavengingvaries seasonally(Figure 2-6 B).BetweenFebruaryandApril,thereisaresourcepulseresulting fromannualanthraxoutbreaksinthelocalzebrapopulation.Theseoutbreaksoccurduring theendofthewetseason[ 49 69 ].Thetimingandseverityofanthraxoutbreaksappearsto bedi!erentbetween2009and2010.Thedi!erenceinseverityprovidesanopportunityto makecomparisonsbetweenthesamemonthsoftheyearbutwithverydi! erentnumbersof 28

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availablecarcasses.InMarchandApril,forexample,theaveragenumberofjackalsobserved atcarcassesdecreasesmarkedlyfrom2009to2010whentherearemorecarcassesavailable.In fact,ineightoutoftheelevenmonthswherepairwisecomparisonsarepossible,theaverage numberofjackalsobservedatacarcassdecreasedwhenmorecarcasseswereavailableinthat month(Figure 2-6 C). Toemployamorequantitativestatisticaltest,wetaPoissongenerallinearmodelwith thenumberofobservedcarcassesasapredictorvariable,andthenumberofjackalsvisiting acarcassastheresponsevariable(availablefromJanuary2009toNovember2010).Wealso includedpredictorvariablesforeachmonthoftheyeartoallowforvariationinenvironmental e ects(e.g.wet/dryseason),populationprocesses(birthpulse,dispersal,etc.)andchallenges indatacollectionthatlikelya! ecttheexpectednumberofjackalsobservedatcarcasses.To beprecise,let y i betheresponsevariableforthenumberofjackalsobservedatacarcasswhen thereare i carcasses.Then log ( y i )= ) 0 + 11 + j =1 ) j { month = j } + ) carc i (25) Forexample,inanAprilwith i totalcarcasses,theexpectednumbervisitorsobservedata carcasswouldbe exp( ) 0 + ) 3 + ) carc i ) .Usingthisstatisticalmodel,wefoundasignicant negativecorrelationbetweenthenumberofobservedcarcassesandtheobservednumberof jackalsatacarcass( ) carc = # 0.025 ,95 % CI: [ # 0.029, # 0.021] ). 2.3.3Therelationshipbetweendefendableterritorysizeandthedistanceof detectionandresponse Thoughtherearethreeparametersinthemathematicalmodel,wefoundthatthereare trulyonlytwodegreesoffreedomintheparameterspace.AsshowninAppendix 2.5.3 ,for everytriplet ( $ # ) ,thereisacorrespondingtriplet (1, # ) ,where # = # / $ and = % $" suchthattheexpectednumbersofencountersforthefocalconsumersarethesame,i.e. E ( $ # )= E (1, # ). 29

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Notably, # and arenondimensionalquantitiesandallformulasintroducedintheprevious sectioncanbeexpressedusingthem: small# approximation: ( 2 # 4 large# approximation: # # 1 peakresourcedensity: ( ( 2 ) # 1 encounterratepeak: ( 2 Bothnondimensionalquantitieshaveinformativebiologicalinterpretations.Whileitis straightforwardtounderstandthesignicanceof # = # / $ (theratiooftheresourcedensity totheconsumerdensity),themeaningof = % $" ismoresubtle.Ifweimaginedividingthe landscapeintoevenpartitions,oneforeachconsumer,theneachconsumerwouldbeallocated aregionofarea 1 / $ .Iftheregionsaresquare,then 1 / % $ wouldbethelengthofeachside andwecanview as = / (1 / % $ ) ,sothatitistheratioofthedistanceofdetectionto thetypicallengthofaconsumer'sspaceallocation.Inbiologicalterms,wemightthinkof theseregionsastheconsumers'defendableterritoriesandtherefore isroughlythenumberof defendableterritoriesaconsumeriswillingtocrossinordertovisitaresource. Ourestimateoftheresourcedensity, # ,isbasedoncarcasssurveillancedatafromthe BerkeleyEtoshaAnthraxProjectduring2009and2010.Theaveragenumberofcarcasses recordedeachmonth(Figure 2-6 B)wasdividedbyfourtogetaweeklynumberofcarcasses available.Sincenotallcarcassesareobserved,wefollowedBellanetal.(2013)[ 11 ]in multiplyingbyascalingfactoroffourtoaccountforexpectedunobservedcarcasses.We thendividedtheexpectedweeklynumberofcarcassesavailableineachmonthbytheareaof thestudyregionfromBellanetal.(2012)[ 10 ],roughly1000km 2 .Thisareacontainsallof thelocationswherecarcasseswereobservedandjackalpositionsrecorded.Theresulting # estimatesrangedfrom0.005km # 2 (AugustandNovember)to0.043km # 2 (April). Assuggestedbythenondimensionalisationargumentabove,weinterpret $ asthedensity ofdefendablejackalterritories.Non-overlappingjackalterritoriesinENPwereestimatedto bebetween4km 2 to12km 2 .Thisiscomparabletoestimatesthatweremadeforjackal populationsincoastalNamibia(0.2-11.11km 2 [ 42 ])andSouthAfrica(3.4-21.5km 2 [ 27 ]). 30

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NotingtheobservationfromBellanetal.(2012)thatjackalsare"unusuallydense"inENP [ 10 ],wesetthetypicaljackalterritorysizetobe5km 2 ,sothat $ =0.2km # 2 Theinterpretationoftheparameter fromthedatarequiressomediscussion.Inthe mathematicalmodel, isthemaximumdistanceatwhichaconsumercandetectandthen respondtoaresource.Wecanthinkofthemodelasassumingthattheprobabilityofdetecting aresourceisonewithinadistance andzerooutsidethatdistance.Ofcourse,inreality,this detectionprobabilitylikelydecreasessteadilyasafunctionofdistance.Ratherthanidentify aspecicvaluethatwedenitivelyclaimtobethebestestimateof ,weusedthejackal movementdatatondarangeofreasonablevalues. InFigure 2-7 ,wedisplayascatterplotofalljackalaveragepositionsrelativetoknown carcassesandmarkeachwithatealdotoragray x dependingonwhetherthejackalvisited thecarcassornot.Jackalswereobservedtovisitknowncarcasssitesasfaras15kmaway,but alargemajorityofcarcassesvisitedwereinarangeof0to4km.Asexpected,theprobability thatajackalvisitedaresourcedecreasedwithdistance,butitisnotknownwhetherthiswas becausethejackalswerenotawareofmoredistantcarcasses,orbecausetherewereother carcassesoralternateresourcesnearerby.InSection 2.3.4 weusethetwovalues =4 and 10 andtheassociatedencounter-ratecurvesaredisplayedinFigure 2-8 .Eachwasgeneratedby averagingtheresultsof10,000simulationsateachof300valuesfor # 2.3.4PlacingmodelresultsinthecontextofDiseaseEcology InSection 2.2.2 ,wedescribedourstochasticsmall-populationmodelforpathogen invasion.Wesaythataninvasionis"successful"ifitachievesapopulationlevelequivalentto whatwouldbetheendemicequilibriumofthedeterministicversionofthemodel.Thereexists anexplicitformulafortheprobabilityofinvasion,butitisdi"culttointerpretintermsofthe parametersofthemodel.So,followingBall&Donnelly[ 8 ],weuseanapproximationforthe truevalue(seeEquation 23 andfurtherdiscussioninAppendix 2.5 ).Thisreducesouranalysis todeterminingwhetherthetotalrateoftransmission(whichisa! ectedbytheresource-driven encounterrate)isgreaterthanthedisease-relatedmortalityrate 31

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Toassesswhetherachangeintheconsumerencounterrateis"large"inthecontextof jackalsandrabies,wefollowedRhodesetal.[ 60 ]inestablishingabackgroundrateofpathogen transmission( b =1 wk # 1 )andadisease-relatedmortalityrate( =1.4 wk # 1 )yieldingthe reproductiveratio R 0 ) 0.7 .Since R 0 < 1 ,rabiesisfoundtobesub-critical.Toconnectthe resource-drivenencounterrateatagiventime t E ( t ) ,tothepathogen-transmissionmodel, werstnotethatnotallencountersinvolvinginfectiousandsusceptibleindividualsleadtoa newinfection.Forexample,inourmodel,aresource-drivenencounterisdenedtooccurif twoindividualsvisitthesameresourcesiteinthesameweek,butthisdoesnotmeantheyvisit concurrently.Eveniftheyvisitconcurrentlythisdoesnotensurepathogentransmission.Dene p inf tobetheprobabilitythataresource-drivenencounterresultsintransmission.Thenour expectednumberofnewinfectionsarisingfromasingleinfectiousindividualis b + p inf E ( t ) and thereproductiveratiois R 0 ( t )= b + p inf E ( t ) (26) Usingthemonth-by-monthencounterratevaluesappearinginFigure 2-8 ,wecalculatedthe time-dependentreproductiveratioforsixscenariosanddisplayedtheminFigure 2-9 .Figure 2-9 AandFigure 2-9 B,correspondtothedistanceofdetectionchoices =4 and =10 respectively.Ineachcase,wevariedtheprobabilityofinfectionparameter p inf todemonstrate itsimpactonthenalresult. When =4 ,theresourcedensityforeachmonthisbelowthecriticalresourcedensity # ,i.e.thedensityforwhichthemaximalencounterrateoccurs.Soanincreaseinresource densityleadstoincreasesintheresource-drivenencounterrateandresultingreproductive ratio,regardlessofthe p inf value.However,becausethepeakoftheencounterratecurveis relativelylow(approximatelyveperweek,seeFigure 2-8 )thereproductiveratioremains belowthecriticalvalueofone.Ontheotherhand,when =10 ,mostofthemonthlyresource densitiesaregreaterthan # .Inthosecases,increasesinresourcedensityleadtodecreasesin theresource-drivenencounterrateandresultingreproductiveratio.Inthisregime,weseethat themonthswithlowcarcassavailabilityaremostvulnerabletopathogeninvasion. 32

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Wenotethatthemagnitudeofchangein R 0 isdirectlydependentontheestimatefor jackalterritorydensity $ (recallEquations 24 ).If p inf =0.02 andif $ =1 insteadof $ =0.2 forexample,thentheApril R 0 for =4 wouldbeapproximately1.07.Thisconstitutes asettingwhereindirectinductionofdiseaseispossible.Asimilarmodicationfor =10 wouldresultinanAugust R 0 of2.86.Theseresultscanreadilybetranslatedtoaprobability ofsuccessfulinvasionoverthecourseofaresourceincreaseofduration T .Asdescribedin Section 2.2.2 ,successfulpathogeninvasionsarriveaccordingtoaPoissonprocesswithrate % spillover p invasion .Assumingthetransmissionrateisconstantovertheperiodofinterest,the probabilityofinvasionis 1 # exp % % spillover (1 # R # 1 0 ) T & 2.4Discussion Inthiswork,wehavedevelopedaframeworkforanalyzingtheimpactofchangesin resourceavailabilityontherateofconspecicencountersamongconsumersandexpress ourresultsinthecontextofdiseaseecology.Givenalandscapeofconsumersandresources weessentiallyaskthequestion:wouldaddingonemoreresourcesiteleadtomoreorfewer encountersamongtheconsumers? Wehaveproposedanovelconsumer-resourceinteractionmodeltoinvestigatethis question.Throughacombinationofnumericalsimulationandmathematicalanalysis,wehave identiedandcharacterizedtwoqualitativelydistinctparameterregimes.Inascarceresource regime,addingmoreresourcesleadstomoreconsumer-consumerencounters;inanabundant resourceregime,addingmoreresourcesleadstofewerconsumer-consumerencounters.The utilityofourmodelisthatitcanbeusedtopredictthequalitativedynamicsofasystemonce certainfundamentalparametersareestimated:theconsumerdensity( $ ),theresourcedensity ( # )andthemaximumdistanceofdetectionandresponse( ). Toworkthroughaspeciccasestudy,weusedlocationdataforapopulationofjackals andthecarcassesuponwhichtheyscavengeinEtoshaNationalPark.Whilesomemodel parameters( # and $ )arefairlystraightforwardtoestimate,othersarenot(seeSection 2.3.3 forourapproachtoestimatingtheparameter inparticular).Onenotablechallengethat 33

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arisesisthatthedenitionofan"encounter"isintrinsicallysubjective,dependingstrongly onthequestionofinterest(alsoseeGurarieetal.[ 32 ]forafulldiscussiononthispoint).To relateourresource-drivenencounterratetoarateofpathogentransmissionfrominfectiousto susceptibleindividuals,weintroducedacorrective"probabilityofinfection"factor p inf .Because pathogentransmissionisessentiallyimpossibletodirectlyobserve,properinferenceforsucha parameterwouldlikelyrequirepopulation-leveldiseaseincidencedatathatdoesnotcurrently exist.Inresponsetothisuncertaintyinparametervalues,wedisplaymodelresultsthatemerge fromarangeofreasonablevaluesforboth p inf and .Thekeytakeawayisthatforcertain combinationsofbiologicallyrelevantparameters,weconrmthatsmallchangesintheresource landscapecanleadtosubstantialchangesinpathogentransmissiondynamics.Infact,weshow thatsuddenscarcityofaresourcecanhavealargere! ectonencounterratesthanaresource pulse. Buildinguponexistinginvestigationsintohowchangesinresourceandconsumerdensities inducechangesindiseasedynamics,ourworksuggeststhattherelationshipbetweenterritory sizeandthedistanceofresourcedetectionplaysacrucialroleindetermininginfectiousdisease outcomes.Tousethepresentcontextforanexample,wenotethatjackalsmayusevisual cuesfromvulturestoidentifycarcasssites([ 40 ]andanecdotalobservationsbyanauthorand colleagues).Ifvulturepopulationsdecline,ashasnowbeendocumentedinbothAsiaand Africa[ 53 ],thedetectiondistanceforjackalscoulddecrease,potentiallycausingapathogen invasionregimeshift.Interestinglythough,thespecicexampleofdecliningvulturepopulations exempliesthecomplexityofconsumer-resourceinteractions.Inanexperimentconductedby Ogadaetal.[ 54 ],theauthorsfoundthattherewereincreasedencountersamongmammalian scavengerswhenvulturescouldnotseeandreacttocarcasses(incontrasttoourFigure 2-5 A). 2.4.1Opportunitiesforintegratingmoredetailedanimalbehavior Thecomplexrelationshipbetweenresourceallocation,consumerbehavior,andpathogen spreaddeservesfurtherstudy.Weconstructedourmodeltobedetailedenoughtoexamineour 34

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primaryquestion,butsimpleenoughtopermitrigorousmathematicalanalysis.Whilethereare manywaystoextendthemodeltoaccountformorenuancedbehavior,wehighlightafew. Resourcedetectionandselection. Thereareothernaturalmodelsfortheconsumer's abilitytodetectresources,aswellasforthealgorithmdeterminingwhichresourceisvisited, ifany.Forexample,onecouldpositthatthereisimperfectdetectionandthattheprobability ofdetectiondecreaseswithaconsumer'sdistancefromtheresource.Also,onecouldrelax therestrictionthattheconsumeralwayspickstheclosestdetectedresource.Aninformal investigationsuggestedthat,aslongasweposeassumptionsconsistentwiththoseoutlinedat thebeginningofSection 2.2.1 ,adoptingalternativemodelspecicationsdoesnotchangethe qualitativedescriptionofourresultsreportedinSection 2.3.1 .Weoptedfortheversionthat yieldsthemostexplicitanalyticalresults,butnotethatchangestomodelassumptionswould likelychangethevalueofthecriticalresourcedensity # aswellastheheightoftheassociated encounterratepeak. Onemajorfactorthatwedidnotconsiderisheterogeneityintheresourcesites.Variation inresourcequality,geographicalcharacteristicsandlocalenvironmentalfactorscana! ectthe modelthroughmultipleparameters.Resourcesitesthatattractvulturesmightbedetectable fromlargerdistancesthanresourcesthatdonot,causingvariationin .Smallcarcassesmay berapidlydepleted,decreasing 1 ,andmaynotsatiateconsumers,decreasing 2 .Interms oftheselectionalgorithm,aconsumermightnotchoosetheclosestavailableresourceifone ofgreaterqualityisjustalittlebitfartheraway.Ourmodelassumesuniformityinresources and,comparedtopredictionsthatwouldfollowfromeachofthesepossiblemodications,it producesalowervarianceinthenumberofvisitorstoagivensite. Thereductioninvarianceissignicantinthefollowingsense.AswereportinAppendix 2.5 ,Equation 29 ,thenumberofvisitorstoagivensiteisPoissondistributedwithamean parameterthatdecreaseswiththetotalnumberofavailableresources.Whilethispredictionfor themeanisconsistentwiththeavailablecarcassvisitationdata(seeFigure 2-6 ),thevariance ofaPoissonrandomvariableismuchlessthanthevarianceoftheobserveddistribution.In 35

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Februaryof2010,forexample,therearevaluesashighasthirtywhenthemeanislessthan ve.However,thisisextremelyunlikelyforaPoissondistribution.Tobeprecise,if { X i } 60 i =1 are independentandidenticallydistributedPois (5) randomvariables,then P (max i X i 20) 0.0001 Modiedbehaviorofinfectiousindividuals. Behavioralchangesassociatedwiththe diseasestatusofanindividualmaya! ectitsexpectedencounterrate.Developingatheoryfor susceptible-infectiousencounterratesthatconsidersbothtypesofindividualswillbeespecially importantforinfectionsthatalterhostbehavior(e.g.rabies).Specically,wenotethatthe mannerinwhicharabidanimaldetectsandselectsresourcesitescouldbemuchdi! erentthan thatofasusceptibleindividual. O!-siteencounters. Atpresent,ourmodelconsiderstherelationshipbetweenresource availabilityandtheconsumerencounterratespecicallyatresourcesites.However,achange inresourceavailabilitywilllikelyinuenceothertypesofencountersaswell.Forexample, whenconsumersareforcedtomakelongtrekstoscarceresources,theymaybeexposedto unfamiliarindividuals.Distinguishingbetweentypicalencounters(e.g.withfamilymembers andterritorialneighbors)anduniqueencounterswithnewindividualscouldbeimportantfor determiningtransmissiondynamics[ 16 64 ]. Dynamicpopulationcounts. Weconsideredaxedpopulationdensity(i.e. $ ,the jackalterritorydensity).However,populationsizeschangeonmultipletimescales.Jackals havebirthpulsesthatwillchangethelocaljackalpopulationsizeonanannualbasis(although, pupsmaynotcontributedramaticallytopathogenspread).Inthelongterm,consumer populationsizemayrespondtoresourceavailability;whenresourcesareabundantmore consumerscanbesupportedinthesamearea.Thisallowsforsmallerterritories(increases in $ ).SpecicallyinEtosha,zebrasareattractedtoagrasslandforagingareasouthofthe saltpan.Thejackaldensityinthisareamaybehigherthanthedensityinotherareasdue togreateraverageresourceavailability.ConsideringFigure 2-4 andEquations 24 ,wesee thatiftheconsumerdensityvarieswiththeresourcedensity,thentherearetwocompeting 36

PAGE 37

e ects:whileincreasing # candecreasethenumberofencountersforxed $ ,asimultaneously increasing $ canovercomethise! ect. 2.4.2Usingseasonalityasatoolforinvestigation Large-scaleecologicalexperimentsareexpensiveandchallengingtoconduct.Itcan thereforebeveryusefultoobserveandcharacterizesystemswithnaturallychangingresource conditions[ 39 73 ].Beingabletoobservethesamesysteminmultiplestatesprovidesthe opportunitytoinvestigateresponsestothealteredsystemcomponentswhilekeepingother characteristicsconstant.Seasonality,inparticular,isfrequentlyobservedintime-seriesdatafor incidenceofdiseaseandhasbeenshowntoa! ectinfectionratesthroughmultiplemechanisms. Seasonalchangescanresultinauctuatingpopulationsizeanda!ectboththequantityand typeofconspecicinteractions.Moreover,periodicchangesinpopulationcountduetobirth pulses[ 59 ]andmigration[ 15 ]havebothbeenshowntoa!ectthepotentialforinfectious diseaseoutbreaks.Webelievethatcouplingtemporallyvaryingenvironmentalinformationin multi-statesystemswithrigorousanalysisofGPSlocationdatacanprovideabasisformore mechanisticmodelsofconsumerresponsetoresourcechange.Ultimatelythiscanleadtomore meaningfulandmoreaccuratepredictionsfortheconsequencesofhabitatalteration. 37

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Table2-1.Parametersusedinthediseaseecologyanalysis. ValueUnitsDenitionSource b 1wk # 1 transmissionrate[ 60 ] 1.4wk # 1 rabiesmortalityrate[ 60 ] # 0.005-0.043km # 2 carcassdensityENPdataanalysis $ 0.2km # 2 jackalterritorydensity[ 10 27 42 ] 4-10kmmaxdistanceofdetectionENPdataanalysis 1wktimescaleofresourceENPdataanalysis availabilityandvisitation 2.5MathematicalAnalysis Thesimplicityoftheresource-drivenencountermodelinvitesarigorousasymptotic analysis.Morethandemonstratingthenon-monotonerelationshipbetweenresourcedensity andtheconsequentencounterrateintheconsumerpopulation,wecanobtaintheexponentsof thepowerlawsthatgoverntherelationship. Inwhatfollows(andinthemaintext)whenwewrite ( x ) + x as x a ,wemeanthat thereexistssomeconstant C $ (0, ( ) suchthat lim x $ a ( x ) x = C Forexample,aresultwewillusebelowisthatif Y + Pois ( & ) forsome & > 0 ,then P { Y > 1 } + & 2 as & 0 .Thisisbecause P { Y > 1 } =1 # P { Y =0 } # P { Y =1 } =1 # e # # # & e # # andusingtheTaylorseriesexpansionfortheexponential(orsimplyL'Hopital'srule),wehave lim # $ 0 1 & P { Y > 1 } =lim # $ 0 1 # e # # # & e # # & = ( ( ( ( ) ( ( ( ( 0, if + < 2 1 2 if + =2 ( if + > 2 (27) ForhigherordertermswewilluseBig-Ohnotation:wesaythat f ( x )= O ( g ( x )) near x = a if thereexistconstants C > 0 and L > 0 suchthatif | x # a | < L ,then | f ( x ) | C | g ( x ) | 38

PAGE 39

Asinthemaintext, # and denotetheresourceintensityandmaximumdistanceof detectionrespectively.Inthepresentationofourresultswewillassumethattheconsumer density $ =1 .InSection 2.5.3 wewilldiscusshowtomodifytheresultswhen $ & =1 Wetakethedomain O tobeacircleofradius R > 3 centeredattheorigin.Thereis a focalconsumer locatedexactlyattheorigin.Resourcesaredistributedthroughout O asa Poissonspatialprocesswithintensity # .Other,non-focalconsumersaredistributedthroughout O asaPoissonspatialprocesswithintensityone.Let x 0 =(0,0) andenumeratethenon-focal consumerlocations { x 1 ,..., x N } where N + Pois ( |O| ) .Furthermorelet z 1 ,..., z Z bethe resourcelocationswhere Z + Pois ( # |O| ) .Foreachpair 1 i N and 1 j Z ,let ij := | x i # z j | .Foreach i $ { 0,..., N } ,let i := { j : ij =min 1 % j % Z ij } .Inotherwords, i is theindexoftheresourcethatisclosesttothe i thconsumer.Fornotationalexpediencywewill writetheindexoftheresourceclosesttothefocalconsumer, 0 ,tosimplybe . Intheabovenotation,wecanexpress ) ,thenumberofresource-drivenencounters experiencedbythefocalconsumer,tobe ) := |{ i $ { 1,..., N } : i = }| and E := E ( ) ) (28) Givenasetofresourcelocations,itisusefultothinkofthelandscapepartitionedaccordingto theassociatedVoronoitessallation.Thatis,neglectingasetofmeasurezero, O = Z i =1 O i where O i := # x $ O : | x # z i | =min j & { 1,..., Z } | x # z j | $ Wesaythat O i isa basinofattraction forresource i :allconsumerslocatedin O i willchoose resource i astheirresourcetovisitifitiswithintheirdetectionradius.Wedene B ( x ; r ) to bethecircleofradius r centeredatthelocation x .Thenthedistributionoftheencounter variable ) conditionedonagivenresourcelandscape L = { z i } Z i =1 is ) | L + Pois % |O $ B ( z $ ; ) | & | % z | % & (29) 39

PAGE 40

wherewerecallthat istheindexoftheresourcechosenbythefocalconsumerand A =1 if theevent A occurs,andiszerootherwise. 2.5.1Smallresourcedensityand/orsmalldetectiondistance Theorem2.5.1. Let E = E ( # ) bedenedasin ( 28 ) .Then E + # and E + 4 as # and gotozero,respectively.Tobeprecise, lim $ 0 1 # E ( # )= ( 2 4 and lim & $ 0 1 4 E ( # )= ( 2 # (210) Proof. Werstintroducesomenotation.Let N ( r ) and Z ( r ) denotethenumberofconsumers andresourceswithinadistance r ofthefocalconsumer.Weproceedbyconditioningonthe numberofresourcesthatarenearthefocalconsumer.Wepartitionthesamplespace as follows: 0 = { Z ( )=0 } 10 = { Z ( )=1, Z (3 ) # Z ( )=0 } 11 = { Z ( )=1, Z (3 ) # Z ( ) 1 } 2 = { Z ( ) 2 } Naturally,itfollowsthat E ( ) ) = i E ( ) | i ) P { i } andwewillndthatthedominantterm istheoneassociatedwith 10 .Lookingattheotherterms,rstobservethat E ( ) | 0 ) =0 since,iftherearenoresourcestoconsume,thefocalconsumerwillnothaveanyencounters. Todealwiththeevent 2 ,weapplyEq 27 above,notingthatthenumberofresources withindetectiondistanceofthefocalconsumerhasthedistribution Z ( ) + Pois ( #(" 2 ) .It followsthatforsmall # P { 2 } + # 2 andforsmall P { 2 } + 4 .Toboundtheconditional expectation E ( ) | 2 ) ,observethatthenumberofresource-drivenencountersexperiencedby thefocalconsumermustbelessthanorequaltothenumberofconsumersthatarelocated withinadistanceof ofthefocalresource(theresourcechosenbythefocalconsumer). Becausethisisaregionofsize (" 2 ,wehave E ( ) | 2 ) (" 2 .Togetherwehavethat E ( ) | 2 ) P { 2 } = O ( # 2 6 ). 40

PAGE 41

Fortheevent 11 weagainexploitthat,whenthedetectiondistanceorresourcedensity issmall,itisunlikelythattherewillbemorethanoneresourcenearthefocalconsumer.By independenceoftheresourcedistributionindisjointregions P { 11 } = P { Z ( )=1 } P { Z (3 ) # Z ( ) 1 } =( #(" 2 e # '(& 2 )(1 # e # 8 '(& 2 ) sincetheareaoftheannuluscoveringtheregionthatisbetweenadistanceof and 3 ofthe originis 8 (" 2 .Itfollowsthat P { 11 } = O ( # 2 4 ) .Meanwhile,usingthesameupperbound onthenumberofresource-drivenencountersexperiencedbythefocalconsumer,wehave E ( ) | 11 ) (" 2 .Therefore E ( ) | 11 ) P { 11 } = O ( # 2 6 ). Turningourattentiontotheevent 10 ,ifthereisonlyoneresourceinthefocal consumer'sdetectionradius,andtheresourceistheonlyoneinthelarger 3 radiuscircle centeredattheorigin,thenallconsumerswithinaradius ofthefocalresourcewillchoosethe sameresourceasthefocalconsumer.Inotherwords,thenumberofencountersconditioned on 10 is ) | 10 + Pois ( (" 2 ) .Whatwasanupperboundinpreviouscasesisnowequality.It followsthat E ( ) | 10 ) = (" 2 .Tocomputetheevent'sprobabilityweargueasbefore, P { 10 } = P { Z ( )=1 } P { Z (3 ) # Z ( )=0 } =( #(" 2 e # '(& 2 )( e # 8 '(& 2 ). (211) Therefore lim $ 0 1 # E ( ) ) =lim $ 0 1 # #( 2 4 e # 9 '(& 2 = ( 2 4 and lim & $ 0 1 4 E ( ) ) =lim & $ 0 1 4 #( 2 4 e # 9 '(& 2 = ( 2 # asclaimed. 41

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2.5.2Analysisinthehighresourceandlargedistanceofdetectionregimes Inthehighresourcedensityandlargedistanceofdetectionweareunabletogetexact results.Thisisduetoafundamentalbarrierintheanalysisthatwewilldescribebelow.In thehighdensityregimewecanprovidewhatappearstobealowerboundon E that,fromthe numerics,seemstoscalewith E as # "( Conjecture: E ( # ) + 1 # as # "( Ourconjectureisbasedonthefollowingheuristic.Recallthat O $ isthebasinof attractionthatcontainsthefocalconsumer.Then: Conditionedonthelandscapeofresources,thenumberofencountersexperiencedby thefocalconsumerisPoissondistributedwithmeanequaltoitscontainingbasinof attraction.Therefore E ( ) ) = E ( |O $ | ) Unconrmedestimate : E ( |O $ || Z = z ) |O| / z E ( |O| Z > 0 / Z ) + 1 / # as # "( ThethirdpartoftheheuristicisestablishedbyLemma 2.5.3 below.Thesecondpartofthe heuristicisjustiedbythefollowing. Lemma2.5.2. Letaresourcelandscapebegivenasdescribedaboveandletthetotalregion O bepartitionedaccordingtoaVoronoidiagramgeneratedusingtheresourcelocations { z 1 ,..., z Z } .Wedenotetheareasofeachofthesebasinsofattraction { A 1 ,..., A Z } Let x + Unif ( O ) bearandomlocationinthelandscapeanddene tobetheindexof thebasinofattractionthatcontainsthispoint.Then E ( A $ | Z = z ) |O| / z Remark1. Unfortunately,atthistime,wearenotabletoextendtheresulttoestablishthe claimthatthebasinofattractionspecicallycontainingtheoriginhasanexpectedareathatis largerthan O / z .Numericsstronglysupportthisconclusion. 42

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Proof. E ( A $ | Z = z ) = z + i =1 A i P { = i } = z + i =1 A 2 i |O| = |O| z + i =1 A i |O| / 2 |O| z 0 z + i =1 A i |O| 1 2 = |O| z where,inthelastline,wehaveusedtheCauchy-Schwarzinequality. Lemma2.5.3. Suppose Y + Pois ( & ) .Then lim # $' & E { Y > 0 } Y / =1 (212) Proof. Recalltheexponentialintegralfunction Ei ( x )= 2 x #' e t t dt for x > 0 ,wheretheintegralistakeninthesenseoftheCauchyprincipalvalue.The exponentialintegralfunctioncanbewrittenintermsoftheseries[ 1 ]. Ei ( x )= % +ln( x )+ + k =1 x k kk Forlargex,Ei ( x ) hastheasymptoticexpansion[ 72 ], Ei ( x ) + e x x 1+ 1 x + 2 x 2 + 3! x 3 +... / (213) FollowingthesuggestionofGrabandSavage[ 30 ],wenotethat E { Y > 0 } Y / = + k =1 1 k & k e # # k = e # # ( Ei ( & ) # % # ln( & )) (214) where % istheEuler-Mascheroniconstant.Combining( 213 )and( 214 )wearriveatthe desiredresult. 43

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Thereisafundamentalmathematicalbarriertomakingmoreprogressonthisproblem. Theprecedinganalysisreducestheproblemtoanalyzingthedistributionofareasofcells generatedbyPoissonVoronoiTessellations,butthisanoutstandingmathematicalproblem[ 55 ]. Inparticular,wethereisnoknownexpressionfor E ( |O $ | ) ,theexpectedareaofthebasinof attractionthatcontainsthefocalconsumer.Thispreventsusfromobtainingaresultforthe highdistance-of-detectionregimethatisexplicitin # Theorem2.5.4. E ( # ) isanincreasingfunctionin and lim & $' E ( # )= E ( |O $ | ) (215) Proof. Foragiven # > 0 ,let L denotealandscapeofresourcesgeneratedbyaspatial Poissonprocesswithintensity # .Foreachsuchlandscape,let ) | L ( ) bethenumberof encountersforthefocalconsumer.AsnotedaboveinEquation( 29 ),thisisthenumberof consumerslocatedinaradius ofthefocalresourcemultipliedbyoneorzerodependingon whetherthefocalresourceiswithinradius ofthefocalconsumer.Foraxedlandscape,note that lim & $' O $ B ( z $ ; )= O $ and lim & $' | % z | % & =1. Assuch, lim & $' E ( ) | L ( ) ) = E ( |O $ ||L ) Becausethisholdsforall L ,thepropositionfollows. 2.5.3Convertingresultsfornon-unitconsumerdensity Alloftheprecedingresultshavebeenexpressedundertheconsumerdensityassumption $ =1 .Similarlyallsimulationswereconductedwith $ =1 .Extendingtheearliernotation,let E ( $ # ) betheexpectednumberofencountersforthefocalconsumerforthegiventriplet ofparameters.Weclaimthat,althoughtherearethreefundamentalparametersinthemodel, thereareonlytwodegreesoffreedomintheparameterspace.Thatistosay,givenatriplet 44

PAGE 45

( $ # ) thereexistsauniquepair ( # ) suchthat E ( $ # )= E (1, # ) .Namely, E ( $ # )= E 1, # $ % $" Toseethis,let N (3 ) and N (3 ) denotethenumberofconsumersinthemodelforthe parametertriplets ( $ # ) and (1, # ) respectively.Let Z (3 ) and Z (3 ) denotethesame forresources.BecauseconsumersandresourcesaredistributedasPoissonspatialprocesses, thesevaluescompletelydenethesystem.Furthermore,becausePoissonrandomvariables arecompletelyparameterizedbytheirmeans,itfollowsthat E ( $ # )= E (1, # , $ ) if E ( N (3 ) ) = E N (3 ) and E ( Z (3 ) ) = E Z (3 ) .Fortherstconstraint, E ( N (3 ) ) = E N (3 ) -. $( 9 2 = ( 9 2 fromwhichitfollowsthat = % $" .Meanwhile E ( Z (3 ) ) = E Z (3 ) -. #( 9 2 = #( 9 2 meaningthat # = #" 2 / 2 = # / $ Wenotethatreducingtheproblemtotwoparametersamountstoanondimensionalization ofthethreeparametermodel.Theunitsof $ and # areboth[ length ] # 2 ,while hasunitsof [ length ].Asaresult, # = # / $ and = % $" arebothdimensionless. Revisitingthetheoremsoftheprevioussectionswehavetheresults: lim $ 0 1 # E ( $ # )= ( 2 $" 4 ,lim & $ 0 1 4 E ( $ # )= ( 2 $# (216) and E ( $ # ) + $ # as # "( (217) 2.5.4Branchingprocessapproximation ThereexistexactsolutionstothehittingprobabilityproblemintroducedinSection 2.2.2 however,suchapresentationmakesitdi" culttounderstandhow p invasion dependson b and .Undertheassumptionthatthesizeofthesusceptiblepoolisverylargewithrespecttothe 45

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initialinfectiouspopulation,itiscommontointroducetheapproximationthat ( N # I ) / N ) 1 [ 2 ].ThemodiedratefunctionsfortheCTMCarelinearandtaketheform & ( i )= bi and ( i )= i TheinfectiouspopulationprocessisthenaGalton-Watsonbranchingprocess.Theonlytwo outcomesforsuchaprocessareextinctionorexplosiontoinnity.Theanalysisreducesto recastingtheCTMCasadiscretetimegeneration-by-generationbranchingprocessthatis denedintermsofthe o springdistribution ,i.e.,thedistributionofthenumberofo!spring anindividualmighthavebeforedying.Inourcase,the"o! spring"aretheinfectionsspawned byasingleindividual.SincetheinfectioneventsoccuraccordingtoaPoissonprocesswith rateparameter b andthedeathoftheindividualoccursatrate ,thenumberofsuccessful infectionsbeforedeathisGeometricallydistributedwithsuccessprobability m := b / ( b + ) Onecanthenshowthat,undertheassumptionthat b > lim t $' I ( t )= ( ) ( ( w.p. 1 # ) b 0 w.p. ) b If b ' ,theprocessgoesextinctwithprobabilityone. 46

PAGE 47

waterhole carcass anthrax positive 0 5 km Figure2-1.CollaredjackalmovementandknowncarcasslocationsinEtoshaNationalPark. GPSlocationsforallcollaredjackalsonFebruary2,2010.Jackalsaredi! erentiated bycolor.EachcoloredlinesegmentwithblackdashesconnectstwoGPSpingsfor thatjackal.Bluecirclesrepresentwaterholes.Bluetrianglesindicatelocationsof knowncarcasses.Whitetriangleinsetsindicatethatthecarcasstestedpositivefor anthraxcausingbacteria.Roadsareindicatedbyblacklinesandtheshadedgray areasarepartofasaltpaninENP. 47

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y y =0 4 =1 =10 A B C Figure2-2.Voronoidiagramsdisplayingtheregionsof"attraction"foreachresource.Each bluetriangleindicatesthelocationofaresourcegeneratedusingresourcedensity # .Theblacksquareandthewhitesquaresindicatethelocationsofthefocal consumerandnon-focalconsumers,respectively.Graycircles,centeredatthe resourceclosesttothefocalconsumer,displaytheregionwhereconsumerscan detecttheresource.A)Thereare2resources.B)Thereare5resources.C)There are50resources. !"#$%&'()%* +,-%$'."#/* spillover I ( t ) I ( t ) I t 1 ODE CTMC A B Figure2-3.Invasioneventsresultingfromspilloverinfections.A)Modeldiagramillustrating spilloverinfectionconcept.B)Numberofinfectiousindividualsresultingfromthe introductionofoneinfectiousindividualinapopulationofsize N =50 .Ten samplepathsforthestochasticSISmodeldenedinAppendix 2.2.2 areplotted (redlines)withthesolutiontotheanalogousordinarydi!erentialequationmodel (blackcurve).Ourrepresentationforhavingachievedtheendemicstateis I (dashedhorizontalblackline),whichisdenedinEquation 21 .Opencirclesare plottedwhen I isreachedbefore0andredpointsindicatewhenthepathogendied outofthepopulationbeforereaching I 48

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Resource intensity Resource driven encounters 0.001 0.01 0.1 1 10 0.01 0.1 1 10 100 1000 Approximation small large ! ! =14 =14 ( ) ( E ) Figure2-4.Encounterratebehavior.Eachdotrepresentstheaverageover1000simulations. Theintersectionofthedottedanddashedlinesistheorder-of-magnitudeestimate describedintheResults,Section 2.3.1 1 2 4 6 8 10 12 14 0.01 0.1 1 10 100 Distance of detection Resource driven encounters 0.001 0.005 0.014 0.091 1.048 10 ( E ) ( ) 1 2 4 6 8 10 14 0.01 0.1 1 10 100 0.01 0.1 1 10 100 Distance of detection Critical resource intensity Max resource driven encounters slope = 1.99 slope = 2.06 ( ) ( ! ) E ( ! ) ! ( E ( ! )) A B Figure2-5.E! ectofdetectiondistanceonresource-drivenencountersandpeaknumbersof resource-drivenencounters.A)Resource-drivenencountersasafunctionof distanceofdetection.Lighter-to-darkershadingcorrespondstoincreasingvaluesof # .B)Inteal(circles),themaximumencounterrateasafunctionof .Inblue (triangles),theresourceintensitythatyieldsthemaximum. 49

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0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 No. carcasses in a month No. jackals at a carcass Jan Apr Jul Oct Avg. carcasses 0 10 30 50 non zebra zebra Avg. carcasses 0 10 20 30 40 50 60 70 0 2 4 6 8 10 12 2009 2010 No. jackals at carcasses No. carcasses in a month A B C Figure2-6.Monthlycarcassavailabilityandjackalvisitationtocarcasssites.A)Foreach month-yearpair,thenumberofobservedcarcassesiscounted(x-axis).Weplotthe numberofjackalsrecordedateachcarcassversusthenumberofobservedcarcasses inthecorrespondingmonth.Jackalcountsatcarcassesarecolor-codedbymonth oftheyearasindicatedbythebarchartintheupperrightcorner.Pointsare shadedsothatdarkershadingindicatesmoreobservations.Regressionlinesare plottedforeachmonthinthecorrespondingcolor.B)Themonthlyaverage numberofobservedtotalcarcasses(solidbars)andzebracarcasses(stripedbars). C)Foreachmonth-yearpair,theaveragenumberofjackalsobservedatcarcasses isplotted.Repeatedmonthsareconnectedbylines. 50

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Visits Non visits 0 4 10 20 km 0 5 10 15 20 Distance from carcass (km) Frequency Visits Non visits 0 100 200 300 A B Figure2-7.Jackalvisitationbasedonaveragedistancefromcarcasssite.A)Relativepositions ofjackalsplottedwithrespecttothelocationofeachknowncarcass.Jackal locationswerecalculatedastheaverageoftheirGPSpingsthatoccurredbetween twodaysbeforeandafterthecarcasswasestimatedtobepresent.B)Stacked histogramforthetimesthatjackalschosetovisitacarcass(tealbars)andthe timesthatjackalsrefrainedfromvisitingacarcass(graybars). 0 10 20 30 40 0.001 0.01 0.1 1 Month ( ) Apr ( 0.043 ) Mar ( 0.042 ) Feb ( 0.034 ) Jun ( 0.018 ) Jul ( 0.013 ) May ( 0.013 ) Oct ( 0.01 ) Sep ( 0.01 ) Jan ( 0.0094 ) Dec ( 0.006 ) Nov ( 0.005 ) Aug ( 0.005 ) Resource driven encounters Resource density ( ) ( E ) =10 =4 Figure2-8.Simulatednumberofresource-drivenencountersfortwochoicesofthethe detectiondistanceparameteroverarangeofresourceintensities.Theverticallines indicatetheestimatedmonth-by-monthcarcassdensitiesobservedintheENPdata set. 51

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0.0 0.2 0.4 0.6 0.8 1.0 1.2 Jan Apr Aug Dec 0.001 0.01 0.02 Reproductive ratio Month 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Jan Apr Aug Dec 0.001 0.01 0.02 Month p inf p inf =4 =10 ( R 0 ) A B Figure2-9.Time-dependentreproductiveratiobasedonthecorrespondingnumberof resource-drivenencountersinFigure 2-8 with b =1 =1.4 .A)Resultsfor detectiondistance =4 .B)Resultsfordetectiondistance =10 52

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CHAPTER3 INVASIONPROBABILITIES Thischapterwillbesubmittedas"Continuumapproximationofinvasionprobabilities forstochasticpopulationmodels"andistheresultofacollaborationbetweenRebecca K.BorcheringandScottA.McKinley. Abstract: InthelastdecadetherehasbeengrowingcriticismofthetheuseofStochastic Di erentialEquations(SDEs)toapproximatediscretestate-space,continuous-timeMarkov chainpopulationmodels.Inparticular,severalauthorshavedemonstratedthefailureof di usionapproximation ,asitisoftencalled,toapproximateexpectedextinctiontimesfor populationsthatstartinaquasi-stationarystate. Inthisworkweinvestigateanotherrelated,butdistinct,populationdynamicsproperty forwhichdi!usionapproximationfails:invasionprobabilities.Weconsiderthesituation inwhichanewindividualisintroducedtoapopulationandaskwhetherthelineageof thisnewindividualcansuccessfullythrivedespitesuchamodestbeginning.Becausethe populationcountissosmallduringthecriticalperiodofsuccessorfailure,theprocessis intrinsicallystochasticanddiscrete.Inadditiontodemonstratinghowandwhydi! usion approximationfailsinthelargepopulationlimit,weprovideamoresuccessfulalternative WKB-likeapproximation.Moreoverweprovideananalysisforhowtheseapproximations performinanimportantintermediateregime.Wendthattherearetimeswhenthedi!usion approximationperformswell:particularlywhenparametersarenear-criticalandthepopulation sizeissmalltointermediate. 3.1Introduction Invasioneventsarefundamentalinpopulationbiology.Inthestudyofdiseasethere areinterestingexamplesbothatthecellularandwhole-organismlevel:whilestudyingthe onsetofinfection,onelooksattheprobabilitythatasinglevirioncaninfectatargetcell andproliferate;inepidemiology,thegoalmightbetoestimatetheprobabilitythatanewly introducedpathogenwillbecomeendemicinanavehostpopulation.Thesamemulti-scale 53

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interestininvasionsappearsinthestudyofpopulationgenetics:atthemulti-organismscale onestudiestheprobabilitythatanovelallelecanxinapopulation,possiblyimprovingthe population'soveralltness;atthecellularlevel,theinvasionofamutationinastemcell populationhasbeenstudiedasanimportantrststepincertaincancers.Thestate-dependent ratesofinteractionsdistinguishthedynamicsatdi!erentscales.Whilemulti-organismmodels tendtofeaturemovement,requiringdirectinteractionamongindividuals,cellulardynamics maybemorespatiallystaticwithmoreindirectinteractions(competitionforresources,or signalingatadistance,forexample). Itisnaturaltomodelthesepopulationdynamicsusingcontinuous-time,discrete-state-space Markovchains.Itiseasytoencodenonlinearinteractionsthroughstate-dependenttransition ratesandtherearenumerousstraightforwardsimulationtechniquesavailablethatcanbeexact, butcostly[ 28 ],orinexact(withrespecttoboundaryinteractions),bute"cient[ 4 ].Itisalso straightforwardtowritedowndi! erenceequationswhosesolutionsdesignatetheprobabilityof invasionfromasmallnumberofindividuals,orthemeanextinctiontimestartingfromalarge populationsize.Oftenitisevenpossibletondexactsolutionsforthesesystemsofdi!erence equations,but,importantly,itisverydi" culttointerprethowtheseexactsolutionsdependon modelparameters. Toovercomebothcomputationalandanalyticalchallenges,ithasbecomecommontouse StochasticDi!erentialEquations(SDEs)asacontinuous-state-spaceapproximationforthe discrete-state-spaceMarkovchains.UsingKurtz'sDi!usionApproximationTheoremasaguide [ 3 47 ],thereisanaturalwaytotranslateMarkovchaintransitionratesintothedriftand di usiontermsofanSDE.(SeeSection 3.3.1 fordiscussion.)Thistechniquehasbeenused tomodelepidemics[ 2 ],neuronalactivity[ 19 29 ]andbranchingprocesseswithlogisticgrowth limits[ 48 ].IntheSDEsetting,solutionsofinvasionprobabilityandmeanextinctiontime questionscanbeobtainedbysolvingcertainODEs(orPDEs,dependingonthedimension). TheexactsameODEscanbeobtainedwithoutappealingtoanSDEmodel,arisingfromthe 54

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applicationofanaveTaylorseriesapproximationoftheKolmogorovEquationsoftheMarkov chainmodel.(SeeSection 3.3.1 againfordiscussion.) However,multipleauthorshaveshownthatthereareimportantdi! erencesbetween Di!usionApproximationsandtheCTMCs.Forexample,Doeringandco-authors[ 20 21 ]have shownthatthemeanextinctiontimeforSDEmodelscandi! ersubstantiallyfromthatof thecorrespondingMarkovchainmodel.NewbyandBresslo!alsodocumentedtheissuefor neuronalmodels[ 17 43 51 ]. Whilethisissueofthemeanextinctiontimehasreceivedconsiderablerecentattention, theprobabilityofinvasionquestionwasstudied,andinmanywaysresolved,muchearlier. FrankBallandcolleagues[ 6 8 ]showedthattheinitialphaseofacertainclassofepidemic modelscanbewell-approximatedbybranchingprocessesforwhichanexactprobabilityof invasioncanbecalculated.Forexample,theprobabilitythatasingleinfectedindividual cancauseanepidemicinthestochasticSusceptible-Infected-Susceptible(SIS)andstochastic Susceptible-Infected-Recovered(SIR)modelsis 1 # d b ,where b istherateatwhichaninfectious individualinfectssusceptiblemembersofthepopulationand d istherateatwhichthedisease isclearedfromaninfectiousindividual.(Throughoutthiswork,wewillusetheletters b and d toevokethewords"birth"and"death"tocorrespondtopopulationincreasesand decreases,regardlessoftheparticularapplication.)Thisformulaisnicebecauseitistrivialto understanditsdependenceonmodelparameters,butwenotethatitreliesontheassumption thatthepopulationsizeisinniteandaccordinglygivesnoguidanceinpredictinghowinvasion probabilitydependsontheoverallsizeofthepopulation.Moreover,thebranchingprocess approximationisonlyvalidwhenthedynamicsare"asymptoticallylinear"inasensethatwill bedescribedafterAssumption 1 below.Heuristically,thismeansthata"birthevent"only dependsononememberofthepopulationofinterest.Thisisoftennotthecaseinchemistry applications(seeDoering,2007[ 21 ]foranexample)andrecentinvestigationintotheimpact ofsensinganddecision-makingonanimalmovementhasrevealednon-linearencounterratesin predator-preymodels[ 37 ]. 55

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Theaboveconsiderationsmotivatethepresentwork,inwhichweanalyzetheperformance ofstandardcontinuumapproximationsfortheprobabilityofinvasioninaMarkovchainmodel. Themathematicalframeworkallowsfornon-lineartransitionratefunctionsandprovides anaturalwaytoestimateprobabilityofinvasionwhenthepopulationsize N isnite.We performasymptoticanalysis( N "( )ontheDi!usionApproximationandaWKB-like ExponentialApproximation,showingthattheformeryieldsanincorrectvalueinessentially allcases.Whenvalid,theExponentialApproximationappearstoyieldthecorrectlarge N limit.Remarkablythough,whenthepopulationsizeissmallorintermediate,andwhenthe parametersoftheproblemareclosetoacriticalvalue,weshownumericallythattheDi! usion ApproximationcapturesinvasionprobabilitiesmuchbetterthantheExponentialApproximation. Ratherthandeclareoneapproximationoranotherthe"winner"wethinkitwouldbemost usefulforpractitionerstokeepinmindthetradeo! sinvolvedinchoosingwhichmethodto implementintheirownwork. 3.1.1MathematicalFramework Foragivenpopulationscale N ,consideracontinuous-timeMarkovchain X N ( t ) thattakes itsvaluesinthesetofintegers n $ N .Weconsidertheclassofmodelswithtransitionsofsize oneandthetransitionratesfromthestate n aregivenby n n +1 atrate & N ( n ) (31) n n # 1 atrate N ( n ) (32) with & N (0)= N (0)=0 .While & N ( n )=0 isallowableforsome n > 0 ,werequirethatthe deathratesatises N ( n ) > 0 forall n $ N .Itfollowsthat0istheonlyabsorbingstate. Assumption1 (Transitionrateshapefunction) Thereexistfunctions & : R + R + and : R + R + suchthat & N ( n )= N & n N and N ( n )= N n N Furthermore, 56

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i) thereexistsan A $ (0, ( ] suchthat & and aredi!erentiableforall x $ (0, A ) ; ii) & (0)= (0)=0 ; & ( x ) 0 forall x $ (0, ( ) ; ( x ) > 0 forall x $ (0, ( ) ;and iii) thereexistsavalue x > 0 suchthat & ( x )= ( x ) andforall x $ (0, x ) ,either & ( x ) > ( x ) or ( x ) < & ( x ) .Intheformercase,thesystemiscalled supercritical ;inthe latter, subcritical Wewillreferto x asthe minimalrate-balancedpoint .Whenthesystemisdiscrete, rate-balancedpointsarenotpreciselyachieved.Foragivensystemsize N ,wedenethe discreteanalogue N asfollows. Denition1. Foragiven N $ N N :=min { n $ N : & N ( n ) N ( n ) } forsupercritical systemsand N :=min { n $ N : & N ( n ) N ( n ) } forsubcriticalsystems.Weevaluate N usingtheceilingfunction,where N = / Nx 0 isthesmallestintegerlargerthanorequalto Nx Ifthetransitionrateshapefunctionsaredi! erentiableatzerowith & ( (0) > 0 and ( (0) > 0 ,thenwecallthesystem asymptoticallylinear Example1 (Stochasticepidemics) Forapopulationsize N ,andconstants b > 0 and d > 0 ,thesizeoftheinfectedpopulationinastochastic(non-densitydependent)SusceptibleInfected-Susceptiblesystemisdenedbythetransitionrates n n +1 atrate bn ( N # n ) N n n # 1 atrate dn Thesedynamicsarecharacterizedbytherateshapefunctions & ( x )= ( ) ( bx (1 # x ), x $ [0,1] 0, x > 1 ( x )= dx forall x $ [0, ( ). Theminimalrate-balancedpoint x =1 # d / b ,andfortheprocess X N ,theratebalance pointis N = / N (1 # d / b ) 0 57

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Example2 (Density-dependentmortality) Inthiscase,thepopulationscale N playstherole ofthecarryingcapacityofanenvironment.Again,let d > 0 andlet b > 0 bethegrowthrate ofthepopulationwhenscarcityofresourcesisnotafactor.Thenweconsiderthedynamicsset bytherateshapefunctions & ( x )= bx and ( x )= dx 2 for x $ [0, ( ) .Wenotethatthisisthestandardlogisticgrowthmodelwiththeintrinsic growthrateparameter b = r anddeathrate d = r / K foracarryingcapacity K .Theresulting transitionratesaresimilartotheLogisticBranchingProcesspresentedbyAmauryLambert (2005)[ 48 ].Therate-balancedvalueforthissystemis x = b / d ,andfortheprocess X N ,we have N = / bN / d 0 Example3 (Resource-constrainedbirth) Let b > 0 d > 0 and a > 0 .Wedeneour dynamicsaccordingtotherateshapefunctions & ( x )= bxe # x / a and ( x )= dx for x $ [0, ( ) .Notethat x = a ln b d ,while N = 3 Na ln b d "4 Example4 (Generalnonlinearsingletermmodel) Let b ) d > 0 with ) & = .Wedene ourdynamicsaccordingtotherateshapefunctions & ( x )= bx and ( x )= dx + for x $ [0, ( ) .If ) wesaythat hasthe leadingorder.Inthiscase, x =( b / d ) # + ,andfortheprocess X N ,theratebalancepointis N = / ( b / d ) # + N 0 Inconductingasymptoticanalysis,wewilloftenpreferanalternativeassumptionforthe transitionratefunctionsthatfeaturesexplicitseriesexpansionsforthebirthandthedeath rates. Assumption2 (Seriesrepresentationfortherateshapefunctions) Let & and beas inAssumption 1 .Additionally,weassumethatthereexistconstants b ) d , > 0 with 58

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{ b n } n & N { d n } n & N 1 R ,andaninteger m $ N ,suchthat & ( x )= bx 1+ + n =1 b n x n / m and ( x )= dx + 1+ + n =1 d n x n / m (33) forall x $ [0, x ] Forsimplicityintheanalysis,wealsoassumethat ) and arerationalnumbersand m is chosensuchthat ) m and m areintegervalues. 3.1.2ApproximationsforInvasionProbabilities TheconditionsofAssumption 1 ensurethatas N "( ,thestochasticprocesses X N ( t ) convergepathwisetoanassociatedODE.Kurtz[ 47 ]denesthesenseofthisconvergence rigorouslyasfollows:Let & and belocallyLipschitzfunctions.Dene Y N ( t )= 1 N X N ( t ), and suppose Y N (0) x 0 .Dene x ( t ) tobethesolutionto x = & ( x ) # ( x ), with x (0)= x 0 Then lim N $' sup s & [0, t ] | Y N ( s ) # x ( s ) | =0, almostsurely. AvisualdemonstrationofthisresultisshowninFigure 3-1 forthestochasticSISepidemic model(Example 1 above).Conditionedonnothittingzero,thesolutionsconvergetothe heteroclinicorbitconnectingzeroandtheminimalrate-balancedpoint x .However,the fractionofsolutionsthatavoidextinction doesnotgottozeroas N "( .Wecallthe probabilityofhitting N before 0 theinvasionprobabilityandadoptthefollowingnotation. Denition2. Fortheprocess X N ( t ) ,wedenethe hittingtime N asfollows: N :=inf 5 t > 0: X N ( t ) $ { 0, N } 6 Wedenetheassociated extinctionprobabilities q N ( k ) k $ { 0,1,..., N } by q N ( k ):= P { X N ( N )=0 | X N (0)= k } 59

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Fromthiswedenetheasymptoticprobabilityofinvasion p invasion ( k ) (conditionedon k individualsattimezero)tobe p invasion ( k )=lim N $' 1 # q N ( k ). Itispossibletoobtainanexactsolutionfortheseinvasionprobabilities,andweprovidea formulainSection 3.2 .In1983,FrankBall[ 6 ]introducedaslightlydi!erentdenitionofan invasionprobability(whichhereferredtoasa trueepidemic )and,intheasymptoticallylinear case,provedthattheprobabilityofatrueepidemicconvergestothesurvivalprobabilityofan associatedbranchingprocess.Toseewhatistherightbranchingprocessdenition,weconsider thenumberofo! springthatanindividualhasbeforeexpiring.Inthelarge N ,asymptotically linearcase,thebirthrateis b whilethedeathrateis d .Therefore,thenumberofbirthsbefore theindividualdieshasaGeometricdistributionwith"successprobability" d / ( b + d ) Denition3 (BranchingProcessApproximation) Supposethatthetransitionrateshape functions & and satisfyAssumptions 1 and 2 andthattheleadingorderexponentssatisfy ) = =1 .Denethediscretetimestochasticprocess { Z n } n 0 tobe Z 0 = k andfor N $ N Z n +1 = Z n + j =1 / n +1, j where / i j + Geom(d/(b+d)) ,( i j $ N ) areindependentandidenticallydistributed. Then, p branch ( k )= P # lim n $' Z n = ( | Z 0 = k $ = ( ) ( 1 # d b k b d 0, b < d (34) Ballprovedthat p invasion ( k )= p branch ( k ) inthissetting.WhattheBranchingProcess Approximationlacksisaclearinterpretationintheasymptoticallynonlinearcaseandasense forwhattheinvasionprobabilityisfornite N .ForthesecasesweconsidertheDi!usion ApproximationandtheExponentialApproximation.Thederivationsoftheseapproximations 60

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Table3-1.Summaryofasymptoticinvasionprobabilitiesasthepopulationsize N "( (see Denition 2 ). Theseresultsareonlyprovedfortheasymptoticallylinearcase(i.e. when ) = =1 ).   Wedenetheexponentialapproximationtobezerofor subcriticalregimesbecausetheasymptoticlimittendstozero,buttheapproachto zeroisthroughnegativevalues. MethodNotation ) <-) = -) = -)>( b d )( b < d ) ExactSolution p invasion ( k )1 # ( d / b ) k 0 BranchingProcess p branch ( k )1 # ( d / b ) k 0 Di!usionApprox. p di usion ( k )1 # exp( # 2 k )1 # exp # 2 k b # d b + d 00 ExponentialApprox.   p exp ( k ) 1 1 # ( d / b ) k 00 aregiveninSections 3.3 and 3.4 ,respectively.Inwhatfollowsweintroducethesmall parameter 0 ,whichcorrespondstotheinverseofthepopulationscaleparameter N Denition4 (Di!usionapproximation) Let 0 > 0 begiven.Thenfor x $ (0, x ) ,let u ( x ) be thesolutiontotheboundaryvalueproblem 0 2 ( & ( x )+ ( x )) u (( ( x )+( & ( x ) # ( x )) u ( ( x )=0 (35) with u (0)=1 and u ( x )=0 Then,setting 0 =1 / N ,wedenethe di usionapproximation forinvasionprobabilitiesto betherelationship p di! usion ( k ):=1 # u ( k 0 ) D ) 1 # q N ( k ). (36) Denition5 (Exponentialapproximation) Let 0 > 0 begiven.Thenfor x $ (0, x ) ,dene w ( x ) tobe w ( x )=exp # 1 0 2 x 0 ln r ( y ) d y / (37) where r ( x ) satises 0 r ( ( x ) % r ( x ) # (1+ $ ( x )) & = r ( x ) % 1+ $ ( x ) & # $ ( x ), (38) 61

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with $ ( x )= & ( x ) / ( x ) .Thenthe exponentialapproximation forinvasionprobabilitiesis denedbytherelationship p exp ( k ):=1 # w ( k 0 ) E ) 1 # q N ( k ). (39) InTable 3-1 wesummarizetheresultsthatweexploreinSections 3.3 and 3.4 .Consistent withasymptoticanalysesoftheDi!usionApproximationforextinctionprobability,wendthat theDi!usionApproximationdisagreeswithBranchingProcessApproximationinthelarge N limit.Remarkably,whentheleadingordersofthetransitionrateshapefunctions & and do notmatch,theDi!usionApproximationreducestotwocases,ignoringalldetailedinformation containedintheratefunctions.Incontrast,theExponentialApproximationdoesprovidethe rightlarge N limit.Wenote,however,thattheExponentialApproximationisonlywell-dened inthesupercriticalcase. InSection 3.5 weinvestigatetheproblemnumericallyandndmixedresults.Inthelarge N limit,theExponentialApproximationagreeswiththeMarkovchainmodel,eveninthe asymptoticallynonlinearcasesweconsider.However,whenthepopulationsizeparameter N is ofsmallorintermediatesize,say N 50 ,itiscommonthattheDi!usionApproximationmore faithfullyrepresentstheMarkovchainmodel,especiallywhentheparametersetisnear-critical. 3.2Exactsolutionforinvasionprobabilities Proposition3.2.1. Forxed N $ N ,andrateshapefunctions & and 5 q N ( n ) 6 ,where n $ { 0,..., N } ,satisesthesystemofdi!erenceequations & N ( n ) q N ( n +1)+ N ( n ) q N ( n # 1) # ( & N ( n )+ N ( n )) q N ( n )=0. (310) with q N (0)=1 and q N ( n )=0 forall n N Proof. Weperformaone-stepanalysison q N ( n ) ,byrstdeningastoppingtimefortherst time X N ( t ) changesstates: T =inf { t > 0: X N ( t ) & = n } 62

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Then,since P { X N ( T )= n } =0 ,bythelawoftotalprobabilityandthenthestrongMarkov property,wehaveforall n $ { 1,..., N # 1 } that P { X N ( N )=0 | X N (0)= n } = P { X N ( N )=0 | X N ( T )= n +1 } P { X N ( T )= n +1 | X N (0)= n } + P { X N ( N )=0 | X N ( T )= n # 1 } P { X N ( T )= n # 1 | X N (0)= n } = P { X N ( N )=0 | X N (0)= n +1 } & N ( n ) & N ( n )+ N ( n ) + P { X N ( N )=0 | X N (0)= n # 1 } N ( n ) & N ( n )+ N ( n ) Applyingthe q N ( n ) notationandmultiplyingthroughby & N ( n )+ N ( n ) ,wehave( 310 ), where q N (0)= P { X ( N )=0 | X 0 =0 } =1 and q N ( N )=0 Thefollowingargumentforndingtheexactsolutionissimilartoonepresentedon birth-deathprocessesinNorris[ 52 ],forexample,andhasasimilarformtotheexactsolutions formeanextinctiontimespresentedbyDoeringetal[ 20 21 ].Weincludetheproofherefor completeness. Theorem3.2.2 (Exactsolutionforextinctionprobability) Let { & N ( k ) } N k =0 and q N (1)=1 # N + n =1 n # 1 7 k =0 1 $ N ( k ) # 1 (311) wherewedene $ N (0)=1 andfor k 1 $ N ( k )= # N ( k ) N ( k ) Proof. Suppressingthedependenceon N ,wewrite q n := q N ( n ), & n := & N ( n ), n := N ( n ), $ n := & n n (312) Rearranging( 310 ),wehave & n ( q n +1 # q n )+ n ( q n # 1 # q n )=0. 63

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Introducing n := q n # 1 # q n ,wecansolvefor n +1 andextrapolatetoadependenceon 1 : n +1 = n & n n = = n n # 1 1 & n & n # 1 & 1 1 Recallingourboundaryconditions q 0 =1 and q N =0 ,wehavethat 1= q 0 # q N = N + n =1 n = 1 + N + n =2 n # 1 7 k =1 1 $ k 1 Nowsolvingfor 1 andimposing q 0 =1 ,wehave 1 # q 1 = 1+ N + n =1 n # 1 7 k =0 1 $ k # 1 which,afterintroducingthenotation $ N (0)=1 yields( 311 ). 3.3Di!usionapproximation 3.3.1MotivationsfortheDi!usionApproximation Therearetwopointsofviewonthederivationofthedi!usionapproximationforinvasion probabilities.Theyyieldthesameresult.Therstpointofview(Motivation1,below)is commoninthephysicsliterature[ 13 ].Itinvolvessubstitutingasmoothfunction u intothe di erenceequation( 310 )andthenconvertingittoadi!erentialequationbywritingout Taylorexpansionsandmatchingterms.Inasecondpointofview(Motivation2,below), wedeneaStochasticDi!erentialEquationwhoseinnitesimalrstandsecondmoments aredenedtomatchthatoftheoriginalbirth-deathchain[ 47 ],AllenandAllen[ 3 ].Then wecomputetheprobabilitythatthisSDEhitstherate-balancedstatebeforegoingextinct, startingfromthevalue 0 =1 / N Taylorseriesapproximation: Tobeginthisapproach,weassumethat u ( x ) isasmooth functionandappealtothefollowingformalargument.Westartwiththedi! erenceequation & n q n +1 # ( & n + n ) q n + n q n # 1 =0 64

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andrewriteitusingtherateshapefunctionsaccordingtoAssumption 1 & n = N & ( n / N ) and n = N ( n / N ) ,alongwith q n = u ( n / N ) : N & n N u n +1 N # N 8 & n N + n N "9 u n N + N n N u n # 1 N =0. Writing x = n / N and 0 =1 / N 0 # 1 & ( x ) u ( x + 0 ) # 0 # 1 : & ( x )+ ( x ) ; u ( x )+ 0 # 1 ( x ) u ( x # 0 )=0. (313) WethenapplyaTaylorseriesapproximationtothe u ( x + 0 ) and u ( x # 0 ) terms: u ( x + 0 )= u ( x )+ 0 u ( ( x )+ 0 2 2 u (( ( x )+ O ( 0 3 ); u ( x # 0 )= u ( x ) # 0 u ( ( x )+ 0 2 2 u (( ( x )+ O ( 0 3 ). Substitutingtheseinto( 313 )andneglectingthehigherorderterms,wehave % & ( x ) # ( x ) & u ( ( x )+ 0 2 % & ( x )+ ( x ) & u (( ( x )=0 whichistheformseeninDenition 4 HittingprobabilitiesforanSDEapproximation: Let 0 =1 / N and y = n / N ,and considerthefollowingcalculationfortheinnitesimalmean: a ( y )=lim h $ 0 1 h % E ( X N ( t + h ) # X N ( t ) | X N ( t )= n ) & =lim h $ 0 1 h % P < X ( t + h ) # X ( t )= 1 N = = = X N ( t )= n > # P < X ( t + h ) # X ( t )= # 1 N = = = X N ( t )= n > + o ( h ) & =lim h $ 0 1 h h N & N ( n ) # h N N ( n )+ o ( h ) = & ( y ) # ( y ). InthelastequalityweusedAssumption 1 : & N ( n )= N & ( n / N ) and N ( n )= N ( n / N ) .Note thattheinnitesimalmeandoesnot,infact,dependon N .Ontheotherhand,afactorof % N 65

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doesappearinthecalculationoftheinnitesimalsecondmoment: 1 2 ( y )=lim h $ 0 1 h % E % X N ( t + h ) # X N ( t ) & 2 | X N ( t )= n & =lim h $ 0 1 h h N 2 ( & N ( n )+ N ( n ))+ o ( h ) = 0 ( & ( y )+ ( y )). Usingthesevalues,wedeneourSDEapproximationasfollows. Denition6 (SDEApproximation) Let 0 > 0 .Wesaythataprocess Y ( t ) isan SDE approximation forthefamilyofCTMCsdenedwithrateshapefunctions & and ifitsolves theItoStochasticDi! erentialEquation: d Y ( t )= a ( Y ( t )) d t + % 01 ( Y ( t )) d W ( t ) (314) where a ( y )= & ( y ) # ( y ) and 1 2 ( y )= & ( y )+ ( y ). (315) Specically,if 0 =1 / N ,thentheSDEassociatedwiththeCTMC X N ( t ) is Y ( t ) Proposition3.3.1. Let Y ( t ) bedenedasin ( 314 ) and ( 315 ) .Let T :=inf { t > 0: Y ( t ) $ { 0, y }} .Then u ( x ):= P { Y ( T )=0 | Y (0)= x } satisestheboundaryvalue problem ( 35 ) Proof. Theprooffollowsquicklyfromstochasticcalculus.Forageneralcontinuoussample-path SDEwithinnitesimalgenerator L ,theprobability u ( y ) thattheprocesshitsavalue a before b startingfrom y $ [ a b ] satisestheboundaryvalueproblem L u ( y )=0, for y $ [ a b ] with u ( a )=1 and u ( b )=0 [ 44 ]. Itremainstonotethatthegeneratorof Y ( t ) is L := a ( y ) d d y + 0 2 1 2 ( y ) d 2 d y 2 66

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3.3.2Analysisof p di! usion ( k ) Dene 2 ( x ):= 2( & ( x ) # ( x )) & ( x )+ ( x ) and ( x ):= 2 x 0 2 ( y ) dy (316) Proposition3.3.2. Let u ( x ) bethedi!usionapproximationfortheextinctionprobabilities, satisfying ( 35 ) ofDenition 4 .Then u ( x )=1 # ? x 0 e # # ( y ) / d y ? x 0 e # # ( y ) / d y (317) where x = min { x : & ( x )= ( x ) } Proof. Itwelet h ( x )= u ( ( x ) thentheODE( 35 )becomes h ( ( x )+ 1 0 2 ( x ) h ( x )=0, whichhasgeneralsolution h ( x )= C 1 e # 1 # # ( x ) .Integratingtoget u ( x )= ? x 0 h ( y ) d y and enforcingtheboundaryconditions u (0)=1 and u ( x )=1 yieldsthesolution( 317 ). WenotethatasimilarformulacanbefoundinPakmadanetal.[ 58 ].Wesummarizethe asymptoticbehaviorofthedi! usionapproximationasfollows. Theorem3.3.3. Supposethatthetransitionrateshapefunctions & and admitaseries expansionasdenedbyAssumption 2 .Iftheleadingorderexponents ) and areequal,then for k $ N p di! usion ( k )= ( ) ( 1 # exp # 2 k ( b # d ) b + d b > d ; 0, b d Otherwise,if ) & = ,then p di! usion ( k )= ( ) ( 1 # e # 2 k ) <0, ) >. Remarkably,thedi! usionapproximationessentiallyignoresthedetailedstructureofthe ratefunctionswhentheyhavedi!erentleadingorders. 67

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Theintegralsthatappearin( 317 )areinaformcompatiblewithapplyingWatson's Lemma[ 71 ].Infact,Watson'sLemmacanbeapplieddirectlytothedenominator.The subtletyinthepresentanalysisisthat,inthenumerator,theupperlimitofintegrationdepends on 0 .Toproceed,wefollowLaplace'smethodandintroduceasubstitution.Supposerstthat weareinthesuper-criticalcase,asdenedbyAssumption 1 ,sothatforall x $ (0, x ) we havethat 2 ( x ) > 0 .Itfollowsthat ( x ) isincreasingonthatintervalandthereforeinvertible. Letting t = "( y ) wehave 2 x 0 e # # ( y ) / d y = 2 # ( x ) 0 f ( t ) e # t / d t (318) where f ( t ):= 1 2 ( # 1 ( t )) = & ( # 1 ( t ))+ ( # 1 ( t )) 2 % & ( # 1 ( t )) # ( # 1 ( t )) & (319) Ontheotherhand,inthesubcriticalcase, 2 ( x ) < 0 when x $ (0, x ) .Writing 2 r ( x )= # 2 ( x ) and r ( x )= # ( x ) ,wesubstitute t = r ( x ) toattain 2 x 0 e # # ( y ) / d y = 2 # r ( x ) 0 f r ( t ) e t / d t (320) notingthattheexponentialinthesecondintegralhaspositiveexponent,where f r ( t ):= 1 2 r ( # 1 r ( t )) = ( # 1 r ( t ))+ & ( # 1 r ( t )) 2 % ( # 1 r ( t )) # & ( # 1 r ( t )) & (321) Theprooffollowsfromacombinationofthreelemmas.InLemma 3.3.6 weassumethat f ( t ) admitsaseriesexpansionandthenperformanasymptoticanalysisoftheratiosthatappear inintegralsoftheform( 318 )and( 320 ).InLemmas 3.3.4 and 3.3.5 weestablishthat 2 ( x ) and f ( t ) admitseriesexpansionswhen & ( x ) and ( x ) do,andprovidethecoe" cientsand powersoftherstfewterms.Afterpresentingtheselemmasandtheirproofs,wecompletethe proofofTheorem 3.3.3 68

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Lemma3.3.4. Supposethatthetransitionrateshapefunctions & and satisfyAssumption 2 Thenthereexistsanexpansionfor 2 ( x ) oftheform 2 ( x )= cx / m 1+ + n =1 c n x n / m (322) where # = ( ) ( min { n 1: b n & = d n } if ) = and b = d ; and 0, otherwise, and c = ( ( ( ( ( ( ( ) ( ( ( ( ( ( ( 2, if ) <; 2( b # d ) b + d if ) = and b & = d ; b # d if ) = and b = d ; and # 2, if ) >. Proof. Substitutingtheexpansions( 33 )infor & and ,wehave 2 ( x )= 2( & ( x ) # ( x )) & ( x )+ ( x ) = 2 bx 1+ n =1 b n x n / m # dx + 1+ n =1 d n x n / m "" bx 1+ n =1 b n x n / m + dx + 1+ n =1 d n x n / m (323) Case1: ) = b & = d Wecanfactorout x = x + fromthenumeratoranddenominator.Then ( 323 )simpliesto 2 ( x )= 2( b # d ) b + d 1+ n =1 3 # n x n / m 1+ n =1 3 + n x n / m where 3 n :=( bb n dd n ) / ( b d ) .FollowingthenotationintroducedinProposition 3.7.2 of theAppendix,wehave 2 ( x )= 2( b # d ) b + d 1+ + n =1 3 # n x n / m "! 1+ + n =1 @ 3 + n x n / m = 2( b # d ) b + d 1+ + n =1 c n x n / m & 69

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sothatinthenotationof( 322 ),wehave c =2( b # d ) / ( b + d ) and # =0 .Theremaining coe" cientscanbeexpressedintermsof { @ 3 + n } and { 3 # n } : c n = 3 # n + n # 1 + j =1 3 # n # j @ 3 + j + @ 3 + n Case2: ) = b = d Inthiscasewecanfactor bx = dx + fromthenumeratorand denominatorof( 323 ),whichleaves 2 ( x )= 2 % n = ( b n # d n ) x n / m & 2+ n =1 ( b n + d n ) x n / m = + n = / # n x n / m "! 1+ + n =1 @ / + n x n / m where / + n =( b n + d n ) / 2 forall n 1 ,and / # n = b n # d n for n # .Theformof @ / + n follows fromProposition 3.7.2 intheAppendix.Itfollowsthat 2 ( x ) canbewrittenintheformof ( 322 )withtheleadingorderexponentbeing # / m and c = b # d and c n = / # n + n # 1 + j = / # j @ / + n # j Case3: ) <. Inthiscasewebeginbyfactoringout bx fromthenumeratorand denominatorofequation( 323 ).Aftercancellationwehave 2 ( x )= 2 1+ n =1 b n x n / m # d b x + # 1+ n =1 d n x n / m "" 1+ n =1 b n x n / m + d b x + # 1+ n =1 d n x n / m FromthisformandProposition 3.7.2 ,weseethat # =0 and c =2 .Furthermore,the existenceofthesequence c n followsfromourassumptionthat ( ) # ) m isaninteger. Case4: ) >. Inthiscasewebeginbyfactoringout dx + fromthenumeratorand denominatorofequation( 323 ).Aftercancellationandsomerearrangementwehave 2 ( x )= # 2 1+ n =1 d n x n / m # b d x # + 1+ n =1 b n x n / m "" 1+ n =1 d n x n / m + b d x # + 1+ n =1 b n x n / m SimilartoCase3,weseethat # =0 and c = # 2 70

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Fromhere,theproofproceedsintwosteps.Firstweneedtorewritetheintegral ( x ) presentedin( 316 )inaformthatiscompatiblewithLemma 3.3.6 Lemma3.3.5. Supposethattheshapefunctions & and satisfyAssumption 2 andwewrite 2 asdescribedinLemma 3.3.4 .Furthermore,let # $ Z + beasdenedinLemma 3.3.4 .Then f ( t ) admitsaseriesexpansionoftheform f ( t )= at 1+ + n =1 a n t n m + " (324) where m isthepositiveintegerdenedinAssumption 2 and a = 1 c cm # + m / ( + m ) + = # # # + m and a 1 = c 1 cm # + m # 1 / ( + m ) # # + m +1 # 1 Proof. Itwillbeconvenienttowritetheseriesexpansionfor 2 intheform 2 ( x )= c + n =0 c n x 1 m ( n + ) withtheconventionthat c 0 =1 .Forthemainpartoftheproof,wewilltakethe c > 0 case. Attheend,wewillexplainhowtheargumentchangeswhen c < 0 .Integratingterm-by-term, wehave ( x )= c + n =0 c n m n + # + m x 1 m ( n + + m ) Itwillbeusefultodene h ( x ):="( x m ) withaseriesexpansionwrittenas h ( x )= + n =0 h n x n + + m where h n = cc n m n + # + m Since c > 0 ( x ) isincreasingon [0, x ] andtherefore h ( x ) isincreasingon [0, x 1 m ] .Setting t = h ( x ) ,byProposition 3.7.3 ,wecanexpresstheinverseof h intermsoftheseries h # 1 ( t )= + n =1 h n t n / ( + m ) 71

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where A h 1 =1 / h 1 / ( + m ) 0 A h 2 = # h 1 / % ( # + m ) h 1+2 / ( + m ) 0 & andsoon. Wenowturnourattentiontotheintegralofinterest,introducingthesubstitution z = y 1 m ,wehave I ( x ):= 2 x 0 e # # ( y ) / d y = 2 x 1 / m 0 mz m # 1 e # h ( z ) d z Then,taking t = h ( z ) ,andobservingthat d d z h ( z )= d d z ( z m )= 2 ( z m ) mz m # 1 ,wehave I ( x )= 2 h ( x 1 / m ) 0 m % h # 1 ( t ) & m # 1 h ( ( h # 1 ( t )) e # t / d t = 2 h ( x 1 / m ) 0 1 2 ( h # 1 ( t ) m ) e # t / d t Todeterminetheseriesexpansionfor f ( t ):=1 / 2 % h # 1 ( t ) m & ,weneedtoanalyzethe expansionforthedenominator: 2 ( h # 1 ( t ) m )= c + n =0 c n ( h # 1 ( t )) n + = c + n =0 c n + j =1 A h j t j / ( + m ) n + (325) Itishelpfultorewrite h # 1 ( t ) asfollows h # 1 ( t )= A h 1 t 1 / ( + m ) 1+ A h 2 A h 1 t 1 / ( + m ) + O ( t 2 / ( + m ) ) 72

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Recallingthat 2 ( x )= cx / m + cc 1 x ( +1) / m + O ( x ( +2) / m ) ,weobserve 2 ( h # 1 ( t ) m )= c ( h # 1 ( t ) )+ cc 1 ( h # 1 ( t ) +1 )+ O ( h # 1 ( t ) ( +2) ) = c A h 1 t / ( + m ) 1+ A h 2 A h 1 t 1 / ( + m ) + O ( t 2 / ( + m ) ) + cc 1 A h +1 1 t ( +1) / ( + m ) 1+ A h 2 A h 1 t 1 / ( + m ) + O ( t 2 / ( + m ) ) +1 + O ( t ( +2) / ( + m ) ) = c A h 1 t / ( + m ) !! 1+ A h 2 A h 1 t 1 / ( + m ) + O ( t 2 / ( + m ) ) + c 1 A h 1 t 1 / ( + m ) 1+ A h 2 A h 1 t 1 / ( + m ) + O ( t 2 / ( + m ) ) +1 + O ( t ( +2) / ( + m ) ). ThenbyapplyingProposition 3.7.1 twice,weobtain 2 ( h # 1 ( t ) m )= c A h 1 t / ( + m ) 1+ # A h 2 A h 1 + c 1 A h 1 t 1 / ( + m ) + O ( t 2 / ( + m ) ) + O ( t ( +2) / ( + m ) ). Proposition 3.7.2 thenyields f ( t )= 1 2 ( h # 1 ( t ) m ) = 1 c A h # 1 t # / ( + m ) 1 # # A h 2 A h 1 + c 1 A h 1 t 1 / ( + m ) + O % t 2 / ( + m ) & + O ( t ( +2) / ( + m ) ). Aftersimplifying,thestatedresultsfor a + and a 1 hold. When c < 0 ,wereplace 2 and with 2 r and r respectively.Theleadingtermof 2 r is then # c ,whichispositiveandexactsameprocedureholds. Lemma3.3.6. Supposethat f : R R admitsaseriesexpansionoftheform 324 with a > 0 + > # 1 { a n } n =1 1 R # $ Z + and m $ N .Furthermore,let g : R R be acontinuouslydi!erentiablefunctionthatismonotonicallyincreasingordecreasingforall t $ (0, t ) ,with g (0)=0 .Dene R k [ f ; g ]:=lim $ 0 ? g ( k ) 0 f ( t ) e t / d t ? g ( t ) 0 f ( t ) e t / d t 73

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Then R + k [ f ; g ]=0; R # k [ f ; g ]=1 # e # kg (0) Proof. Ourmethodistoreplace f ( t ) withitsseriesexpansionandconsidertheintegration term-by-term.Ourresultswillbeexpressedintermsoftheincompletegammafunction % ( s x ):= 2 x 0 t s # 1 e # t d t (326) Nearzero,theincompletegammafunctionhasthebehaviorthat lim x $ 0 % ( s x ) x s = 1 s Asusual,wedene # ( s )=lim x $' % ( s x ) .Notethatforany s > # 1 and A > 0 ,underthe substitution y = t / 0 ,wehave 2 A 0 t s # 1 e # t / d t = 0 s 2 A / 0 y s # 1 e # y d y = 0 s % ( s A / 0 ). Therefore, lim $ 0 ? A 0 t s # 1 e # t / d t 0 s = #( s ). Writing G for g ( k 0 ) and g ( t ) respectively,itfollowsthatthenumeratorandthedenominator appearingin R # k [ f ; g ] havetheform 2 G 0 f ( t ) e # t / d t = a 2 G 0 ( t + a 1 t +1 / m +...) e # t / d t = a 0 +1 % ( + +1, G / 0 )+ O % 0 +1+1 / m & Intakingtheratio,thecoe" cients a 0 +1 cancel,andneglectinghigherorderterms,wehave R # k [ f ; g ]=lim $ 0 % ( + +1, g ( k 0 ) / 0 ) % ( + +1, g ( t ) / 0 ) = % ( + +1, kg ( (0)) # ( + +1) When + =0 ,thistakestheform R # k [ f ; g ]=1 # e # kg (0) 74

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Fortheplus-signcase,wehave R + k [ f ; g ]=lim $ 0 ? g ( k ) 0 t e t / d t ? g ( t ) 0 t e t / d t Since g ( iscontinuous,let M :=max t & [0, t / 2] g ( ( t ) .ThenbytheTaylorRemainderTheorem, since g (0)=0 wehave g ( k 0 )= k 0 g ( ( / ), forsome / $ (0, t / 2) .Itfollowsthat g ( k 0 ) kM 0 forall t $ [0, t / 2] .Soforthenumerator wehave lim $ 0 2 g ( k ) 0 t e t / d t lim $ 0 2 kM 0 t e kM d t =0. Tostudythedenominator,weagainmakethesubstitution y = t / 0 andobtain lim $ 0 2 g ( t ) 0 t e t / d t =lim $ 0 0 2 g ( t ) / 0 ( 0 y ) e y d y =lim $ 0 0 +1 2 g ( t ) / 0 y e y d y Noticingthattheintegralontherighthandsidedivergestoinnity,werewritetheexpression andapplyl'Hopital'sruletondthatthedenominatordivergestoinnityas 0 0 .Itfollows thattheratiotendsto0as 0 0 Weintroducedthenotation and g toemphasizethatthecontributiontothenalvalue comesfromthelimitofintegration,notthefunction f intheintegrand. ProofofTheorem 3.3.3 First,recallingDenition 4 andtheresultfromProposition 317 ,we ndthat p di! usion ( k )= ? x 0 e # # ( y ) / d y ? x 0 e # # ( y ) / d y Noticingthefamiliarformofthisexpression,weapplyLemma 3.3.5 with g setto torewrite p di! usion ( k ) inaformcompatiblewithLemma 3.3.6 75

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When ) >or ) = with b d Lemma 3.3.6 implies p di! usion ( k )= R + k [ f ;"]=0. Alternatively,if ) d ,Lemma 3.3.6 showsthat p di! usion ( k )= R # k [ f ; "]=1 # e # k # (0) Since ( (0)= 2 (0) ,byDenition 316 ,theresultfollowsfromLemma 3.3.4 3.4Exponentialapproximation Doeringetal[ 20 21 ]demonstratedthatmakingaWKB-typeansatzoftheform q n ) 1 n e # V n forsomefunctions 1 and V ,canbeanaccuratemethodforconstructingacontinuum approximationforsolvingKolmogorovequations.Inthemotivationthatfollowsweshowhow asimilar,ande!ective,approximationcanbereachedbytransformingthesystemofequations denedby( 310 )intoanequationforratiosinsteadofdi!erences,thenapplyingaTaylor SeriesexpansiontechniquesimilartowhatispresentedasMotivation2fortheDi!usion Approximation. 3.4.1MotivationfortheExponentialApproximation For n =1,2,..., N ,dene a N ( n ):= q N ( n # 1) / q N ( n ) (327) Since q N (0)=1 ,wecanwrite q N ( n )= n 7 k =1 1 a N ( k ) = n 7 k =1 exp( # ln a N ( k )). where & N ( n ) a N ( n +1) # ( & N ( n )+ N ( n ))+ N ( n ) a N ( n )=0. (328) 76

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Wecanrewritetheexponentontheright-handsidetolooklikeaRiemannsum: q N ( n )=exp 0 # N n + k =1 ln % a N ( k ) & 1 N 1 (329) MimickingthecontinuumapproximationsandTaylorexpansionsintroducedfortheDi!usion Approximation,weintroducethefunction r ( x ) .For x = n / N and 0 =1 / N ,wewillwrite a N ( n ) ) r n N = r ( x ). (330) Then, n + i =1 ln % a N ( i ) & 1 N ) 2 x 0 ln r ( y ) d y Assuch,weassertthat q N ( n ) ) exp # 1 0 2 x 0 ln r ( y ) d y / andnotethatthisreplacementwithanintegralintroduceserrorintoourexpressionfor q N ( n ) whichmayaccountforsomeoftheerrorweobservelater.Itremainstocharacterizethe function r ( x ) Indeed,wewillnowshowthat,followingaTaylorSeriesexpansiontechnique,wehave r ( x )= $ ( x )+ 0 $ ( ( x ) $ ( x ) # 1 + 0 2 $ (( ( x ) $ 2 ( x ) # 2( $ ( ( x )) 2 2( $ ( x ) # 1) 3 + O ( 0 3 ). (331) Dividing( 328 )throughby N ( n ) andmultiplyingby a N ( n +1) wehave & N ( n ) N ( n ) + a N ( n ) a N ( n +1)= & N ( n ) N ( n ) +1 / a N ( n +1). Let $ ( x ):= & ( x ) / ( x ), (332) usingthedenitionofthetransitionrateshapefunctions & and & N ( n ) N ( n ) = N & % n N & N % n N & = $ n N 77

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Thus, $ n N + r n N r n +1 N / = $ n N +1 r n +1 N / and,using x and 0 for n / N and 1 / N ,wearriveat $ ( x )+ r ( x )+ r ( x + 0 ) # (1+ $ ( x )) r ( x + 0 )=0. Neglectinghigherorderterms,wemakethesubstitution r ( x ) ) r 0 ( x )+ 0 r 1 ( x )+ 0 2 r 2 ( x ), andobtain $ ( x )+( r 0 ( x )+ 0 r 1 ( x )+ 0 2 r 2 ( x ))( r 0 ( x + 0 )+ 0 r 1 ( x + 0 )+ 0 2 r 2 ( x + 0 )) # (1+ $ ( x ))( r 0 ( x + 0 )+ 0 r 1 ( x + 0 )+ 0 2 r 2 ( x + 0 )). NowweperformaTaylorexpansionofthe r 0 ( x + 0 ), r 1 ( x + 0 ), and r 2 ( x + 0 ) terms.Thenby organizingthetermsbypowersof 0 ,wendthefollowingsystemofequations: 0 0 : $ ( x )+ r 2 0 ( x ) # (1+ $ ( x )) r 0 ( x )=0, 0 1 : r 0 ( x ) r ( 0 ( x )+2 r 0 ( x ) r 1 ( x ) # (1+ $ ( x ))( r ( 0 ( x )+ r 1 ( x ))=0, 0 2 : 1 2 r 0 ( x ) r (( 0 ( x )+ r 0 ( x ) r ( 1 ( x )+2 r 0 ( x ) r 2 ( x )+ r 1 ( x ) r ( 0 ( x )+ r 2 1 ( x ) # 1 2 (1+ $ ( x )) r (( 0 ( x ) # (1+ $ ( x )) r ( 1 ( x ) # (1+ $ ( x )) r 2 ( x )=0. Therstoftheseequationsyieldstwosolutionsfor r 0 ( x ) : r 0 ( x )= $ ( x ) and r 0 ( x )=1 .To showthattheformermustbetrue,considerthatif r 0 ( x ) 2 1 ,them r ( 0 ( x ) 2 0 .Substituting intothe 0 1 equation,itfollowsthat r 1 ( x )=0 .Continuinginthisway,wend r i ( x )=0 for all i 1 .Thiscorrespondstothesolution q N ( n )=1 forall n $ { 0,..., N } ,whichcannot matchtheboundarycondition q N ( N )=0 .Therefore,weadoptthesolution r 0 ( x )= $ ( x ) Feedingthisintotheorder 0 1 equationallowsustosolvefor r 1 ( x ) .Similarly,substitutingthese 78

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solutionsintothe 0 2 equationyieldsasolutionfor r 2 ( x ) andsoon: r 0 ( x )= $ ( x ), r 1 ( x )= $ ( ( x ) $ ( x ) # 1 r 2 ( x )= $ (( ( x ) $ 2 ( x ) # 2( $ ( ( x )) 2 2( $ ( x ) # 1) 3 3.4.2Analysisof p exp ( k ) Theorem3.4.1 (FirstOrderExponentialApproximationforSupercriticalSystems) Suppose thatthetransitionrateshapefunctions & and formasupercriticalsystem,asdened inAssumption 1 .Furthersupposethat & and admitaseriesexpansion,asdenedby Assumption 2 .Let w ( x ) bearstorderapproximationoftheforminDenition 5 ,wherewe set r e p ( x ):= r 0 ( x ) ,andfor k $ N ,weset p exp ( k ):=lim $ 0 w ( k 0 ) Iftheleadingorderexponents ) and areequal,then p exp ( k )=1 # d b k when b d (333) Otherwise,when ) <, p exp ( k )=1 Theprooffollowsdirectlyfromtaking 0 0 inEquation 334 inthestatementofLemma 3.4.2 ,whichfollows. Remark2. When & and formasubcriticalsystemtheexponentialapproximationisnotwell dened.Thisisbecause,inthisregime,thefunction w ( k 0 ) isoftengreaterthanone,meaning that 1 # w ( k 0 ) isnegativeandnotareasonableestimateforaprobability.Forexample, notethatthisiswhathappensto ( 333 ) when ) = and b < d .Theproposedvaluefor p invasion ( k ) isnegativeandcannotbeaninvasionprobability.Apractitionercouldtakethe exponentialapproximationtobezerowheneverthesystemofstudyissubcritical(seeTable 3-1 ). Inthefollowinglemma,wecalculatetherstorderexponentialapproximationforthe extinctionprobability.ThisquantityformsthefoundationfortheresultsinTheorem 3.4.1 79

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Lemma3.4.2. Supposethattheshapefunctions & and admitaseriesexpansionasdened byAssumption 2 ,thentherstorderexponentialapproximationfortheextinctionprobability hastheform: exp # 1 0 2 k 0 ln $ ( y ) d y = d b k e k 0 ( # + ) k exp d 1 # b 1 1+1 / m k ( k 0 ) 1 / m + O (( k 0 ) 2 / m ) (334) Proof. Firstwendaseriesexpansionfor $ ( x ) .Fromthedenitionof $ inequation( 332 ) andtheseriesexpansionsfor & and assumedinAssumption 2 ,wehave $ ( x )= b d x # + 1+ + n =1 b n x n / m "! 1+ + n =1 d n x n / m # 1 ByapplyingProposition 3.7.2 andseparatingouttheleadingorderterms,weobtain $ ( x ) ) b d x # + 1+ b 1 x 1 / m + + n =2 b n x n / m "! 1 # d 1 x 1 / m + + n =2 d n x n / m + + n =1 d n x n / m 2 Collectingorderedtermsyieldstheexpansion $ ( x )= b d x # + (1+( b 1 # d 1 ) x 1 / m + O ( x 2 / m )). (335) Toconsidertherstorderexponentialapproximationweuseonetermof r ( x ) ,i.e. r ( x )= $ ( x ) .Thenbypluggingequation( 335 )intoDenition 5 ,wehavethefollowing exponentialapproximationfortheprobabilityofextinction: w ( x )=exp # 1 0 2 x 0 ln b d y # + (1+( b 1 # d 1 ) y 1 / m + O ( y 2 / m ) d y / whichcanthenbeexpressedintermsofhaving k individualsintroduced w ( k 0 )=exp # 1 0 2 k 0 ln( b / d ) d y exp # 1 0 2 x 0 ln( y # + ) d y 3 exp # 1 0 2 k 0 ln(1+( b 1 # d 1 ) y 1 / m + O ( y 2 / m ) d y (336) 80

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Consideringthersttermintheproductofequation( 336 ), exp # 1 0 2 k 0 ln( b / d ) d y =exp # 1 0 k 0 ln( b / d ) = d b k (337) Thenusinganexpansionfor ln(1+ x ) andintegratingtheresult,thethirdtermintheproduct becomes 2 k 0 ln(1+( b 1 # d 1 ) y 1 / m + O ( y 2 / m )) d y = 2 k 0 ( b 1 # d 1 ) y 1 / m + O ( y 2 / m ) d y = b 1 # d 1 1+1 / m ( k 0 ) 1+1 / m + O (( k 0 ) 1+2 / m ). Thusthethirdtermintheproductofequation( 336 )canberewrittenas exp # 1 0 2 k 0 ln(1+( b 1 # d 1 ) y 1 / m + O ( y 2 / m ) d y =exp d 1 # b 1 1+1 / m k ( k 0 ) 1 / m + O (( k 0 ) 2 / m ) (338) Rearrangingandintegratingtheexponentfromthesecondterminequation( 336 ),wehave # 1 0 2 k 0 ln( y # + ) d y = # ) # 0 2 k 0 ln y d y = # ) # 0 ( y ln y # y ) = = = k 0 d y = # ( ) # )( k ln( k 0 ) # k ). Therefore,thesecondtermbecomes exp # 1 0 2 x 0 ln( y # + ) d y = e k 0 ( # + ) k (339) Theresultfollowsbyrewritingequation( 336 )usingequations( 337 ),( 338 ),and( 339 ). InLemma 3.4.2 ,weshowthattheinitialproportionofthepopulationintroducedplaysa nontrivialroleindeterminingtheapproximatedvalue,evenwhenconstrainedtotherstorder exponentialapproximationregime.Weexplicitlyincludedthenumberofindividualsintroduced astheproportionofthepopulation( k 0 )intheerrortermtorecordthaterrortermsmaybe 81

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nontrivialanddonotnecessarilyapproachzerowith 0 ifthenumberofindividualsintroduced isalsoallowedtovary.Theerrorattributedbyapproximatingseriesalongthewaytothenal expressiondependsontheproportionofthepopulationthatisinitiallyintroduced. 3.5Numericalobservationsforinvasionprobabilityapproximations Intheprecedingsections,ourmainanalyticalresultsforthebranchingprocess,di! usion, andexponentialapproximationsfocusonthelargepopulation( N "( )limit.Inthis section,wevalidateourasymptoticresultsnumericallyandthenshiftourattentiontoconsider thebehavioroftheapproximationswhenthepopulationsizeissmallorintermediate.We challengethemethodsintheirabilitytobestapproximatetheexactsolutionasdenedand calculatedinSection 3.2 .Wefocusourinvestigationonthespecicexamplespresented intheintroduction,andobtainexactsolutionsfortheapproximationswheneverpossible. Weusenumericalintegrationwhenitisnotpossibletoobtainanexactsolutionforan approximation.Inthisway,weverifyouranalyticalresultsandexplorehowtheconditionsof asystem(e.g.populationsizeandsubcriticalvs.supercriticaldynamics)haveanimpacton whetheraparticularapproximationmethodshouldbedeemedtforuse.Wesetthedeath rateparameter ( d ) tobeonebydefault. 3.5.1Di!usionapproximationmethodsfailforlargepopulationswhendynamicsare supercritical Tocomplementouranalysisintheprevioussection,weusedexactsolutionswhenpossible andnumericalintegrationwhennecessary(Simpson'smethodcodedinR)toevaluatethe di usionapproximationandexponentialapproximationforinvasionprobabilities.InFigure 3-2 theresultsareshownforExamples 1 2 ,and 3 forarangeofpopulationsizesandparameter valuechoices.Wechosetheexponentsandcoe"cientssothatthedynamicsaresupercritical andtheexponentialapproximationiswelldened.Ineachpanel,wehighlightthatasthetotal populationsizebecomeslarge,thevaluescalculatednumericallyapproachtheircorresponding analyticallydeterminedlimit(indicatedby"*"). 82

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ForthesupercriticalsystemsdisplayedinFigure 3-2 thedi! usionapproximationyieldsa di erentanswerthantheexactsolutioninthelargepopulationsizelimit.Thisdiscrepancy betweenthedi!usionapproximationandtheexactsolutionconrmsourasymptoticanalysis andismostapparentwhentheleadingcoe"cientsofthebirthanddeathratesaredissimilar. AsdisplayedinFigure 3-2 C, 3-2 F, 3-2 I,andforpopulationsizesgreaterthan100inFigure 3-3 ,thedi!usionapproximationfailstomatchtheexactsolutionwhenthedynamicsarefar fromcritical.Theinabilityofthedi! usionapproximationtoapproachoneispredictedbyour asymptoticresultthat p di usion ( k )=1 # e # 2 k when ) <.ThiscanbeseeninFigure 3-2 D, 3-2 E, 3-2 F,andinFigure 3-3 A, 3-3 B, 3-3 C, 3-3 D.Theseplotsalsohighlightthephenomenon that,iftheleadingorderexponentssatisfy ) <,theninthelargepopulationlimit,the di usionapproximationcompletelyignorestheparametersoftheratefunctions. Bycontrast,thedi!usionapproximationsucceedsincharacterizingthelargepopulation sizebehaviorwhenthedynamicsaredominatedbythedeathrate( isleadingorder).Thisis seeninFigure 3-3 E, 3-3 F, 3-3 G, 3-3 H,aspredictedbyourresultthat p di usion ( k )=0 when ) >.(SeeTheorem 3.3.3 .) 3.5.2Di!usionapproximationmethodscanworkwellforsmallpopulationsthat exhibitnearcriticaldynamics Intheexampleswehavestudied,thedi! usionapproximationconsistentlyoutperformsthe exponentialapproximationwhenthepopulationissmallandtheparametersetisnearcritical. ThisresultisrstdemonstratedinFigure 3-2 .Movingfromlefttorightinthismulti-panel gure,theparametercharacterizingthebirthrate( b )movesfartherawayfromthedeathrate parameter( d )whichissettoone.Assuch,Figure 3-2 A, 3-2 D, 3-2 G,showsthatthedi! usion approximationcloselyapproximatestheexactsolutionfortheinvasionprobabilityforsmall populationsizes. Thedi!usionapproximation'stransitionfromexceptionaltopoorperformanceiseven moreclearlydemonstratedwhenstudyingExample 4 .InFigure 3-3 A, 3-3 B, 3-3 C, 3-3 D,we seethatthedi! usionapproximationcapturesnon-monotonicfeaturesoftheexactsolution 83

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thattheexponentialapproximationmissesentirely.Inthisexample,thedynamicsaredictated bythebirthratesince & featuresthelowerleadingorderterm.Whenthedynamicsarenear critical b =0.7 and b =1.1 ,thereisarangeofsmallpopulationsizeswherethedi!usion approximationtrackstheexactsolution. 3.5.3Whenleadingordertermsmatch,higherordertermsmatter:forsmall,but notlargepopulationsizes Whentheleadingorderofthe"birth"and"death"ratesarethesameandtheirleading coe" cientsareequal,subtletiesintheoutcomesaredeterminedbytherstpairofmismatched coe" cients.ThisresultisdisplayedprominentlyinFigure 3-4 forwhichwechosebothrate functionstobeasymptoticallylinear( ) = =1 ),tosharethesameleadingcoe"cient ( b = d =1 ),buttohavedi!erentvaluesforthecoe"cients b 1 and d 1 (seeAssumption 2 ). InFigure 3-4 A, b 1 > d 1 andthesystemissupercritical(asdenedinAssumption 1 ). Inthiscase,boththedi! usionapproximationandrstorderexponentialapproximationare welldened.Asthepopulationsizebecomeslarge,theapproximationscorrectlypredict thattheinvasionprobabilityapproacheszero.Thecorrespondinganalyticalresultsare presentedinTheorems 3.3.3 and 3.4.1 ,respectively,withtheirasymptoticlimitsindicated intheplotasblackandgray"*"s.Limitationsofthenumericalintegrationprocedureprevent usfromdisplayingthedi! usionapproximationforthefullrangeofpopulationsizes(i.e.up to10,000).Themaindi! erencebetweenthetwoapproximationsinthisregimeistherate ofconvergencetozero,withrespecttoincreasesinpopulationsize.Consistentwiththe di usionapproximation'ssuccessforrelativelysmallpopulationsizes,itinitiallytracksthe exactsolution. InFigure 3-4 B, b 1 < d 1 andthedynamicsaresubcritical.InTheorem 3.4.1 ,wenoted thattheexponentialapproximationdoesnotholdwhen b < d .Asadirectconsequenceof equation( 334 ),wefurtherobservethattheexponentialapproximationwillbeinvalidfor su cientlysmallpopulationsizes.Inparticular,when d = b and ) = ,wehave w ( k 0 )=exp d 1 # b 1 1+1 / m ( k 0 ) 1 / m + O (( k 0 ) 2 / m ) > 1. 84

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Sincetheexponentialapproximationis 1 # w ( k 0 ) ,thisreturnsaninvalidvaluelessthanzero. Fromapracticalpoint-of-view,onecouldsimplydenetheexponentialapproximationtobe zeroinsuchacircumstance. 3.5.4Approximationsuccessdependsontheinitialnumberofindividualsintroducedinthepopulation Whenmorethanoneindividualisinitiallyintroducedinapopulation,theprobabilityof invasionincreases.InFigure 3-5 ,wedisplayresultsforeachapproximationmethodalongwith theexactsolutionfortheprobabilityofextinctionwhen 1 k < Nx individualsareinitially introducedinthepopulation.Bydenition,foralllargervaluesof k ,theinvasionprobabilityis one. AsshowninFigure 3-5 A, 3-5 B, 3-5 C,whentheleadingordertermsmatch,thedi!usion approximationperformswellandtrackstheexactsolutionoverthefullrangeofinitial numbersofindividualsintroduced.Typicallythesecondorderexponentialapproximation (darkgraydashedline)betterapproximatestheexactsolutionthantherstorderexponential approximation(lightgraydashedline).However,when k iscloseto Nx thesecondorder exponentialapproximationsharplyturnsup,awayfromtheexactsolution.Thisnumerical resultisinlinewithexpectationsfromouranalyticalresultsinequation( 331 )sincethe additionalhigherordertermisundenedfor $ ( x )=1 ,i.e.when & ( x )= ( x ) Inspecialcases,itispossibletocomputetheexponentialapproximationbyhand usingDenition 5 .Wevalidatedtheexponentialapproximationfortheepidemicextinction probability(Example 1 )bycomparingtheexactresultfortherstorderapproximation (denoted p exp,1 ( k ) ), p exp,1 ( k )= d b (1 # k 0 ) k withthenumericalintegrationresultplottedinFigure 3-5 .Therewasnotavisibledi!erence betweenthevaluefoundbyhandandthenumericallyobtainedvalue. Wealsocomparedtheexactandnumericalvaluesfortheexponentialapproximationwith theanalyticalapproximationfoundinLemma 3.4.2 .Duringthisinvestigation,wefoundthat 85

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sincetherangeofpotentialnumberofintroducedindividualsscaleswiththetotalpopulation size,itisimportanttokeeptrackofthe k parameterintheerrorterminequation( 334 ).As k approaches Nx ,theerrortermbecomessignicant(evenforlargepopulationsizes).For xed k ,thepopulationsizecanbechosenlargeenoughtoyieldanapproximationwiththe desiredlevelofaccuracy. 3.6Discussion MathematicalmodelersuseavarietyoftechniquestoapproximateMarkovchainmodels forpopulationprocesses.ThewidespreaduseofSDEs,inparticular,raisesthequestionofhow di erenttheresultsareforbasicprobabilityquestions:inparticularhittingtimesandhitting probabilities.Ouranalysisofthemostprominentapproximationmethodsyieldsmixedresults andweassertthatitremainsanopenquestiontocompletelyunderstandwhichapproximation isbestforwhichcircumstances. Inthismanuscript,wehaveshownthatbothpopulationsizeandthestateofsub-versus super-criticalityplayanimportantroleindeterminingwhichapproximationmethodmostly faithfullyreproducestheprobabilityofinvasionforagivenMarkovchainmodel.Whenthe dynamicsarenearcriticalandthepopulationofinterestissmall,thedi! usionapproximation bestapproximatestheexactsolution.Bycontrast,forsupercriticalsystemsthatarefarfrom critical,theexponentialapproximationyieldsthecorrectexactsolutionforlargepopulations whilethedi! usionapproximationvisiblymissesthemark. Thisanalysisraisesseveralquestionsgoingforward.Firstamongtheseisdetermining howtheinclusionofhigher-ordertermsa!ectstheexponentialapproximation.Aspointedout inSection 3.4.2 ,therstorderexponentialapproximationisinvalidwhenthedynamicsare subcriticalbecausethemethodreturnsnegativevalues.Thisisincontrasttothedi! usion approximation,whichalwaysreturnsvaluesbetween0and1.Itispossible,butnotatallclear, thattakinghigherorderapproximationsoftheexponentialapproximationorafullyformulated WKBapproximationwouldenabletheexponentialfamilyofapproximationmethodstohandle subcriticalcases. 86

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Itisalsoanimportantchallengetobetterunderstandthebehavioroftheapproximations innear-criticalparameterregimes.Asacontinuousapproximationmethod,itisanunexpected resultthatthedi!usionapproximationsometimesperformswellforsmallpopulations(lessthan 100individuals)whileyieldingthewronganswerinthelargepopulationlimit. InTheorem 3.4.1 ,weshowthatforasymptoticallylinearsupercriticalsystemsthe exponentialapproximationapproachesthebranchingprocessapproximationvalueinthelarge populationlimit.ConsideringExample 1 andExample 3 ,weverifythisresultnumericallyin Figure 3-2 andFigure 3-5 .Whilewewerenotabletocalculatethelimitoftheexactsolution analytically,ournumericalresultsindicatethatthisisthecorrectlimitingvalue.Assuch,itis animportantconsequenceofourworkthatapproximationscanbutdonotalwaysimproveas thepopulationsizeincreases. Moreinvestigationisneededtofullycharacterizetheexponentialapproximation, particularlywhentheinitialnumberofindividualsintroducedisclosetotheinvasionthreshold populationsize.InTheorem 3.4.1 ,weshowthatforxed k ,theexponentialapproximation approachesthebranchingprocessapproximationinthelargepopulationlimit.However, aswesawinSection 3.5.4 ,thevaluefortheexponentialapproximationinLemma 3.4.2 dependsstronglyontheinitialproportionofthepopulationintroduced.Eachtimethatan approximationoftheexactexponentialapproximationismade,evenforlargepopulationsizes, iftheinitialnumberofindividualsislargerelativetothepopulationsize,theincrementalerrors contributedateachapproximationstepmaynotbetrivial.Intheanalysispresentedhere, wefocusedonthe"rstorder"exponentialapproximationwhereweconsideronlytherst termof r ( x ) .Figure 3-5 illustratesthataddingevenonehigherordertermtotheexponential approximationcansignicantlychangetheapproximatedvaluewhenthenumberofintroduced individualsisnearthecriticalpopulationsize. 87

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3.7SeriesFormulas Proposition3.7.1 (SeriesExpansionI) Foragivenpowerserieswith % 0 and > 0 ,there existcoe" cients a n suchthat + n =0 a n x n / m + ) = + n =0 a n x n / m + !) wheretherstfewcoe" cientsare a 0 = a ) 0 , a 1 = a ) # 1 0 a 1 and a 2 = a ) # 1 0 a 2 + ( # 1) a 2 1 2 a 0 Proof. Theproofbeginsbyfactoringouttheleadingterminthegivenseries + n =0 a n x n / m + ) =( a 0 x ) ) 1+ + n =1 a n a 0 x n / m ) Thenmakethesubstitution y = n =1 a n a 0 x n / m andndtheTaylorexpansionof (1+ y ) ) at1, (1+ y ) ) =1+ y + ( # 1) 2 y 2 + O ( y 3 ) Theybypluggingintheseriesfor y andcollectingtermsofthesamepower,wehave 1+ + n =1 a n a 0 x n / m ) =1+ a 1 a 0 x 1 / m + a 0 a 2 + ( # 1) a 2 1 2 a 0 x 2 / m + O ( x 3 / m ). Theformofthenewseriesisfoundbymultiplyingeachtermbytheoriginalleadingorderterm ( a 0 x ) ) Proposition3.7.2 (SeriesExpansionII) Supposeapowerseriesisgivenintheform, n =0 a n x n / m ,with a 0 & =0 a n $ R for n 1 ,and m > 0 .Whenever x issuchthat = = = n =1 a n x n / m = = = < | a 0 | ,then 1 n =0 a n x n / m = + n =0 @ a n x n / m where @ a 0 = 1 a 0 @ a 1 = # a 1 a 0 and @ a 2 = a 2 1 # a 2 a 0 88

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Proof. Supposethereisagivenpowerseriesthatsatisestheconditionsoftheproposition. Furthersupposethattheinequality = = = n =1 a n x n / m = = = < | a 0 | holds.Byrearrangingouroriginal expression,wendafamiliarform 1 n =0 a n x n / m = 1 a 0 1+ 1 a 0 n =1 a n x n / m Sincewehaveassumedthat = = = n =1 a n x n / m = = = < | a 0 | 1 n =0 a n x n / m = 1 a 0 + j =0 ( # 1) j + n =1 a n x n / m j = 1 a 0 1 # ( a 1 x 1 / m + a 2 x 2 / m +...)+( a 2 1 x 2 / m +...)+... = 1 a 0 1 # a 1 x 1 / m +( a 2 1 # a 2 ) x 2 / m +... Therstfewcoe"cientsoftheresultingseriesareobtainedbydistributingthe ( a 0 ) # 1 inthe lastline. Proposition3.7.3 (SeriesInversion) Suppose f ( x ) isincreasingon [0, x ) .Furtherassume that f (0)=0 and f ( x ) canbewrittenasaformalpowerseries: f ( x )= + n =0 a n x n + ) Let t = f ( x ) .Thentheinverseoftheseries f canbeexpressedas x = + n =1 A a n t n / ) as t 0 + where A a 1 = 1 a 1 / ) 0 and A a 2 = # a 1 a 1+2 / ) 0 Proof. Werstrewrite f ( x ) asfollows f ( x )= x ) + n =0 a n x n 89

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Setting t = f ( x ) ,wehave t 1 / ) = xg ( x ), where g ( x ):= + n =0 a n x n 1 / ) Then,byProposition 3.7.1 ,wehave g ( x )= + n =0 a n x n where a 0 = a 1 / ) 0 , a 1 = a 1 / ) # 1 0 a 1 , a 2 = a 1 / ) # 1 0 (1 / # 1) a 2 1 2 a 0 + a 2 Then, t 1 / ) = a 1 / ) 0 x 1+ a 1 a 0 x + 1 a 0 (1 / # 1) a 2 1 2 a 0 + a 2 x 2 + O ( x 3 ) and t 2 / ) = a 2 / ) 0 x 2 1+ 2 a 1 a 0 x + 2 a 0 (1 / # 1) a 2 1 2 a 0 + a 2 + a 1 a 0 2 x 2 + O ( x 3 ) Choose A a 1 = a # 1 / ) 0 sothatitcancelstheleadingcoe" cientof t 1 / ) .bewrittenasFromhere weusepowersof t 1 / ) tondanexpansionoftheform x + + n =1 A a n t n / ) Bycollectingorderedterms,wendthecorrespondingtwotermapproximation, A a 1 t 1 / ) + A a 2 t 2 / ) = x + a 1 a 0 + A a 2 a 2 / ) 0 x 2 + O ( x 3 ) Wethenchoose A a 2 = # a 1 a 1+2 / ) 0 sothatthecoe" cientforthe x 2 termiszero.Increasinglypreciseapproximationsof x can befoundbykeepingtrackofthehigherorder t termsandchoosingeach A a n toappropriately cancelouttheseterms. 90

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! ! 1 1 1 N N N N =10 N =100 N =1000 t t t 5 5 5 5 7 3 0 0 0 N N N A B C Figure3-1.Numberofinfectiousindividualsresultingfromtheintroductionofoneinfectious individualinapopulationofsize N .TensamplepathsforthestochasticSISmodel denedbyExample 1 areplotted(redlines)withthesolutiontotheanalogous ordinarydi!erentialequationmodel(blackcurve).Ourrepresentationforhaving achievedtheendemicstateis N (dashedhorizontalblackline),whichisdenedin Denition 1 .Opencirclesareplottedwhen N isreachedbefore0andredpoints indicatewhenthepathogendiedoutofthepopulationbeforereaching N .Forall simulations b =2 and d =1 .A) N = 10.B) N = 100.C) N = 1000. 91

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0.0 0.2 0.4 0.6 0.8 1.0 b= 2 N Invasion probability 2 10 100 1000 * 0.0 0.2 0.4 0.6 0.8 1.0 b= 4 N Invasion probability 2 10 100 1000 * 0.0 0.2 0.4 0.6 0.8 1.0 b= 10 N Invasion probability 2 10 100 1000 * Exact Solution Exponential Approx Diffusion Approx 0.0 0.2 0.4 0.6 0.8 1.0 b= 1 N Invasion probability 2 10 100 1000 * 0.0 0.2 0.4 0.6 0.8 1.0 b= 2 N Invasion probability 2 10 100 1000 * 0.0 0.2 0.4 0.6 0.8 1.0 b= 4 N Invasion probability 2 10 100 1000 * 0.0 0.2 0.4 0.6 0.8 1.0 b= 1.1 N Invasion probability 2 10 100 1000 * 0.0 0.2 0.4 0.6 0.8 1.0 b= 2 N Invasion probability 2 10 100 1000 * 0.0 0.2 0.4 0.6 0.8 1.0 b= 4 N Invasion probability 2 10 100 1000 * Invasion probability Population size Epidemics Density dependent mortality Resource constrained births b =1 1 b =1 b =2 b =2 b =2 b =4 b =4 b =4 b =10 ( x )= bx ( x )= dx 2 ( x )= bxe x/a ( x )= dx ( x )= bx (1 x ) ( x )= dx A B C D E F G H I Figure3-2.Invasionprobabilitycontinuumapproximationsandtheexactsolutionforthe introductoryexamples:Example 1 stochasticepidemics( ) = =1 ),Example 2 densitydependentmortality( ) =1 < =2 ),andExample 3 resource-constrained birth( ) = =1 and a =1 ).Inallpanels,wesetthedeathrateparameter d =1 Numericalintegrationwasusedtocalculatethevaluesforbothapproximation techniquesforExample3.A-C)Example 1 .D-F)Example 2 .G-I)Example 3 92

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0.0 0.2 0.4 0.6 0.8 1.0 b= 0.7 N Invasion probability 2 10 100 1000 0.0 0.2 0.4 0.6 0.8 1.0 N Invasion probability 2 10 100 1000 0.0 0.2 0.4 0.6 0.8 1.0 b= 1.1 N Invasion probability 2 10 100 1000 0.0 0.2 0.4 0.6 0.8 1.0 N Invasion probability 2 10 100 1000 0.0 0.2 0.4 0.6 0.8 1.0 b= 2 N Invasion probability 2 10 100 1000 0.0 0.2 0.4 0.6 0.8 1.0 N Invasion probability 2 10 100 1000 0.0 0.2 0.4 0.6 0.8 1.0 b= 4 N Invasion probability 2 10 100 1000 Exact Solution Exp Approx 1 term Diffusion Approx 0.0 0.2 0.4 0.6 0.8 1.0 N Invasion probability 2 10 100 1000 Invasion probability Population size = # 1 0 5 1 0 3 3 1 = # 1 3 0 3 1 0 5 1 A B C D E F G H ! ! b =0 7 b =1 1 b =2 b =4 Figure3-3.InvasionprobabilitycontinuumapproximationsandtheexactsolutionforExample 4,inwhichthebirthanddeathratesaresingletermswithmismatchedexponents. Inallpanels,wesetthedeathrateparameter d =1 .A-D)Theleadingorderisin thebirthrateequation & ( x ) .E-H)Theleadingorderisinthedeathrateequation ( x ) .Sincetheexponentialapproximationisinvalidinthesubcriticalcase,we haveomittedtheapproximation,where ) >. 93

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0.0 0.2 0.4 0.6 0.8 1.0 2 10 100 1000 10000 * 0.0 0.2 0.4 0.6 0.8 1.0 2 10 100 1000 10000 Invasion probability Population size Exact Solution Exp Approx 1 term Diffusion Approx !"#$%&'(")*+(%#,+-,%.')/+"&.,&+ ( = =1) ( x )= x (1+ x ) ( x )= x (1 x + x 2 ) b 1 =1 ,d 1 = 1 0.0 0.2 0.4 0.6 0.8 1.0 2 10 100 1000 10000 * 0.0 0.2 0.4 0.6 0.8 1.0 2 10 100 1000 10000 Invasion probability Population size Exact Solution Exp Approx 1 term Diffusion Approx ( x )= x (1 x + x 2 ) ( x )= x (1+ x ) b 1 = 1 ,d 1 =1 A B Figure3-4.Invasionprobabilitycontinuumapproximationsandtheexactsolutionwhenboth rateequationsareasymptoticallylinear( ) = =1 )andtheleadingcoe"cients match( b = d =1 ).Theinvasionprobabilitytendstozeroasthepopulationsize N becomeslarge.Di!erencesbetweenthepanelsaredrivenbytheleadingpairof termswithmismatchedcoe"cients.A)Supercriticalcasewith b 1 > d 1 .B) Subcriticalcasewith b 1 < d 1 94

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! 8 6 4 2 0 N = 25 1 2 3 4 5 6 7 8 9 11 log extinction probability Number of individuals introduced 8 6 4 2 0 N = 50 1 3 5 7 9 12 15 18 21 24 8 6 4 2 0 N = 100 1 5 9 14 20 26 32 38 44 Exact Solution Exp Approx 1 term Exp Approx 2 term Diffusion Approx Branching Process 20 15 10 5 0 Number of individuals introduced log extinction probability 1 3 5 7 9 11 14 17 20 20 15 10 5 0 Number of individuals introduced 1 3 5 7 9 11 14 17 20 20 15 10 5 0 Number of individuals introduced 1 3 5 7 9 11 14 17 20 N =25 ! N =50 N =100 A E D C B F Figure3-5.Invasionprobabilitycontinuumapproximationsandtheexactsolutionfordi! erent numbersofinitiallyintroducedindividuals.Resultsareshownforrateparameters b =2 and d =1 .A-C)Example 1 stochasticepidemics.D-F)Example 2 density dependentmortality. 95

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CHAPTER4 ELIMINATIONPROBABILITIES Abstract: Inthisproject,weextendonourworkoninvasionprobabilitiesinChapter 3 toconsidertheprobabilityofre-invasionofapathogeninapopulationafteravaccination campaignends.Thegoalofavaccinationcampaignistotemporarilyalterthesystemstate fromsupercriticaltosubcritical.Inourmodelswebasethetimethatthevaccinationcampaign endsonanimperfectobservationprocess.Wethenseektodeterminehowlikelyitisthatthere areremainingunobservedinfectiousindividualsandifso,howlikelyisitthatthediseasewill reboundinthepopulationafterthecampaignends.Weproposetwostochasticmodelsanda deterministicframeworktoinvestigatethesequestions.Throughacombinationofanalytical andnumericalanalysiswecharacterizetheprobabilityofre-invasion. 4.1Context Introduction: Thehighestbenchmarkofsuccessininfectiousdiseasemanagementis eradication.Onceapathogenisdrivenextinct(eradicated),thatpathogennolongerposes athreattopotentialhosts.Ofcourse,fullyeradicatingapathogenisverydi"cult,evenwhen ane! ectivevaccineisavailable:uniformvaccinationcoveragerequiresreachingeventhemost remotepopulationsandmaintainingvaccinationenthusiasmisnoteasywhenonlyasmall numberofreportedcasesremain.Vaccinationcampaignsthatcontroldiseaseoccurenceto levelslowenoughtoachieveeliminationaregenerallyveryexpensiveandourobservationof infectedindividualsisalwaysimperfect. Inthischapter,wewilluseamathematicalmodeltoinvestigatetherelationshipsbetween vaccinationcoverageandobservationratesintermsoftheire!ectsontheprobabilityof elimination.Themodelcharacteristicsaremotivatedbyvaccinationandsurveillancestrategies fortheprospectofeliminatingrabiesfrompopulationsofsmallcarnivoresinthenortheastern UnitedStates.Uncertaintyinlevelsofvaccinationcoverageanddiseaseprevalencewill beparticularlyimportantinthissystem,giventhewaythathumansinteractwithwildlife populations.Unlikedomesticcaninerabiesvaccinationcampaigns,raccoonrabiesvaccinationis 96

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indirectandreliesonraccoonseatingvaccinatedbait.Ourabilitytoobserverabidraccoonsis alsoimpairedandmayvaryseasonally. Fromamathematicalpoint-of-view,eliminationprocessesconstituteamulti-scale systemthatrequiresexibleanalysis:whenpopulationnumbersarelarge,thedynamicsare largelydeterministic;butnearextinction,thenumbersaresmallandintrinsicallystochastic. Understandinghowtobridgethesetwoscalesisalong-standingprobleminappliedprobability. Ourcurrentfocusisonthemathematicalanalysisofsimplestochasticmodelsthatdescribe pathogenextinctionwhentheobservationprocessisimperfect.Ultimately,makinga meaningfulcontributiontoactualeliminatione! ortswillrequire1)usingdatatoinform parameterscharacterizingtheproportionvaccinatedandprobabilityofobservation,2)using dataandsimulationstoidentifythesourcesofheterogeneitythatsubstantiallyaltertheresults, and3)developinginnovativewaystoaccountforthesesourcesofheterogeneity. 4.1.1RabiesvaccinationofwildlifeintheeasternUnitedStates RabiesiscurrentlyendemicinwildlifepopulationsacrosstheUnitedStates[ 24 70 ]. Rabiesvaccinationprogramsforwildlifeareinherentlydi! erentfromtraditionalcaninerabies vaccinationcampaigns.Thecaninerabiesvaccineistypicallyadministeredintheformofa shotdirectlybytechnicians.Indevelopingcountries,caninerabiesvaccinationcampaignsare generallythoughtofasdiscretepulsedevents[ 14 34 67 ]. Particularlywhenvaccinatedbaitsareusedtopreventrabiesfromspreadingtonew areas,forexampleacrossa cordonsanitaire [ 62 ],wildliferabiesvaccinationcampaignsmaybe consideredtooccuronamorecontinuousbasisthancaninerabiesvaccinationcampaignsin thedevelopingworld.Notably,caninerabiesvaccinationprogramsinsomedevelopedcountries areadministeredonamorecontinuousbasisoverextendedperiodsoftime[ 12 ].This,along withhighvaccinationcoverageandhighcasedetectionrates,likelycontributestotheessential eliminationofcaninerabiesindevelopedcountries[ 12 33 ]. Inaddition,theunmonitoreduptakeofvaccinatedpelletsposesaconsiderablechallenge toestimatingvaccinationcoverageinwildlifepopulations.Itisalsoverydi"culttoobserve 97

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infectiousindividualsinthissetting.Weincorporateanobservedinfectiousindividualsclass sothatthedecisiontoendavaccinationcampaigncanbebasedontheperceivedcumulative rabiesincidenceoveradenedperiodoftimeratherthantheunknowabletrueincidence. Observedinfectiousindividualsareconsideredremovedfromthesystemanddonotcontribute tonewinfections(consistentwiththerabiestestingprocedure,whichrequiresthatthesuspect individualbekilled). 4.1.2Pathogeneliminationsettingsandmodeling Sofar,smallpoxandrinderpestaretheonlypathogensthathavebeensuccessfully eradicated[ 25 ](althoughguineawormmaybeveryclose[ 65 ]).Thechallengespresented duringeliminatione! ortsareinherentlydi! erentfromthoseassociatedwithcontrolling endemicdiseases[ 45 46 ].Di"cultiesassociatedwithvaccinatione!ortstargetingelimination includeresourcelimitation(dosesofvaccineandphysicianstodistributethem),waning immunity,andvaccinerefusal.Ofnote,theproportionofthepopulationthatisinfectiousdi! ers greatlybetweenthestagesofendemiccirculationandlimitedtransmissionpreceding elimination.Thelastfewinfectiousindividualsmaybehardtoobserveandareoftenthe mostdi" culttoreachduringvaccinationcampaigns[ 46 ]. Extensivesimulationstudieshavebeenusedtostudytheprobabilityofeliminationand expectedtimeuntileliminationofcaninerabiesinbothepidemic[ 26 66 67 ]andendemic settings[ 26 ].Byincorporatingspatialstructurebothintheinfectionprocessesandvaccination strategies,thesestudiesfoundthatspatialheterogeneityinvaccinationcoveragemakesit moredi" culttoeliminaterabies.Sometheoreticalstudieshaveshownthatcontrolmeasures (suchasvaccination)needtobemaintained"throughatransitionperiod"beforeelimination becomesstable[ 63 ]. Arguably,stochasticprocessesareneededtofullycharacterizethedynamicsdirectly precedingelimination.Whiletimetoextinctionhasbeenapproximatedinastochastic framework[ 50 ],thesestudiestypicallyassumethattheinitialnumberofsusceptibleand infectiousindividualsisinaquasi-stationarystate.Incontrast,wewillbeconcernedwiththe 98

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dynamicswhencontrolisinplace(i.e.theinfectionanddeathratesaresuchthatthebasic reproductionnumberislessthanoneandthediseaseisindecline),andimmediatelyfollowing theendofacontrole! ort.Giventhathomogeneouscoverageispreferable[ 67 ]andchallenging toimplement,havinganideaofhowlongcoveragewouldneedtobemaintainedwouldbevery valuable(particularlytofundingagenciesandcampaignstrategists). 4.1.3QuestionsandAims Inthelargerprojectourprimaryresearchquestionsareasfollows: 1. Whatcanwelearnaboutunobservedcasesbasedonourknowledgeoftheimperfect detectionprocess? 2. Howlongshouldvaccinationcoveragebemaintainedtoensureaspecied high probabilityofelimination? Concretely,wewillattempttoanswerthesequestionsbyaddressingthefollowingaims. Aim1. Tocharacterizetheprobabilitythatthereareremainingunobservedinfectious individualswhenthevaccinationcampaignends. Aim2. Tocharacterizethedistributionofunobservedinfectiousindividualsattheendofthe vaccinationcampaign. Aim3. Todeterminethedurationoftimethatacontrolshouldcontinuetobeimplemented afterobservingzerocases,toensureaspecied low probabilitythatre-invasionwilloccur. Theseaimsrelatetotheprecedingchapteroninvasionprobabilities.Inpractice,the numberofunobservedinfectiousindividualswhencontrole! ortsstopisthesameasthe numberofintroducedindividualsweconsideredintheprobabilityofinvasionsetting. ResultsfromAim 1 andAim 2 willbeusedtoaddressAim 3 .Oncetheprobabilitythat unobservedinfectiousindividualsremainandtheexpectednumberofunobservedinfectious individualswhenthevaccinationcampaignendsisknown,wecandeterminehowchanging thedurationofcontrolimplementationafterobservingzerocaseswillhaveanimpacton theprobabilityoftrueelimination(i.e.eliminationwithoutre-invasionresultingfromthe unobservedinfectiousindividuals).Wecanthenidentifyhowlongextendedcontrolwouldhave beenneededtoreducetheexpectednumberofunobservedindividualsbelowacertainnumber. 99

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Usingtheresultsfromtheinvasionprobabilitieschapter,wethendeterminetheprobabilityof invasiongiventhenumberofintroducedindividuals(inthiscase,thenumberofunobserved infectiousindividuals). InSection 4.2 ,wedeveloptwostochasticmodelsthataccountforimperfectdetection ofcases.Therstmodelbasesthetimetoendthevaccinationcampaignonthetimes betweenobservationsofinfectiousindividuals.Thismodelisintuitive,butcannotbeanalyzed analytically.Thesecondmodelislessintuitive,butisconducivetoanalysisaddressingAim 1 InSection 4.3 ,wewilladdressallthreeaimsusingthedeterministicmodel.Given accomplishmentofAim 2 ,addressingAim 3 primarilyreliesontheanalysisfromChapter 3 Aim 1 determinesanupperboundfortheprobabilityofre-invasionandisspecicallyincluded heresincethereisamathematicalstrategythatdirectlyaddressesthatquestion.Aims 2 and 3 donothaveanobviousmathematicalmethodthataddressesthem. InSection 4.4 ,weanalyzethe2DmodelproposedinSection 4.2 toaddressAim 1 .Each aimpresentedabovecanbeaddressedwiththedeterministicmodel,buttheresultsdependon thethresholdparameterneededtoapproximatezeroobservedinfectiousindividuals.Forthe stochasticmodelanalysis,thereisatradeo! .Wearenotabletoanalyticallyaddressallofthe aims,butwedoobtainaresultthatdoesnotdependontheintroducedboundaryparameter 4 InSection 4.5 ,wepresentnumericalresultsforeachaimusingthestochasticmodelsin Section 4.2 andthedeterministicmodelfromSection 4.3 .Weusesimulationstodemonstrate qualitativesimilaritybetweenthetwostochasticmodelsandtoillustratelimitationsofthe deterministicmodel. 4.2Stochasticmodeldevelopment Ourmainfocuswillbeonstochasticmodels,sinceeliminationdynamicsareinherently stochastic.AdiagramoftheconceptualframeworkofourproblemispresentedinFigure 4-1 Thevaccinationcampaignbeginsattimezero,sowetaketheendemicequilibriumstatefrom asystemwithoutvaccinationasourinitialcondition.Below,wepresenttwowaysofchoosing thetimetoendthevaccinationcampaign.Thenumberofinfectiousindividualsremaining 100

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whenthevaccinationcampaignendsisthenumberofinvadingindividualsweconsiderwhen evaluatingtheprobabilityofre-invasion( k inthenotationofChapter 3 ). Forthesakeofsimplicity,webeginbyconsideringasimpleSISMarkovchainmodel.The numberofinfectiousindividuals, X ( t ) ,inapopulationofsize N ,changesaccordingtothe followingtransitions X X +1 atrate c (1 # p v ) X ( N # X ) / N X X # 1 atrate ( + $ ) X (41) where c isacontactrateand isthedisease-relatedremovalrate.Weincorporatethe additionalremovalrate $ tocaptureanobservationprocess,wherebyinfectiousindividualsare observedandremovedfromtheinfectiouspopulation.Wereducetherateofnewinfections bytheproportionofthenon-infectiouspopulationthatisvaccinated( p v ).Forallmodel realizations,werequirethat c > + $ .Otherwise,theoriginalsystemwouldalreadybe subcriticalandtheimplementationofavaccinationcampaignwouldnotbenecessaryto achieveelimination. 4.2.1Traditionalobservationmodel:timesinceobservation Inthismodel,theobservationprocessisdeterminedbytheremovalrate $ ,atwhich unobservedinfectiousindividualsareobserved.Withinthisframework,webasethedecision toendthevaccinationcampaignonthetimesincethelastobservationevent.Foragiven durationofcontinuedcoverageafterthemostrecentobservation,weusesimulationsto ndthersttimethatobservationeventswereseparatedbythatamountoftime.Therst observationeventinthatpairisconsideredthe"last"observation.Wethenchecktoseehow manyunobservedinfectiousindividualsarepresentafterthespecieddurationofcontinued coverage(startingfromthestateofthesystematthetimeofthelastobservation). 4.2.2Unconventionalobservationmodel:Markovmodelfor"forgetting"observed cases Itisnotclearhowtoanalyzetheobservationmodelpresentedabove,sowepropose alessconventionalalternative.Weaddaseparateclassforobservedinfectiousindividuals totheone-dimensionalCTMCmodel.Weintroduceaparameter todescribetheaverage 101

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"forgetting"timeforanobservedindividual.Inotherwords,eachobservedindividualwillbe "remembered"(i.e.remainintheobservedclass)foranaverageof weeks. Theresultingtwo-dimensionalMarkovchainmodelisasfollows.Inapopulationofsize N wehavethestatevariables X ( t )= thenumberofunobservedinfectiousindividualsattime t Y ( t )= thenumberofobservedinfectiousindividualsattime t withthefollowingtransitions ( X Y ) ( ( ( ( ( ( ( ) ( ( ( ( ( ( ( ( X +1, Y ) atrate c (1 # p v )( X )( N # X ) / N ( X # 1, Y ) atrate X ( X # 1, Y +1) atrate $ X ( X Y # 1) atrate (1 / ) Y (42) whereagain c isacontactrate, p v istheproportionofthenon-infectiouspopulationthatis vaccinated, isthedisease-relatedremovalrate, $ isthecasedetectionrate,and isthe averagetimethatadetectedcaseremainsintheobservedinfectiousindividualclass.Rather thanimmediatelyremovingindividualsfromtheobservedinfectiousclass,incorporatingthe removalrate( 1 / )allowsustocharacterizetheperiodoftimethatacontrolwillcontinue tobeimplementedafterthemostrecentobservationevent.Inthefollowinganalysis,this parameterwillbeimportantfordeterminingtheamountoftimethatacontrolshouldremain inplaceafterthemostrecentobservationinordertoensureaspecied"high"probabilityof elimination(alternativelya"low"probabilityofre-invasion). 4.2.3Fundamentalmathematicalquestion Wewilldene trueelimination asthecasewhen X ( t )=0 and apparentelimination as thecasewhen Y ( t )=0 but X ( t ) > 0 .Ourgoalistomathematicallydescribetheprocess ofmovingfromapparenteliminationtotrueeliminationintermsofthesystemparameters, withtheultimategoalofprovidingguidanceforchoosinghowlongtomaintainvaccination campaignsbeyondthelastdetectedcase. 102

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Mathematically,thisproblemamountstothereductionofatwo-dimensionalsystemtoa one-dimensionalsystem.The Y =0 statefallsontheboundaryofthe2Dsysteminvolving both X and Y .Oncethesystemdimensionisreducedinthisway,theresultsofourworkon invasionprobabilitiesapply.Soitisashortsteptomakeinferencesaboutrecovery/re-invasion probabilitiesafter Y =0 .If T isthersttimesuchthat Y ( T )=0 ,thenthenumberof invadingindividuals( k inpreviousnotation)is X ( T ) .SincethesystemisMarkovian,wecan simplyrestarttimeandthenapplytheresultsofChapter 3 todeterminetheprobabilityof (re-)invasion. Inourmodel,thevaccinationcampaignendsonceapparenteliminationisachieved.We areinterestedintheprobabilitythatthereareremainingunobservedinfectiousindividualswhen thevaccinationcampaignends.Inwhatfollows,wemathematicallyanalyzetheprobabilityof apparent(versustrue)elimination.Later,wewillconsidertheprobabilitythattrueelimination followsapparenteliminationgiventhesystemparameters. InordertoaddressAim 1 ,werstdenethehittingprobabilityproblemforhittingthe Y =0 boundaryinthetwo-dimensionalsysteminSection 4.2.2 Denition4.2.1. Fortheprocess { X N ( t ), Y N ( t ) } t > 0 wedenethe hittingtime T N as follows: T N :=inf 5 t > 0: X N ( t )=0 or Y N ( t )=0 6 Wedenetheassociated apparenteliminationprobabilities q N ( i j ) ( i j ) $ { 0,1,... } by q N ( i j ):= P { Y N ( T N )=0 | ( X N (0), Y N (0))=( i j ) } Wethendenetheasymptoticprobabilityofapparentelimination p elimination ( i j ) (conditioned on i unobservedand j observedinfectedindividualsattimezero)tobe p elimination ( i j )=lim N $' q N ( i j ). InordertoaddressAim 2 weuseDenition 2 fromChapter 3 todenetheprobabilityof re-invasionasfollows. 103

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Denition4.2.2. Fortheprocess X N ( t ) ,wedenethehittingtime N asfollows: N :=inf 5 t > T N : X N ( t ) $ { 0, N } 6 Wedenetheassociatedtrueelimination(extinction)probabilities q N ( k ), k $ { 0,1,..., N } by q N ( k )= P { X N ( N )=0 | X N ( T N )= k } Theasymptoticprobabilityofre-invasionalsofollowsfromDenition 2 inChapter 3 : p re-invasion ( N )= p invasion ( X N ( T N )). FinallytoaddressAim 3 ,werstdenetheadditionalparameters p and ! asfollows. Denition4.2.3. Let p bethedesired(low)probabilityofre-invasion.Wethendenethe desiredextendeddurationofcontrolapplicationtobe ! :=min { : p re-invasion ( ) p } Heuristically, ! istheshortestexpecteddurationofextendedcontrolsuchthatthe probabilityofre-invasionasdenedinDenition 4.2.2 (andinvestigatedinAim 2 )isless than p .ThisisthesameasthevaluesoughtinAim 3 .Futureworkwillfocusonndingan expressionfor ! asafunctionoftheparametersofthesystemand p .Below,weconsider thedeterministicanaloguesoftheabovestochasticsystemsincasethatanexplicitsolutionis possibleinthedeterministicsetting. 4.3Deterministicframeworkformodelingeliminationandre-invasionphenomena. SincetrueextinctioncannotoccurinODEmodels,wedeneathresholdparameter 4 tobethevalueforobservedinfectiousindividualsatwhichthediseasewillnolongerbe consideredtobepresentinthepopulation.Atthistimethevaccinationcampaignwillbe terminated. 104

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4.3.1ODEmodelforSISdiseasedynamics AnalogoustotheCTMCmodelin( 41 )wehaveforlargepopulationsizes,thefollowing simpleone-dimensionaldeterministicsystem: x = c (1 # p v ) x (1 # x ) # ( + $ ) x where,asbefore, c isacontactparameter, p v istheproportionofthepopulationthatis vaccinated,and isthedisease-relatedmortalityrate.Theadditionalremovalrate $ isthe rateatwhichinfectiousindividualsaredetected.Onceanindividualisobserveditisremoved fromtheinfectiouspopulationandreplacedwithanon-infectiousindividual.Incontrastto thegeneralSusceptible-Infectious-Susceptible(SIS)model,here 1 # x istheproportionof non-infectiousindividualsratherthantheproportionofthepopulationthatissusceptible.Only aproportion( 1 # p v )ofthenon-infectiousindividualsaresusceptibletoinfection. Thediseasefreeequilibrium ( x =0) alwaysexists.Solvingfortheendemicequilibrium,we ndthat x =1 # 1 R 0 with R 0 = c (1 # p v ) / ( + $ ) .Theendemicequilibrium x > 0 existswhen R 0 > 1 .A linearizationargumentcanbeusedtoshowthat x islocallyasymptoticallystable(l.a.s.)if andonlyif c (1 # p v ) > + $ .Thus, x isl.a.s.wheneveritexists.Thediseasefreeequilibrium isstablewhen R 0 < 1 andunstablewhen R 0 > 1 4.3.22DODEmodelanalogueto2DCTMCmodel AnalogoustotheCTMCmodelin( 42 )wehaveforlargepopulationsizes,thefollowing two-dimensionaldeterministicsystem: x = c (1 # p v ) x (1 # x ) # ( + $ ) x y = $ x # y / (43) wheretheparametersarethesameasthosein( 42 ). 105

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Inordertodene"extinction"inthedeterministicsetting,weintroduceathreshold parameter 4 .Whenthenumberofobservedindividualsfallsbelowthisvalue,thevaccination campaignends.Inordertoidentifyhowmanyunobservedinfectiousindividualsremainatthe endofthevaccinationcampaignwemustrstdeterminethetimethatthenumberofobserved infectiousindividuals y fallsbelowthecampaignendingthreshold.Formally,wedeneapparent extinctioninthedeterministicsettingasfollows. Denition4.3.1. Foragiven 4 4 1 andpopulations x and y withdynamicsspeciedby ( 43 ) wedenethe apparentextinctiontime T tobethersttimethattheobservedclass valuereaches 4 ,thatis T :=min 5 t > 0: y ( t )= 4 } (44) Thecorresponding apparentextinctionstate ofthetwo-dimensionalsystemis ( x ( T ), y ( T )) Inotherwords, T denotesthetimethatthevaccinationcampaignendsand 4 isthe observedinfectiousclassvaluethattriggerstheendofthevaccinationcampaign.Oncethe vaccinationcampaignends,weset p v =0 andthesystemreturnstoasupercriticalstate. Weacknowledgethatundermostcircumstancestheproportionvaccinatedwoulddecline graduallyovertimeaftertheendofavaccinationcampaign.Whiletheassumptionthat p v immediatelyreturnsto0attheendofthecampaignisunrealistic,wearestillabletoestimate anupperboundforthere-invasionprobability.Inthisregime,weapplyresultsfromChapter 3 oninvasionprobabilities.Thenumberofintroducedindividualsistakentobethenumber ofunobservedinfectiousindividualsatthetimewhenthevaccinationcampaignends, Nx ( T ) Theroleof Nx ( T ) indeterminingthere-invasionprobabilityinthedeterministicsettingis madeexplicitinthefollowingformaldenition. Denition4.3.2. Forthedeterministicsystem ( x ( t ), y ( t )) ,wedenethere-invasion probabilitystartingwith Nx ( T ) unobservedinfectiousindividualsasfollows p determinisitic :=1 # 1 R 0 Nx ( T ) 106

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wheretheformofthisexpressionisbasedonthebranchingprocessapproximationforthe invasionprobabilityinChapter 3 Denition 3 Remark3. Wechoosetousethebranchingprocessapproximationsinceitallowsforfractional numbersofintroducedindividuals.Itisonlypossibletoevaluatetheexactsolutionforthe invasionprobabilitywhenthenumberofintroducedindividualsisanonnegativeinteger.In ordertoavoidroundingindividualsweoptedforanapproximation.Wechosethebranching processapproximationinparticularsinceitsexpressioniseasytointerpretintermsofthe modelparameters. Stabilityanalysis: Thesystemin( 43 )hasatrivialequilibriumat (0,0) .When + $ < c (1 # p v ) ,thefollowingendemicequilibriumexists x =1 # + $ c (1 # p v ) y = $! 1 # + $ c (1 # p v ) (45) Wealwayssettheinitialconditionofoursystemtobetheendemicequilibriumvalueinthe absenceofvaccination,i.e.with p v =0 .Thus,from( 45 )wehave x 0 =1 # + $ c and y 0 = $! 1 # + $ c Wenotethat R 0 = c (1 # p v ) / ( + $ ) inboththeone-dimensionalandtwo-dimensionalmodels sinceonlythe x classofindividualscontributestonewinfections.Beforethevaccination campaignbegins R 0 > 1 .Wealwayschoose p v $ 1 # ) + c ,1 9 sothatduringthevaccination campaignwehave R 0 < 1 .Inthefollowingproposition,wedeterminethestabilityofeachof theequilibria. Proposition4.3.3. Thetrivialequilibriumofsystem ( 43 ) at(0,0)islocallystablewhen c (1 # p v ) < + $ andunstablewhen c (1 # p v ) > + $ .Whenthenontrivialequilibrium ( x y ) exists,i.e.when c (1 # p v ) > + $ ,itisstable. 107

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Proof. Inordertoevaluatethestabilityoftheequilibriaofthesystem,werstndthe Jacobian J = B C D c (1 # p v )(1 # 2 x ) # ( + $ )0 $ # 1 / E F G (46) Since( 46 )islower-triangular,theeigenvaluesarejustthediagonalentries, & 1 = c (1 # p v )(1 # 2 x ) # ( + $ ) & 2 = # 1 / Wenotethat > 0 ,so & 2 isalwaysnegative.Atthetrivialequilibrium,wehave & 1 = c (1 # p v ) # ( + $ ). Itfollowsthat(0,0)islocallystableifandonlyif c (1 # p v ) < + $ .Atthenontrivial equilibrium, & 1 = c (1 # p v ) 1 # 2 1 # + $ c (1 # p v ) "" # ( + $ ). Itfollowsthat ( x y ) islocallystableifandonlyif c (1 # p v ) > + $ Modelparameterization: Wechosevaluesthatreectourapplicationtorabiesvaccination campaignsinsmallcarnivorepopulations.WewillalsoutilizeparametersfoundbyTownsendet al.[ 66 67 ]fortheircaninerabieseliminationmodelingstudieswhichalsomotivatethiswork. OurparameterchoiceinformationissummarizedinTable 4-1 Wesetthediseaserelatedmortalityrate =1 perweekbasedontheestimateforthe raccoonrabiesinfectiousperiodduration4-5days[ 35 ]andtobeconsistentwithChapter 3 Wechoseourobservationrate $ basedonthedetectionprobability0.1usedinTownsendetal. 2013[ 66 ].FromtheMarkovchainmodel 42 weknowthattheprobabilitythataninfectious individualisobservedis ) + ,whichroughlyyieldsareportingrateof $ =0.1 Toobtainavalueforthecontactparameter, c ,weuseitsrelationtothebasicreproduction number.Thereproductionnumberforrabiesisoftenfoundtobeslightlyabove1,following theliterature[ 34 66 67 ]weset R 0 =1.2 .Weassociatethisreproductionnumberwith 108

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Table4-1.Parametersusedintheeliminationprobabilityanalysis. ValueUnitsDenitionSource R 0 1.2,1.4reproductionnumber(withoutvaccination) 1wk # 1 diseaserelatedmortalityrate4-5daysin[ 35 ] $ 0.1wk # 1 observationrateofinfectiousindividuals[ 67 ] c R 0 ( + $ ) wk # 1 contactrate p v 0.3proportionvaccinatedassumption 1/7to8wksavg.timeuntilforgettingobservation asystemwithoutvaccination(i.e.when p v =0 ).Since R 0 = c / ( + $ ) ,wehavethat c = R 0 ( + $ ) Thekeyconsiderationinchoosingtheproportionvaccinatedparameter p v isthatitmust belargeenoughsothatthesystemissubcriticalduringthevaccinationcampaign.Wechose aparticularvalueof p v =0.3 tosatisfythisconditionandreectthatvaccinationcoverageis likelylowinwildlifepopulations.Theratethatobservedinfectiousindividualsare"forgotten" 1 / istheinverseoftheaveragedexpectedtimethattheyremainintheobservedclass.Inour analysis,wevary between1/7and8weekstoinvestigatethee!ectthatthisparameterhas ontheprobabilityofelimination. Inordertodeterminethenumberofremainingunobservedinfectiousindividualswhenthe vaccinationcampaignends, x ( T ) ,ournextstepistondasolutionto( 43 ).Inthefollowing proposition,wendanexplicitsolutionfor x anddemonstratethatthereisnotanobvious explicitsolutionfor y .InSection 4.5 wedisplayresultsbasedonnumericalintegrationofthe di erentialequationsystem,sincewedonothaveananalyticalsolutionatthistime. Proposition4.3.4. Let ( x (0), y (0))=( x 0 y 0 ) .Thenthesolutiontosystem ( 43 ) is x ( t )= x 0 e # at (1+ bx 0 ) # bx 0 e # at y ( t )= e # t / y 0 + $ b 1 / ( a ) ab 2 b be # at u # 1 / ( a ) 1 # u d u wherewehavedened a :=( + $ ) # c (1 # p v ), b := c (1 # p v ) ( + $ ) # c (1 # p v ) and b := bx 0 1+ bx 0 109

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Proof. Since x doesnotdependon y wecansolvetheODEdirectlybyseparationofvariables. Werewrite x in( 43 )as x = # ax (1+ bx ), andnotethatwhen c (1 # p v ) < + $ ,theparameters a and b arepositive.Then d x x (1+ bx ) = # a d t IntegratingtheRHS,weobtain 2 t 0 # a d t = # at (47) DecomposingtheLHS,wehave 1 x (1+ bx ) = 1 x # b 1+ bx ThenintegratingtheLHSweobtain 2 1 x (1+ bx ) d x =ln x 1+ bx + C (48) Combiningequations( 47 )and( 48 ),wehave ln x 1+ bx + C = # at Evaluatingat t =0 wendtheintegrationconstant C = # ln x 0 1+ bx 0 Substitutinginourvaluefor C wehave ln x 1+ bx = # at +ln x 0 1+ bx 0 Aftersomealgebraicmanipulation,wendthesolution x ( t )= x 0 e # at 1+ bx 0 # bx 0 e # at (49) 110

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Turningourattentionto y ( t ) ,weusetheintegratingfactor e t / ,tointegrate y in( 43 ) andnd y ( t )= y 0 e # t / + $ 2 t 0 e # ( t # s ) / x ( s ) d s Incorporatingthesolutionfor x ( t ) from( 49 ),wehave y ( t )= e # t / y 0 + $ 2 t 0 e s / x 0 e # as 1+ bx 0 # bx 0 e # as d s Aftersomerearranging,wehave y ( t )= e # t / y 0 + $ x 0 1+ bx 0 2 t 0 e (1 / # a ) s 1 # be # as d s Wenoticethat b < 1 sotheintegralisdened.Inanattempttoevaluatetheintegral,we makethesubstitution u = be # as y ( t )= e # t / y 0 + $ x 0 1+ bx 0 2 be # at b u 1 # 1 / a b 1 # 1 / a (1 # u ) # 1 au d u = e # t / y 0 + $ x 0 b ab (1+ bx 0 ) b 1 # 1 / a 2 b be # at u # 1 / a 1 # u d u andthenalresultisobtainedbyrecallingthedenitionof b Remark4. InProposition 4.3.4 wefoundthatthesolutionfor y ( t ) canbereducedto anintegral,butitisunclearhowtoproceedanalyticallyfromthisstagesincetheresulting integrandisnotanamedfunction.Theequationfor y ( t ) yieldsanimplicitequationfor T when y ( t ) isreplacedwith 4 onthelefthandsideand t isreplacedwith T ontherighthand side.However,thestructureofthesolutionfor y ( t ) preventsusfromobtaininganexplicit solutionfor T .Itremainsanopenquestionwhetherthereisane!ectiveapproximationfor T overabiologicallyrelevantrangeofparameters. 4.3.3InterpretingtheODEmodelforre-invasionprobabilities. Recallthat istheaverageamountoftimethatanobservedinfectiousindividual remainsintheobservedclass.Attime T when y ( t ) fallsbelowthethresholdvalue 4 ,the vaccinationcampaignends.InFigure 4-2 ,wedisplaynumericalsolutionsfor x and y fortwo 111

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parameterizationsoftheODEsystemin( 43 ).As increases,ittakeslongerforthe y class tofallbelowthethreshold(i.e. T increases).Thesizeoftheunobservedinfectiousclass x ( t ) doesnotdirectlydependon andismonotonicallydecreasing,soas increases x ( T ) decreases. InFigure 4-3 ,weseethattheproportionofthepopulationthatisunobservedand infectiousattheendofthecampaigndecreasesastheaveragedurationofmemoryfor(time untilforgetting)anobservedinfectiousindividualincreases.Wealwayschoosetheinitial conditionofthedeterministicsystemin( 43 )tobetheendemicequilibriumvaluewhenthe proportionvaccinatediszero, ( x 0 y 0 )=( x y )= 1 # + $ c $! 1 # + $ c "" Thevaccinationcampaignendswhen y ( t ) < 4 .Forthelarger 4 valuesof0.01and0.02in Figure 4-3 ,weobserveinitialatsegmentsbecausethereisarangeof > 0 suchthatthe startingcondition y 0 < 4 .Inthiscase,thecampaignendsattimezeroandtheremaining proportionofunobservedindividualsisjustthestartingcondition, y 0 4.4Stochasticanalysisoftheapparenteliminationprobability. Inthissectionweanalyzethetwo-dimensionalstochasticmodel( 42 )presentedin Section 4.2.2 .Inparticular,weapproximatesolutionstothethehittingprobabilityproblem developedinDenition 4.2.1 ,thatconcerns q N ( i j ) theprobabilityofapparenteliminationina populationofsize N startingwith i unobservedinfectiousindividualsand j observedinfectious individuals.Wepresentadi! usionapproximationforthisproblemanddemonstratethatarst orderexponentialapproximationfailstoyieldadditionalinsight.WethenapplyWKBmethods tothedi! usionapproximationandndasolutionfortheapparenteliminationprobability. 112

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Forourstochasticanalysis,rstwerecalltherateequationsin( 42 ).Foreaseofanalysis, werelabeltheratesasfollows ( i j ) ( ( ( ( ( ( ( ) ( ( ( ( ( ( ( ( i +1, j ) atrate & 1 ( i ):= c (1 # p v ) i ( N # i ) / N ( i # 1, j ) atrate 1 ( i ):= i ( i # 1, j +1) atrate & 2 ( i ):= $ i ( i j # 1) atrate 2 ( j ):=(1 / ) j (410) anddenotethesumoftheratestobe + ij := & 1 ( i )+ 1 ( i )+ & 2 ( i )+ 2 ( j ). Suppressingthepopulationscalingparameter N ,theapparenteliminationprobability (formallydenedinDenition 4.2.1 )satisesthefollowingsystemofdi!erenceequations, q ( i j )= q ( i +1, j ) & 1 ( i ) ij + q ( i # 1, j ) 1 ( i ) ij + q ( i # 1, j +1) & 2 ( i ) ij + q ( i j # 1) 2 ( j ) ij + ij q ( i j )= q ( i +1, j ) & 1 ( i )+ q ( i # 1, j ) 1 ( i )+ q ( i # 1, j +1) & 2 ( i )+ q ( i j # 1) 2 ( j ), (411) withboundaryconditions q (0, j )=1 and q ( i ,0)=0 Denition4.4.1 (Di!usionapproximation) Let 0 > 0 begiven.Thenfor x $ (0,1) and y $ (0,1) ,let u ( x y ) bethesolutiontotheboundaryvalueproblem 0=( & 1 ( x ) # 1 ( x ) # & 2 ( x )) 5 x u +( & 2 ( x ) # 2 ( y )) 5 y u + 0 2 ( & 1 ( x )+ 1 ( x )+ & 2 ( x )) 5 xx u + 0 2 ( & 2 ( x )+ 2 ( y )) 5 yy u # 0& 2 ( x ) 5 xy u , (412) with u (0, y )=1 and u ( x ,0)=0 .Thenwesay u ( x y ) isthe di usionapproximation for theprobabilityofapparenteliminationand u ( x y ) D ) q N ( i j ), where x = i / N and y = j / N 113

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Motivationforthedi!usionapproximation: Let x = i / N y = j / N ,and 0 =1 / N .Using ourdenitionofrateshapefunctioninAssumption 1 fromChapter 3 wehavethefollowing relationships & 1 ( i )= N & 1 ( x ), 1 ( i )= N 1 ( x ), & 2 ( i )= N & 2 ( x ), 2 ( j )= N 2 ( y ) and + ij = N + xy Let u ( x y )= q ( i j ) ,thenusingTaylorexpansionsoftheform u ( x + 0 y ) ) u ( x y )+ 05 x u ( x y )+ 0 2 2 5 xx u ( x y ) u ( x # 0 y ) ) u ( x y ) # 05 x u ( x y )+ 0 2 2 5 xx u ( x y ) u ( x # 0 y + 0 ) ) u ( x y ) # 05 x u ( x y )+ 05 y u ( x y ) + 0 2 ( 1 2 5 xx u ( x y )+ 1 2 5 yy u ( x y ) # 5 xy u ( x y )) u ( x y # 0 ) ) u ( x y ) # 05 y u ( x y )+ 0 2 2 5 yy u ( x y ), thenwehave q ( i +1, j ) & 1 ( i )=( u + 05 x u + 0 2 2 5 xx u ) N & 1 ( x ) = 1 0 u + 5 x u + 0 2 5 xx u & 1 ( x ) q ( i # 1, j ) 1 ( i )= 1 0 u # 5 x u + 0 2 5 xx u 1 ( x ) q ( i # 1, j +1) & 2 ( i )= 1 0 u # 5 x u + 5 y u + 0 1 2 5 xx u + 1 2 5 yy u # 5 xy u "" & 2 ( x ) q ( i j # 1) 2 ( j )= 1 0 u # 5 y u + 0 2 5 yy u 2 ( y ). Noticingthatthe u termscancel,wehave 0= 5 x u + 0 2 5 xx u & 1 ( x )+ # 5 x u + 0 2 5 xx u 1 ( x )+ # 5 y u + 0 2 5 yy u 2 ( y ) + # 5 x u + 5 y u + 0 1 2 5 xx u + 1 2 5 yy u # 5 xy u "" & 2 ( x ), whichafterrearrangingyieldsthePDEinDenition 4.4.1 InChapter 3 ,wefoundthatmakinganexponentialansatzyieldedthecorrecthitting probabilityforlargepopulationsexperiencingsupercriticaldynamics.Weattemptthis 114

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procedureforthe2Dhittingprobabilityproblem,butareunabletomakeuseoftheexponential forminthiscase.Forthesakeofcompleteness,weincludethecalculationhereandshow wherethebarrierarises. Exponentialapproximationinvestigation: Again,let x = i / N y = j / N with 0 =1 / N Assumethattheapparenteliminationprobabilitycanbewritteninthefollowingform q ( i j ) ) e # $ # ( x y ) forsomefunction $ .Substitutingthisforminoursystemofdi!erenceequations( 411 ),we obtain N + xy e # $ # ( x y ) = N & 1 ( x ) e # $ # ( x + , y ) + N 1 ( x ) e # $ # ( x # , y ) + N & 2 ( x ) e # $ # ( x # , y + ) + N 2 ( y ) e # $ # ( x y # ) WeproceedbyndingTaylorexpansionsfortheexponentsontheRHS, $ ( x + 0 y )=$ ( x y )+ 05 x $ ( x y )+ 0 2 2 5 xx $ ( x y )+ O ( 0 3 ) $ ( x # 0 y )=$ ( x y ) # 05 x $ ( x y )+ 0 2 2 5 xx $ ( x y )+ O ( 0 3 ) $ ( x # 0 y + 0 )=$ ( x y ) # 05 x $ ( x y )+ 05 y $ ( x y ) + 0 2 2 ( 5 xx $ ( x y )+ 5 yy $ ( x y ) # 2 5 xy $ ( x y ))+ O ( 0 3 ) $ ( x y # 0 )=$ ( x y ) # 05 y $ ( x y )+ 0 2 2 5 yy $ ( x y )+ O ( 0 3 ). Neglectinghigherordertermsanddividingby exp( # $ ( x y )) ,wehave + xy = & 1 ( x )exp # 05 x $ ( x y ) # 0 2 2 5 xx $ ( x y ) + 1 ( x )exp 05 x $ ( x y ) # 0 2 2 5 xx $ ( x y ) + & 2 ( x )exp 05 x $ ( x y ) # 05 y $ ( x y ) # 0 2 2 ( 5 xx $ ( x y )+ 5 yy $ ( x y ) # 2 5 xy $ ( x y )) 2 ( y )exp 05 y $ ( x y ) # 0 2 2 5 yy $ ( x y ) Atthisstage,itisunclearhowtomakeuseoftheexponentialformtosolvetheabove equation.Inwhatfollows,wedemonstratethatitisnothelpfultoinsertatwotermTaylor 115

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approximation,sincethissubstitutionultimatelyleadsbacktothedi! usionapproximation equation.ReplacingeachexponentialtermwiththecorrespondingtwotermTaylorapproximation, weobtain + xy = & 1 ( x ) 1 # 05 x $ ( x y ) # 0 2 2 5 xx $ ( x y ) + 1 ( x ) 1+ 05 x $ ( x y ) # 0 2 2 5 xx $ ( x y ) + & 2 ( x ) 1+ 05 x $ ( x y ) # 05 y $ ( x y ) # 0 2 2 ( 5 xx $ ( x y )+ 5 yy $ ( x y ) # 2 5 xy $ ( x y )) + 2 ( y ) 1+ 05 y $ ( x y ) # 0 2 2 5 yy $ ( x y ) Wethensubtract xy frombothsidesoftheequationandobtain 0= & 1 ( x ) # 05 x $ ( x y ) # 0 2 2 5 xx $ ( x y ) + 1 ( x ) 05 x $ ( x y ) # 0 2 2 5 xx $ ( x y ) + & 2 ( x ) 05 x $ ( x y ) # 05 y $ ( x y ) # 0 2 2 ( 5 xx $ ( x y )+ 5 yy $ ( x y ) # 2 5 xy $ ( x y )) + 2 ( y ) 05 y $ ( x y ) # 0 2 2 5 yy $ ( x y ) Afterorganizingtermsbypartialderivativeswehave 0= 0 ( 1 ( x )+ & 2 ( x ) # & 1 ( x )) 5 x $ ( x y )+ 0 ( 2 ( y ) # & 2 ( x )) 5 y $ ( x y ) # 0 2 2 ( & 1 ( x )+ 1 ( x )+ & 2 ( x )) 5 xx $ ( x y ) # 0 2 2 ( & 2 ( x )+ 2 ( y )) 5 yy $ ( x y )+ 0 2 5 xy $ ( x y ). Wethendividethroughby 0 andobservethatwehaverecoveredthePDEforthedi! usion approximationinDenition 4.4.1 Thus,theexponentialapproximationdoesnotreadilyprovideanalternativeapproximation fortheapparenteliminationprobability.However,WKBmethodscanbeusedtomotivatean approximationbasedonthedi! usionapproximation,asdemonstratedbelow.Wenowpresent ourformalapproximationoftheapparenteliminationprobabilityinDenition 4.2.1 Investigationintotheapparenteliminationprobabilityapproximation: Werstrewrite oursystemofdi! erenceequations( 411 )intermsofthecorrespondingdi!usionapproximation 116

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u ( x y ) inDenition 4.4.1 + xy u ( x y )= u ( x + 0 y ) & 1 ( x )+ u ( x # 0 y ) 1 ( x )+ u ( x # 0 y + 0 ) & 2 ( x )+ u ( x y # 0 ) 2 ( y ), (413) andlet u ( x y )= e # Q # ( x y ) / , where Q ( x y )= + n =0 S n ( x y ) 0 n Thenfrom( 413 ),wehave + xy e # Q # ( x y ) / = & 1 ( x ) e # Q # ( x + , y ) / + 1 ( x ) e # Q # ( x # , y ) / + & 2 ( x ) e # Q # ( x # , y + ) / + 2 ( y ) e # Q # ( x y # ) / (414) Bydenition, Q ( x + 0 y )= + n =0 S n ( x + 0 y ) 0 n = S 0 ( x + 0 y )+ S 1 ( x + 0 y ) 0 + S 2 ( x + 0 y ) 0 2 + O ( 0 3 ) Q ( x + 0 y )= S 0 ( x y )+ 05 x S 0 ( x y )+ 0 2 2 5 xx S 0 ( x y ) + 0 S 1 ( x y )+ 0 2 5 x S 1 ( x y )+ O ( 0 3 ) + 0 2 S 2 ( x y ) . Q ( x + 0 y )= Q ( x y )+ 05 x S 0 ( x y )+ 0 2 1 2 5 xx S 0 ( x y )+ 5 x S 1 ( x y ) + O ( 0 3 ) Q ( x # 0 y )= Q ( x y ) # 05 x S 0 ( x y )+ 0 2 1 2 5 xx S 0 ( x y ) # 5 x S 1 ( x y ) + O ( 0 3 ) Q ( x y # 0 )= Q ( x y ) # 05 y S 0 ( x y )+ 0 2 1 2 5 yy S 0 ( x y ) # 5 y S 1 ( x y ) + O ( 0 3 ) Q ( x # 0 y + 0 )= Q ( x y )+ 0 ( 5 y S 0 ( x y ) # 5 x S 0 ( x y ))+ 0 2 1 2 5 xx S 0 ( x y ) + 1 2 5 yy S 0 ( x y ) # 5 xy S 0 ( x y )+ 5 y S 1 ( x y ) # 5 x S 1 ( x y ) + O ( 0 3 ) 117

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Returningto( 414 )anddividingbothsidesby e # Q # ( x y ) / ,weobtain + xy = & 1 ( x )exp # 5 x S 0 # 0 1 2 5 xx S 0 + 5 x S 1 + O ( 0 2 ) + 1 ( x )exp 5 x S 0 # 0 1 2 5 xx S 0 # 5 x S 1 + O ( 0 2 ) + & 2 ( x )exp 5 x S 0 # 5 y S 0 # 0 1 2 5 xx S 0 + 1 2 5 yy S 0 # 5 xy S 0 + 5 y S 1 # 5 x S 1 + O ( 0 2 ) + 2 ( y )exp 5 y S 0 # 0 1 2 5 yy S 0 # 5 y S 1 + O ( 0 2 ) WereplaceeachexponentialtermwithitsTaylorexpansiontond + xy = & 1 ( x ) 1 # 5 x S 0 # 0 1 2 5 xx S 0 + 5 x S 1 + O ( 0 2 ) + 1 ( x ) 1+ 5 x S 0 # 0 1 2 5 xx S 0 # 5 x S 1 + O ( 0 2 ) + & 2 ( x ) 1+ 5 x S 0 # 5 y S 0 # 0 1 2 5 xx S 0 + 1 2 5 yy S 0 # 5 xy S 0 + 5 y S 1 # 5 x S 1 + O ( 0 2 ) + 2 ( y ) 1+ 5 y S 0 # 0 1 2 5 yy S 0 # 5 y S 1 + O ( 0 2 ) andaftersubtracting xy frombothsidesthisreducesto 0= & 1 ( x ) # 5 x S 0 # 0 1 2 5 xx S 0 + 5 x S 1 + O ( 0 2 ) + 1 ( x ) 5 x S 0 # 0 1 2 5 xx S 0 # 5 x S 1 + O ( 0 2 ) + & 2 ( x ) 5 x S 0 # 5 y S 0 # 0 1 2 5 xx S 0 + 1 2 5 yy S 0 # 5 xy S 0 + 5 y S 1 # 5 x S 1 + O ( 0 2 ) + 2 ( y ) 5 y S 0 # 0 1 2 5 yy S 0 # 5 y S 1 + O ( 0 2 ) Thenbycollectingtermswehave 0 0 :( 1 ( x )+ & 2 ( x ) # & 1 ( x )) 5 x S 0 +( 2 ( y ) # & 2 ( x )) 5 y S 0 =0 0 1 : # 1 2 ( & 1 ( x )+ 1 ( x )+ & 2 ( x )) 5 xx S 0 # 1 2 ( & 2 ( x )+ 2 ( y )) 5 yy S 0 + & 2 ( x ) 5 xy S 0 # ( & 1 ( x )+ 1 ( x ) # & 2 ( x )) 5 x S 1 # ( & 2 ( x ) # 2 ( y )) 5 y S 1 =0. (415) Ourchallengenowistodeterminewhatisanappropriateboundaryconditionfor S 0 .For reasonsthatwillbecomeclearlater,wewillmodifyourproblemtoaskwhethertheprocess hits x = 4 or y = 4 rst,where 4 > 0 isasmall.Supposewechoosetheboundaryconditions 118

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of S 0 sothat S 0 ( x 4 )=0 and S 0 ( 4 y )=1. Thennotethattheleadingordertermof u ( x y ) behavesasfollows: lim $ 0 e # S 0 ( / y ) / =0 and lim $ 0 e # S 0 ( x / ) / =1, (416) whichsatisesthedesiredboundaryconditions. WhilethisisnotasolutiontothePDE,webelievethatitisanimportantobjectofstudy. Findingasolutionfor S 0 : Inwhatfollows,weusethemethodofcharacteristicstonda systemofparametricequationsforthecharacteristiccurvesofthesolution d x d s =( + $ ) x # c (1 # p v ) x (1 # x ), x (0)= x 0 d y d s =(1 / ) y # $ x y (0)= y 0 SincetherighthandsideofthePDEfortheequationof S 0 iszero( 415 ), u ( x ( s ), y ( s ))= u ( x 0 y 0 ). SincethecharacteristiccurvesarederivedfromlocallyLipschitzODEs,thesolutions areunique.Thereforethecurvescannotcross.Thus,analysisofthesystemreduces tocharacterizingthecharacteristicemanatingfrom( 4 4 ).Allpointsbelowthe( 4 4 ) characteristicwilllieoncharacteristiccurvesemanatingfromthehorizontalboundaryand allpointsabovethe( 4 4 )characteristicwilllieoncharacteristiccurvesemanatingfromthe verticalboundary. WenotethesimilaritybetweenthecharacteristiccurvesresultingfromthePDEin Denition 4.4.3 andthedeterministicsystemin( 43 ).Thecalculationneededtondthe characteristiccurvescloselyfollowsthecalculationpresentedintheproofofProposition 4.3.4 119

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Proposition4.4.2. Let ( x (0), y (0))=( x 0 y 0 ) .ThenthecharacteristiccurvesofthePDEfor theapparenteliminationprobabilityapproximationinDenition 4.4.3 taketheform x ( s )= x 0 e as 1+ bx 0 # bx 0 e as y ( s )= e s / y 0 # $ 2 s 0 x 0 e ( a # (1 / )) r 1+ bx 0 # bx 0 e ar d r wherewehavedened a :=( + $ ) # c (1 # p v ) and b := c (1 # p v ) ( + $ ) # c (1 # p v ) Proof. Werstplugintherateequationsfrom( 42 )inDenition 4.4.3 toobtainourequation fortheapparenteliminationprobability: ( x + $ x # c (1 # p v ) x (1 # x )) 5 x w ( x y )+( y / # $ x ) 5 y w ( x y )=0, withboundaryconditions w ( x ,0)=1 and w (0, y )=0. Wendtheshapeofthecharacteristiccurvesgivenbytheparametricequations ( x ( s ), y ( s )) thatsatisfy d x d s =( + $ ) x # c (1 # p v ) x (1 # x ), x (0)= x 0 d y d s = y / # $ x y (0)= y 0 Rewritingthederivativeof x intermsof a and b weobtain d x d s = ax (1+ bx ), andsolvefor x ( s ) usingacalculationsimilartothatpresentedintheproofofProposition 4.3.4 Insummary,thesolutionto S 0 takesoneoftwovalues,dependingonwhetherthe characteristicoriginatesfromthehorizontalorverticalaxis.Aswesawfrom( 416 ),whenever S 0 =0 theleadingordervalueof u =1 andwhenever S 0 =1 theleadingordervalueof u =0 Wethereforedeneourapproximationfortheprobabilityofapparenteliminationasfollows. 120

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Denition4.4.3 (Apparenteliminationprobabilityapproximation) Let 4 > 0 begiven.For x $ ( 4 x ) and y $ ( 4 ( ) ,let w / ( x y ) bethesolutiontotheequation ( 1 ( x )+ & 2 ( x ) # & 1 ( x )) 5 x w / ( x y )+( 2 ( y ) # & 2 ( x )) 5 y w / ( x y )=0 (417) with w / ( 4 y )=0 and w / ( x 4 )=1 Thenweproposethat w / ( x y ) approximatesthe probabilityofapparenteliminationwhere w / ( x y ) W ) q N ( i j ) with x = i / N and y = j / N 4.5Numericalinvestigationoftheprobabilityofre-invasion. Asdemonstratedabove,itisnotalwayspossibletoobtainanalyticalresultsevenfor greatlysimpliedmodelsofeliminationandinvasiondynamics.Weproposedatwo-dimensional Markovchainmodeltoapproximateamoreintuitiveimperfectobservationprocess.Inthis section,wewillusesimulationstotestwhetherthelessconventionalmodelservesasagood approximationtothemoreintuitiveprocess.Wealsoproposedaframeworkforinterpreting ananalogousODEmodeltoapproximateeliminationandre-invasiondynamics.Wewere unabletondananalyticalsolutionforthedeterministicmodelin( 43 )andresorttousing numericalintegrationtosolvethesystem.Bycomparingtheseresultstothesimulationsresults forourstochasticmodels,weinvestigatecircumstancesunderwhichthedeterministicmodel canbeconsideredtoprovideareasonableapproximationoftheprobabilityofre-invasion.The parametervaluesusedforournumericalanalysisaresummarizedinTable 4-1 4.5.1Thetwo-dimensionalmodelthat"forgets"observedindividualsbehaveswell comparedtothemodelbasedonobservationtimes. TotestwhetherourMarkovmodelforforgettingobservedinfectiousindividuals yieldsreasonableresults,wecomparesimulationsofthismodelwithsimulationsofamore conventionalobservationmodelbasedonobservationtimes.Forthiscomparisonmethod,we runeachsimulationuntiltrueelimination,i.e.until X ( t )=0 oruntilthenaltimeisreached. 121

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SimilartothemethodologyappliedinTownsendetal.2013[ 66 ],wechoseanaltimeof twoyearsandinterpretfailuretoreacheliminationbythenaltimeasafailedvaccination campaign.Foreachvalueof wefoundtherstinstancethataconsecutiveperiodoftime ofminimumlength wasexperiencedbetweenobservationsofinfectiousindividuals.Wethen recordedthenumberofunobservedinfectiousindividualsremainingafter weeks,X( T + ), where T hereisthetimeoftherstobservationinthepairofobservationtimesseparatedby atleast weeks. InFigures 4-5 and 4-6 ,overlaidhistogramsforthenumberofremainingunobserved individualsfoundusingeachoftheobservationmodelsinSection 4.2 arepresentedfor populationsizesof N =1000 and N =100 respectively.Wenotethatthedistributions resultingfrom100simulationsofeachmodelforeach valuearequalitativelysimilar, particularlywhen N =1000 (Figure 4-5 ).Forlarge bothmodelspredictthattherewillbe noremaininginfectiousindividualswhenthecampaignends.When N =100 (Figure 4-6 ), weseethatforlowtointermediate values,thetwo-dimensionalmodelfrom 4.2.2 tendsto yieldfewerremaininginfectiousindividualsthanthemoretraditionalmodelfromSection 4.2.1 FutureworkwillinvestigatewhethertheMarkovmodelresultsconvergetothereferencemodel resultsasthepopulationsizebecomeslargeandwilldevelopmethodsforquantifyingtheerror thatisintroducedbytheMarkovmodelwhenthepopulationsizeisnite. Figure 4-7 usesthenumberofremainingunobservedinfectiousindividualsandappliesthe resultsfromChapter 3 tondthecorrespondingprobabilityofre-invasion.Theexactsolution (seeSection 3.2 )isusedforthetwostochasticmodelssincetheyreturnintegervaluesand thebranchingprocessapproximation(seeDenition 3 )isusedforthedeterministicmodelas speciedinDenition 4.3.2 .Overall,theforgettingmodelseemstofairlywellapproximate themodelbasedonobservationtimes.InFigure 4-7 A,weseethatforapopulationsizeof N =100 andareproductionnumberwithoutvaccinationclosetoone,thetwo-dimensional Markovforgettingmodelpredictsalowerprobabilityofre-invasionthanthetraditional stochasticobservationmodel.Thelowerre-invasionprobabilityresultsfromalowernumberof 122

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remaininginfectiousindividuals.Soonaveragethevaccinationcampaignisendinglaterinthe Markovforgettingmodelinthisparameterregime.ThissuggeststhattheMarkovmodelcan beusedtoidentifyaconservativestoppingcriterion. 4.5.2Thedeterministicpredictionforthere-invasionprobabilitydoesnotperform well. TheerrorbetweentheODEapproximationofthere-invasionprobabilityandthe stochasticmodelscanbesmallforsome 4 values(seeFigure 4-8 ),buttheredoesnotappear tobeaclearrelationshipbetween 4 and N thatwouldallowidenticationofthebest 4 value touseforaparticularsystem.Thus,itwouldbedi"culttousetheODEapproximationin practicebecauseitisnotclearwhatthresholdvalueforobservedinfectiousindividualsshould beusedinordertodecidewhentoendthevaccinationcampaign. InFigure 4-8 ,wedisplaythemeansquareerror(MSE)betweentheODEapproximation andthetwostochasticmodelsforpopulationsizes N =100,200,316,and1000.Foreach value,theaveragere-invasionprobabilityforeachstochasticmodelwastakenovertheresults of100simulations.TheODEapproximationofthere-invasionprobabilitywascalculated asdenedinDenition 4.3.2 .Wethenfoundtheaveragedi!erencebetweentheODE approximationandeachstochasticmodelovertherangeof investigated(between0.14and 8weeks).WedisplaythesquareofthoseerrorvaluesinFigure 4-8 .Theoptimal 4 valueisthe valueoftheobservationthresholdthatminimizestheMSE.Forbothofthestochasticmodels, weobservethatsometimestheoptimal 4 isbelow 1 / N andsometimesitisabove 1 / N .It wouldbedi"culttousetheODEapproximationinpractice,sincethereisnotaconsistent patternbetweenthepopulationsizeandtheoptimalthresholdcuto!value. InFigure 4-7 ,weseethatforeach 4 theODEapproximation,thesuccessofthe approximationappearstovaryovertherangeof underinvestigation.Partoftheutility ofapproximatingthere-invasionprobabilityistoinformhowlongvaccinationcampaignsshould extendafterobservinganinfectiousindividual.TheODEapproximationcannotbeusedforthis purpose,ifthereisarelationshipbetween andtheoptimal 4 123

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4.5.3Usingourapproximationfortheapparenteliminationprobability. InProposition 4.4.2 ,wefoundasystemofparametricequationsforthecharacteristic curvesofthePDEfortheapparenteliminationprobabilityinDention 4.4.3 .InFigure 4-9 wedisplaycharacteristiccurvesthatemanatefromapoint ( 4 4 ) ,justinsidetherstquadrant boundary,sincethe y -axisisacharacteristicofthesystem.Foreach value,weplotthe characteristiccurveandthecorrespondinginitialconditionwhichwetaketobetheendemic equilibriumvalueoftheanalogousdeterministicsystemintheabsenceofvaccination, ( X (0), Y (0))= 1 # ( + $ ) / c $! (1 # ( + $ ) / c ) Thepositionoftheinitialconditionrelativetothecorrespondinglineofdemarcation determineswhetheritislikelythatapparenteliminationortrueeliminationwilloccur.In thelargepopulationlimit,withprobabilityone,pathsstartingfrominitialconditionsabove thelineofdemarcationwillhitthe y -axisrstandpathsstartingfrombelowthethelineof demarcationwillhitthe x -axisrst. Usingashootingmethod,weidentiedvaluesof thatproducedacharacteristiccurve emanatingfrom ( 4 4 ) andpassingthroughthecorrespondinginitialcondition.Wedenote thesevalues ( 4 ), anddisplaytheminFigure 4-10 .Itappearsthatas 4 approacheszero,the valuesareconsistentandapproachavaluearound3.625.Itremainstofurthercharacterizethis limit,butthepresentanalysissuggeststhatanaveragememorytimeofjustunderamonth mightbeanappropriatechoice. 124

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T N t =0 vaccination vaccination ends t =0 vaccination starts N X Y ? N Figure4-1.Diagramoftheframeworkunderlyingthetrueeliminationvs.re-invasionproblem. Thevaccinationcampaignbeginsattimezeroandendsattime T N ,thetimeof apparentelimination(i.e.thetimethatthenumberofobservedinfectious individuals Y ( t )=0 ).Theendemicequilibriumvalues, X and Y fromthesystem with p v =0 aretakentobetheinitialconditions.Time N isthersttimethat thenumberofunobservedinfectiousindividualsreturnstoanendemicstateorhits zero.Ourgoalistocharacterizethechancethatthenumberofunobserved infectiousindividualsreturnstoanendemicstate( p re-invasion )ratherthanachieving true-elimination. 125

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0 2.85 5 10 15 20 0 0.02 0.1 0.17 0.2 0 5 10 13.95 20 0 0.07 0.1 0.17 0.2 T T x ( t ) y ( t ) x x y y t t x ( T ) x ( T ) A B Figure4-2.Time-seriesofunobservedinfectiousclass x ( t ) andobservedinfectiousclass y ( t ) valuesaredisplayedasgrayandblackcurvesrespectivelyforapopulationofsize N =100 T isthetimethat y ( t ) crossesthethreshold 4 =1 / N (displayedasa horizontalblackdashedline). x ( T ) isthenumberofunobservedinfectious individualswhenthevaccinationcampaignends.Inbothpanels: R 0 =1.2 without vaccinationand R 0 =0.84 withvaccination.A)Theaveragetimeuntilforgetting anobservationis =1 week.B)Theaveragetimeuntilforgettinganobservation is =4 weeks. 126

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0 2 4 6 8 0.00 0.05 0.10 0.15 0.20 threshold cutoff 0.001 0.002 0.005 0.01 0.02 remaining unobserved proportion average duration of memory in weeks !"#$%&$'()*%$#*!+' Figure4-3.Theproportionofthepopulationremainingintheunobservedinfectiousclassat theendofthevaccinationcampaign, x ( T ) ,basedontheODEmodelin( 43 )asa functionoftheaverageamountoftimeittakesto"forget"anobservedinfectious individual, .TheODEsystemwassolvednumericallyforvedi!erentvaluesof thethresholdparameter 4 asindicatedinthelegend.Forthesesimulations N =100 R 0 =1.2 withoutvaccination,and R 0 =0.84 withvaccination.The initialconditionofthesystemwastakentobetheendemicequilibriumvalueinthe absenceofvaccination ( x 0 y 0 ) ) (0.17,0.017 ) 127

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0 50 100 150 200 0 10 20 30 40 50 tau= 0.14 * 0 50 100 150 200 0 10 20 30 40 50 tau= 0.34 * 0 50 100 150 200 0 10 20 30 40 50 tau= 0.55 * 0 50 100 150 200 0 10 20 30 40 50 tau= 0.75 * 0 50 100 150 200 0 10 20 30 40 50 tau= 0.95 * 0 50 100 150 200 0 10 20 30 40 50 tau= 1.96 * 0 50 100 150 200 0 10 20 30 40 50 tau= 3.97 * 0 50 100 150 200 0 10 20 30 40 50 tau= 8 * Y(t) X(t) * initial condition (x(T),y(T)) Y 100 ( t ) ! ! A B C D E F G H X 100 ( t ) Figure4-4.Phaseplotsofthenumberofunobservedinfectiousindividuals X N ( t ) vs.the numberofobservedinfectiousindividuals Y N ( t ) forapopulationofsize N =100 for between0.14and8weeks,where istheaveragetimeuntilforgettingan observation. T isthersttimethateither Y 100 ( t )=0 or X 100 ( t )=0 (see Denition 4.2.1 ).Aredstarindicatestheinitialstateofthesystemandabluestar indicatesthenalstateofthesystem ( X 100 ( T ), Y 100 ( T )) .Inallpanels, R 0 =1.2 withoutvaccinationand R 0 =0.84 withvaccination.A) =0.14weeks.B) =0.34weeks.C) =0.55weeks.D) =0.75weeks.E) =0.95weeks.F) =1.96 weeks.G) =3.97weeks.H) =8weeks. 128

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tau= 0.14 X.T.sigma[i, ] Frequency 0 50 100 150 200 0 20 40 60 80 100 tau= 0.34 X.T.sigma[i, ] Frequency 0 50 100 150 200 0 20 40 60 80 100 tau= 0.55 X.T.sigma[i, ] Frequency 0 50 100 150 200 0 20 40 60 80 100 tau= 0.75 X.T.sigma[i, ] Frequency 0 50 100 150 200 0 20 40 60 80 100 tau= 0.95 X.T.sigma[i, ] Frequency 0 50 100 150 200 0 20 40 60 80 100 tau= 1.96 X.T.sigma[i, ] Frequency 0 50 100 150 200 0 20 40 60 80 100 tau= 3.97 X.T.sigma[i, ] Frequency 0 50 100 150 200 0 20 40 60 80 100 tau= 8 X.T.sigma[i, ] Frequency 0 50 100 150 200 0 20 40 60 80 100 forgetting comparison Frequency X(T) "#$%&'&'(!)'*+,#"-#.!!!!!! !! /"#0)#'12! remainingunobservedfor N =1000( X 1000 ( T )) =0 14 =0 34 ! ! =0 55 =0 75 A B D E F G H ! ! C ! Figure4-5.Comparisonofthedistributionofthenumberofremaininginfectiousindividualsat theendofthevaccinationcampaignwithpopulationsize N =1000 X 1000 ( T ) ,for thestochasticmodelspresentedinSection 4.2 .Forthesesimulations R 0 =1.2 withoutvaccination,and R 0 =0.84 withvaccination.A) =0.14weeks.B) =0.34weeks.C) =0.55weeks.D) =0.75weeks.E) =0.95weeks.F) =1.96 weeks.G) =3.97weeks.H) =8weeks. tau= 0.14 X.T.sigma[i, ] Frequency 0 10 20 30 40 50 0 20 40 60 80 100 tau= 0.34 X.T.sigma[i, ] Frequency 0 10 20 30 40 50 0 20 40 60 80 100 tau= 0.55 X.T.sigma[i, ] Frequency 0 10 20 30 40 50 0 20 40 60 80 100 tau= 0.75 X.T.sigma[i, ] Frequency 0 10 20 30 40 50 0 20 40 60 80 100 tau= 0.95 X.T.sigma[i, ] Frequency 0 10 20 30 40 50 0 20 40 60 80 100 tau= 1.96 X.T.sigma[i, ] Frequency 0 10 20 30 40 50 0 20 40 60 80 100 tau= 3.97 X.T.sigma[i, ] Frequency 0 10 20 30 40 50 0 20 40 60 80 100 tau= 8 X.T.sigma[i, ] Frequency 0 10 20 30 40 50 0 20 40 60 80 100 forgetting comparison Frequency X(T) "#$%&'&'(!)'*+,#"-#.!!!!!! !! remainingunobservedfor N =100( X 100 ( T )) /"#0)#'12! ! =0 14 =0 34 ! ! =0 55 =0 75 A B D E F G H ! ! C ! Figure4-6.Comparisonofthedistributionofthenumberofremaininginfectiousindividualsat theendofthevaccinationcampaignwithpopulationsize N =100 X 100 ( T ) ,for thestochasticmodelspresentedinSection 4.2 .Forthesesimulations R 0 =1.2 withoutvaccination,and R 0 =0.84 withvaccination.A) =0.14weeks.B) =0.34weeks.C) =0.55weeks.D) =0.75weeks.E) =0.95weeks.F) =1.96 weeks.G) =3.97weeks.H) =8weeks. 129

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0 2 4 6 8 0.0 0.2 0.4 0.6 0.8 1.0 re invasion proability 0 2 4 6 8 0.0 0.2 0.4 0.6 0.8 1.0 Markov forgetting model stochastic observation model deterministic approximation average duration of memory in weeks 1 ! 1 R 0 Nx ( T ) 1 (1 /R 0 ) Nx ( T ) A B Figure4-7.Comparisonoftheprobabilityofre-invasionforthetwostochasticmodelsandthe deterministicapproximationwithpopulationsize N =100 .TheODEresultsare displayedforsixthresholdparameter ( 4 ) values.Fromtoptobottomtheblack linesindicatetheODEapproximationofthere-invasionprobabilityfor 4 =0.01,0.008,0.006,0.004, and 0.002 .Thenaltimewastakentobe104 weeks.A) R 0 =1.2 withoutvaccination,and R 0 =0.84 withvaccination.B) R 0 =1.4 withoutvaccination,and R 0 =0.98 withvaccination. 130

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0.00 0.04 0.08 N= 100 log10(theta.vals) temp 0.001 0.003 0.01 0.032 ODE approx error^2 observed threshold stochastic forgetting model stochastic comparison model 0.00 0.04 0.08 N= 200 log10(theta.vals) temp 0.001 0.003 0.01 0.032 0.00 0.04 0.08 N= 316 log10(theta.vals) temp 0.001 0.003 0.01 0.032 0.00 0.04 0.08 N= 1000 log10(theta.vals) temp 0.001 0.003 0.01 0.032 N =100 N =200 N =316 N =1000 A B C D !"#$%&$'()*%$#*!+' ,-.(/00%!123/)2!4($%%!% 5 Figure4-8.ThemeansquareerrorbetweentheODEapproximationforthere-invasion probabilityinDenition 4.3.2 andthetwostochasticmodelspresentedinSection 4.2 .Foragivenobservedthreshold 4 ,theaverageapproximationerroristaken over40 -valuesbetween0.14and8weeks.Forthesesimulations R 0 =1.2 withoutvaccination,and R 0 =0.84 withvaccination.A) N =100.B) N =200. C) N =316.D) N =1000. 131

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0.00 0.05 0.10 0.15 0.00 0.02 0.04 0.06 0.08 0.10 x.vec y.vec 0.00 0.05 0.10 0.15 0.00 0.02 0.04 0.06 0.08 0.10 x.vec y.vec 0.00 0.05 0.10 0.15 0.00 0.02 0.04 0.06 0.08 0.10 x.vec y.vec y(s) x(s) A B C y ( s ) x ( s ) 0.00 0.05 0.10 0.15 0.00 0.02 0.04 0.06 0.08 0.10 x.vec y.vec tau 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 ! Figure4-9.CharacteristiccurvesforthesolutiontothePDEcharacterizingtheprobabilityof apparenteliminationinDenition 4.4.3 .Curvesaredisplayedfor between1and 6weeks.Sincethe y -axisisacharacteristicofthesystem,wepresentcurves emanatingfromthepoint( 4 4 )justinsidetheboundaries.Foreachvalueof ,we alsoplotthecorrespondinginitialcondition ( X (0), Y (0)) whichwetaketobethe endemicequilibriumvalueoftheanalogousdeterministicsystemintheabsenceof vaccination.A) 4 = 0.001.B) 4 = 0.005.C) 4 = 0.01. 3.595 3.600 3.605 3.610 3.615 3.620 log10(c(0.001, 1e 04, 1e 05, 1e 06, 1e 07)) c(3.595, 3.613, 3.6202, 3.62236, 3.62297) 1e 07 1e 06 1e 05 1e 04 0.001 theta tau_* "#$%"&$'()%"*+,-',.'/$/,%0'+-'1$$23 ( ) ( ) 4,)-("%0'5)*,..'#"6)$ Figure4-10.Averagedurationofmemory( )valuescorrespondingwithcharacteristiccurves forthesolutiontothePDEcharacterizingtheprobabilityofapparentelimination inDenition 4.4.3 thatemanatefrom( 4 4 )andpassthroughthecorresponding initialcondition.Theinitialconditionistakentobetheendemicequilibriumvalue oftheanalogousdeterministicsystemintheabsenceofvaccination. 132

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BIOGRAPHICALSKETCH RebeccaBorcheringisfromAlva,Florida.SheearnedanInternationalBaccalaureate diplomaandgraduatedfromFortMyersHighSchoolin2007.RebeccaattendedArizona StateUniversityasaNationalMeritScholar.Herinterestindiseaseecologybeganwhileshe workedonanundergraduatehonorsthesismodelingcross-speciestransmissionofrabiesinbats andskunksinTexas.ShewasawardedtheCharlesWexlerMathematicsPrizeandgraduated withconcurrentdegreesinMathematicsandPhilosophyin2011.Shethenmovedbackto FloridatopursueadoctoraldegreeinmathematicsattheUniversityofFlorida.Shecontinued pursuingherinterestinmathematicalbiologyasaNationalScienceFoundationQuantitative SpatialEcology,Evolution,andEnvironmentIntegrativeGraduateResearchTraineeshipFellow. RebeccawasawardedtheMathematicsDepartmentEleanorEwingEhrlichAwardin2015and aCollegeofLiberalArtsandSciencesDissertationFellowshipin2016.Ingraduateschool, RebeccaservedasajuniorfacultymemberfortheInternationalClinicsonInfectiousDisease DataProgram.ShereceivedherPh.D.fromtheUniversityofFloridainthespringof2017. 139