A Simulation Model for the Spread of Citrus Greening via Transmission between Flush Shoots and Diaphorina Citri

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Title:
A Simulation Model for the Spread of Citrus Greening via Transmission between Flush Shoots and Diaphorina Citri
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1 online resource (159 p.)
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english
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Lee, Jo Ann
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Doctorate ( Ph.D.)
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University of Florida
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Mathematics
Committee Chair:
Keesling, James E
Committee Members:
Shabanov, Sergei
Singer, Burton H
Pilyugin, Sergei S
Ritter, Gerhard

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Subjects / Keywords:
citri -- citrus -- diaphorina -- flush -- greening -- huanglongbing -- model -- psyllid -- spatial -- transmission -- vector-transmitted
Mathematics -- Dissertations, Academic -- UF
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Abstract:
Huanglongbing (HLB, citrus greening) is an insect-transmitted bacterial disease of citrus trees that has impacted the citrus industry worldwide. We develop a transmission model that can mechanistically account for the rapid proliferation through groves of previously uninfected citrus trees. Based on the results of this simulation model, we present a method to estimate the time from infection to the appearance of symptoms in a citrus tree. In the last section of this thesis we consider assessment of the extent to which the spread of HLB deviates from a spatial random point process in the plane. To this end, we present an algorithm for estimating the Hausdorff dimension of self-similar fractal sets using a common tool in spatial analysis, the Ripley K-function.
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by Jo Ann Lee.
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Thesis (Ph.D.)--University of Florida, 2013.
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Adviser: Keesling, James E.
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ASIMULATIONMODELFORTHESPREADOFCITRUSGREENINGVIATRANSMISSIONBETWEENFLUSHSHOOTSANDDiaphorinacitriByJOANNLEEADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2013

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c2013JoAnnLee 2

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Tomyparentswhoneverfailtoshowmetheirperfectlove 3

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ACKNOWLEDGMENTS First,IwanttogivethankstoGodwhoinHisgraceandmercyhasprovidedforallofmyneeds,isthesourceofmyknowledge,andhasblessedmethroughoutmylife.ThenImustthankmyadvisor,JamesE.Keesling.Frominspiringmewithyourknowledgeinareasbeyondmathematicstothededicationthatyouhavetousemathtosolvereal-worldproblems,youhavebeenaconstantsourceofsupportandencouragement.Thankyouforyourpatienceduringourlongdiscussionsonhowtosolvetheseproblemsthatwehavebeenpresentedwith.IwouldalsoliketothankBurtonH.Singer,whohasdevotedcountlesshourstoimprovingmywriting,aswellasprovidinginvaluablediscussionsthatledtothecompletionofthiswork.Myprogressionintoamathematicalmodelerwouldnothavebeenpossiblewithoutthebothofthem.ToSusanE.Halbert,yourinsightsandknowledgeonthehistoryofcitrusgreeninginFloridawereaconstantsourceofinspirationtoguidethedirectionofmyresearch.ToBillDawson,forthecollaborationanddatathatwasusedtostrengthenthismodel.Iwouldalsoliketothankmycommitteemembers,SergeiShabanov,SergeiPilyugin,andGerhardRitter,fortheirquestions,suggestions,andinput.IwouldnotbewhereIamtodaywithouttheloveandsupportofmyfamily.Myparentshavebeenconstantsourcesoflove,support,andencouragementthroughoutmylifeandespeciallyduringthislastyearofmydegree.Mysiblingswhohavealwaysbeenthereformeandprovidedmewithopportunitiestorelaxandenjoytimewithfamily.Tomychurchfamily,Iwouldnothavebeenabletodothisaloneandamthankfulfortheirloveandprayers.IwouldliketospecicallythankJinahandDanielforbeingthereforme.Finally,IamgratefultoPastorPettitforremindingmethatChrististhesourceofmyeverything. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 8 LISTOFFIGURES ..................................... 9 ABSTRACT ......................................... 11 CHAPTER 1INTRODUCTION ................................... 12 1.1OverviewofCitrusGreening ......................... 12 1.1.1Huanglongbing ............................. 12 1.1.2BiologyofDiaphorinacitri ....................... 13 1.1.3TransmissionofHLB .......................... 13 1.1.4ControlStrategiesandGroveManagement ............. 15 1.2LiteratureReview ................................ 15 1.3Objectives .................................... 18 2TRANSMISSIONMODEL .............................. 21 2.1ModelDescription ............................... 21 2.1.1LocalTransmission ........................... 22 2.1.2TransmissionAcrossaGrove ..................... 26 2.1.3DemographyofPsyllidsandFlush .................. 28 2.1.4InitialConditions ............................ 30 2.1.5GroveLayout .............................. 31 2.1.6DeningtheInfectiousPeriod ..................... 31 2.1.7ModelforAppearanceofSymptoms ................. 32 2.2ParameterEstimation ............................. 33 2.2.1SurvivalProbabilities .......................... 33 2.2.2BirthProcessforPsyllidPopulation .................. 34 2.2.3BirthProcessforFlushPopulation .................. 35 2.2.4TransmissionProbabilities ....................... 36 2.2.5PsyllidMovement ............................ 38 3SIMULATIONMODEL ................................ 39 3.1ApproximationforTransmissiontoEmergingAdults ............. 41 3.2ApproximationforDailyTransmissiontoFlushShoots ........... 41 4RESULTS ....................................... 43 4.1InitialStagesofInfection ............................ 43 4.2ParameterVariation .............................. 50 5

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4.3ControlStrategies ............................... 50 4.3.1EfcacyofPsyllidControl ....................... 50 4.3.2TimingofApplyingControl ....................... 50 5ESTIMATIONOFINCUBATIONPERIOD ..................... 54 5.1Background ................................... 54 5.2RecentlySymptomaticGrove ......................... 55 5.2.1SpreadofInfectionandSymptoms .................. 55 5.2.2IncubationPeriodEstimate ...................... 56 5.3ExtensivelySymptomaticGrove ........................ 58 6USINGTHERIPLEYK-FUNCTIONTOESTIMATETHEFRACTALDIMENSIONOFASELF-SIMILARFRACTALSET ........................ 61 6.1Motivation .................................... 61 6.2Background ................................... 62 6.3Notation ..................................... 63 6.4HausdorffMeasureandDimension ...................... 65 6.5TheRipleyK-Function ............................. 66 6.6RandomDistributionofPointsonaFractal .................. 68 6.7TheTwo-LinesTheorem ............................ 68 6.8TheTwoLinesAppliedtotheRipleyK-function ............... 71 6.9APracticalAlgorithm .............................. 73 6.10ComputationalExamples ........................... 74 6.11FurtherDirections ............................... 75 7DISCUSSION ..................................... 78 7.1ImplicationsoftheTransmissionModel .................... 78 7.2Limitations ................................... 79 7.3SuggestionsforFutureResearch ....................... 80 7.4Conclusion ................................... 81 APPENDIX ASIMULATIONMODELDAILYACTIVITIES ..................... 83 A.1SpringFlushPeriod .............................. 83 A.2SummerFlushPeriod ............................. 123 A.3FallFlushPeriod ................................ 139 BSUPPORTINGEVIDENCE ............................. 143 B.1MechanismforFlushtoPsyllidTransmission ................ 143 B.2LocalDispersalofPsyllidsinSarawak,Malaysia .............. 145 B.3AppearanceofHLBinaCitrusOrchardinColima,Mexico ......... 148 B.4PresenceofInfectedPsyllidsPriortoInfectedTreesinFlorida,USA ... 149 6

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REFERENCES ....................................... 152 BIOGRAPHICALSKETCH ................................ 159 7

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LISTOFTABLES Table page 4-1Sensitivityofmodeltovariationsinselectedparameters ............. 51 4-2Sensitivityofmodeltovariationsinlevelsofpsyllidcontrol ............ 52 4-3Sensitivityofmodeltovariationsintimingandlevelsofpsyllidcontrol ..... 53 B-1Infectionofpsyllidprogeny ............................. 144 8

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LISTOFFIGURES Figure page 2-1Diagramofqualitativedistinction .......................... 22 2-2Transitionbetweenstates .............................. 24 2-3Transitionsinspace ................................. 26 2-4Annualappearanceofush ............................. 29 2-5Survivalprobabilitiesforthevector ......................... 33 2-6Probabilityofemergingasinfectedadult ...................... 37 3-1Modelapproximationfortransitionbetweenstates ................ 40 4-1ProportionofinfectedD.citriadultsateveryushpatch ............. 43 4-2ProportionofinfectedD.citriadultsatdesignatedcoordinates ......... 44 4-3Populationofushshoots .............................. 44 4-4Proportionofinfectedadultpsyllidsateachushpatchstartingwith4randomlyselectedpatchesontherighthandedgeofthegrove ............... 45 4-5Proportionofinfectedadultpsyllidsateachushpatchstartingwith17selectedpatchesdistributedalongthebottomandrighthandedgesofthegrove .... 46 4-6Progressionofinfectedandsymptomaticushpatchesforthesouthwestcornerconguration ..................................... 47 4-7Progressionofinfectedandsymptomaticushpatchesforthesouthwestcornercongurationafter90%reduction .......................... 47 4-8Progressionofinfectedandsymptomaticushpatchesforthesouthwestcornercongurationaftertwo90%reductions ....................... 48 4-9Spreadofinfectioninagrovewithremovedtrees ................. 49 6-1K(h)andthetwolines ............................... 72 6-2K(h)foratypicalexamplefromSection 6.10 ................... 72 6-3K(h)andthetwolinesif<=dimH(M) .................... 73 6-4SierpinskiCarpet ................................... 74 6-5vonKochCurve ................................... 75 6-6Self-similarfractaldimH(M)=log7 log31.77124 ................... 75 9

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6-7Self-similarfractaldimH(M)=log6 log31.63093 ................... 76 6-8MengerSponge ................................... 76 B-1SpreadofD.citriinacitrusorchardatJemukan ................. 147 B-2DailyinfestationofD.citriinasimulatedgrove .................. 147 B-3MonthlyinfestationofD.citriinasimulatedgrove ................. 148 B-4ProportionofinfectedD.citriadultsateveryushpatch ............. 149 B-5SymptomatictreesinsimulatedColimagrove ................... 150 B-6DistributionoftreeswithHLBsymptomsinColima,Mexico ........... 151 10

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyASIMULATIONMODELFORTHESPREADOFCITRUSGREENINGVIATRANSMISSIONBETWEENFLUSHSHOOTSANDDiaphorinacitriByJoAnnLeeAugust2013Chair:JamesKeeslingMajor:MathematicsHuanglongbing(HLB,citrusgreening)isaninsect-transmittedbacterialdiseaseofcitrustreesthathasimpactedthecitrusindustryworldwide.Wedevelopatransmissionmodelthatcanmechanisticallyaccountfortherapidproliferationofinfectionthroughgrovesofpreviouslyuninfectedcitrustrees.Basedontheresultsofthissimulationmodel,wepresentamethodtoestimatethetimefrominfectiontotheappearanceofsymptomsinacitrustree.InthelastsectionofthisthesisweconsiderassessmentoftheextenttowhichthespreadofHLBdeviatesfromaspatialrandompointprocessintheplane.Tothisend,wepresentanalgorithmforestimatingtheHausdorffdimensionofself-similarfractalsetsusingacommontoolinspatialanalysis,theRipleyK-function. 11

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CHAPTER1INTRODUCTIONOneofthemostseriousproblemsofcitrusworldwideishuanglongbing(HLB)orcitrusgreening.HLBhasbeenresponsiblefortheneardestructionofthecitrusindustriesinAsiaandAfrica[ 11 ],anditisseriouslyimpairingtheindustryinFlorida.ThecitrusindustryinFloridaisresponsibleforcreatingover75thousanddirectandindirectjobsandthetotaloutputeconomicimpactin2007-08exceeded$8.91billion[ 36 ].TheearliestdescriptionofsymptomsresemblingHLBwasfromcentralIndiainthe1700s[ 13 ].Themainsymptomsareyellowshoots,earlyfruitdrop,leaveswithblotchymottle,andsmalllopsidedgreenfruit.Theinfectedbrancheseventuallydiebackandthetreedies.AreportfromthePunjab[ 38 ]wasthersttoassociatetheproblemwithaninsect,theAsiancitruspsyllid(ACP),DiaphorinacitriKuwayama(D.citri),whichwenowrecognizeasthemajorinsectvectorofthediseaseeverywhereexceptAfrica.TheputativecausalagentofHLBisanalpha-proteobacterium,CandidatusLiberibacterasiaticus(CLas),thatresideswithinthephloemandistransmittedbythepsyllidD.citri.ThehighestconcentrationofCaLiberibacterininfectedtreesareinstemandmidribsofush[ 16 ].Theushisanewlydevelopingclusterofyoungleavesontheexpandingedgeofaterminalshoot.Thebacteriummultipliesinboththepsyllidandthetrees,butthepsyllidsareessentialforthenaturalspreadofthedisease. 1.1OverviewofCitrusGreening 1.1.1HuanglongbingItislikelythatHLBwasrstestablishedinIndiainthe1700s[ 13 ]andthenspreadtoChina.ItalsowasreportedinSouthAfricain1929,wherethevectorisTriozaerytreae(delGuercio),andthebacteriainvolvedisarelatedspeciesinthesamegenus[ 53 75 ].DiaphorinacitrihasbeenpresentinBrazilfor60yearsandwasrstdiscoveredinFloridain1998.IntheAmericas,HLBrstwasidentiedinBrazilin2004andFlorida 12

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in2005[ 11 ],spreadingrapidlyinbothplaces.MorerecentlyHLBhasappearedinTexasandCalifornia[ 66 69 76 ]. 1.1.2BiologyofDiaphorinacitriAsthemainvectorofthediseaseintheUnitedStates,itisimportanttounderstandthelifecycle,movement,andfeedingbehaviorofDiaphorinacitri.Thepsyllidhassevendistinctnon-overlappingstages,egg,venymphalinstars,andadult,whichformodelingpurposeswillbegroupedintothreepsyllidstagesofegg,nymph,andadult.Onaverage,theeggstagemaylastfrom2to4days,thevenymphalstagesarecompletedin11to15days[ 28 ].TherehavebeenmanystudiesconductedontheeffectsoftemperatureonthelifecycleandsurvivalofD.citriaswellasondifferentvarietiesofcitrus.Theoptimumdevelopmenttemperaturerangewasfoundtobe25to28Candthemaximumaveragenumberofeggsproducedperfemalewasat28C[ 48 ].Inthesametemperaturestudy,theyfoundthatthemaximallongevityofindividualfemalesrangedbetween51and117daysat30Cand15Crespectively.Psyllidsmovebyadiversityofnaturalandhuman-inducedmethods.Forexample,inastudytoestimatethelocaldispersalkernelforthepsyllid,Koborietal.estimatedthediffusioncoefcientforaportionofpsyllidstobe7.23m[ 44 ].Acrosstheliterature,theestimatedrangeofpsyllidmovementdependinguponmethodisfromunder10m[ 4 44 ],tobetween500mand4km[ 3 4 9 49 ],andasfaras90to124km[ 26 ].Psyllidscanbetransportedlongdistancesthroughhuman-aidedmovementsuchasonpottedplantsorcarriedonunprocessedfruitintrailers[ 29 ].Inadditiontheycanbecarriedfurtherdistanceswiththeassistanceofwind[ 4 9 11 ]. 1.1.3TransmissionofHLBTransmissionofHLBoccurseitherbygraftingofinfectedplantmaterialorcitruspsyllids.Sincegraftingcanbecontrollediftheregulationsonproducingdisease-freenurserystockforcitrusproductioninFloridaisfollowed,wefocusonthetransmissionofHLBbythevector,Diaphorinacitri.Manystudieshaveestimatedparametersfor 13

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transmissionfromD.citritocitrustreesaswellasacquisitionofCLasbyadultsandnymphsfrominfectedtrees.Howevertheseparametersvarywidelyanddependonthelengthoftimepsyllidsfeed[ 15 57 ].Inadditiontotheinconsistentratesofpathogenacquisition,inthesestudies,adultpsyllidsthatacquirethepathogenasadultsrequirealatentperiodof1to25daysbeforetheyareabletotransmittoatree[ 28 57 ].However,nymphsraisedoninfectedplantsareabletotransmitimmediatelyasadults.Therearealsoconictingconclusionsfromstudiesonwhethertransovarialtransmissionoccurswhichrangefromnotransmission[ 37 ]toalowpercentage.Pelz-Stelinksietal.[ 57 ]showedtransovarialtransmissionoccurredfromfemaleadultstooffspringandfoundthat2.0,6.3,and2.4%successfultransovarialtransmissionwasdetectedinoffspringconsistingof49eggs,48nymphs,and42adults,respectively.Recently,anexperimentalstudyhasshownthatpsyllidsmaypasstheinfectionfromonegenerationtothenextthroughfeedingonushshootsinaperiodasshortas15days[ 46 ].Withinthisperiodof15days,nymphsraisedonhealthyplantswiththepresenceofinfectedadultsresultedintransmissionrangingfrom0to83%.Inthisstudy,theydidnotdistinguishbetweentransovarialtransmissionandtransmissionthroughushshoots,howeverthehighpercentagefromtheushshootexperimentcombinedwiththelowpercentagefromtransovarialtransmissionaloneleadsustoconcludethataviabletransmissionpathwaytothenextgenerationofpsyllidsisthroughfeedingonushshoots.Duetothemechanismfortransmission,researchonfeedinghabitsofthepsyllidisimportanttounderstand.Therehavebeenafewstudiesonthefeedingbehaviorofadults[ 2 15 ],andonlyonethatconsidersbehaviorofnymphsandthefeedingstructureofnymphs[ 2 ].Thestudy[ 2 ],determinedthatpsyllidshavenopreferencebetweenhealthyandinfectedleaves.However,thereisnoquantitativeinformationaboutthemovementofpsyllidsbetweenushshootsorthefrequencyoffeedingsthroughouttheday. 14

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1.1.4ControlStrategiesandGroveManagementCitrusgreeningisanextremelydestructivediseasethatcausescitrustreestodeclinewithin5to8yearsafterplanting,whileithasbeenshownthatcitrusgrovesmustliveforaminimumof10yearsbeforetheybecomeprotable[ 64 ].Thedominantcontrolmeasurescurrentlyinuseatthegrove-levelarethecombinationofinsecticidalsprayingtolimitthepsyllidpopulationsandtheremovaloftreestheinoculumsourcewhentheyaresymptomatic.Thisstrategyhashadonlymarginaleffectbecausesymptomsappearanywherefrom1to2years,orperhapsmore[ 67 ],afteraninitialinfection,longafterthetreeshavebeenactiveinthetransmissionprocess.Pre-symptomatictreescanserveasasourceofinoculumforpsyllids,thelengthoftimefrominitialinfectionofatreeuntilitisinfectiveasasourceofinoculumhasrecentlybeenshowntobelessthan15days[ 46 ].Thispreviousknowledgegaphaslimitedthedevelopmentofdefensibletransmissionmodelsthatcanaccountfortherapidproliferationofinfectioninagroveofcitrustreesandalsoprovideusefulguidanceaboutnewcontrolstrategies.Therealsohasbeenalackofadequatemethodologyforlowcostsamplingofasymptomatictreestoascertainwhetherornottheyareinfectedand,thereby,inneedofapplicationofcontrolmeasurestolimitspread. 1.2LiteratureReviewAreportbytheNationalResearchCouncil(NRC)statedthatMathematicalmodelsthatincorporatedatasuchas[...]behaviorofACPadultsandnymphs;interactionsofACP,CLas,citrustrees,otherorganismsandtheenvironment,provideatoolthroughwhichtheresultsofbehavioralecologycanbeappliedtodiseasemitigation.[...][The]constructionofamathematicalmodelfortheCLas/ACP/citrus-HLBsystemshouldbealong-termgoalofHLBresearch.Itisrecognizedthatthecomplexityofpathogentransmissionbyaninsectvectorwillmakethetaskverychallenging[ 55 ,p.124].Thechallengefromthecomplexityofthisvector-transmittedpathogenisuniquelysuitedtomathematicalandcomputationalmodeling.Theobjectivesofthese 15

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modelsaretodeterminethedynamicsbetweenvectorandhostinordertoprovideinsightsintodevelopingeffectivemanagementstrategies.Ourmodelisdevelopedasanagent-basedmodelinordertoincorporatetheessentialcomponentsinvolvedinthetransmissionofHLBamongtheushshootsbytheACPadultsandnymphs.Wecombineimportantaspectsofpreviousmodelstogethertogainunderstandingofthediseasedynamicswithinagrove.Theaspectsweincludearevectorpopulationdynamicsdependentonushshoots[ 5 16 ],spatiallyexplicitlocations[ 5 45 ],andtemperaturedependency[ 5 ].Wewillalsodiscusstheimpactofcontrolstrategiessuchasinsecticidespraying[ 16 19 45 ],removalofush[ 16 ],androguingandreplantingtrees[ 19 40 45 ].Understandingtherapidspreadofpsyllidsupto30,000psyllidsonasingletree[ 1 ]accumulatingduringa60dayushperiodandcorrelativelyinfection,throughoutgrovesisfacilitatedbytheuseofspatiallyexplicittransmissionmodelswithinwhichpsylliddemography,pathwaysofinfectionbetweenpsyllidsandyoungush,andinter-treemigrationaretakenintoaccount.HLBtransmissionmodelsto-datehavebeendeterministiccompartmentalmodelswheresystemsofordinarydifferentialequationsrepresentthedynamics[ 19 40 ].Thegeometryofgrovesandthefociofpsyllidentryarenottakenintoconsiderationinthesemodels,despitethefactthattheseclearlyinuencetheinitialspreadofHLBandthedistributionofsymptomatictreesonthelongertimescaleof1to5years[ 67 ].TwoformsofdelaysnaturallyenterthemodelingofthespreadofHLBacrossagrove.Theseare:(i)thetimefrominitialinfectionofyoungushonatreeuntiltheonsetofdiseasesymptoms,and(ii)thetimefrominitialinfectionofyoungushbyadultpsyllidsuntiltheushbecomesinfectious.AninstructivestudyoftheimpactofsuchdelaysontransmissionforsoiltransmittedplantpathogenshasbeengivenrecentlybyCunniffeetal.[ 18 ].Theyextendthetypicalsusceptible,exposed,infected,andremoved(SEIR)compartmentalmodelbysplittingthelatentandinfectioncompartmentstoallowfortime-varyinginfectionrates,therebyallowingformorerealisticrepresentationof 16

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thesedelays.However,theirformulationisnotspatiallyexplicitandwouldneedtobeaugmentedbyequationsforvectordynamicsandinteractionsofvectorsandplantstobecomesuitableforHLBmodeling.ThedelayintimefrominitialinfectionofatreeuntiltheonsetofdiseasesymptomshasbeenincludedinpreviousHLBtransmissionmodelsbyincorporatingacompartmentforinfected,asymptomatictrees[ 19 40 ].Withourspatiallyexplicitmodel,weareabletodescribethespreadofinfectionthroughagroveunderseveralinitialconditionsthatconsequentlyproducedifferentpatternsofspread.Weareabletocomparetheinitialspreadofinfectionwiththerecordedspreadofsymptomsinordertoapproximatetherstformofdelay.Currently,therehasonlybeenonepreviousmathematicalmodeltoidentifytheushshootsastheprimaryhostofinfectiontobestudied.Biologically,itisknownthattheushshootsarethelocationswheretheeggsarelaidandthetendernessoftheshootsprovideoptimalfeedingforthenymphalstagesforconsecutivedays.Chiyakaetal.[ 16 ]presentamodelthatfocusesonushtoushtransmissioninasingletree.Intheirmodeltheushtoushtransmissionoccursthroughinternalmovementandrequiresalatentperiodofunknownlengthwhichtheyvaryfrom30to180days.Ourmodelaccountsformultipletreesinagrovewhereushtoushtransmissionoccursduetothepsyllids.Themaindifferenceduetothismodelingassumptionisthatthislatentperiodisnowasshortas15days[ 46 ].Whiletheypresenttheinteractionsbetweenthepathogen,vector,andtree,thereisnospatiallyexplicitinteractionbetweentreesinagrovewhichisanimportantpartinunderstandinghowthediseasespreads.ForspatialmodelsofthespreadofHLB,thereisanindividual-basedsimulationmodelduetoKoborietal.[ 45 ].Intheirsimulationmodel,atreemovesthroughthestagesofhealthy,infectedandasymptomatic,symptomatic,andremoved.Thereisnodistinguishingbetweenlatentandincubationperiodswhichmeansthattheirtreesareinfectiousandsymptomaticatthesametime.Withthisassumptiontheyshowedthatremovingtheinfectioustreeswithin6to9monthsofbecominginfectiousresulted 17

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inadecreaseofinfectedtrees.Thiswouldbetrue,howevertheyaremissinganimportantfactthatwithHLB,thetreebecomesinfectiousmuchearlierthanthedetectionofsymptomsinatree.Thereisanotherspatiallyexplicitmodelthatdescribesthedynamicsbetweenthehostandvectorbasedonrainfallandtemperature[ 5 ].Theyconsiderthepsyllidonacountry-widescaletohighlighttheimpactofclimateonhostsandthereforevectorpopulations.However,thisstudydoesnotincludetransmissionofHLB.Beforecontinuinganyfurther,Itisimportanttounderstandthedenitionsoftermssuchaslatentperiodandincubationperiod. Latentperiod:thetimefrominfectiontoinfectiousness. Incubationperiod:thetimefrominfectiontoonsetofsymptoms.Dependingonthespecicdisease,theincubationandlatentperiodaretypicallythesame.Howeverwithcitrusgreening,itisimportanttodistinguishbetweenthetwoperiodssincetheyareofqualitativelydifferentduration.AnothertermmisusedintheliteratureisthephraseincidenceofHLB.Incidenceistypicallyusedinepidemiologyastherateofacquitsitionofsomeconditionperunittimeperindividualatrisk.InmostofthecaseswheredataonHLBsymptomsispresented,diseaseincidence[is]calculatedasthenumberoftreesexpressingsymptomsdividedbythetotalnumberoftreesintheorchard[ 47 ,p.2].Thisisactuallyaprevalencerateforsymptomsandwewillusethisterm. 1.3ObjectivesWiththecomplexitiesinvolvedinthisdisease,furtherquantitativestudytounderstandthedynamicsbetweenthehostandvectorisneeded.Gottwaldetal.statedthatrelativelyfewquantitativeepidemiologicalstudieshavebeenconducted.Thisisduetotheperennialnatureofthediseaserequiringadedicationtodatacollectionovermultipleyears...Inaddition,therehasbeendifcultyinlocatingstudysiteswherethediseaseisallowedtoprogressunimpeded[ 26 ,p.8].Withouragentbasedmodel, 18

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weaccountfordailydetailsofegg-laying,survivalofthepsyllid,ushemergence,transmissiontoushshootsandemergingnymphs,andlocalmigrationbetweentrees.Weapplyinterventionstrategiestosimulaterealisticconditions,andwechoosetofocusonpsyllidcontroloverroguingoftreesduetotheshortlatentperiodofthediseaseintrees.Wedevelopmethodstoestimatetheincubationperiodinatreethroughtheuseofcurrentdataontheappearanceofsymptomatictrees.Inaddition,surveyresultsfoundinfectedD.citrinymphsandadultsingroves6yearsbeforeCLaswasdetectedinasymptomaticorangetrees[ 29 50 67 ].Thisdoesnottwiththecurrentepidemiologicalunderstandingofthedisease,wherethecitrustreescanonlypassontheinfectiontothepsyllidafteralatentperiod.Duetothisobservation,oneofthegoalsofourmodelistopresentamodelthatisabletoexplainthepresenceofinfectednymphsbeforesymptomaticallyinfectedtreesappear.Intherstpartofthisdissertation,wedevelopanddescribeatransmissionmodelthatcanmechanisticallyaccountfortheobservedrapidproliferationofCLasinfectionthroughgrovesofpreviouslyuninfectedcitrustrees.Wedemonstrate,viamodeling,thepotentialimpactofnewsurveillanceandinterventionstrategiesthattakeaccountoftheexperimentalevidenceontheelapsedtimefrominitialinfectionofatreeuntilitisasourceofinoculumfornewpsyllids.Thesecondhalfofthisdissertationpresentsapplicationsthatstemfromourtransmissionmodel.InChapter 5 weusetheresultsoftherapidspreadofHLBindicatedbyourmodeltodevelopamethodtoapproximatethetimetosymptomsforindividualtreesfromelddata.Thisestimatefortheincubationperiodcannotbetestedinalabandisnotcurrentlyknown.Theabilitytoknowthetimetosymptomsiscrucialtodevelopingmanagementpracticesforcontrollingthedisease.Inthenalchapter,ourlastresultstemsfromthespatialanalysisofthespreadofcitrusgreeningusingtheRipleyKfunction.TheRipleyKfunctionhasbeenusedtoanalyzespatialpatternsoftrees[ 58 70 ],herbaceousplants[ 68 ],anddiseasecases[ 20 ].Ripley'sKanalysisisastatisticalanalysisofspatialpointprocesses.Itsmain 19

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useistodeterminedeparturefromacompletespatialrandom(CSR)pointprocess.Forspatialanalysisofcitrusgreening,Gottwaldetal.[ 27 ]usedaRipleyKanalysistodemonstrateshortrangeandregionalcomponentsinvolvedinthespreadofHLB.Theyestimatedthattheaveragedispersaldistancefromaregionalpointofviewis1.58km.Inordertodeterminethelongerdistancemovementofpsyllids,thespreadthroughthecitrusgrovesinFloridaareconsidered.ThecitrusgrovesformasubsetoftheplaneinR2andthereforethedeviationfromCSRwouldbeduetothespatialarrangementofthegrovesandnotofthedistributionofsymptomatictrees.ThisbecamethemotivationforapplyingRipleyK'sanalysistoaspatialpointprocessthatmayliveonafractalset.WeusetheRipleyKfunctionasthebasisforanalgorithmtoestimatetheHausdorffdimensionofaself-similarsetusingarandomdistributionofpointsontheset. 20

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CHAPTER2TRANSMISSIONMODELOurmodelforthetransmissionofHLBdiffersfrompreviousmodelsthatrequiresa2to6monthlatentperiod[ 16 ]forthetreetobecomeinfectiousbeforenewpsyllidgenerationscanbecomeinfected.InourmodelthetransmissionoccurslocallybetweenpsyllidsandushshootsasseenfromtheresultsfromDawson'sexperiments[ 46 ].ThatexperimentshowsthatpsyllidprogenyacquiredLasfromplantsthat30daysearlierwerenotinfectedwithHLB.Thistimeframewassufcientfortheplantstobecomeefcientdonorhosts.ItisknownthatpsyllidshavehighertitersofLaswhenthebacteriumisacquiredbynymphs[ 37 ].Thus,theassayedadultprogenypsyllidswouldhavehadtoacquirethebacteriumearlierthan30days,likelywithin10to20days.Psyllidsdidnotlayeggsonallplantsineachcage.Usuallyabout1/3to1/2oftheplantswereinfestedwithnymphs.Nymphsgenerallystayinthesameareawhereeggsarelaid.ThustheplantsamplesthatwereanalyzedbyPCRforLassequencesweretakenfromlocationswherenymphsdeveloped.Plantsamplesthatweretakenat10to15daysalreadytestedpositiveforLas.TheexperimentalresultsconrmthattheareasoftheplantswherepsyllidprogenydevelopedcouldbecomeinfectedwithLaswithinthisshorttimeperiodandtherebyserveasasourceofinoculumforthedevelopingnymphs. 2.1ModelDescriptionWefocusontransmissionofthebacteriaC.Liberibacterbetweenthepsyllidvector,D.citri,andyoungushshootsonthetrees.Patchesofushandgroupsofpsyllidsareidentiedwithpointsonatwo-dimensionalgrid,representingagroveofcitrustrees.Incontrasttopriorindividual-basedsimulationmodelsofHLBspread[ 45 ],weconsiderthechangeintreestatusovertimeaccordingtothefollowingstepsshowninFigure 2-1 .ThequalitativedistinctionbetweenthispathwayandthoseintheextantliteratureistheveryshortlatencyperiodassupportedbytheexperimentdescribedinAppendix B.1 andtheconsequentialrapidinfectionofnymphsandpreviouslyuninfectedadultpsyllids 21

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. . Healthytree Infectedandinfectiousush Symptomatictree Deathoftree TransmissionofHLBbyinfectedpsyllids [LatentPeriod<15days;asymptomatic] [1)]TJ /F9 7.97 Tf 7.11 0 Td[(2.5yearsafterinitialinfection] Figure2-1. Qualitativediagramthatshowsthetreestatusandtimeperiodsconsideredinthismodel. onthenewlyinfectedush.Thelongasymptomaticperiodofthetreeisaccompaniedbytherapidspreadofinfectionamongpsyllidsandnewush,therebyleadingtolargepercentageseven100%oftreesinfectedinagrovebeforeanysymptomsbecomemanifest.Thisplacesapremiumonintensivepsyllidcontrolandthenecessityofsurveillanceofpsyllidpopulationsinasymptomaticgroves.Formodelspecication,werstsummarizethedynamicsoveradayatapatchofushasrepresentedbythetransitiondiagraminFigure 2-2 .Spatialtransmission,whereinfectiondispersesfrompoint-to-pointonthegrid,isdescribedinSection 2.1.2 2.1.1LocalTransmissionAush,orhost,patchisrepresentedbythecompartmentsshowningreeninFigure 2-2 .Thestatesofthepsyllidvectorsareidentiedbythebluecompartments.Transitionsthroughtheage-dependentsusceptibleandinfectedstates,SjiandIji,consideredinthemodelareshowninFigure 2-2 ,whereidenotesthevector(v)orhost(h)andjdenotestheagecategories.LetSv(k,d)andSh(k,d)representthepopulationofsusceptiblepsyllidvectorsandushshoothostsondaykwithaged.Let Sv(k,d)=8>>>>>><>>>>>>:Sev(k)if0d
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and Sh(k,d)=8>><>>:Syh(k)if0d
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. Syh Iyh Sev Snv Sav Iev Inv Iav Soh Ioh + + . e n a e n a h h h . Sv Iv v Figure2-2. Transitionbetweenstates,blackarrowsindicatetransitionbetweenstatesbyaging,redarrowsindicatetransitionbetweenstatesbytransmission,dashedarrowsindicatestateswithaninuenceonoviposition,dashedredarrowindicatestransovarialtransmission,anddottedredarrowsindicatestateswithaninuenceontransmission. simplyadelayfromthenymphstagetoadultemergence,withapproximately7.9%ofrstinstarnymphssurvivingtoemergeasadults.Onanygivenday,Iav(k)recordsthenumberofinfectedadultpsyllidsataushpatch.Thispopulationconsistsofadultsthatemergeinfectedfromthenymphstageaswellasthosethatfeedoninfectedush.Sav(k)countsthenumberofuninfectedadultpsyllidsatthepatch.WewilloftenrefertothetotalpopulationofhostsandvectorswhereNjh(k)=Sjh(k)+Ijh(k)istotalnumberofushinstatej=y,oandNjv(k)=Sjv(k)+Ijv(k)isthetotalpsyllidpopulationineachagecompartmentj=e,n,a.Oneachday,thereisastate-dependentprobability, 24

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sj=1)]TJ /F7 11.955 Tf 12 0 Td[(j,ofsurvivingtothenextday.Thetransitionbetweenstatesoccurforsurvivingpsyllidsthatageintothenextstate,detailsofthesetransitionsandprobabilitiesinthegeneralizedbirth-deathprocessareinSection 2.2.1 and 2.2.2 .Leth(k)bethenumberofnewushshoots, h(k)=h(k)1k2F(2)where1istheindicatorfunctionandFistheperiodofdayswherenewushisproducedandh(k)isthenumberofushperdaythatisallowedtovarydependingontheseason.LetSv(k)andIv(k)bethedensity-dependentbirthrateofsusceptibleandinfectedeggs.LetMe(k)bethenumberofeggslaidondayk. Me(k)=min1 2Nav(k)ea,Nyh(k)eh(2)whereeaandehisthenumberofeggsperfemaleadultandushperday,respectively.Then Iv(k)=pvIav(k) Nav(k)Me(k)(2) Sv(k)=Me(k))]TJ /F7 11.955 Tf 11.95 0 Td[(Iv(k)(2)whereistheprobabilityoftransovarialtranmissionandpvistheproportionofinfectiveadults.WedeneinfectivepsyllidsasabletotransmitLaswith100%efciency.ItisimplicitintheexpressionsofSv(k)andIv(k)thatalimitonthecarryingcapacityisduetoeitherNyh(k)orNav(k).Forlocationswithbothpopulationsofpsyllidandushshoots,transitionsfromhealthytoinfectedstatesareallowed.Leth(k)betheexpectednumberofyoungandoldhostinfectionsduetopsyllids,assumingthatallushhaveanequalprobabilityofbeingfedonwehave h(k)=Sh(k) Nh(k)mXj=1jXi=1j()]TJ /F6 11.955 Tf 9.3 0 Td[(1)j)]TJ /F14 7.97 Tf 6.59 0 Td[(iNh(k)jjii Nh(k)B(k)(2) 25

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whereB(k)isthenumberofinfectivefeedings,Nh(k)=Nyh(k)+Noh(k)isthetotalpopulationofush,andm=min(B(k),Nh(k)).Letv(k)betheexpectednumberofacquistionsofHLBbypsyllidnymphs.Thetransitionisassumedtooccuronlyinthenymphalstageasithasbeenshownthatputativelyinfectiveadults(rearedonLas-infectedplants)transmitmoreeffectively[ 57 ], v(k)=f(~!,Inv(k),Iav(k),Iyh(k),Ioh(k))(2)where~!isavectorcontainingparametersfortheprobabilityoftransmissionthroughdifferentpathways.Thesepathwaysoftransmissionincludeinfectedadults-to-nymphsandinfectednymphs-to-nymphsthroughfeedingonushshoots.WewilldescribethepathwaysoftransmissionandtheirprobabilitieslaterinSection 2.2.4 2.1.2TransmissionAcrossaGroveAgroveoftreesismodeledasagridwheretheintersectionsareassociatedwithpatchesofush.Spacingofpatchesiscloserwithinarowthenbetweenrows.Inrealgroves,withinrowspacingisapproximately5meters,andbetweenrowspacingis7.62meters[ 73 ].Figure 2-3 showsthreecategoriesofushpatchlocationswithinagroveandthenewlocationstowhichpsyllidsareallowedtomoveonasingledayinoursimulations. AInterior BEdge CCornerFigure2-3. Transitionsinspace.ThreedifferentsetupsareshownwherethestartingpositioninthegroveisA)Locatedintheinteriorofthegrove,B)Locatedonanedgeofthegrove,andC)Locatedonacornerofthegrove.Thegreencirclerepresentsthecurrentlocationandbluecirclesarethepossiblenextlocations 26

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Atthebeginningofthesimulation,psyllidsaredistributedamongthegroveinanygiveninitialconguration.ThedetailsfordeterminingtheinitialcongurationwillbeinSection 2.1.4 .Weassumethatthepopulationofushshootsisindependentofthelocationandthereisnopreferenceshownbythepsyllids.Anothermodelassumptionisthataproportionofpsyllidsmovefromthetreeonadailybasis[ 43 ],buttheageatwhichtheymoveisvariable,theymayonlystarttomovetwoormoredaysafteremergingasadults[ 43 52 ].Wefocusonthedynamicswithinthegrovethroughagrowingpopulationandassumethatnochangesoccurtothispopulationthroughmigrationawayfromthegrove.Localmovementfromoneclusterofushtoanotheroccursdaily.Weassumethateachofthepsyllidsattheushpatchhasaprobabilityof.40thattheywillmoveawayfromthepatchonagivenday.[ 43 ].Conditionaluponmovingtoanotherushpatch,thelocationtowhichtheymoveisgivenbyawithinrowandbetweenrowprobability,pwandpbrespectively.ForthecongurationinFigure 2-3 Ainteriortothegroveconditionalonamovementawayfromtheoriginalpatch,within-rowmovementsoccurequallylikelytothetwonearestneighbors,andeachoccurswithprobabilitypw.Between-rowmovementsarealsoequallylikely,buteachoccurswithprobabilitypb,wherepw>pb.Valuesusedinoursimulationsarepw=0.475andpb=0.025.ForcongurationFigure 2-3 B,within-rowmovementoccurswithprobabilitypw,andthesinglebetweenrowmovementoccurswithprobabilitypb,withthenumericalvaluesbeingthesameasforcaseFigure 2-3 A.Formovementfromacornerpatch(congurationFigure 2-3 C),within-rowandbetween-rowprobabilitiesarethesameasfortheothercongurations.TheparametersforthemodelofpsyllidmovementandderivationofmovementprobabilitieswillbediscussedfurtherinSection 2.2.5 .Intypicalgroves,notalltreemaybepresentateachintersectiononagrid.Thetreelosscanbeattributedtoanumberoffactors,including,butnotlimitedto,citrustristezavirus,blight,footrot,insects,andvariousenvironmentalabioticfactors[ 25 ].Movement 27

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inagrovewithremovedtreesoccursinthefollowingmanner.Eachtreereceivesa41vectorcontaininginformationaboutwhetherthereisatreetothecardinaldirections.Thisvectorisgivenbythevectorfunction,F,whereXisthecurrenttreeF(X)=[N(X)S(X)E(X)W(X)]andisanindicatorfunctionandis0ifthereisnotreeinthedirectionand1ifthereisatreeinthedirection.Withthesameprobabilitiesgivenaboveformovementwithinandbetweenrows,thepsyllidsmovewithprobabilitiesdependentonthenumberofneighboringtreespresent.Supposethenorthandsouthdirectionarethewithinrowandtheeastandwestdirectionareacrossarow.ForagiventreewithvectorF=[1011],thepsyllidsmovingfromthecurrenttreewillmovetothenorthandsouthwithprobabilityF1pw F1+F2=pwandF2pw F1+F2=0respectively.ThemovementtotheeastandwestwithprobabilityF3pw F3+F4=pb 2andF4pw F3+F4=pb 2.Inthecasewherethereisnotreetothenorthandsouthoreastandwest,thepsyllidsremainonthecurrenttree.Thisgeneralizesthemovementsallowedinastandardgrovesotherecanberemovedtreesinthesimulatedgroves. 2.1.3DemographyofPsyllidsandFlushWeassume,asjustiedinSection 2.2.3 ,that20ushpertreearegeneratedeachdayduringaushperiod,andthattheyareregardedasyoung(meaningacceptableforegglayingbyfemalepsyllids)for16days.Thisdenesushgenerationforasinglepatchinthesimulations.Agivenushcanacceptatmost40eggsperday.Weacknowledgethatushmayoccurthroughouttheyear,howeverinthismodelweonlyconsidertheperiodswhereanabundanceofnewushshootsareproduced.UsingagraphshowninFigure 2-4 [ 59 ],therearenoticeablepeaksinthenumberofushshootsperm2duringthemonthofApril,JuneandOctoberofvaryingheights.Werefertotheseperiodsasspring,summer,andfallandassumethatthenumberofdaysineachperiodis60,30,and15daysrespectively. 28

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TheperiodsoftimeinbetweeneachushperiodarealsoestimatedfromFigure 2-4 whereweassumethatthereare60daysbetweenspringandsummer,70daysbetweensummerandfall,and130daysbetweenfallandspring. AFlushingPatternsin4GrovesinPuertoRico(AdaptedfromPluke,R.W.Hetal.2008.Citrusushingpatterns,Diaphorinacitri(Hemiptera:Psyllidae)populationsandparasitismbyTamarixiaradiata(Hymenoptera:Eulophidae)inPuertoRico.FloridaEntomologist(Page40,Figure2A).) BSimulatedFlushFigure2-4. Numberofushpresentannually.A)Numberofyoungushesperm2[ 59 ]at4differentgrovesinPuertoRicofromOctober2004toAugust2005.B)Thenumberoftotalushpresentonasingletreeinthemodel.Thisoccursduringthreeperiodsinthespring,summer,andfallwherenewushshootsareproducedduring60,30,and15daysrespectively. Adultfemalepsyllidsareassumedtolay10eggsperday[ 56 ].Asinglefemalepsyllidcanlayfrom500to800eggsinherlifetime[ 51 72 ],dependingonthetemperature.Neweggshatchin3daysafterlayingandtake11daystodevelopthroughveinstar 29

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stagesintoadultsat28C[ 48 ].Approximately8%[ 52 ]oftheeggslaidleadtosuccessfulemergenceofadultpsyllids.Thehalflifeoftheadultpsyllidsisassumedtobe45days[ 56 ]inoursimulations.However,thelifetimeisquitesensitivetotemperature,rangingfromashighas117daysat15Cdownto50daysat30C[ 48 ]andstilllowerabovethattemperature,withlimitedsurvivaltimeattemperaturesabove34C.Inmodelwewillassumethattheadultpsyllidshavevaryingmaximumlifespansdependingonthetemperature.Inthespringandfallseasons,weassumethattheadultpsyllidsliveamaximumof75days,inthesummeramaximumof51days,andadultpsyllidsmayoverwinterwithamaximumlifespanof117daysinthewinterperiodwithnoush.Newadultswait2to3daysbeforefeedingonyoungush.Theyreachreproductivematuritywithin2or3daysafteremerging,andovipositionbegins1to2daysaftermating. 2.1.4InitialConditionsForaninitiallyuninfectedgrove,westartsimulationofthetransmissionprocessbyplacing200psyllidsoneitherafewtreesinacornerofthegroveoronalltreesalonganedgeofthegrove.Thelatterspecicationisconsistentwithconsiderableevidenceabouthownewwavesofpsyllidsarriveatagrove,frequentlydriveninbythewind.HallandHentzusedstickytrapstocapturepsyllidmovementinandoutofgrovesduringarbitrarytimesintheyear,apeaktimebeinginthespring[ 31 ].Boinaetal.documentedmovementinbothdirectionsbetweenmanagedandunmanagedgroves[ 9 ].Weassumethatapproximately30%oftheinitialpsyllidpopulationisinfected.Newlyarrivedfemalepsyllidsinitiateegglaying,andbothmaleandfemalepsyllidsfeedonush.ThisinitiatesthedynamicsshowninFigure 2-2 .Ingeneral,wemayvarythenumberofinitialpsyllidsateachushpatchinthefollowingmanner.Eachtreeisallowedtostartwithacertainnumber,I0,ofinfectedadultpsyllidsorwithoutinfectedpsyllids.Weplaceaninitialnumberofadults,N0,ateachtree,withIav(0)2f0,I0gandSav(0)=N0)]TJ /F4 11.955 Tf 12.95 0 Td[(Iav(0).Therearetwowayswedeterminethetreesthatstartwithapopulationofinfectedpsyllids.Therstistochoose 30

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aninitialconguration,corneroredge,thatrepresentshowaninfectioninaspecicgrovemayhavestarted.ThesecondistosimulateamigrationofpsyllidsintothegroveandtoassigneachtreeacertainprobabilityofstartingwithapopulationofIav.Weallowmigrationintothegroveonlyontherstdaytodetermineaninitialconguration.Thisisdoneinordertoanalyzethedynamicsofspreadinthegrovewithoutattributingthespreadtoinfectedpsyllidsthatmigrateintothegrove.Itisassumedthatifpsyllidsmigrateintothegrovewithaidfromthewind,thenalltreeshaveanequalprobabilityofreceivingpsyllids.ThetreesstartingwithapopulationofIavarechosenusingaspatialPoissonprocesswithintensitytoreectcompletespatialrandomnessinthegrove. 2.1.5GroveLayoutWespecifyagrovecontaining275trees(=275ushpatches),wherethereare11rowsand25treesineachrow.Weassumethatthespacingofthetreesis10ft.withinarowand20feetbetweenrows.ThisstructureisqualitativelysimilartoanorganicgroveinCitrawhichhasspacingof15feetbetweentreesinarowand20feetbetweenrows.Thedimensionsofthegroveare115mX165m,resultinginapproximately650trees[ 67 ]. 2.1.6DeningtheInfectiousPeriodAtreeisdenedtobeinfectiousatthepointintimewherethebacteriumcanbepassedfromthetreetothepsyllidvector.AsithasbeenshowninBillDawson'sexperiments,thisperiodoftimeisequivalenttothatofthegenerationtimeofthepsyllid[ 46 ].Toindicateinfectioninourmodel,werequirethataninfectioustreemusthaveenoughinfectedpsyllidsinorderforthemtomovetoothertreesinthegrove.Hereweonlyconsidertheolderadultpsyllidsinvolvedinmovementandtransmission.Thisresultsindeterminingathresholdfortheminimumnumberofinfectedpsyllidsneededfortheprobabilitytobegreaterthan95%thattheinfectionfromthistreewillmovetoanother.LetTvdenotethethresholdofinfectedpsyllidvectorsneededtoclassifyatreeasinfectious.Theeventthatapsyllidsmovesandthatapsyllidisable 31

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totransmitisindependent,thustheprobabilitythatapsyllidmovesandinfectsisgivenbytheproductofthetwoprobabilities,p=p2pv.LetXbethenumberofsuccessfulinfectionsatanewtree.TodetermineTv,theprobabilityofhavingTvindependenteventsthatresultsinatleast1successfulinfectionwithprobabilityp,givenby P(X1)=1)]TJ /F4 11.955 Tf 11.95 0 Td[(P(X=0).95.(2)Withtheparametersweusedinthemodel,p=(.3)(.4)=.12andP(X=0)=Tv0p0(1)]TJ /F4 11.955 Tf 11.96 0 Td[(p)Tv,wenowsolveforTvfromEquation 2 1)]TJ /F6 11.955 Tf 11.96 0 Td[((1)]TJ /F6 11.955 Tf 11.96 0 Td[(.12)Tv.95(.88)Tv.05 (2) togetthethresholdnumberofinfectedpsyllidsisTv=24.Nowwedeneaninfectedtreetobeatreewithatleastoneinfectedushshoot.Infectionoftheushshootsoccurswhenaninfectivepsyllidfeeds,thereforetheprobabilityofsuccessistheprobabilitythatapsyllidisinfective,pv=.3.Byasimilaranalysisusingthebinomialdistribution,wehavethatathresholdof6infectedpsyllidswouldresultinan88%probabilitytoresultinatleastoneinfectedushshoot.TheresultspresentedinChapter 4 usethisthresholdof6infectedpsyllidspresenttoidentifyinfectedtrees. 2.1.7ModelforAppearanceofSymptomsThereisnoconsensusonhowtheinfectionandthereforethesymptomsprogressinatree.AmodelfortheincidenceofsymptomatictreesofdifferentageshasbeenproducedbyBassaneziandBassanezi[ 8 ]whichsuggeststhatHLBincidencereaches50%inyoungtreeslessthantwoyearsaftertherstappearanceofsymptomatictrees.Asaresultofhighuncertaintyandvariabilityinwhatisreportedasthetime 32

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tosymptoms,wedenoteTsasthetimeuntilsymptomsappearandallowTstobearandomvariable,normallydistributedwithmeanandvariance2.Inthesimulationmodelweassumethattheincubationperiodisnormallydistributedwithmean1.5yearsandvariance(.25)2.InChapter 5 wedevelopamethodforestimatingthedistributionoftheincubationperiodundervariousconditions.Scoutingforsymptomatictreesisdonebyhumaninspectors,thereforetheremaybesomeerrorintheidenticationofsymptomatictrees.Inourmodel,allsymptomatictreesareidentiedaccuratelyandtoreectthescoutingprocessinthegroves,wewillcountthecumulativenumberofsymptomatictreesatmonthlyintervals. 2.2ParameterEstimationHerewewillgointotheestimationsandcalculationsfromtheliteraturewherewearriveatprobabilitiesforsurvival,birthprocessesforvectorandhost,transmissionprobabilities,initialconditions,andmovementparameters. 2.2.1SurvivalProbabilitiesThetransitionbetweenpsyllidstatesdescribedinEquation 2 andthedailysurvivalbetweenvectorstatesisshowninFigure 2-5 .Theratesareage-dependentandweassumetheyareindependentofbeinginstateSorI.Weusedelddatafromtwo Figure2-5. Survivalprobabilitiesforthevector,distheageofthevectorindays,adultsintheNa)]TJ /F14 7.97 Tf -5.22 -10.11 Td[(vstatedonotcontributetotransmissionoroviposition.ThisreectstheparametersusedinthemodelwithE(k)=3andN(k)=14. cohortsinMichaud'spapertoestimatethesurvivalprobabilityfromnymphstoadults.Thepercentageofrstinstarnymphsthatsurvivetoadultsis7.91%[ 52 ].Thereisnoeldestimateforthesurvivalofeggs,soweassumethesurvivalisthesameasthe 33

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nymphstagestoaccountforthehighsurvivalratefoundinlabs[ 72 ]andthepredationthatoccursintheeld.Thedailysurvivalratesforeggsandnymphsisse=sn=(0.0791)1=N(k).8343.Thedailysurvivalratefortheadultpsyllidisestimatedwiththeassumptionthathalfoftheadultpopulationremainafter45days[ 10 72 ].Thuswehavesa=(0.5)1=45.9847.Inadditiontothedailysurvivalforthepsyllid,weintroduceapseudo-adultcompartmentinFigure 2-5 ,Na)]TJ /F14 7.97 Tf -5.22 -10.11 Td[(v=Sa)]TJ /F14 7.97 Tf -5.36 -10.11 Td[(v+Ia)]TJ /F14 7.97 Tf -5.88 -10.11 Td[(v,todescribethedelayinmovementandovipositionthatoccurswhenadultsareatleasttwodaysold[ 52 77 ].ForthetransitionbetweenhoststatesdescribedinEquation 2 ,thedailysurvivalisindependentofbeinginstateSorI.Wedonotincludeanyabscissionofushshootsfromfeedingorovipositionandassumethatallushshootssurvivedaytodaywithprobabilityequalto1. 2.2.2BirthProcessforPsyllidPopulationForsimplicity,wewillcontinuetouseE(k)=3andN(k)=14whichareestimatesobtainedfrom[ 71 72 ].Egg-layingisinuencedbythepresenceofnewushgrowthandistypicallysuspendedwhentreesaredormant[ 71 ].ThebirthprocessforpsyllidswiththisresourceconstraintisdescribedinEquations 2 and 2 .Onaveragethenumberofeggsonasingleushshoot,ne,isthesumoftheeggsthathavesurvivedoveraperiodofE(k)days,whichisgivenbyne=E(k))]TJ /F9 7.97 Tf 6.59 0 Td[(1Xj=0ehsje,whereehisthenumberofeggsperushshootperday.TheeggstagelastsforE(k)=3dayswithanestimateddailysurvivalrateofse=.8343.ThesevaluesinadditiontopersonalobservationbySusanHalbertthatonaveragene=100whenexaminedintheeldisusedthistoestimatethevalueofehinEquation 2 .Thuswehave100=2Xj=0eh(.8343)j 34

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)eh=100 P2j=0(.8343)j40.Halletal.[ 32 ]foundthatthemeannumberofeggsperushshootwas16to26andthehighestnumberofeggspershootwas316to777.Theestimateofanaverageof100eggspershootishigherthanthemeannumberof16to26,howeversincethedatainHall'spaperoftenincludesperiodswheretherearenoeggsperushshoot,themeannumberof16to26ismostlikelylowerthantheaveragenumberofeggsperushwhenconsideringonlytheushshootswitheggs. 2.2.3BirthProcessforFlushPopulationConsiderasingleush,whereeheggsarelaiddailyforY(k)days.Thetotalnumberofadultsperush,na,atagedisgivenby na(d)=8>>>>>>>>><>>>>>>>>>:0if0d14ehd)]TJ /F9 7.97 Tf 6.59 0 Td[(14Xj=1s3es11nsjaif14
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theushperiodwherenoadultsemerge.ThenforthespringushperiodwithF=60,h=40,000 60na20. 2.2.4TransmissionProbabilitiesFirstweexplaininmoredetailthenumberofhostinfectionsfromEquation 2 .Transmissionoccursonadailybasis,soweconsiderthepopulationineachstateondaykandtosimplifynotation,wewilldroptheargumentkinwhatfollows.ThemodelassumptionisoneinfectivefeedingleadstoastatechangefromSjhtoIjh.Themodeldoesnotseparatethelocationsoftheushonthetree,soweviewthehostpopulationatonetreeasawhole.Tocalculatethenumberofhostinfections,weviewtheushasbinsandinfectivefeedingsasballs.ThisisnowaclassicalprobabilityproblemofdistributingBballsintoNhbins.LetXbethenumberofnonemptybins,thenP[X=j]=jXi=1()]TJ /F6 11.955 Tf 9.3 0 Td[(1)j)]TJ /F14 7.97 Tf 6.59 0 Td[(iNhjjii NhB E[X]=mXj=1jP[X=j].(2)OnlyinfectivefeedingsonSjhresultinnewinfections,sowemultiplyEquation 2 bySh NhtoobtainEquation 2 .Theexpectednumberofinfectivefeedings,B,istheproductofthenumberoffeedingsperpsyllideachday,vandtheexpectednumberofinfectiveadultsi.e.B=vpvIav.NextwediscussthetransmissionfromhosttovectorinEquation( 2 ).Theassumptionforvectoracquisitionisthenymphcanpickupbacteriathroughfeedingbythefollowingpathways:infectedushorushwithinfectivenymphswithprobability1and2respectively.TheprobabilityofthistransmissionisshowninFigure 2-6 .Wevieweachnymphasanindependenttrialwithprobabilityofsuccessp.LetYbethenumberofeventswhere 36

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. ushshootinfective feedingtrans P(Inv)=1 yes 1 P(Snv)=(1)]TJ /F7 11.955 Tf 11.95 0 Td[(1) no 1)]TJ /F7 11.955 Tf 11.95 0 Td[(1 yes femaleparentinfective transovarialtrans P(Inv)=(1)]TJ /F7 11.955 Tf 11.96 0 Td[() yes feedingtrans P(Inv)=(1)]TJ /F7 11.955 Tf 11.96 0 Td[()(1)]TJ /F7 11.955 Tf 11.96 0 Td[()2 yes 2 P(Snv)=(1)]TJ /F7 11.955 Tf 11.95 0 Td[()(1)]TJ /F7 11.955 Tf 11.95 0 Td[()(1)]TJ /F7 11.955 Tf 11.95 0 Td[(2) no 1)]TJ /F7 11.955 Tf 11.96 0 Td[(2 no 1)]TJ /F7 11.955 Tf 11.95 0 Td[( yes ushshootinfested feedingtrans P(Inv)=(1)]TJ /F7 11.955 Tf 11.96 0 Td[()(1)]TJ /F7 11.955 Tf 11.96 0 Td[()^2 yes 2 P(Snv)=(1)]TJ /F7 11.955 Tf 11.95 0 Td[()(1)]TJ /F7 11.955 Tf 11.95 0 Td[()^(1)]TJ /F7 11.955 Tf 11.95 0 Td[(2) no 1)]TJ /F7 11.955 Tf 11.96 0 Td[(2 yes ^ P(Snv)=(1)]TJ /F7 11.955 Tf 11.95 0 Td[()(1)]TJ /F7 11.955 Tf 11.95 0 Td[()(1)]TJ /F6 11.955 Tf 12.81 0 Td[(^) no 1)]TJ /F6 11.955 Tf 12.81 0 Td[(^ no 1)]TJ /F7 11.955 Tf 11.95 0 Td[( no 1)]TJ /F7 11.955 Tf 11.95 0 Td[( Figure2-6. Probabilitytreeforemergenceasaninfectedorhealthyadultondayk;=PZ(k)d=15Ih(k,d) PZ(k)d=15Nh(k,d)istheproportionofinfectedhostswithemergingadults,=pvIav=Navistheproportionofinfectiveadultpsyllids,and^istheproportionofhostswithinfectivenymphs. anymphacquiresLasfromthehost,thenP[Y=j]=Iv(k,N(k))jpj(1)]TJ /F4 11.955 Tf 11.96 0 Td[(p)Iv(k,N(k)))]TJ /F14 7.97 Tf 6.59 0 Td[(jE[Y]=Iv(k,N(k))pwhereIv(k,N(k))isthenumberofready-to-emerge5thinstarnymphsondayk.Sincetheinfectedshootsaredeterminedbydistributinginfectivefeedings,weassumethatallhealthyshootsdonothaveanyvisitsfrominfectiveadults.Underthisassumptionthenymphsonhealthyshootsdonothaveinfectivefemaleparentsand 37

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therearenoinfectivenymphspresent,so=0and^=0inFigure 2-6 .Theprobabilityoftransmissionfromhosttovectorsimpliestop=1andv(k)=Inv(k,N(k))p. 2.2.5PsyllidMovementTheestimatedrangeofpsyllidmovementvariesfromunder10m[ 4 44 ],tobetween500mand4km[ 3 4 9 49 ],andasfaras90to124km,thereisnotaconsensusonmovementofthepsyllidonitsownability,human-aided,orwind-aided.Duetothewiderangeofspeculationandalackofconsensusonhowpsyllidsmove,wemodelpsyllidmovementinagroveasarandomwalkonagrid.Weallowmovementtothenearestneighborswithinrowandbetweenrows,andassumethatamigratingpsyllidvisitsonly1treeperday.Fortheindividualpsyllid,weassumethatmovementoneachdayisindependentofwhatoccurredonthepreviousday.Wecanvarythemovementprobabilitieswithinandbetweenrowsoftreestoarriveatpatternsthatmaybeobservedintheeld.Evidenceofwithin-rowaggregation[ 49 ]ofHLBsymptomatictreesledtotheassumptionthatpw=.475andpb=.025.Mostoftheelddataavailabledoesnottrackpsyllidmovementinagroveexplicitlybetweentrees,thusmanyofourestimateshereformovementaretheoretical.However,thereisoneeldstudyfromacitrusgroveinSarawak,Malaysia[ 47 ]thatrecordedthenumberofinfestedtreesovertime.AdiscussionofthecomparisonofoursimulationtotheinfestationofthatgroveisprovidedinAppendix B.2 38

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CHAPTER3SIMULATIONMODELInthischapterwemakeapproximationsthatdonotqualitativelyimpactthebiologicalassumptionsmadeinthemodel.Groupsofushshootsontreesinterpretedasushpatchesandpsyllidsinegg,nymph,andadultinthesameageclassareidentiedandtreatedascohorts.Flushpatchesarelocatedattheintersectionsofagridwithintegercoordinates,c=(c1,c2).Thesecoordinatesdonotrepresenttheactualspacingofushpatchesinarealgrove,astheyarestructuredintheformofrowsofcloselyspacedtrees,withoverlappingcanopies,separatedbyawideropenspace.WepresentthedailydetailsofaspatiallyexplicitmicrosimulationofthisapproximationmodelinalgorithmicfashioninAppendix A.1 .Therstapproximationoccursinthemodelingoftheacquisitionofthebacteriumbythepsyllids.Wehaveassumedthattheadultswhowereinfectedasnymphsarethemainsourceofthetransmission,basedontransmissionparametersfromStelinski[ 57 ].Thisallowsustofocusonthepathwaysoftransmissionforthenymph.Inthesimulationmodel,ushshootsareinfecteduponreceivingavisit(orfeeding)fromasingleinfectiveadultwithprobabilitypv=.30.Transovarialtransmissionmayonlyoccurineggslaidbyinfectivefemaleadults.Thustheseinfectedeggswillbelaidonlyonushshootsthatareorwillbecomeinfected.Wemaketheassumptionthatthelowpercentageoftransovarialtransmission[ 57 ]iscountedtogetherwiththetransmissionfromDawson'sexperiments[ 46 ]andistheparameter1inthemodel.Thisfollowstheresultsoftheexperimentaldata[ 46 ]wheretherewasnoabilitytodistinguishbetweentransovarialtransmissionandlocalizedfeedingtransmission.Wemakethisapproximationbasedonthebiologicalinformationavailableforthetransmissionmechanism.Asaresult,thediagramfromFigure 2-2 simpliestothediagram,shownhereinFigure 3-1 .Thenoticeablechangesarethatwenowdonotincludecompartmentsforinfectedeggsor 39

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. Syh Iyh Sev Snv Sav Iav Soh Ioh + . . e n a a h h h v v Figure3-1. Modelapproximationfortransitionbetweenstates,blackarrowsindicatetransitionbetweenstatesbyaging,redarrowsindicatetransitionbetweenstatesbytransmission,dashedarrowsindicatestateswithaninuenceonoviposition,anddottedredarrowsindicatestateswithaninuenceontransmission. nymphs.Also,thetransmissionisnowdependentontheproportionofinfectedushshootswithemergingadults,,andtheprobabilityofinfectionthroughfeeding,1.Inaddition,therearenospecicmeasurementsforthefrequencyandlocationoffeedingforapsyllidwithinatree.Thisleadstothesecondapproximationsinthemodelwherewedonotassignthepsyllidstospeciclocationsofushshoots.Weassumethebehaviorofpsyllidsamongtheushshootsisassumedtoberandomanduniformlydistributedatavailablelocationsinthetree.Thisapproximationimpactsboth 40

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thecalculationofthetransmissiontoemergingadultsandthetransmissiontoushshoots. 3.1ApproximationforTransmissiontoEmergingAdultsAsaresult,wewillapproximatethenumberofinfectedemergingadultsbydeterminingtheprobabilitythattheyareonaninfectedushshoot.Tondthenumberofinfectedemergingadults,wedeterminetheprobabilitythattheyareonaninfectedushshoot.Werestrictthepopulationofushshootstothosethatareoldenoughtohaveemergingadultsandrecordtheproportion,,oftheseushshootsthatareinfected.Wemaketheassumptionthateachemergingadultcomesfromaninfectedushshootwithprobabilityof.Welabeleachcohortofushshootsas(c,(t1,t2),20)=20ushshootsatlocationconDayt1sinceaninitialpsyllidinvasionandDayt2sinceemergenceoftheushshoots.TheushshootswithemergingadultscorrespondtoE=(c,(t1,t2),20)witht215.Oneachday,wedeterminethetotalnumberofushshootswithemergingadults,E,andthenumberofinfectedushshootswithemergingadults,EI=I(c,(t1,t2),20).Thentheprobability,p,thatanemergingadultbecomesinfectedisobtainedbymultiplyingtheproportionofinfectedushshootswithemergingadults,=EI Eandtheprobabilityoftransmissionfromushshoottoemergingadult,1,sop=1=1EI E.Eachemergingindividualisassigneda1(success)withprobabilitypanda0withprobability1)]TJ /F4 11.955 Tf 11.95 0 Td[(p. 3.2ApproximationforDailyTransmissiontoFlushShootsTocomputethedailynumberofnewlyinfectedushshoots,weconsiderthetotalnumberofinfectivefeedingsandhowthesearedistributedamongthepopulationofush.Thenumberofinfectivefeedingsisdeterminedbytakingthenumberofinfective 41

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adults(infectedadultshavea30%probabilityofbeinginfective)andmultiplyingthisbythenumberoffeedingsperadultperday,whichweassumetobe1.Thisnumberoffeedingsperadultisanestimatethatisnotvariedthisinthemodel.Itisanimportantparameterthatneedsfurtherinvestigation,butcurrentlythereisnodataonhowoftenapsyllidmightfeedonyoungushandhowmanydifferentushshootspertreeitmayvisitinthecourseofaday.Thisnumberoffeedingsperadultmayalsodependonthenumberofadultspresentatthetree.Itisunknownwhetherovercrowdingatatreewouldleadtoadultspreferringtofeedonmatureleavesratherthanushshoots.Tondthenumberofushshootsthatreceiveaninfectivefeeding,wethentakethetotalnumberofinfectivefeedings,B,andthetotalnumberofushshootsNandcalculatetheexpectednumber,h=mXj=1jXi=1j()]TJ /F6 11.955 Tf 9.29 0 Td[(1)j)]TJ /F14 7.97 Tf 6.58 0 Td[(iNhjjii NhB,ofushshootsthatcontainafeedingbyplacingallBballsintoNbins.Evenifaushshootreceivesmorethanoneinfectivefeeding,theprobabilityoftranmissionremainsthesamebecauseoftheassumptionthatoneinfectivefeedingsresultsininfectionofaushshoots.Nowsinceaportionoftheushshootsmayalreadybeinfected,thenumberofnewinfection,h,isobtainedbymultiplyinghbytheproportionofhealthyushtototalushandwehaveh=Sh Nhh.Youngandoldushhaveequalprobabilityofreceivinganinfectivefeedingsowechoosehushshootsrandomlyfromthehealthypopulationofushandrecordtheageofeachushshootandmoveitfromthepopulationofhealthytoinfectedushshoots. 42

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CHAPTER4RESULTS 4.1InitialStagesofInfectionFigures 4-1 showshowtheinfectedpsyllidpopulationspreadsinagrovewith11rowsand25trees(ushpatches)perrowstartingwith4treesinthesouthwestcornerofthegrovethatcontains200adultpsyllids,60ofwhichareinfected.Thekeyinthepanelontherightindicatestheproportionofinfectedpsyllidsonthedesignatedpsyllidpatch. ADay1 BDay45 CDay90 Figure4-1. ProportionofinfectedadultpsyllidsateachushpatchafterA)1,B)45,andC)90daysstartingfrom4infectedpatcheswhere30%ofthepsyllidsareinfected. Figure 4-2 showstheproportionofinfectedadultpsyllidsatselectedpatcheswithcoordinatelocations(i,j)overtherst90daysfollowingtheinitialinfectionof4southwestcornerpatchesforDay1showninFigure 4-1 .Usuallytherstinfectionsarefoundinthesoutheast,howeverthisinitialspreadisforvisualizationpurposes.Complementarytothisinformation,Figure 4-3 showsthecountofinfectedushatthesamecoordinatelocationsusedtoconstructFigure 4-2 .Therapidincreaseintheproportionofinfectedpsyllidsandthenumberofinfectedushisaconsequenceofthenearlyinstantaneoustransmissionofinfectionfrompsyllidtoushfollowedbyatmostafewdayselapsedbeforethenewlyinfectedushbecomeinfectioustothepsyllids. 43

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Figure4-2. ProportionofinfectedD.citriadultsatdesignatedcoordinatepositionsinthegroveshowninFigure 4-1 Figure4-3. Numberofhealthy(resp.infected)ushatcoordinatelocationsspeciedinthepanelofFigure 4-2 .Solidlinesdenotehealthyush.Dashedlinesdenoteinfectedush. 44

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Asweplacetheinitiallyinfectednumberofushpatchesalongbordersofthegrove,theinfectedadultpsyllidpopulationspreadsmorerapidlyandmoreextensivelythroughoutthegrove.Figure 4-4 showstheresultofstartingwith4randomlyselectedushpatchesattherighthandedgeofthegrove.Eachselectedpatchwasoccupiedby60infectedand140uninfectedadultpsyllids.Figure 4-5 showsthemuchmorerapidandintensivespreadofinfectionwhen17treesdistributedalongthesouthernandeasternboundariesofthegroveareinitiallyinfectedwith60infectedand140uninfectedadultpsyllids.Fifthinstarnymphsoninfectedushareinfected. ADay1 BDay45 CDay90 Figure4-4. Proportionofinfectedadultpsyllidsateachushpatchstartingwith4randomlyselectedpatchesontherighthandedgeofthegrove.Outof200initialadultpsyllidsoneachselectedtree,30%areinfected. UnderthescenarioofFigure 4-5 ,theentiregrovecontainsinfectedadultpsyllidsafter90daysand,inaccordancewithDawson'sexperimentalresults,infectedush[ 46 ].Thiscompletecoverageofthegrovewithinfectedushoccurswellinadvanceoftheappearanceoftherstsymptomatictrees.Estimatesintheliteraturefortimetoonsetofsymptomsvaryfrom1to5years[ 67 ]underaninvasionscenariosuchasindicatedinFigure 4-5 .Thisplacesapremiumontheneedforearlysurveillanceandcontrolofthepsyllidpopulation,andclearlyindicatestheprimacyofpsyllidcontrolforextendingtheproductiveperiodoftreesinagrove. 45

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ADay1 BDay45 CDay90 Figure4-5. Proportionofinfectedadultpsyllidsateachushpatchstartingwith17selectedpatchesdistributedalongthebottomandrighthandedgesofthegrove.Oneachofthe17patches,200adultpsyllidswereputinplaceonday1,30%ofwhichwereinfected. TofurtherclarifythedistinctionbetweentimetoinfectionofanentiregroverelativetotimetoappearanceofsymptomsofHLBinagrove,Figure 4-6 showaminimalistintroductionof200psyllids30%ofwhichareinfectedoneachof4treesinthesouthwestcornerofagrove.Thetreeswithinfectedushareshownin30dayintervalsstartingfromDay30.OnDay180,thegroveistotallyinfectedbutentirelyasymptomatic.SymptomatictreesstarttoappearonDay390,andtheyareshownat60dayintervalsthroughDay810,atwhichtimeessentiallyallofthegroveissymptomatic.Thewidelypracticedinterventionofroguingsymptomatictrees[ 4 9 49 ]wouldnotbeginuntil210daysaftertheentiregroveisinfected.TheconsequencesofpsyllidreductionofvaryingintensitiesisshowninFigure 4-7 forthesameminimalistintroductionschemethatwasusedinFigure 4-6 .AcomparisonofFigures 4-6 4-7 ,and 4-8 showsthat: (i) After180daysalmosttheentiregrovehasinfectedushunderthescenariowithnoadultpsyllidcontrolandone90%reduction,whereasinfectedushhasmovedhalfwaydowntherowsandtherearethreefullrowswithnoinfectedushatthattimeunderthetwo90%reductionsduringeachushseason. (ii) After810days,thereare172asymptomaticushpatchesoutof275intheeldunderthetwo90%reductionscheme,whereas,theentiregrovewiththe 46

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AInfected(andasymptomatic)ushpatches BSymptomatictreesFigure4-6. TheprogressionofA)infected(andasymptomatic)ushpatchesvs.B)symptomatictreesfollowingintroductionof200adultpsyllidsofwhich30%wereinfectedonfourushpatchesinthesouthwestcornerofthegrove.Red=recentlyinfected(resp.symptomatic)ushpatch;Blue=establishedinfected(symptomatic)ushpatch AInfected(andasymptomatic)ushpatches BSymptomatictreesFigure4-7. TheprogressionofA)infected(andasymptomatic)ushpatchesvs.B)symptomatictreesfollowingintroductionof200adultpsyllidsofwhich30%wereinfectedonfourushpatchesinthesouthwestcornerofthegrove.Thereisa90%reductionoftheadultpsyllidpopulationduringeachofthethreeushperiodsoftheyear.Red=recentlyinfected(resp.symptomatic)ushpatch;Blue=establishedinfected(symptomatic)ushpatch exceptionof5and29ushpatchesissymptomaticwithoutadultpsyllidcontrolandwithonly1initialreductioninpsyllidpopulation. 47

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AInfected(andasymptomatic)ushpatches BSymptomatictreesFigure4-8. TheprogressionofA)infected(andasymptomatic)ushpatchesvs.B)symptomatictreesfollowingintroductionof200adultpsyllidsofwhich30%wereinfectedonfourushpatchesinthesouthwestcornerofthegrove.Therearetwo90%reductionsoftheadultpsyllidpopulationatthebeginningandendofeachofthethreeushperiodsoftheyear.Red=recentlyinfected(resp.symptomatic)ushpatch;Blue=establishedinfected(symptomatic)ushpatch Thisimpliesthatmorethanhalfofthegroveisstillprovidinggoodfruitafter810daysunderthispsyllid-onlytwo-spraycontrolscenario.Sincethepatchesthatareasymptomaticat810daysallhaveinfectedushandcontainontheorderof24,000adultpsyllidseach,anintensivepsyllidcontrolprogramisstillnecessarytofurtherdelaytheonsetofsymptoms.TheedgeinvasionscenarioshowninFigure 4-5 leads,ofcourse,tomuchmorerapidinfectionofushpatchesintheentireeldand,correlativelymorerapidspreadofsymptomatictrees.Ingeneral,thepatternofspreadisquitesensitivetothelocalitiesoftheinitialinfectedpsyllids.ToseethisinamoredramaticfashionconsidertheirregularinitialdistributionofpsyllidsshowninFigure 4-9 andthecorrespondingspreadofsymptomatictrees.Weshowthisfor30%initiallyinfectedpsyllidsof200totaladultspertreeinfourscenarios:A)nocontrolapplied,B)90%reductionatthebeginningoftheushperiod,C)90%reductionatthebeginningandtheendofeachushperiod,andD)90%reductionatthebeginningandonthe15thdayofeachushperiod.Comparing 48

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Figure 4-9 AandFigure 4-9 B,theresultsofaone-time,90%reductionintheinitialpopulationofpsyllids,isonlyadelayof30daysbeforetheentiregroveisinfected.Applyingtwosprays,thenoticeablereductionininfectedushoccursunderthescenariowherethesecondcontrolmeasureisappliedearlierintheushperiod. AInfected(andasymptomatic)ushpatcheswith30%initialinfectionof200psyllids BInfected(andasymptomatic)ushpatcheswith30%initialinfectionof20psyllids CInfected(andasymptomatic)ushpatcheswith30%initialinfectionof20psyllidsand90%controlmeasureappliedonthelastdayushemerges DInfected(andasymptomatic)ushpatcheswith30%initialinfectionof20psyllidsand90%controlmeasureappliedonDay15oftheushperiodFigure4-9. Spreadofinfectionfollowinginitiallyinfectedpatchesinthe11X25groveshownat30dayswithA)30%of200psyllidsinfected,B)30%of20psyllidsinfected,C)30%of20psyllidsinfectedwith90%controlmeasuresappliedonDay60,andD)30%of20psyllidsinfectedwith90%controlmeasuresappliedonDay15.Red=recentlyinfectedushpatch;Blue=establishedinfectionofushpatch Theslowerrateofproliferationofinfectionunderthetwosprayscenarioindicatestheimportanceofpsyllidcontrolingroves.Varyingthenumberofinitiallyinfected 49

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psyllidsdidnotaffecttherapidproliferationofinfectioninthegrove.Figure 4-9 AtoCshowsthatmaintainingalowpopulationofpsyllidsintheentiregroveslowsdowntheinfectionmorethanareductionofthenumberofinfectedpsyllids.Thisphenomenonimpliesthatpsyllidcontrolataregionalratherthanjustgrove-specicleveliscriticalforsuppressingHLBtransmission.OrganizedregionallevelpsyllidcontroliscurrentlyanintensifyingfeatureofthemanagementofHLBinFlorida[ 74 ]. 4.2ParameterVariationTable 4-1 showsvariationin:(i)durationoftimeuntil50%oftheushpatchesareinfected;(ii)fractionofushpatchesthatareinfectedafter90days;and(iii)prevalenceofinfectedpsyllidsinthegroveafter90daysunderavarietyofconditions.TheoutcomesinTable 4-1 weregeneratedbyvaryingtheparametersoneatatime,holdingallotherparametervaluesxedatthesettingsusedtogeneratethesimulationsinFigures 4-1 to 4-9 .Theseare:N0=200,I0=60,1=0.8,pv=0.3,po2=0.4,py2=0.1,andpb=0.05. 4.3ControlStrategies 4.3.1EfcacyofPsyllidControlPsyllidcontrolisappliedonceperushingperiodontherstdayushshootsemerge.Althoughinsecticidesareknowntohavelingeringeffectsonovipositionofadultsthatsurvive,wehaveaone-dayreductioninpopulation.Lettheinsecticidespray,reducethepsyllidpopulationbyr,theneachpsyllidpresenthasaprobabilityof1)]TJ /F4 11.955 Tf 12.48 0 Td[(rofsurvivingthatspray.Wepresenttheresultsofourmodelintermsofpercentagereductionandnotethattheseresultsdependmoreonthenumberofpsyllidsleftratherthanthepercentreduction. 4.3.2TimingofApplyingControlAsseeninFigure 4-9 ,aone-timeapplicationofinsecticidesprayingisneitherthemostefcientuseofresourcesnorisitapracticewidelyusedbygrowersofcitrus.Forveryyoungtrees,theyreceiveasoildrenchthatprotectsthemfromthepsyllids. 50

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Table4-1. Sensitivityofoutcomestatisticstovariationinselectedparameterswheninitialinfectioninthegroveison4ushpatchesinthesouthwestcorner. ParameterRangeD50a%patchesinfectedPrevalenceofinfectedafter90dayspsyllidsat90daysminmaxminmaxminmaxminmax N01002400577084.097.1.287.332I05200607278.994.6.239.3680.61.0657081.189.5.199.394pv0.20.7607474.995.6.219.412po20.20.6608162.996.7.190.381py200.4646883.694.6.279.336pb0.010.1648060.493.5.200.325 aTheminimumvalueofD50correspondstothemaximumvalueofeachparameter.Analogously,themaximumvalueofD50occursfortheminimumvalueofeachoftheparameters.Legend:D50=#daysuntil50%ofthegroveisinfectedN0=#ofpsyllidsperinitiallyinvadedushpatchI0=#ofinfectedpsyllidsperinitiallyinvadedushpatch1=probabilitythat5thinstarnymphsacquireinfectiononhostushpv=proportionofinfectedadultpsyllidswhoareinfectivepo2=dailyproportionofolderthan2-dayadultpsyllidsmovingawayfromtheircurrenttreepy2=dailyproportionof1and2-dayoldadultpsyllidsmovingawayfromtheircurrenttreepb=probabilityofmovingbetweenrowsconditionalonchangingtrees Differentcountrieshavedifferentregulationsconcerningsprayinginsecticidesinordertonotallowthepsyllidpopulationtobecomeresistant.Insomecases,grovesreceivesprayingduringperiodsofnoush,orsprayeverytwoweeks.Wewillsimulatetheeffectofmultiplesprayingandcomparethetimingofthesesprays.Ifitisaninsecticidethatcanbesprayedovermultipledays,thebestpracticemaybetosprayeachdayforaperiodof15days.Thereasonfortheperiodof15dayswouldbetocapturetheemergingadultsastheyfeedontheleavesandthiswouldgreatlyreducethepopulationofpsyllids.InTable 4-3 wequantifytheresultsofvaryingthetimingofspraysandshowthatthisholdsforvariousefcacyofthecontrolsprays. 51

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Table4-2. Spreadofinfectedpsyllidpopulationwithvaryinglevelsofpsyllidcontrolappliedatthebeginningofeachushperiodwheretheinitialinfectiononfourtreesinthesouthwestcorner ReductionaI0N0D50bD100cInfectedPercentdPrevalencee (%)90df365d90d365d 0602006716086.6100.0.297.8057515507814473.1100.0.217.7958012408725055.6100.0.158.800859308725454.2100.0.151.798906208926351.3100.0.134.770 AllvaluesforD50,D100,infectedpercent,andprevalencecorrespondto=1,pv=.3,p2=.4,pb=.01aThepsyllidpopulationisreducedbythispercentageatthebeginningofthespring,summer,andushperiods.bTherstdaywhere50%ofthetreesareinfected.cTherstdaywhere100%ofthetreesareinfected.dPercentageofthe275treesinthesimulatedgroveafterthegivennumberofdays.eProportionofinfectedpsyllidsinthewholegroveattheendofthespeciednumberofdays;prevalenceonindividualtreesmayrangefrom0to1.fd=daysafterinitialinfection InTable 4-3 ,thezerointheprevalencecolumnattheendoftherstyearshowsthatnoinfectedpsyllidssurvivetotheendoftheyearunderthescenariowith90%controlappliedtwiceduringeachushingperiod.Inpractice,noamountofpsyllidcontrolisabletoeradicatethepsyllidpopulationusuallyduetotheimmigrationofpsyllidsfromoutsideofthegrove,howeverinoursimulationwedonothaveincominginfectedpsyllidsalthoughthisislikelyasourceforreinfectionwithingroves.FromTable 4-3 ,inthecaseswherethecontrolefcacyis85%,thenonzeroprevalenceindicatesthatinfectedpsyllidsarestillpresentinthegrove.Howeversincetheinfectedpercentoftreesis0%,thismeansthatthenumberofinfectedadultsateachtreeislessthanthethresholdof6infectedpsyllids.Thispopulationalsoeventuallydiesoutwhencontinuingwiththetwo-sprayprogramintothenextspringushseason.ThetimingofthesecondsprayismoreeffectiveatDay15ratherthanthelastdayoftheushperiod. 52

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Table4-3. Spreadofinfectedpsyllidpopulationwithvaryinglevelsofpsyllidcontrolappliedtwiceineachushingperiodatspeciedtimes.Theinitialinfectionbeginsonfourtreesinthesouthwestcorner. ReductionaFirstSecondD50bInfectedPercentcPrevalenced(%)ControlControl90de365d90d365d 7511515339.682.6.114.653751lastf8753.191.6.182.7348511526.20.063.058851lastf16638.90.137.0909011522.90.0200901lastf35.30.0770 AllvaluesforD50,infectedpercent,andprevalencecorrespondtoI0=60,N0=200,=0.8,pv=.3,po2=.4,po2=.1,pb=.05aThepsyllidpopulationisreducedbythispercentageatthespecieddaysinthecorrespondingushperiod.bTherstdaywhere50%ofthetreesareinfected.cPercentageofthe275treesinthesimulatedgroveafterthegivennumberofdays.dProportionofinfectedpsyllidsinthewholegroveattheendofthespeciednumberofdays;prevalenceonindividualtreesmayrangefrom0to1.ed=daysafterinitialinfectionfThelastdayofeachushperiodwhichcorrespondstoday60,30,and15forthespring,summer,andfallperiodrespectively. Thisisduepartlytothefactthatinoursimulations,theemergingadultsemergeonDay15andthiswouldbetherstdayforanincreaseintheinfectedpopulation.FromTable 4-3 ,foralllevelsofcontrolefcacy,thebesttimingforpsyllidcontrolisearlierintheushperiod.The15thdayistherstdayinwhichadultpsyllidsemerge,sothetimingofthissprayreducesthenumberofadultsthatcanlayeggs. 53

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CHAPTER5ESTIMATIONOFINCUBATIONPERIOD 5.1BackgroundMonitoringsymptomsofHLBintreesisdifcultduetosymptomsoftenmasqueradingasnutrientdeciencies,environmentalconditions,treeageandspecies,andmanagementpractices[ 27 ].TheGompertzmodelfortheprogressofHLBispresentedbyGottwaldetal.[ 27 ]inananalysisofanHLBepidemicinFlorida.TheydeterminedthataGompertztemporalmodelwasabettertcomparedwiththelogisticmodeltodescribetheincreaseinHLBthroughtime.Gottwaldetal.collecteddataontreeswithvisualsymptomsofHLBovera2-yearperiodfrom11citrusblocksinFloridaandtransformedthedatausinglogisticlineartransformation,logit(y)=lny 1)]TJ /F4 11.955 Tf 11.96 0 Td[(yandGompertzlineartransformation,Gompit(y)=)]TJ /F6 11.955 Tf 11.29 0 Td[(ln()]TJ /F6 11.955 Tf 11.29 0 Td[(ln(y))wherey=diseaseincidence.ThenthetransformeddatawasttedvialinearregressionanalysisandtheGompertzmodelbestrepresentedthechangeindiseaseincidencein10ofthe11citrusblocks.Understandingthetimetosymptomsisanimportantfactorwhenconsideringtheeconomicimpactsofthedisease.InastudytoestimatetheimpactofHLBoncitrusyields,BassaneziandBassanezi[ 8 ]usetheGompertzmodelfortheprevalenceofHLBcombinedwithamodelfordiseaseseveritytoestimatetheexpectedyieldcurvesforHLBinfectedgroveswithoutanydiseasecontrol.Recallthatthetermincubationperiodisdenedtobethetimefrominitialinfectiontotheappearanceofsymptoms.Muchoftheearlydatathathasbeencollectedinthestudyofcitrusgreeningcontainsinformationonwhenatreeissymptomaticandonlylatertheywareabletodetermine 54

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throughaPCRtestwhetheratreewasinfectedbeforeitbecamesymptomatic.Itisimportanttoconsideranapproachthatwillestimatetheincubationperiodbasedonthecurrentdata.Inthischapterweprovideanestimateforthetheincubationperiodbasedupontheinfectionstatusoftwodifferentgrove-types.Intherstcase,weconsideragrovethatisbeginningtoshowsymptomsinafewtrees.Wewillcomparetherateofinfected(andasymptomatic)treeswiththerateofsymptomatictreesandthenpresentamethodforestimatingtheincubationperiodinthisgrovebyrestrictingtheinfectiontotherowswithsymptomatictrees.Inthesecondcase,weconsideragrovewheretheinfectionhaspersistedforalongperiodoftimeandmanyofthetreesinthegrovehavealreadyshownsymptoms.Wewillprovideanestimationforthedistributionoftheincubationperiodbydividingthegroveintosmallerpatchesoftreeswhereweassumethatthetreesineachpatchwereinfectedsimultaneously. 5.2RecentlySymptomaticGroveConsidertheinfectionwithinarowoftreesinagrove.Wewillassumethatfromaninitialstartingpositionintherow,theinfectionmayonlypasstotheadjacenttreeintherow.LetthenumberofinfectedtreesinagroveattimetbegivenbyN(t)andsupposethatN(0)=0.Onceatreeisinfecteditremainsinfectedandisabletopasstheinfectionontotheadjacenttreeatarate.Thiseliminatesthepossibilityofsimultaneousinfections.Wealsoassumethenumberofinfectionsintwodisjointtimeintervalsisindependentandthenumberofinfectionsinanintervaldependsonlyonthelengthoftime.ThuswemodeltheinfectionprocessbyaPoissonprocesswithparameter. 5.2.1SpreadofInfectionandSymptomsInthescenariowheretheinfectionprocessinarowisPoissonwithparameter,weassumethattheappearanceofsymptomsoccursatrate,whichisexponentiallydistributed.ThusweassumetheappearanceofsymptomssatisestheMarkovpropertyandisindependentofthetimethelastsymptomappearanceoccurred.Theappearance 55

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ofsymptomsonlydependsonthenumberofinfectedtrees,N(t).Attimet,thenumberofinfectedtrees,N(t),isequaltothenumberofasymptomatic,A(t),plusthenumberofsymptomatic,S(t),trees. Claim5.1. LetTsrepresenttheincubationperiod,soTs=1=.Bothandarerandomvariables,exponentiallydistributed.Thentherateofspreadofsymptomsfollowstherateofspreadofinfectionatsometimeinthepast. ProofofClaim 5.1 Wewillsupposethatthereareaninnitenumberoftreesalongaline.SinceI(t)=A(t)+S(t),wewillshowthatthenumberofasymptomatictreesapproachesaconstant,thenfortlargeenough,therateofinfectionisequaltotherateofsymptomatictrees.Theinfectionbeginsatthersttreeinthelineandpassestothenexttreeatarate.ConsiderthenumberofinfectedasymptomatictreesasthestatesinaMarkovchain,wherethenexttreeisinfectedatrate,sowemovefromstatenton+1withrate.Weassumethatthereisnoremovaloftrees,soadecreaseinthestatesoccurswhenatreeshowssymptoms.TheappearanceofsymptomsforindividualtreesoccursatrateandisMarkovian.Thuswhenthereareninfectedtrees,therateatwhichsymptomsappearisn,sowemovefromstatenton)]TJ /F6 11.955 Tf 12.33 0 Td[(1withraten.Nowthesteady-stateprobabilityofbeinginstatenisgivenbypn=( )n n!e)]TJ /F29 5.978 Tf 7.78 3.25 Td[( ,whichisthePoissondistributionwithparameter= .Weassumethattheincubationperiodismuchlongerthanthetimebetweeninfection,sotherateofinfectionismuchlargerthantherateofsymptoms,>>.With large,thePoissondistributionwillbeapproximatelynormalsotheexpectednumberofasymptomatictreesisn= .NowsinceA(t)!n,d dtA(t)!0thussinceS(t)=I(t))]TJ /F4 11.955 Tf 11.95 0 Td[(A(t)wehaved dtS(t)!d dtI(t). 5.2.2IncubationPeriodEstimateWeassumethatinthistypeofgrove,theinfectionisstillprogressingthroughthegrove.Wealsoassumedetectinginfectioninthetreesisaccomplishedbytestingthe 56

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nymphsonthetreesforthepresenceofCLas.Firstconsidertherowinwhichthesymptomatictreehasappeared.Withoutlossofgenerality,letthetimeofdetectionofthesymptomsbet=0anddenethepositionofthesourceintherowoftreestobex=0.WewillassumethattheinfectionalongarowisaPoissonprocesswithparameter.Wewillestimatetheincubationperiodforbothcaseswheretheinfectionprocessisahomogeneousandnon-homogenousPoissonprocess.Atthetimesymptomsaredetected,ndthefurthesttreeintherowthatisinfected.Denotethisdistancebetweentheinfectedtreesandthesourceattimet=0byR0.Letaperiodoftime,t,passbeforethenextsurveyfordetectionofinfectionalongtherowoftreeoccurs.Supposeattimet=t,theinfectionisdeterminedtobeRtawayfromthesource.Herealsoassumethatthetreesarespacedaunitdistanceapartandthereforethedistancefromthesourcecorrespondstothenumberoftreesinfected.FirstsupposethattheinfectionisahomogeneousPoissonprocess,thenthattherateofspreadofinfection,,alongarowisconstantandtheexpectednumberofnewinfectionsinaperiodoftimetist.Fromthetwodetectionsurveysforinfectioninthegrove,thenumberofnewlyinfectedtreesisRt)]TJ /F4 11.955 Tf 11.88 0 Td[(R0,thuswehavethattherateofinfectionintherowisonaverage =Rt)]TJ /F4 11.955 Tf 11.95 0 Td[(R0 t.(5)Inordertodeterminethetimeofinitialinfection,ti,weusethisaveragerateofinfectionandtheinitialconditionofthegrovestatedearlier.Weknowtheinfectionmovedfromtheinitialposition,0,toR0inaperiodfromthetimeofinitialinfection,ti,to0.Sinceisconstant,wehavethat=R0)]TJ /F6 11.955 Tf 11.95 0 Td[(0 0)]TJ /F4 11.955 Tf 11.95 0 Td[(tiandsolvingfortiwehaveti=)]TJ /F4 11.955 Tf 9.29 0 Td[(R0 57

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ThissaysthattheinitialinfectionoccurredR0 unitsoftimebeforesymptomsappeared.Thustheincubationperiod,T,forthetreesthatshowedrstsymptomsisT=R0 whereisfromEquation 5 .Thiscanbedoneforeachrowwherethereisasymptomatictree,providinganestimatefortheincubationperiod.Inaddition,oncetherateandtheinitialtimeofinfectiontiisdeterminedwithinarow,theinitialtimeofinfectionforalltreesinthatrowareapproximatelydetermined.Assymptomscontinuetoappearinothertreeswithintherow,theincubationperiodwouldcorrespondtothesymptomatictimeminustheinitialinfectionofthetree.Thenumberofestimatesfortheincubationperiodwouldcontinuetogrowandabetterestimatecouldbeprovided.Nowsupposethatthetherateofinfectionvariesduetotheushingpatternofthetreesaswellastheresultingpopulationuctuations,whichwecanassumeiscyclical.Let(t)beperiodicwithperiod,thentheaveragerateofinfection,,intheperiodis =1 Z0(t)dt.(5)Afteraperiodofunitsoftime,supposethattheinfectionhasprogressedtoR.ThenumberofnewinfectionsinthisperiodisR)]TJ /F4 11.955 Tf 12.97 0 Td[(R0,so=R)]TJ /F14 7.97 Tf 6.59 0 Td[(R0 .Inthisscenario,sincetheinfectionrateis-periodic,theestimatefortheinitialtimeofinfectioncanbedeterminedonlytowithinaperiodofthetrueinitialinfection.Inoneperiod,thereareexpectedinfections,soifR0
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theassumptionthattheyweresimultaneouslyinfected.Foronesubsection,supposeNtreesareinfectedattimeti.Weassumealltreesareindependentandidenticalandthattheincubationperiod,Ts,isarandomvariable.Supposethatthecumulativedistributionfunction(CDF)ofbecomingsymptomaticduringaperiodoflengthdaysafterinfectionisgivenbyP(Ts).Nowattime,ifweconsidertheappearanceofsymptomsasasuccesswithprobabilityP(Ts)withNtrials,weusethebinomialdistributiontondtheprobabilityofktreesshowingsymptomswhichisgivenby Nk(P(Ts))k(1)]TJ /F4 11.955 Tf 11.96 0 Td[(P(Ts))N)]TJ /F14 7.97 Tf 6.59 0 Td[(k.(5)Thentheexpectednumberofsymptomatictreesdaysafterinitialinfectionis P(Ts)N.(5)Supposethedatacollectedonwhentreesbecamesymptomatic,isfromanareawheretheinfectionoccurredatthesametime.WewillderivetheCDFforwhenthetreesbecomesymptomatic.Obtainn(),thenumberoftreesthathaveHLBsymptomsbyday,fromthedata.FromEquation 5 wealsohavethattheexpectednumberoftreesshowingsymptomsisNP(Ts).Thus Ps(Ts)=E[n()] Nh.(5)Astandardwaytoapproximatethecumulativedistributionfunctionistousetheempiricaldistributionfunction,n() NhPs(Ts).Thus,ifallthetreeswereinfectedatthesametimeandtheincubationperiodwasarandomvariable,thenwewouldbeabletousethismethodalongwithmonthlydataincludingthenumberofsymptomatictreestoestimatepartoftheCDFfortheincubationperiod.ThetimeoftheinitialinfectionwouldbeneededinordertohavethecompleteCDFfortheincubationperiod.Currentlythereisnomethodtodeterminethetimeof 59

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initialinfectionfortheextensivelysymptomaticgrove.Inthiscase,sincetherearemoretreesinthesubsection,wewouldexpectthattheincubationperiodislessthanorequaltotheincubationperiodfoundintherstcase. 60

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CHAPTER6USINGTHERIPLEYK-FUNCTIONTOESTIMATETHEFRACTALDIMENSIONOFASELF-SIMILARFRACTALSET 6.1MotivationAquestionoftenaskedaboutthespatialspreadofdiseasesiswhatarethefactorsthatinuencethisspread.Forexample,inthestudyofthespreadofcitrusgreening,aquestionarisesastowhetherthediseaseappearedduetolongdistancemigrationorduetolocalmovementofthepsyllid.Therstmethodwouldresultinarandomdistributionofinfectionwhereasthesecondmethodwouldresultinclustersofinfection.Inpractice,theRipleyKisaspatialanalysistoolthatcanbeusedtoanswersuchquestions.TheRipleyKfunctionhasbeenusedtoanalyzespatialpatternsoftrees[ 58 70 ],herbaceousplants[ 68 ],anddiseasecases[ 20 ].Ripley'sKanalysisisastatisticalanalysisofspatialpointprocesses.Itsmainuseistodeterminedeparturefromacompletespatialrandompointprocess.Inapreviousinvestigativemodel,wewantedtousetheRipleyKfunctiontoestimatethedispersalofthepsyllidsbyconsideringtheappearanceofsymptomatictreesanddeterminingwhetherthedistributionofthesetreesfollowedaCSR(completespatialrandomness)orwhethertherewasnoticeableclusteringatadistancethatwouldbeconsideredthestandarddeviationinthenormaldistributionofthepsyllids'movementkernel.Fromaregionalpointofview,aRipleyKanalysisdemonstratedshortrangeandregionalcomponentsinvolvedinthespreadofHLBanditwasestimatedthattheaveragedispersaldistancefromaregionalpointofviewis1.58km[ 27 ].ThenextstepwastodeterminethelongerdistancemovementofthepsyllidsbyfocusingoncitrusgrovesthroughoutFlorida.ThereisaproblemwithapplyingtheRipleyKanalysisonsuchalargescalebecausethecitrusgrovesthemselvesformasubsetoftheplaneinR2andthereforethedeviationfromCSRwouldbeduetothespatialarrangementofthegrovesandnotofthedistributionofsymptomatictrees.ThisbecamethemotivationforapplyingRipleyK'sanalysistoaspatialpointprocessthatmaylive 61

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onafractalset.WethendevelopatheoryusingthebasisoftheRipleyKanalysistoestimatethefractaldimensionofself-similarsets. 6.2BackgroundThetheoryoffractalsetsisavitalandgrowingeldofgeometryinpuremathematics.Itconnectswithgeometry,topology,measuretheory,anddynamics.Thetheoryoffractalshasnumerousapplicationsaswell.Theapplicationsrangefrompercolationanddiffusionlimitedaggregationtofracturedmaterials[ 12 ].TheworkbyMichaelBarnsleydemonstratedtheusefulnessoffractalsincompressingandmanipulatingimages[ 6 ].Seealso[ 7 ]and[ 24 ].Weassumesomeknowledgeofself-similarfractals.ThepaperbyJohnHutchinsonhasbecomeastandardreferenceonthesubject[ 39 ].Agoodtextbookreferencethatcarefullydevelopsthetheoryofself-similarfractalsalongthelinesofHutchinson'spaperisthebookbyGeraldEdgar[ 21 ].Edgar'scompanionbook,ClassicsonFractals[ 22 ]hasreprintsortranslationsofmanyoftheclassicpapersthathelpedframeanddevelopthetheoryinitsearlystages.ThetextbyKennethFalconercoversawiderangeoftheoryandapplicationsforfractalsetsis[ 23 ].TheRipleyK-functionwasintroducedbyB.D.Ripleyinaseriesofpapers[ 60 62 ]asameasureofthespatialdistributionofpoints.Ithasbecomeoneofthestandardtoolsforthispurpose.OneofthemainapplicationsistoshowhowthedistributionofrandompointsinEuclideanspacemaydeviatefrombeingPoisson.MorewillbesaidaboutthisinSection 6.5 .Fractals,AUser'sGuidefortheNaturalSciencesbyHastingsandSugiharadevelopsthetheoryoffractalsfromthestandpointofapplicationstothenaturalsciences[ 33 ].Therearetwopapersthatillustratethemethodsofthebook.ThepapersHastings,Schneider,Schreiber,Gorray,Maytal,andMaimon[ 34 ]andSchneider,Hastings,andMaytal[ 65 ]showthattheIsletsofLangerhansinthepancreaslieonasetofprescribedfractaldimensionstrictlybetweentwoandthree.Thereareother 62

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circumstanceswherepointsmaylikelylieonafractalset.InthischapterweshowthattheRipleyK-functioncanbeadaptedtoestimatethedimensionofsuchafractalset.WedevelopthealgorithmforestimatingtheHausdorffdimensionofasetbyusingtheRipleyK-functionasthelaunchingpointforitsunderstanding.Wearealsoabletogivecompleteproofsforcertainpartsofthetheorywhenitisappliedtoself-similarsets.ThemainresultofthischapterisgivingthebasisforanalgorithmtoestimatetheHausdorffdimensionofaself-similarsetusingarandomdistributionofpointsontheset.TherearesomedistinctionsbetweenaRipleyK-functiononafractalandtheclassicRipleyK-functionforrandompointsinEuclideanspace.ItwouldbeinterestingtodevelopafulltheoryfortheRipleyK-functiononself-similarfractals.However,thishasamorelimitedpurpose.ThealgorithminSection 6.9 forself-similarfractalswillapplytofractalsMthatarenotnecessarilyself-similar.IfEquations 6 and 6 holdforM,thenthealgorithmtoestimatetheHausdorffdimensionwillbevalidiftherandompointsaredistributedaccordingtoHausdorffmeasure.However,wecanprovethatthemethodworksforself-similarfractals.Thevalueoftheapproachisthatitgivesaconceptualframeworkandgraphicalconrmationusingastandardtechniqueusedintheanalysisofspatialdistributions.Inthelastsectionwegivecomputationalexamplestodemonstratetheaccuracyofthemethod.Sincethemethodisbasedonprobability,thereisrandomnessinthecalculatedresults.However,withsufcientlymanysamplepoints,themethodcanbeseentobereliable. 6.3NotationLetf:Rn!Rn.Wesaythatfisasimilitudeifthereisaconstantc>0suchthatd(f(x),f(y)=cd(x,y)forallx,y2Rn.Wesaythatitisacontractionsimilitudeif0
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similitudesff1,f2,...,fNgwitheachfi:Rn!Rn.ThecontractionfactorsforeachfunctionmaybedifferentinanIFS.Aself-similarfractalisanon-emptycompactsubsetMofRnsuchthatthereisanIFS,ff1,f2,...,fNg,suchthatM=N[i=1fi(M).FromthisdenitionitisclearthatforeveryopensetUwithU\M6=;,therewillbeasmallcopyofM,callitL,withLU\M.OnemaynotbeabletoseethesesmallcopiesofMiftheoverlapoffi(M)andfj(M)istoogreatforsomei6=j.AconditionthatlimitstheoverlapoftheseimagesiscalledtheOpenSetCondition(OSC).Inthischapteritisassumedthatallself-similarsetsMlieinsomeEuclideanspaceRnandthateachisgivenbysomeIFS,ff1,f2,...,fNg.TheOSChasgeometricimplicationsforM.Thesetismoreobviouslyself-similarwhentheOSCissatised.However,theOSCisvaluableforotherreasonsaswell.Iftheself-similarsetMsatisestheOSC,thentheHausdorffdimensionofMcanbedeterminedbyasimpleformula.Itisgivenbytheuniquesatisfyingtheequation NXi=1ci=1(6)wheretheci'sarethecontractionfactorsofthefunctionsintheIFS.Furthermore,ifthesetMliesinEuclideanspace,asweareassuming,thentheHausdorffmeasureinthisdimensionisalsoknowntobepositiveandnite,0
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fromthecodespacetoM,f:!M.Itisdenedbyf(ni)=limk!1fn1fn2fnk(M).ThelimitinEquation 6.3 willbeapointinM.Thereisaspecialmetricon.If(ni)1i=1and(mi)1i=1aretwopointsin,thenthedistancebetweenthemisgivenbythefollowing.((ni),(mi)=8>>>>>><>>>>>>:1n16=m1c1ckn1=m1,...,nk=mkandnk+16=mk+10ni=miforalliWiththismetricdim()=whereisgivenbyEquation 6 .ThisisalwaystrueevenwhenMdoesnotsatisfytheOSC.ItisalsoalwaystruethatH()=1.See[ 21 ]formoredetails. 6.4HausdorffMeasureandDimensionTherearevariousnotionsoffractaldimension.Theyallproduceanumberthatlayssomeclaimtobeingthemetricdimensionoftheset.Hausdorffdimensionisperhapsthemostcomplicatedofthese.However,itistheonethathasthemostmathematicallyrichandvaluabletheory.Itisbasedonmeasuretheory.ForeachseparablemetricspaceXandeachnon-negativenumberthereisameasureontheBorelsetsofX.HausdorffbasedhistheoryonatheoryofoutermeasureanditspropertiesdevelopedbyConstantinCaratheodoryin[ 14 ].Hausdorffobservedthatthemeasuresexistedforallnon-negativevaluesofandinfactthenaturaldimensionofthespaceXbasedonthesemeasurescouldbeanynon-negativerealnumber[ 35 ].WedenotetheHausdorff-measureofXbyH(X).WithoutgoingintothedetailedconstructionofHausdorffmeasure,itcanbeshowntohavethepropertythatifH(X)>0,and<,thenH(X)=1.Furthermore,ifH<1and>,thenH=0.So,thereisabreakatauniquesuchthatH(X)=1forall
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>.WecallthistheHausdorffdimensionofXanddenoteitbydimHX=.Inthecaseofaself-similarfractalMinRnsatisfyingtheOSC,thisisgivenbyEquation 6 andhasthepropertythat00anddeneIh(dij)=8><>:1dijh0dij>hInthespatialanalysisofoccurrencesofrandompoints,oneisoftenconcernedwithwhethertheyaretrulyPoissondistributed.Therearevariousstatisticalmethodstodeterminethis.TheRipleyK-functionisusefulinshowinghowclosethepointsmaybe 66

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tobeingPoissondistributed.Theformulaforthecomputationis Kn(h)=nvuut kXi=1kXj6=iIh(dij) k(k)]TJ /F6 11.955 Tf 11.96 0 Td[(1).(6)Lethbexed.ItiseasytoseethatAi(h)=kXj6=iIh(dij) k)]TJ /F6 11.955 Tf 11.96 0 Td[(1isthenumberofpointsfromfxjgkj=1thatlieinaclosedballofradiushaboutthepointxinotincludingxi.Thus,(Kn(h))n=kXi=1Ai(h) kistheaverageofAi(h)overiforthekpointsintheset.Ifthesetofpointsfxigki=1werefromaPoissondistributionwithrateperunitvolume,thentheexpectednumberofpointsintheclosed(oropen)ballofradiushwouldbeEh(m)=Ln( B(h))whereLnisn-dimensionalLebesguemeasureand B(h)isaclosedballofradiush.NowLn( B(h))=Ln(vn)hnwherevnisthevolumeofaclosedballofradiusone.So,wehaveEh(m)=np Ln(vn)hwhosegraphislinearinh.ThePoissonprocesswasassumedtobeonaboundedregionofRn.So,forthenitenumberofpointsinhand,thegraphwouldonlybeapproximatelylinearneartheorigin.ObservethatforthesekpointsKn(h)k)]TJ /F6 11.955 Tf 12.02 0 Td[(1forh>maxfd(xi,xj)g.ThefunctionKn(h)providesavisualtestforwhetherthepointscamefromaPoissondistribution.IfthegraphofKn(h)formsastraightlineneartheorigin,thenthepointslikelycomefromaPoissondistribution. 67

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6.6RandomDistributionofPointsonaFractalItisnaturaltoaskiftheRipleyK-functioncouldalsobeusefulinanalyzingpointsonafractalsetcomingfromaPoissondistributiononthatset.Inthenextfewsectionswewillanswerthisintheafrmative.Foraself-similarfractalMinRnsatisfyingtheOSCwehavetheHausdorffmeasureonthatsetwith0
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Corollary1. (TheTwo-LinesTheorem)Foragivenself-similarfractalMinRnsatisfy-ingtheOSCasabovethereareconstantsC1,C2,andC3suchthatforall0
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ThisleavesusdeterminingavalueforC2.WereferthereadertothetheproofofTheorem6.3.12in[ 21 ,p.161-2].InthatproofitisshownthatthereisaconstantDsuchthatifAM,then H(A)Ddiam(A).(6)LetA= Bh(x0)\M.Thendiam(A)2h.ThenEquation 6 givesusthefollowingH(Bh(x0)\M)D2nh.So,wehavethat C2=D2n(6)So,ourtwoconstantsaregivenbyEquation 6 andEquation 6 AsanillustrationoftheresultsinTheorem6.1anditsCorollaryconsiderthefollowingself-similarfractalMnRnforn>2.Letfi:R!Rbedenedbyf0(x)=x 3,f1(x)=x 3+1 3,andf2(x)=x 3+2 3.UsingthesefunctionswedenetheIFSthatdenesMn.LetF(m1,...,mn)=fm1fm2fmnwith0mi2.LettheIFSdeningMnbegivenbyIFS=fF(m1,...,mn)j9mi6=mjg.SeeFigure1inthenextsectionforthestandardtwo-dimensionalSierpinskiCarpet.ThisMn.NotethatMnsatisestheOSC.Theopenn-cell,(0,1)n,istherequiredopenset.So,dimMn=log(2n)]TJ /F9 7.97 Tf 6.58 0 Td[(1) 3.InsteadoftheusualmetriconRnusethetaxicabmetric,d(x,y)=max0infjxi)]TJ /F4 11.955 Tf 11.95 0 Td[(yijg.Underthismetric,thesimilitudesintheIFSdeningMnarestillsimilitudeswithcontractionfactor1 3andMnisstillthelimitset.TheOSCstillholdswiththesameopenset.So,thedimensionofMnwillstillbe=log(3n)]TJ /F9 7.97 Tf 6.58 0 Td[(1) log3.Theessentialfeatures 70

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ofMnhavenotchanged,buttheballBh(x)isnowann-cellwithsidesoflength2h.ConsideroneofthecornersofMn.Letx0bethiscornerandleth=1 3n.ThenitisthecasethatH(Bh(x0)\M)=)]TJ /F9 7.97 Tf 8.64 -4.97 Td[(1 3nH(M)sinceBh(x0)\M=fn(M)wherefisthesimilitudecontractingMntox0.Ontheotherhand,letx12MnacorneroftheremovedcentralsquareinMn.ForthesamevaluesofhwehavethatH(Bh(x0)\M)=2n)]TJ /F9 7.97 Tf 6.59 0 Td[(1)]TJ /F9 7.97 Tf 8.64 -4.98 Td[(1 3nH(M).ThisisbecauseBh(x1)\Mnconsistsof2n)]TJ /F9 7.97 Tf 6.59 0 Td[(1copiesoffn(M)ratherthanjustone.Inthisexample,theC1mustbelessthanH(M).TheactualestimateintheproofofTheorem 6.1 giveninEquation 6 isC1=H(M) 2n.Ontheotherhand,C2ofthetheoremmustbegreaterthan2n)]TJ /F9 7.97 Tf 6.58 0 Td[(1H(M).ItisdifculttocomparethisC2withtheestimategiveninEquation 6 withoutgoingintodetailinthederivationofDinthatequation.Wewillleavethattotheinterestedreader. 6.8TheTwoLinesAppliedtotheRipleyK-functionLetMbeaself-similarfractalwithHausdorffmeasureH(A)denedonthemeasurablesubsetsAM.Letfxigki=1MbearandomsetofpointsfromaPoissonprocessonthisself-similarfractaldistributedatratewithrespecttotheHausdorffmeasure.Let Bh(xi)\MbetheclosedballofradiushcenteredatxiintersectedwithM.LetE(mi)betheexpectednumberofthepointsfromfxjgkj=1thatareexpectedtobefoundin Bh(xi)\M.CombiningEquation 6 withEquation 6 wehavethat E(mi)=H)]TJ ET q .478 w 242.53 -444.9 m 251.73 -444.9 l S Q BT /F4 11.955 Tf 242.53 -454.88 Td[(Bh(xi)\M.(6)FromEquation 6 wehavethatthereareconstantsD1andD2suchthatthefollowingholds. D1h


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K-function,Kn(h),wouldtoclusteraroundalineneartheoriginifthepointswerefromaPoissonprocess.WerepresentthisinthefollowinggraphicaldiagramFigure 6-1 .WealsoincludeagraphoftheRipleyK-functionthatarisesinthecomputationalexamplesinSection 6.10 .ThisisgivenasthegraphinFigure 6-2 .NotethatthegraphismuchclosertoalinethantheTheorem 6.1 andCorollary 1 mightsuggest.WecangraphthispreciselysincedimH(M)=isknowntheoretically. Figure6-1. K(h)andthetwolines Figure6-2. K(h)foratypicalexamplefromSection 6.10 WenowshowhowtousethetwolinestoestimatetheHausdorffdimensionofM.Supposethat=dimH(M).Suppose,however,thatwedonotknowthisdimensionandguessittobe<.ThenthediagrammaticgraphinFigure 6-1 becomesthegraphinFigure 6-3 .Notethatthetwolinesforcethedatatobetangenttothehorizontalaxis.Similarly,ifweguess>,thenthegraphofK(h)willbeforcedbythetwolinestobetangenttotheverticalaxis.So,wecantellif>or
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Intheory,byvisualinspectionofthesegraphswecouldhomeinon=dimH(M)bybisectionbythismeans.Inpractice,thiswouldbelaborintensivetimeconsuming,buttheobservationgivesusatheoreticalreasontounderstandwhythetwolinesareimportantindemonstratingthatthedataintheRipleyK-functionallowsustoestimatethedimension. Figure6-3. K(h)andthetwolinesif<=dimH(M)Thegraphistangenttothehorizontalaxis.If>,thenthegraphistangenttotheverticalaxis. 6.9APracticalAlgorithmThediscussionabovesuggestsapracticalalgorithmforcomputingtheHausdorffdimensionofaself-similarfractal.WeusetheRipleyK-functiontoestimatethedimensioninthefollowingway.LetK(h)=K1(h).Thisgivesagraphusingtherawdistances.Letg(x)=axbbethefunctionwhichistheleastsquaresttoK(h)neartheorigin.Theestimateofthedimensionistheb.Alternatively,wecouldtakethegraphoflog(K(log(x))andgettheleastsquarestofh(x)=ax+b.Inthatcasetheestimateofthedimensionwillbea.Throughexperimentswithbothapproaches,therstseemstogivethebestestimatewiththeleastvariance.TheTwo-LinesTheoremshowsthateitheroftheseestimatesshouldgiveavalidestimateoftheHausdorffdimensionoftheself-similarfractalM.Inthenextsectionwedosomesamplecalculationstoshowtheaccuracyofthemethod. 73

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6.10ComputationalExamplesInthissectionweillustratethetheorywehavedevelopedintheprevioussectionsbyproducingsomecomputationalexamples.Weproduceseveralself-similarfractals.Foreachoneagraphofthefractalisgiven.ForeacheachoftheseweestimatetheHausdorffdimensionusing1000randomlydistributedpoints.Weproduce200suchestimatesforeachofthefractals.AhistogramoftheestimatesisproducedandsuperimposedonthehistogramisaGaussianfunctionwiththesamemeanandstandarddeviationasthe200estimates.Asyoucansee,theestimatescloselyapproximatethedimension,butwithsomestatisticalvariationaswouldbeexpected.Theexperimentgivessomeindicationoftheamountofvariationthatcanbeexpectedinthemethod.Themethodworksequallywellforself-similarfractalsembeddedinhigherdimensionalEuclideanspace.ThelastexampleistheMengerSponge.ThecomputationsandgraphicsweredoneusingMathematicac. AGraphofSierpinskiCarpet BHistogramof200estimateseachusing1000randompointsFigure6-4. A)GraphofSierpinskiCarpet=dimH(M)=log8 log31.89279.B)Histogramof200estimateseachusing1000randompointscomparedwithNormalDistributionwiththesamemeanandstandardDeviation=1.8788and=.0513796 74

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AGraphofthevonKochCurve BHistogramof200estimateseachusing1000randompointsFigure6-5. A)GraphofthevonKochCurvedimH(M)=log4 log31.26186.B)Histogramof200estimateseachusing1000randompointscomparedwithNormalDistributionwiththesamemeanandstandardDeviation=1.26759and=.0320847 AGraphofself-similarfractaldimH(M)=log7 log31.77124 BHistogramof200estimateseachusing1000randompointsFigure6-6. A)Graphself-similarfractaldimH(M)=log7 log31.77124.B)Histogramof200estimateseachusing1000randompointscomparedwithnormaldistributionwiththesamemeanandstandarddeviation=1.77449and=.0476043 6.11FurtherDirectionsThetwo-linesestimateinTheorem 6.1 andCorollary 1 iscrude.Thereisapossibilitythatforalmostallx2MthatH)]TJ ET q .478 w 236.04 -561.63 m 245.23 -561.63 l S Q BT /F4 11.955 Tf 236.04 -571.6 Td[(Bh(x)\M=ChforsomeC.TheexampleafterCorollary 1 usesexceptionalpointsintheboundaryofM.Ofcourse,C1
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AGraphofself-similarfractaldimH(M)=log6 log31.63093 BHistogramof200estimateseachusing1000randompointsFigure6-7. A)Graphself-similarfractal,dimH(M)=log6 log31.63093.B)Histogramof200estimateseachusing1000randompointscomparedwithnormaldistributionwiththesamemeanandstandarddeviation=1.59555and=.0423772 AGraphoftheMengerSponge BHistogramof200estimateseachusing1000randompointsFigure6-8. A)GraphoftheMengerSpongedimH(M)=log20 log32.72683.B)Histogramof200estimateseachusing1000randompointscomparedwithnormaldistributionwiththesamemeanandstandarddeviation=2.6577and=.0736642 Thereareotherself-similarmeasuresonM,oneforeachproductmeasureonthecodespacecomingfromaprobabilitymeasureonf1,2,...,Ngwhereff1,f2,...,fNgistheIFSforMsatisfyingtheOSC.TowhatextentcantheRipleyK-functionbeusedtoanalyzePoissondistributionofpointsonMusingoneofthesemeasures? 76

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Lastly,thevalueofthisapproachisthatitindicatesthatifMisanyfractalsetwithHausdorffdimension,0
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CHAPTER7DISCUSSIONThischapterdiscussesandinterpretstheresultspresentedinChapter4aswellasthenewmethodfordeterminingtheincubationperiod.Fromtheresultsandanalysisofthemodel,wedrawconclusionsandimpactsthismodelhasonthenewdirectionsforresearch.Limitationsofthestudyaswellassuggestionsforfurtherstudyarealsopresentedinthischapter. 7.1ImplicationsoftheTransmissionModelIthasbeenshownexperimentallythatthelatencyperiodfromnewinfectionbyinfectedadultpsyllidstoinfectiousnessinyoungushislessthan15days[ 46 ].Usingthisinformationtogetherwithevidencethatpathwaysofinfectionfrominfectedadultfemalepsyllidscangoviatransovarialinfectiontoeggsandthentonymphsandviafeedingonyoungushtonymphsalreadyontheushandtopreviouslyuninfectedadultsfeedingonthesameush,weshowedinamicrosimulationmodelthatentiregrovescanbecomeinfectedinamatterofafewmonths.Theresultinginfectedtreescanallbeasymptomatic,andcaneachbecomehometoontheorderof20,000to30,000psyllids,alargefractionofwhichareinfected,duringasingleushperiod.Sincetreesdonottendtoshowsymptomsforanywherefrom1to2.5yearsandpossiblylongerafterinitiallybecominginfected,apremiummustbeplacedonongoingsurveillanceandcontrolofpsyllids.ThedatafromthetransmissionexperimentprovidethemechanismtoexplainobservationsmadebyManjunathetal.[ 50 ],Halbertetal.[ 29 30 ],andShenetal.[ 67 ].Ineverycase,positivepsyllidswerefoundmonthstoasmuchas6yearspriortodiseasedevelopment.Thesedatastronglysupportaggressivepsyllidcontrol,andalsoregulationsthatensureabsenceofD.citrionplantsforsale.Oursimulationsindicatethat90%reductionsinthepsyllidpopulationasaresultofcontrolstrategiescarriedoutduringallushperiodsinayear,candelaytheappearance 78

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ofsymptomatictreesbyatleast240daysandinmanyinstancesoneyearormorebeyondwhatwouldensuewithoutsuchpsyllidcontrol.Ifitisfeasibletoattain95%reductionsinthepsyllidpopulationfromwhatwouldbepresentintheabsenceofpsyllidcontrol,grovescouldbeproducingfruitfreeofHLBfor2+yearsbeyondthetimeofonsetofsymptomsinuncontrolledgroves.Thequestionofhowtoreach90%to95%reductionsinpsyllidpopulationsrelativetotheuncontrolledsettingisstartingtogetseriousconsiderationfromgrowerscooperatives[ 74 ].Acriticalrststepissynchronizationofinsecticidesprayingschedulesonaregionalbasistodrasticallyreducemovementofpsyllidsfromonegrovetothenextaswellastoreducetheirprevalencewithingroves.Inaddition,theuseofaluminizedmulch[ 17 ]toprotectnewlyplantedtreesfortheroughlytwoyearsittakesbeforeacanopypreventseffectiveutilizationofthismethodology,hasthepotentialtodelaytheintroductionofpsyllidstoagroveduringthisperiodofearlydevelopment.Otherpsyllidcontroltoolsarecurrentlyunderdevelopment,andsuggestthatthestringenttargetswehaveindicatedfordelayingtheonsetofsymptomsshouldbewithinreach.Anotherpathwayofinfectionthatwehavenotconsideredinourmodelisdegradationoftherootsofinfectedtrees[ 41 ].WhetherornotthisfacilitatessubstantialsoiltransmissionofCLas,asspeculatedinthemodelingpaperofJacobsenetal.[ 40 ],isanempiricalquestionthatneedsstudy.However,wehaveprivilegedpsyllid-ushtransmissionasthedominantmodeofdispersalofCLasinagrovebecauseofthedocumentedlargenumbersofpsyllidsthatoccupytreesininfectedgroves,aswellasthemultipleroutesofrapidpropagationofthisbacteriumbetweenpsyllidsandush.Bettersurveillancetoolsareneededtohelpquantifythewaveofadvanceofnewinfectioninpreviouslyuninfectedgroves. 7.2LimitationsAlthoughtherehasbeenalotofresearchconductedonthepsyllid,thereisstillalotthathasnotbeendiscoveredsuchasfeedingpreferencesonyoungversusold 79

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shootsandthefrequencyoffeedingandmovementwithinatreeaswellasbetweentrees.Inaddition,oursimulatedgroveissmallerthantypicalgroves.Tosimulatespreadinalargergrove,wewouldneedtoallowforlongdistancemovementperhapsintermsofadiffusionmodelwithajumpkerneltomodelthespreadofthepsyllid.Inaddition,migrationintoandoutofthegrovehasnotbeenconsideredinthiscurrentmodel.Ideally,theresearchbeingconductedonpsyllidbehaviorwouldhelptunethefeedingandmovementparametersinourmodel.Wehaveverygenerallyestimatedparametersforthenymph,ushcycle,withoutexplicitlytakingintoaccountageandcultivartypesofthetreesinthegrove.Wehaveestimatedparameterswheretheyarenotspecied,howeverwithcontinuedresearchonthebehaviorofthepsyllidandtheinteractionofD.citriwithinfectedtrees,weexpectwedonotexpectthatourmodelreectscompletereality,howeverwedescribethespreadandincludeimportantparametersthatmayneedfurtherinvestigationandresearch. 7.3SuggestionsforFutureResearchThemodelhasbeensetuptoincorporatedailysurvivalratesthatmaychangeasaresultofdailyvariationsintemperature.Asushingperiodsuctuateduetotheageofthetreesandtherainfall,incorporatingstochasticityinthemodelforushgrowthmaychangethedynamicsofthespread.Duetomanycitrusgrowersreplantingtreesandthefactthatyoungertreestendtoushthroughouttheyear,thiswouldprovidemanysitesforpsylliddevelopmentaswellascontinuedtransmissionduringthetypicalnoushingperiods.Wehavebroadlyimplementedtheeffectoftemperatureonthepsyllidsurvivalrate,andhavenotincorporateditintotheappearanceofush.Thiswouldbeanextstepforthemodelinordertodepictthedependenceofthebacterium,ushingpatterns,andthepsyllidtouctuationsintemperatureandrainfall.Controlstrategiessuchasinsecticidetreatmentsmaybebetterrepresentedashavinglingeringeffects.Acombinationofinsecticidesandnutritionaltreatmentsmaybeabetterwaytocombatthedisease.Thisalsotendstoshowthattreesmayhavesome 80

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resistancetoshowingsymptomsimmediately,thecausesofwhichareunknown.Ifearlydetectioncanpinpointtreeswithinfectednymphs,thentheremovalofthesenymphsandnotnecessarilythewholetreecouldbeapossiblereductionininoculum.ThetimingofthespraysalsoneedstobeinvestigatedfurthertodecidehowoftenandhoweffectivetheyneedtobeinordertoslowthespreadofHLB.DetectionofHLBmayalsochangebecausetherealtimePCRtestsaretypicallydeterminedfromsymptomatictreematerial,nowthedetectionofinfectednymphsontreeswouldbeabletodeterminewhetherthediseaseispresentfarearlierthanthepresenceofsymptoms.Earlydetectionwouldallowformorepreventativemeasurestobetakeninordertoensuretheproductivityofthegrove.AsCoca-colahasdemonstratedtheirfaiththatthecitrusindustrywillsurvivebybuyinggrovesinFlorida,ourmodelwithmorerealisticparameterswouldbeabletoprovidecomparisonsintermsofmanagementpracticessuchastimingandefcacyofinsecticidesprays.AstrongresultfromourmodelintermsofcitrusmanagementwouldbetofurthercompareourmodelsforthespreadofsymptomswithdatacollectedfromgrovesinFlorida.Beingabletoestimatetheprobabilitydistributionfortimetosymptomswouldgreatlyimpactthedecisionsthatgointoreplantingagroveandtheeconomicconsiderationsneededtodetermineifthecitrusindustrywillbeprotable. 7.4ConclusionItisourgoaltocomparevariouscontrolstrategiesandtoguidefutureresearchdirectionswiththismodel.Currentlytherearelongtermprojectsthatfocusongeneticallymodiedpsyllid,thenuPsyllid,thatwillnottransmitCLas,althoughthereneedstobefurtherstudyontheinteractionanddynamicsbetweentheACPandnuPsyllid.Withourmodel,theconsequencesoftheseprogramsshouldbeconsideredwhethertheywillbeviable.Ifinfacttheinfectionspreadsrapidly,itmaybemorebenecialtoconsidertreetreatmentsthatwillprolongthelifeofthetree'sabilitytoproducefruit.Wehaveamodelthatdescribesthespreadofcitrusgreeningthrough 81

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ushshootsinaspatiallyexplicitgrove,wheretheprimarymeansofinfectioninatreebeginattheushshoots.Theresultsofthismodelimplythattheappearanceofsymptomsmimicsthespreadofinfection.Asaresultswepresentedamethodthatreliesondatathathasbeencollected,aswellasimplementinganewdetectiontechniquewhichwouldleadtoabetterestimateoftheincubationperiod,whichiscurrentlyunknown.Thisestimatewillimpacttheeconomicaspectsconcerningcitrus,mainlywhetheritisviabletomaintainthegroveandwhatstepscanbetakentoprolongthelifeofaninfectedtree.ArecommendationfromourmodelistosaythattheremovalofasymptomatictreesdoesnotremovetheinoculumfromthegrovebecausethepsyllidsareabletopassCLastothenextgenerationwithoutthetreesbecomingsymptomaticforaslongas6years. 82

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APPENDIXASIMULATIONMODELDAILYACTIVITIES A.1SpringFlushPeriodThedetailsofthemicrosimulationoverthespringushperiodaredetailedinthefollowingsections.Wepresentthedaysonwhichactivitiesbeginorceaseandusethesedaysasthebasisforwhatoccursinbetweenthesedays.Webeginbyoutliningthedailyactivitiesthattakeplaceduringtherstushingperiodof60days.Day1ActivitiesFlushshootinitialconditions.Welabeleachcohortofushshootsas(c,(t1,t2),20)=20ushshootsatlocationconDayt1sinceaninitialpsyllidinvasionandDayt2sinceemergenceoftheushshoots.Forthesimulations,weassumethatinaninitialushperiod20ushareavailableateachushpatchatthestartofapsyllidinvasion.Weadopttheconventionthatthesearenewlyemergedontheinitialday,andthereforesett1=t2=1ontherstdayofinvasion.Onallsubsequentdays,20newushshootsareassumedtoemergeateachpatch.Thus,forexample,onDay2,wehave(c,(2,2),20)and(c,(2,1),20)asthecollectionof40ushatlocationc.Emergingushwillbecalledyoungfor16days,afterwhichtheywillbedesignatedold.Egglayingbyadultfemalesonlyoccursonyoungush,butCLasinfectioncanbetransmittedtoyoungandoldushbyinfectiveadultpsyllids.Onceaushshootisinfected,weassumethatitremainsinfecteduntilitdies.Infectedyoungandoldushshootsareassumedtobecomeinfectiousimmediately,howeverforyoungushshootsthiswillnotproduceanyinfectedadultsuntilt2=15.Allnymphsthatfeedonaninfectedushshootemergeasaninfectedadultwithprobability1.Psyllidinitialconditions.Weassumethat200adultpsyllidsofage17daysareinitiallyplacedatasetofdesignatedlocalities,c,onDayt1=1.Theage17 83

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ischosentoallowtheadultstocontributeimmediatelytolayingeggs,transmission,andmovement.Thereisnothingspecialaboutthenumber200,aswecouldstartwithawiderangeofpopulationsizesthatcouldvarybetween,say30,and2000.Theinitialpopulationisdividedintofourgroupsconsistingof60a1infectedfemales(F),60(1)]TJ /F4 11.955 Tf 10.97 0 Td[(a1)infectedmales(M),140a2uninfectedfemales(F),and140(1)]TJ /F4 11.955 Tf 10.97 0 Td[(a2)uninfectedmales(M).Here,0
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pinf=.3oftransmittingCLasduringaninfectivefeeding.Thusthenumberofinfectivefeedings,B,andthetotalnumberofushpresent,20,isobtainedandthentheexpectednumberofinfectedushofaget2(t2daysafteremergence)onthet1thdayafterinitialinfection,h(t1,t2)iscalculated.Allushonthisdayareinthesamecohort,thereforeh(1,1)oftheushmovetotheinfectedushand20-h(1,1)remaininthehealthyushpopulation.AninventoryofushispreparedforDay2accordingtotheformalismS(c,(2,2),20)]TJ /F7 11.955 Tf 12.73 0 Td[(h(1,1)))andI(c,(2,2),h(1,1)))forhealthyandinfectedushrespectively.Summarystatistics.SincethereisnomigrationofpsyllidsawayfromtheiroriginallocationsonDay1andweassumethattheinitialnumberofpsyllidssurvivedthroughDay1,wecanoutputthesummarystatisticsthatcorrespondtothecompartmentsinFigure 3-1 ofthemaintext.Theseare: Ia(1)=#ofinfectedadultpsyllidsattheendofDay1.WesuppressthedependenceofIaonctosimplifytheterminology.ThecompartmentlabeledIainFigure1ofthemaintextisspeciedintermsoffemaleandmalepsyllidsviaIa=IFa+IMa=#infectedadultfemales+#infectedadultmales. Sa(1)=#ofuninfectedadultpsyllidsattheendofDay1 Se(1)=#ofuninfectedeggsattheendofDay1 Iy(1)=#ofinfectedyoungushattheendofDay1 Sy(1)=#ofuninfectedyoungushattheendofDay1TheremainingthreecompartmentsinFigure 3-1 pertaintooldushandnymphs,neitherofwhichexistonDay1.Eggsdonotbecomerstinstarnymphsuntil3daysaftertheyarelaid.Day2ActivitiesMigration.Eachpsyllidhasprobability.4ofmigratingtoanewushpatch,callitc0.Thus,wescanthepsyllidsinsequenceassigningeachthevalue1withprobability.4andthevalue0withprobability.6.Weassumethattheinitialinvasionofpsyllidswasrestrictedtofourushpatchesinthesouthwestcornerofthegrove.Theyhave 85

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coordinates,c,givenby(1,1),(1,2),(2,1),and(2,2).Thiscornerinvasionisusedformanyofthesimulationsshowninthemaintext.However,arbitraryinitiallocationscanbehandledbythesamemethodologythatweshowhere.Now,weassumethatmigrationfrom(2,2)isto: (2,3)w.p..495(2,1)w.p..495(1,2)w.p..005(3,2)w.p..005Migrationfrom(1,1)isto: (1,2)w.p..99(2,1)w.p..01Migrationfrom(1,2)isto: (2,2)w.p..01(1,1)w.p..495(1,3)w.p..495Migrationfrom(2,1)isto: (2,2)w.p..99(1,1)w.p..005(3,1)w.p..005TherationaleforthehighprobabilityofwithincolumnmigrationrelativetobetweencolumncanbeseenfromthespacingoftreesshownthegroveinFigureS1.Especiallyastheushseasondevelopsinfullgrowntrees,thecanopycontainsushfromonetreethatareliterallyadjacenttoushfromtheneighboringtree.Whenpsyllidsarriveatanewtree,weassumethattheytakepartintheremainingactivitiesonDay2and 86

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combinethepopulationofarrivingpsyllidswiththosethatremainedminusthosethatmigratedawayfromushpatchc.Emergenceofnewushshoots.Inadditiontothe20ushavailableattheendofDay1,anewsetof20ushshootsemergesateachlocation,c.Theseusharedesignatedas(c,(2,1),20).Witht1=2andt2=1,thismeanswearerepresentingtheseushshootsatlocationconthe2nddaysincethepsyllidinvasionandontherstdaythatthisushemerged.The20ushfromtheinitialinvasiondayarenowdesignatedby(c,(2,2),20),meaningthatthisisthe2nddaysinceinvasionandthe2nddaysincetheseushemerged.Thereareatotalof40ushpresentateachpatchonDay2.Psyllidaging.Afterthemigrationofpsyllidstotheirnewushpatches,eachadultpsyllidhasprobabilitysaofsurvivingfromDay1.Thus,wescantheadultpsyllids,applyingprobabilitysatoeachofthemtodeterminetheirsurvivorstatus.Forvaluesoft215,thenumberofsurvivingadults,sa()=sa(P(c,(1,t2),n)),isrecordedintoP(c,(2,t2+1),sa())foractivitiesonDay2.Egglayingandsurvivalofpreviouslylaideggs.ThenumberofeggslaidonDay2,E2,ateachushpatchislimitedeitherbyasmallnumberoffemaleadultsorbythenumberofyoungushshoots.AllfemalepsyllidsthatsurvivedDay1layanother10eggs,thuswehave10(Fa)eggs.ThecapacityforeggsonDay2is40((c,(2,2),20)+(c,(2,1),20))=1600.Thenumberofeggslaidistheminimumofthesetwovalues.TheneweggsarenowenteredintoalistofeggsasP(c,(2,1),E2)andtheeggsfromthepreviousdayareP(c,(2,2),se(E1)).EachoftheeggshasprobabilityseofsurvivingDay2.ThesurvivalprocessismodeledasasetofindependenteventsacrosseggsandthenumberofsurvivingeggsareretainedinthepopulationmatrixP(c,(3,3),s2e(E1))andP(c,(3,2),se(E2))forDay3activities.Hereweusethenotationsketoindicatetheeggsthathavesurvivedthroughtothe(k+1)thday. 87

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Transmissiontoushshoots.h(t1,t2)isthenumberofnewinfectedushofaget2ondayt1.OnDay2,transmissionofCLascanonlybefrominfectedpsyllidtoushshoot,astherearenoemergingadultsfromtheinfectedushshootsinfectedonDay1.Thus,eachinfectedpsyllidisscannedandhasprobabilitypinf=.3oftransmittingCLasduringaninfectivefeeding.Thusthenumberofinfectivefeedings,B,andthetotalnumberofushpresent,40,isobtainedandthentheexpectednumberofinfectedushonDay2,h(2,),iscalculated.Foreachoftheh(2,)infectedush,wethendecidewhatage(t2=2,1)isassignedtoeachush.Theprobabilityisequaltothecurrentproportionofhealthyushshootsofthatage.ThenumberofhealthyushshootsonDay2are20)]TJ /F7 11.955 Tf 12.61 0 Td[(h(1,1)fromDay1andthe20newushshootsemergingonDay2.Theprobabilityofinfectionofeachushshootwitht2=2and1is20)]TJ /F5 7.97 Tf 6.59 0 Td[(h(1,1) 40)]TJ /F5 7.97 Tf 6.59 0 Td[(h(1,1)and20 40)]TJ /F5 7.97 Tf 6.58 0 Td[(h(1,1)respectively.AninventoryofushispreparedforDay3accordingtotheformalismS(c,(3,3),20)]TJ /F7 11.955 Tf 13.12 0 Td[(h(1,1))]TJ /F7 11.955 Tf 13.12 0 Td[((2,2))),S(c,(3,2),20)]TJ /F7 11.955 Tf 13.12 0 Td[(h(2,1))),I(c,(3,3),h(1,1)+h(2,2))),andI(c,(3,2),h(2,1)))forhealthyandinfectedushrespectively.Summarystatistics.ThesummarystatisticscorrespondingtothecompartmentsinFigure 3-1 atushpatchcarerecorded.ThesearethenumberofpsyllidsthatwerepresentandtakingpartinactivitiesonDay2atushpatchcandincludethenewphysicaldistributionofpsyllids.Theseare: Ia(2)=#ofinfectedandsurvivingadultpsyllidsattheendofDay2.WesuppressthedependenceofIaonctosimplifytheterminology.ThecompartmentlabeledIainFigure1ofthemaintextisspeciedintermsoffemaleandmalepsyllidsviaIa=IFa+IMa=#infectedadultfemales+#infectedadultmales. Sa(2)=#ofuninfectedandsurvivingadultpsyllidsattheendofDay2 Se(2)=#ofuninfectedandsurvivingeggsattheendofDay2 Iy(2)=#ofinfectedyoungushattheendofDay2 Sy(2)=#ofuninfectedyoungushattheendofDay2 88

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TheremainingthreecompartmentsinFigure 3-1 pertaintooldushandnymphs,neitherofwhichexistonDay2.Eggsdonotbecomerstinstarnymphsuntil3daysaftertheyarelaid.Day4ActivitiesTheactivitiesonDay3,includingmigration,emergenceofnewushshoots,psyllidegglaying,aging,survival,andtransmissiontoushshoots,areidenticaltothoseonDay2.Thenewactivityonthisdayisthateggswitht2=3onDay3survivetonymphsonDay4.Migration.Eachpsyllidhasprobability.4ofmigratingtoanewushpatch,callitc0.Thus,wescanthepsyllidsinsequenceassigningeachthevalue1withprobability.4andthevalue0withprobability.6.Foralltheushpatchesinthegrovewithapopulationofadultpsyllids,themethodologyshownisappliedtoeachushpatchwithcoordinates,c=(c1,c2).Now,weassumethatmigrationfrominteriorpatches(c1,c2)isto: (c1,c2+1)w.p..495(c1,c2)]TJ /F6 11.955 Tf 11.95 0 Td[(1)w.p..495(c1)]TJ /F6 11.955 Tf 11.95 0 Td[(1,c2)w.p..005(c1+1,c2)w.p..005Migrationfromcornerpatches(1,1),(1,25),(11,1),or(11,25)isto: (1,2),(1,24),(11,2),or(11,24)w.p..99andto(2,1),(2,25),(10,1),or(10,25)w.p..01respectivelyMigrationfrompatchesonthewestedge(1,c2)forc2=2,...,24isto: (2,c2)w.p..01(1,c2)]TJ /F6 11.955 Tf 11.96 0 Td[(1)w.p..495(1,c2+1)w.p..495Migrationfrompatchesonthesouthedge(c1,1)forc1=2,...,10isto: 89

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(c1,2)w.p..99(c1)]TJ /F6 11.955 Tf 11.95 0 Td[(1,1)w.p..005(c1+1,1)w.p..005Migrationfrompatchesontheeastedge(11,c2)forc2=2,...,24isto: (10,c2)w.p..01(11,c2)]TJ /F6 11.955 Tf 11.95 0 Td[(1)w.p..495(11,c2+1)w.p..495Migrationfrompatchesonthenorthedge(c1,25)forc1=2,...,10isto: (c1,24)w.p..99(c1)]TJ /F6 11.955 Tf 11.95 0 Td[(1,25)w.p..005(c1+1,25)w.p..005Whenpsyllidsarriveatanewtree,weassumethattheytakepartintheremainingactivitiesonday4andcombinethepopulationofarrivingpsyllidswiththosethatremainedminusthosethatmigratedawayfromushpatchc.Emergenceofnewushshoots.Anewsetof20ushshootsemergesateachlocation,c.Theseusharedesignatedas(c,(4,1),20).Witht1=4andt2=1,thismeanswearerepresentingtheseushshootsatlocationconthe4thdaysincethepsyllidinvasionandontherstdaythatthisushemerged.Theushshootsfrompreviousdaysaredesignatedby(c,(4,t2),20),wheret2=2,3,4.Thereareatotalof80ushshootspresentateachpatchonDay4.Psyllidaging.Afterthemigrationofpsyllidstotheirnewushpatches,eachadultpsyllidhasprobabilitysaofsurvivingfromDay3.Thus,wescantheadultpsyllids,applyingprobabilitysatoeachofthemtodeterminetheirsurvivorstatus.Forvaluesoft215,thenumberofsurvivingadults,sa()=sa(P(c,(3,t2),n)),isrecordedintoP(c,(4,t2+1),sa())foractivitiesonDay4.Thenymphstageforthepsyllidiswhen4t214.Eggssurvivetobecomenymphswithprobabilityse.OnDay4,we 90

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recordthenumberofeggsthatsurvivetonymphs,s3e(E1)=se(P(c,(3,3),s2e(E1)))intoP(c,(4,4),s3e(E1))foractivitiesonDay4.Egglayingandsurvivalofpreviouslylaideggs.ThenumberofeggslaidonDay4,E4,ateachushpatchislimitedeitherbyasmallnumberoffemaleadultsorbythenumberofyoungushshoots.AllfemalepsyllidsthatsurvivedDay3layanother10eggs,thuswehave10sa(Fa(3))eggs.ThecapacityforeggsonDay4is 40((c,(4,4),20)+(c,(4,3),20)+(c,(4,2),20)+(c,(4,1),20))=404Xi=1(c,(4,i),20)=3200Thenumberofeggslaidistheminimumofthesetwovalues.TheneweggsarenowenteredintoalistofeggsasP(c,(4,1),E4)andtheeggsfromthepreviousdaysareP(c,(4,2),se(E3))andP(c,(4,3),s2e(E2)).EachoftheeggshasprobabilityseofsurvivingDay4.ThesurvivalprocessismodeledasasetofindependenteventsacrosseggsandthenumberofsurvivingeggsareretainedinthepopulationmatrixP(c,(5,3),s2e(E3))andP(c,(5,2),se(E4))forDay5activities.Aspreviouslymentioned,theeggsP(c,(4,3),s3e(E2))thatsurvivetoDay5,becomenymphsandthereforearenotcountedinoureggcount.Hereweusethenotationsketoindicatetheeggsthathavesurvivedthroughtothe(k+1)thdaysinceegg-laying.Transmissiontoushshoots.h(t1,t2)isthenumberofnewinfectedushofaget2ondayt1.OnDay4,transmissionofCLascanonlybefrominfectedpsyllidtoushshoot,astherearenoemergingadultsfromushshoots.Thus,eachinfectedpsyllidisscannedandhasprobabilitypinf=.3oftransmittingCLasduringaninfectivefeeding.Thenumberofinfectivefeedings,B,andthetotalnumberofushpresent,80,isobtainedandtheexpectednumberofinfectedushonDay4,h(4,),iscalculated.Foreachoftheh(4,)infectedush,wethendecidewhatage(t2=4,3,2,1)isassignedtoeachush.Theprobabilityisequaltothecurrentproportionofhealthyushshootsofthatage.ThenumberofhealthyushshootspriortoinfectiononDay4ofage 91

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t2=4arethe20)]TJ /F7 11.955 Tf 11.4 0 Td[(h(1,1))]TJ /F7 11.955 Tf 11.39 0 Td[(h(2,2))]TJ /F7 11.955 Tf 11.4 0 Td[(h(3,3)thatemergedonDay1,ofaget2=3arethe20)]TJ /F7 11.955 Tf 10.82 0 Td[(h(2,1))]TJ /F7 11.955 Tf 10.81 0 Td[(h(3,2)thatemergedonDay2,ofaget2=2arethe20)]TJ /F7 11.955 Tf 10.82 0 Td[(h(3,1)thatemergedonDay3,andthe20newushshootsemergingonDay4.ThetotalnumberofhealthyushonDay4isT=80)]TJ /F9 7.97 Tf 18.38 14.95 Td[(3Xi=1iXj=1h(i,j).Theprobabilityofinfectionofeachushshootwitht2=4,3,2and1is20)]TJ /F5 7.97 Tf 6.59 0 Td[(h(1,1))]TJ /F5 7.97 Tf 6.59 0 Td[(h(2,2))]TJ /F5 7.97 Tf 6.59 0 Td[(h(3,3) T,20)]TJ /F5 7.97 Tf 6.59 0 Td[(h(2,1))]TJ /F5 7.97 Tf 6.59 0 Td[(h(3,2) T,20)]TJ /F5 7.97 Tf 6.58 0 Td[(h(3,1) Tand20 Trespectively.AninventoryofushpresentonDay4isgivenbytheformalismS(c,(4,4),20)]TJ /F7 11.955 Tf 11.95 0 Td[(h(1,1))]TJ /F7 11.955 Tf 11.95 0 Td[(h(2,2))]TJ /F7 11.955 Tf 11.95 0 Td[(h(3,3))]TJ /F7 11.955 Tf 11.95 0 Td[(h(4,4))),S(c,(4,3),20)]TJ /F7 11.955 Tf 11.95 0 Td[(h(2,1))]TJ /F7 11.955 Tf 11.95 0 Td[(h(3,2))]TJ /F7 11.955 Tf 11.95 0 Td[(h(4,3)),S(c,(4,2),20)]TJ /F7 11.955 Tf 11.95 0 Td[(h(3,1))]TJ /F7 11.955 Tf 11.95 0 Td[(h(4,2)),S(c,(4,1),20)]TJ /F7 11.955 Tf 11.95 0 Td[(h(4,1)),I(c,(4,4),h(1,1)+h(2,2)+(3,3)+(4,4)),I(c,(4,3),h(2,1)+h(3,2)+h(4,3)),I(c,(4,2),h(3,1)+h(4,2)),andI(c,(4,1),h(4,1))forhealthyandinfectedushrespectively.Summarystatistics.ThesummarystatisticscorrespondingtothecompartmentsinFigure 3-1 atushpatchcarerecorded.ThesearethenumberofpsyllidsthatwerepresentandtakingpartinactivitiesonDay4atushpatchcandincludethenewphysicaldistributionofpsyllids.Theseare: Ia(4)=#ofinfectedandsurvivingadultpsyllidsattheendofDay4. Sa(4)=#ofuninfectedandsurvivingadultpsyllidsattheendofDay4 Sn(4)=#ofuninfectedandsurvivingnymphsattheendofDay4 Se(4)=#ofuninfectedandsurvivingeggsattheendofDay4 Iy(4)=#ofinfectedyoungushattheendofDay4 Sy(4)=#ofuninfectedyoungushattheendofDay4 92

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TheremainingtwocompartmentsinFigure 3-1 pertaintooldush,whichdonotexistonDay4.Flushdonotbecomeolduntilday17whennoeggscanbelaidonthem.Day15ActivitiesTheactivitiesthatoccuronDay4continueinthesamemannerfromDay5toDay14.Migration,emergenceofnewushshoots,psyllidegglaying,aging,andsurvival,transmissiontoushshoots,andtherecordingofthesummaryofpsyllidsthatparticipatedineachdays'activitiesproceedsasdescribedinSection A.1 .Theadditionofthenymphcompartment,t2=4,5,...,14,doesnotchangetheactivitiesandtheagingthroughthenymphstagesisrecordedintoP(c,(t1,t2),#).OnDay15,theeggslaidonDay1thatsurvivetothisdayemergeasadults.Migration.Eachpsyllidhasprobability.4ofmigratingtoanewushpatch,callitc0.Thus,wescanthepsyllidsinsequenceassigningeachthevalue1withprobability.4andthevalue0withprobability.6.Foralltheushpatchesinthegrovewithapopulationofadultpsyllids,themovementtonewushpatchesfollowsthemethodologydescribedinSection A.1 .Whenpsyllidsarriveatanewtree,weassumethattheytakepartintheremainingactivitiesonday15andcombinethepopulationofarrivingpsyllidswiththosethatremainedminusthosethatmigratedawayfromushpatchc.Emergenceofnewushshoots.Anewsetof20ushshootsemergesateachlocation,c.Theseusharedesignatedas(c,(15,1),20).Witht1=15andt2=1,thismeanswearerepresentingtheseushshootsatlocationconthe15thdaysincethepsyllidinvasionandontherstdaythatthisushemerged.Theushshootsfrompreviousdaysaredesignatedby(c,(15,t2),20),wheret2rangesfrom2to15.Thereareatotalof300ushshootspresentateachpatchonDay15.Psyllidaging.EachnymphhasprobabilitysnofsurvivingfromDay14.Eachnymphisscanned,applyingprobabilitysntoeachofthemtodeterminetheirsurvivorstatus.Thenumberofsurvivingnymphs,sn()=sn(P(c,(14,t2),n)),isrecordedintoP(c,(15,t2+1),sn())foractivitiesonDay15.Inparticular,thenymphswith 93

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t2=14thatsurvive,emergeasadults,denotedbyP(c,(15,15),s11n(s3e(E1))),thesearetheemergingadultsthatwillbeconsideredinthetransmissiontopsyllids.Afterthemigrationofpsyllidstotheirnewushpatches,eachadultpsyllidhasprobabilitysaofsurvivingfromDay14.Thus,wescantheadultpsyllids,applyingprobabilitysatoeachofthemtodeterminetheirsurvivorstatus.Forvaluesoft215,thenumberofsurvivingadults,sa()=sa(P(c,(14,t2),n)),isrecordedintoP(c,(15,t2+1),sa())foractivitiesonDay15.Egglayingandsurvivalofpreviouslylaideggs.ThenumberofeggslaidonDay15,E15,ateachushpatchislimitedeitherbyasmallnumberoffemaleadultsorbythenumberofyoungushshoots.Femaleadultswhoemergeonthisdaydonotimpactoviposition.Toaccountforthefactthatemergingadultsdonotdepositeggsduringtheirrst24to48hours,allfemalepsyllidswitht217thatsurvivedDay14layanother10eggs,thuswehave10sa(Fa(14))eggs.ThecapacityforeggsonDay15is 4015Xi=1(c,(15,i),20)=12000Thenumberofeggslaidistheminimumofthesetwovalues.TheneweggsarenowenteredintoalistofeggsasP(c,(15,1),E15)andtheeggsfromthepreviousdaysareP(c,(15,2),se(E14))andP(c,(15,3),s2e(E13)).EachoftheeggshasprobabilityseofsurvivingDay15.ThesurvivalprocessismodeledasasetofindependenteventsacrosseggsandthenumberofsurvivingeggsareretainedinthepopulationmatrixP(c,(16,3),s2e(E14))andP(c,(16,2),se(E15))forDay16activities.Transmissiontoemergingadults.OnDay15,thepopulationofushshootswithemergingadultsisE=(c,(15,15),20).WeconsideronlytheushshootsthathavebeeninfectedonorbeforeDay14.Thus,onDay15thereareEI=P14k=1h(k,k)infectedushshootswithemergingadults.Theprobability,p,thatanemergingadultbecomesinfectedisobtainedbymultiplyingtheproportionofinfectedushshootswithemergingadults,=EI Eandtheprobabilityoftransmissionfromushshoottoemerging 94

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adult,1,sop=1=1EI E.Eachemergingindividualisassigneda1(success)withprobabilitypanda0withprobability1)]TJ /F4 11.955 Tf 12.67 0 Td[(p.Itisimportanttonotethattheseemergingadultsdonotcontributetoegglayingortotransmissiontoushshootsonthedaytheyemerge.Also,themovementprobabilitiesforemergingadultsislowerthanof3-dayorolderadultsMichaud2004,Kobori.Transmissiontoushshoots.h(t1,t2)isthenumberofnewinfectedushofaget2ondayt1.OnDay15,transmissionofCLastoushshootscanonlybefrominfectedpsyllidswitht216.Thus,eachage-appropriateinfectedpsyllidisscannedandhasprobabilitypinf=.3oftransmittingCLasduringaninfectivefeeding.Thenumberofinfectivefeedings,B,andthetotalnumberofushpresent,300,isobtainedandtheexpectednumberofinfectedushonDay15,h(15,),iscalculated.Foreachoftheh(15,)infectedush,wethendecidewhatage(t2=15,...,2,1)isassignedtoeachush.Theprobabilityisequaltothecurrentproportionofhealthyushshootsofthatage.ThenumberofhealthyushshootspriortoinfectiononDay15foraget2=2,...,15is20)]TJ /F14 7.97 Tf 11.96 15.05 Td[(t2)]TJ /F9 7.97 Tf 6.59 0 Td[(1Xj=1h(j+t1)]TJ /F4 11.955 Tf 11.96 0 Td[(t2,j).Thereare20newemergingushshootsofaget2=1onDay15.ThetotalnumberofhealthyushonDay15isT=15(20))]TJ /F9 7.97 Tf 16.16 14.95 Td[(14Xi=1iXj=1h(i,j).Theprobabilityofinfectionofeachushshootofaget2is20)]TJ /F20 7.97 Tf 6.59 5.98 Td[(Pt2)]TJ /F21 5.978 Tf 5.76 0 Td[(1j=1h(j+15)]TJ /F14 7.97 Tf 6.59 0 Td[(t2,j) T.AninventoryofhealthyandinfectedushpresentonDay15istaken.Summarystatistics.ThesummarystatisticscorrespondingtothecompartmentsinFigure 3-1 atushpatchcarerecorded.Thesearethenumberofpsyllidsthatwere 95

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presentandtakingpartinactivitiesonDay15atushpatchcandincludethenewphysicaldistributionofpsyllids.Theseare: Ia(15)=#ofinfectedandsurvivingadultpsyllidsattheendofDay15. Sa(15)=#ofuninfectedandsurvivingadultpsyllidsattheendofDay15 Sn(15)=#ofuninfectedandsurvivingnymphsattheendofDay15 Se(15)=#ofuninfectedandsurvivingeggsattheendofDay15 Iy(15)=#ofinfectedyoungushattheendofDay15 Sy(15)=#ofuninfectedyoungushattheendofDay15TheremainingtwocompartmentsinFigure 3-1 pertaintooldush,whichdonotexistonDay15.Flushdonotbecomeolduntilday17whennoeggscanbelaidonthem.Day16ActivitiesMigration.Eachpsyllidwitht217hasprobability.4ofmigratingtoanewushpatch,callitc0.Fortherecentlyemergedadults,eachpsyllidwitht2=15or16hasprobability.1ofmigratingtoanewushpatch.Thus,wescantheolder(resp.younger)adultsinsequenceassigningeachthevalue1withprobability.4(.1)andthevalue0withprobability.6(.9).Foralltheushpatchesinthegrovewithapopulationofadultpsyllids,themovementtonewushpatchesfollowsthemethodologydescribedinSection A.1 .Whenpsyllidsarriveatanewtree,weassumethattheytakepartintheremainingactivitiesonDay16andcombinethepopulationofarrivingpsyllidswiththosethatremainedminusthosethatmigratedawayfromushpatchc.Emergenceofnewushshoots.Anewsetof20ushshootsemergesateachlocation,c.Theseusharedesignatedas(c,(16,1),20).Witht1=16andt2=1,thismeanswearerepresentingtheseushshootsatlocationconthe16thdaysincethepsyllidinvasionandontherstdaythatthisushemerged.Theushshootsfrompreviousdaysaredesignatedby(c,(16,t2),20),wheret2rangesfrom2to16.Thereareatotalof320ushshootspresentateachpatchonDay16. 96

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Psyllidaging.EachnymphhasprobabilitysnofsurvivingfromDay15.Eachnymphisscanned,applyingprobabilitysntoeachofthemtodeterminetheirsurvivorstatus.Thenumberofsurvivingnymphs,sn()=sn(P(c,(15,t2),n)),isrecordedintoP(c,(16,t2+1),sn())foractivitiesonDay16.Inparticular,thenymphswitht2=14thatsurvive,emergeasadults,denotedbyP(c,(16,15),n)),thesearetheemergingadultsthatwillbeconsideredinthetransmissionofCLastopsyllids.Afterthemigrationofpsyllidstotheirnewushpatchesiscompleted,eachadultpsyllidhasprobabilitysaofsurvivingfromDay15.Thus,wescantheadultpsyllids,applyingprobabilitysatoeachofthemtodeterminetheirsurvivorstatus.Forvaluesoft215,thenumberofsurvivingadults,sa()=sa(P(c,(15,t2),n)),isrecordedintoP(c,(16,t2+1),sa())foractivitiesonDay16.Egglayingandsurvivalofpreviouslylaideggs.ThenumberofeggslaidonDay16,E16,ateachushpatchislimitedeitherbyasmallnumberoffemaleadultsorbythenumberofyoungushshoots.Femaleadultswhoemergeonthisdayandthepreviousdaydonotimpactoviposition.Toaccountforthefactthatemergingadultsdonotdepositeggsduringtheirrst24to48hours,allfemalepsyllidswitht217thatsurvivedDay15layanother10eggs,thuswehave10sa(Fa(15))eggs.ThecapacityforeggsonDay16is 4016Xi=1(c,(16,i),20)=12800Thenumberofeggslaidistheminimumofthesetwovalues.TheneweggsarenowenteredintoalistofeggsasP(c,(16,1),E16)andtheeggsfromthepreviousdaysareP(c,(16,2),se(E15))andP(c,(16,3),s2e(E14)).EachoftheeggshasprobabilityseofsurvivingDay16.ThesurvivalprocessismodeledasasetofindependenteventsacrosseggsandthenumberofsurvivingeggsareretainedinthepopulationmatrixP(c,(17,3),s2e(E15))andP(c,(17,2),se(E16))forDay17activities. 97

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Transmissiontoemergingadults.OnDay16,thepopulationofushshootswithemergingadultsisE=(c,(16,16),20)+(c,(16,15),20).WeconsideronlytheushshootsthathavebeeninfectedonorbeforeDay15.Thus,onDay16thereareEI=P15k=1h(k,k)+P14k=1h(k+1,k)infectedushshootswithemergingadults.Theprobability,p,thatanemergingadultbecomesinfectedisobtainedbymultiplyingtheproportionofinfectedushshootswithemergingadults,=EI Eandtheprobabilityoftransmissionfromushshoottoemergingadult,1,sop=1=1EI E.Eachemergingindividualisassigneda1(success)withprobabilitypanda0withprobability1)]TJ /F4 11.955 Tf 12.18 0 Td[(p.Itisimportanttonotethatthenewlyemergedadultsdonotcontributetoegglayingortotransmissiontoushshootsonthedaytheyemerge.Also,themovementprobabilitiesforemergingadultsislowerthanof3-dayorolderadultsMichaud2004,Kobori.Transmissiontoushshoots.h(t1,t2)isthenumberofnewinfectedushofaget2ondayt1.OnDay16,transmissionofCLastoushshootscanonlybefrominfectedpsyllidswitht216.Thus,eachage-appropriateinfectedpsyllidisscannedandhasprobabilitypinf=.3oftransmittingCLasduringaninfectivefeeding.Thenumberofinfectivefeedings,B,andthetotalnumberofushpresent,320,isobtainedandtheexpectednumberofinfectedushonDay16,h(16,),iscalculated.Foreachoftheh(16,)infectedush,wethendecidewhatage(t2=16,...,2,1)isassignedtoeachush.Theprobabilityisequaltothecurrentproportionofhealthyushshootsofthatage.ThenumberofhealthyushshootspriortoinfectiononDay16foraget2=2,...,16is20)]TJ /F14 7.97 Tf 11.96 15.06 Td[(t2)]TJ /F9 7.97 Tf 6.59 0 Td[(1Xj=1h(j+t1)]TJ /F4 11.955 Tf 11.96 0 Td[(t2,j). 98

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Thereare20newemergingushshootsofaget2=1onDay16.ThetotalnumberofhealthyushonDay16isT=16(20))]TJ /F9 7.97 Tf 16.16 14.95 Td[(15Xi=1iXj=1h(i,j).Theprobabilityofinfectionofeachushshootofaget2is20)]TJ /F20 7.97 Tf 6.59 5.98 Td[(Pt2)]TJ /F21 5.978 Tf 5.76 0 Td[(1j=1h(j+16)]TJ /F14 7.97 Tf 6.59 0 Td[(t2,j) T.AninventoryofhealthyandinfectedushpresentonDay16istaken.Summarystatistics.ThesummarystatisticscorrespondingtothecompartmentsinFigure 3-1 atushpatchcarerecorded.ThesearethenumberofpsyllidsthatwerepresentandtakingpartinactivitiesonDay16atushpatchcandincludethenewphysicaldistributionofpsyllids.Theseare: Ia(16)=#ofinfectedandsurvivingadultpsyllidsattheendofDay16. Sa(16)=#ofuninfectedandsurvivingadultpsyllidsattheendofDay16 Sn(16)=#ofuninfectedandsurvivingnymphsattheendofDay16 Se(16)=#ofuninfectedandsurvivingeggsattheendofDay16 Iy(16)=#ofinfectedyoungushattheendofDay16 Sy(16)=#ofuninfectedyoungushattheendofDay16TheremainingtwocompartmentsinFigure 3-1 pertaintooldush,whichdonotexistonDay16.Flushdonotbecomeolduntilday17whennoeggscanbelaidonthem.Day17ActivitiesMigration.Eachpsyllidwitht217hasprobability.4ofmigratingtoanewushpatch,callitc0.Fortherecentlyemergedadults,eachpsyllidwitht2=15or16hasprobability.1ofmigratingtoanewushpatch.Thus,wescantheolder(resp.younger)adultsinsequenceassigningeachthevalue1withprobability.4(.1)andthevalue0withprobability.6(.9).Foralltheushpatchesinthegrovewithapopulationofadultpsyllids,themovementtonewushpatchesfollowsthemethodologydescribedinSection A.1 .Whenpsyllidsarriveatanewtree,weassumethattheytakepartinthe 99

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remainingactivitiesonDay17andcombinethepopulationofarrivingpsyllidswiththosethatremainedminusthosethatmigratedawayfromushpatchc.Emergenceofnewushshoots.Anewsetof20ushshootsemergesateachlocation,c.Theseusharedesignatedas(c,(17,1),20).Witht1=17andt2=1,thismeanswearerepresentingtheseushshootsatlocationconthe17thdaysincethepsyllidinvasionandontherstdaythatthisushemerged.Theyoungushshootsfrompreviousdaysaredesignatedby(c,(17,t2),20),wheret2rangesfrom2to16.Day17istherstdaywherethereareushshootspresentintheoldushshootcompartment.Theseushconsistoft217andonDay17aredesignatedby(c,(17,17),20).Thereareatotalof340ushshootspresentateachpatchonDay17.Psyllidaging.EachnymphhasprobabilitysnofsurvivingfromDay16.Eachnymphisscanned,applyingprobabilitysntoeachofthemtodeterminetheirsurvivorstatus.Thenumberofsurvivingnymphs,sn()=sn(P(c,(16,t2),n)),isrecordedintoP(c,(17,t2+1),sn())foractivitiesonDay17.Inparticular,thenymphswitht2=14thatsurvive,emergeasadults,denotedbyP(c,(17,15),n)),thesearetheemergingadultsthatwillbeconsideredinthetransmissionofCLastopsyllids.Afterthemigrationofpsyllidstotheirnewushpatchesiscompleted,eachadultpsyllidhasprobabilitysaofsurvivingfromDay16.Thus,wescantheadultpsyllids,applyingprobabilitysatoeachofthemtodeterminetheirsurvivorstatus.Forvaluesoft215,thenumberofsurvivingadults,sa()=sa(P(c,(16,t2),n)),isrecordedintoP(c,(17,t2+1),sa())foractivitiesonDay17.Egglayingandsurvivalofpreviouslylaideggs.ThenumberofeggslaidonDay17,E17,ateachushpatchislimitedeitherbyasmallnumberoffemaleadultsorbythenumberofyoungushshoots.Femaleadultswhoemergeonthisdayandthepreviousdaydonotimpactoviposition.Toaccountforthefactthatemergingadultsdonotdepositeggsduringtheirrst24to48hours,allfemalepsyllidswitht217thatsurvivedDay16layanother10eggs,thuswehave10sa(Fa(16))eggs.Thecapacityforeggson 100

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Day17isdependentonthenumberofyoungushshootsavailablewhichisgivenby 4016Xi=1(c,(17,i),20)=12800Thenumberofeggslaidistheminimumofthesetwovalues.TheneweggsarenowenteredintoalistofeggsasP(c,(17,1),E17)andtheeggsfromthepreviousdaysareP(c,(17,2),se(E16))andP(c,(17,3),s2e(E15)).EachoftheeggshasprobabilityseofsurvivingDay17.ThesurvivalprocessismodeledasasetofindependenteventsacrosseggsandthenumberofsurvivingeggsareretainedinthepopulationmatrixP(c,(18,3),s2e(E16))andP(c,(18,2),se(E17))forDay18activities.Transmissiontoemergingadults.OnDay17,thepopulationofushshootswithemergingadultsisE=(c,(17,17),20)+(c,(17,16),20)+(c,(17,15),20).WeconsideronlytheushshootsthathavebeeninfectedonorbeforeDay16.OnDay17thenumberofinfectedushshootsofaget2ist2)]TJ /F9 7.97 Tf 6.58 0 Td[(1Xj=1h(j+17)]TJ /F4 11.955 Tf 11.96 0 Td[(t2,j).Inorderfortheushshoottohaveemergingadults,wehavetherestrictiont215.ThusonDay17thereare EI=17Xt2=15t2)]TJ /F9 7.97 Tf 6.59 0 Td[(1Xj=1h(j+17)]TJ /F4 11.955 Tf 11.96 0 Td[(t2,j)=16Xj=1h(j,j)+15Xj=1h(j+1,j)+14Xj=1h(j+2,j)infectedushshootswithemergingadults.Theprobability,p,thatanemergingadultbecomesinfectedisobtainedbymultiplyingtheproportionofinfectedushshootswithemergingadults,=EI Eandtheprobabilityoftransmissionfromushshoottoemergingadult,1,sop=1=1EI E. 101

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Eachemergingindividualisassigneda1(success)withprobabilitypanda0withprobability1)]TJ /F4 11.955 Tf 12.18 0 Td[(p.Itisimportanttonotethatthenewlyemergedadultsdonotcontributetoegglayingortotransmissiontoushshootsonthedaytheyemerge.Also,themovementprobabilitiesforemergingadultsislowerthanof3-dayorolderadultsMichaud2004,Kobori.Transmissiontoushshoots.h(t1,t2)isthenumberofnewinfectedushofaget2ondayt1.OnDay17,transmissionofCLastoushshootscanonlybefrominfectedpsyllidswitht216.Thus,eachage-appropriateinfectedpsyllidisscannedandhasprobabilitypinf=.3oftransmittingCLasduringaninfectivefeeding.Thenumberofinfectivefeedings,B,andthetotalnumberofushpresent,340,isobtainedandtheexpectednumberofinfectedushonDay17,h(17,),iscalculated.Foreachoftheh(17,)infectedush,wethendecidewhatage(t2=17,...,2,1)isassignedtoeachush.Theprobabilityisequaltothecurrentproportionofhealthyushshootsofthatage.ThenumberofhealthyushshootspriortoinfectiononDay17foraget2=2,...,17is20)]TJ /F14 7.97 Tf 11.95 15.06 Td[(t2)]TJ /F9 7.97 Tf 6.58 0 Td[(1Xj=1h(j+17)]TJ /F4 11.955 Tf 11.95 0 Td[(t2,j).Thereare20newemergingushshootsofaget2=1onDay17.ThetotalnumberofhealthyushonDay17isT=17(20))]TJ /F9 7.97 Tf 16.16 14.95 Td[(16Xi=1iXj=1h(i,j).Theprobabilityofinfectionofeachushshootofaget2is20)]TJ /F20 7.97 Tf 6.59 5.98 Td[(Pt2)]TJ /F21 5.978 Tf 5.76 0 Td[(1j=1h(j+17)]TJ /F14 7.97 Tf 6.59 0 Td[(t2,j) T.AninventoryofhealthyandinfectedushpresentonDay17istaken.Summarystatistics.ThesummarystatisticscorrespondingtothecompartmentsinFigure 3-1 atushpatchcarerecorded.ThesearethenumberofpsyllidsthatwerepresentandtakingpartinactivitiesonDay17atushpatchcandincludethenewphysicaldistributionofpsyllids.Theseare: 102

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Ia(17)=#ofinfectedandsurvivingadultpsyllidsattheendofDay17 Sa(17)=#ofuninfectedandsurvivingadultpsyllidsattheendofDay17 Sn(17)=#ofuninfectedandsurvivingnymphsattheendofDay17 Se(17)=#ofuninfectedandsurvivingeggsattheendofDay17 Iy(17)=#ofinfectedyoungushattheendofDay17 Sy(17)=#ofuninfectedyoungushattheendofDay17 Io(17)=#ofinfectedoldushattheendofDay17 So(17)=#ofuninfectedoldushattheendofDay17Day31ActivitiesTheactivitiesthattakeplaceonDay18to30followthesamedescriptionofactivitiesthatoccuronDay17.WecontinuewithadescriptionoftheactivitiesthatoccuronDay31,thenewactivitypresentonDay31istheremovalofoldush(t2>30)fromthesystem.Migration.Eachpsyllidwitht217hasprobability.4ofmigratingtoanewushpatch,callitc0.Fortherecentlyemergedadults,eachpsyllidwitht2=15or16hasprobability.1ofmigratingtoanewushpatch.Thus,wescantheolder(resp.younger)adultsinsequenceassigningeachthevalue1withprobability.4(.1)andthevalue0withprobability.6(.9).Foralltheushpatchesinthegrovewithapopulationofadultpsyllids,themovementtonewushpatchesfollowsthemethodologydescribedinSection A.1 .Whenpsyllidsarriveatanewtree,weassumethattheytakepartintheremainingactivitiesonDay31andcombinethepopulationofarrivingpsyllidswiththosethatremainedminusthosethatmigratedawayfromushpatchc.Emergenceofnewushshootsandremovalofoldush.Anewsetof20ushshootsemergeateachlocation,c.Theseusharedesignatedas(c,(31,1),20).Witht1=31andt2=1,thismeanswearerepresentingtheseushshootsatlocationconthe31stdaysincethepsyllidinvasionandontherstdaythatthisushemerged. 103

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Theyoungushshootsfrompreviousdaysaredesignatedby(c,(31,t2),20),wheret2rangesfrom2to16.Theoldushshootsfrompreviousdaysaredesignatedby(c,(31,t2),20),wheret2rangesfrom17to30.OnDay31,weremovefromthesystemanyushshootsthathavet231astheynolongerimpactanyofthedailyactivitiesatthisage.StartingonDay31,20ushshootsareremovedandcombinedwiththe20newushshootsthatemerge,wehaveaconstantnumberof600ushshootsateachpatchfromDay30toDay60.Psyllidaging.EachnymphhasprobabilitysnofsurvivingfromDay30.Eachnymphisscanned,applyingprobabilitysntoeachofthemtodeterminetheirsurvivorstatus.Thenumberofsurvivingnymphs,sn()=sn(P(c,(30,t2),n)),isrecordedintoP(c,(31,t2+1),sn())foractivitiesonDay31.Inparticular,thenymphswitht2=14thatsurvive,emergeasadults,denotedbyP(c,(31,15),n)),thesearetheemergingadultsthatwillbeconsideredinthetransmissionofCLastopsyllids.Afterthemigrationofpsyllidstotheirnewushpatchesiscompleted,eachadultpsyllidhasprobabilitysaofsurvivingfromDay30.Thus,wescantheadultpsyllids,applyingprobabilitysatoeachofthemtodeterminetheirsurvivorstatus.Forvaluesoft215,thenumberofsurvivingadults,sa()=sa(P(c,(30,t2),n)),isrecordedintoP(c,(31,t2+1),sa())foractivitiesonDay31.Egglayingandsurvivalofpreviouslylaideggs.ThenumberofeggslaidonDay31,E31,ateachushpatchislimitedeitherbyasmallnumberoffemaleadultsorbythenumberofyoungushshoots.Femaleadultswhoemergeonthisdayandthepreviousdaydonotimpactoviposition.Toaccountforthefactthatemergingadultsdonotdepositeggsduringtheirrst24to48hours,allfemalepsyllidswitht217thatsurvivedDay30layanother10eggs,thuswehave10sa(Fa(30))eggs.ThecapacityforeggsonDay31isdependentonthenumberofyoungushshootsavailablewhichisgivenby 4016Xi=1(c,(31,i),20)=12800 104

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Thenumberofeggslaidistheminimumofthesetwovalues.TheneweggsarenowenteredintoalistofeggsasP(c,(31,1),E31)andtheeggsfromthepreviousdaysareP(c,(31,2),se(E30))andP(c,(31,3),s2e(E29)).EachoftheeggshasprobabilityseofsurvivingDay31.ThesurvivalprocessismodeledasasetofindependenteventsacrosseggsandthenumberofsurvivingeggsareretainedinthepopulationmatrixP(c,(32,3),s2e(E30))andP(c,(32,2),se(E31))forDay32activities.Transmissiontoemergingadults.OnDay31,thepopulationofushshootswithemergingadultsisE=P30t2=15(c,(31,k),20).WeconsideronlytheushshootsthathavebeeninfectedonorbeforeDay30.OnDay31thenumberofinfectedushshootsofaget2ist2)]TJ /F9 7.97 Tf 6.58 0 Td[(1Xj=1h(j+31)]TJ /F4 11.955 Tf 11.96 0 Td[(t2,j).Inorderfortheushshoottohaveemergingadults,wehavetherestriction15t230.ThusonDay31thereare EI=30Xt2=15t2)]TJ /F9 7.97 Tf 6.58 0 Td[(1Xj=1h(j+31)]TJ /F4 11.955 Tf 11.96 0 Td[(t2,j)infectedushshootswithemergingadults.Theprobability,p,thatanemergingadultbecomesinfectedisobtainedbymultiplyingtheproportionofinfectedushshootswithemergingadults,=EI Eandtheprobabilityoftransmissionfromushshoottoemergingadult,1,sop=1=1EI E.Eachemergingindividualisassigneda1(success)withprobabilitypanda0withprobability1)]TJ /F4 11.955 Tf 12.18 0 Td[(p.Itisimportanttonotethatthenewlyemergedadultsdonotcontributetoegglayingortotransmissiontoushshootsonthedaytheyemerge.Also,themovementprobabilitiesforemergingadultsislowerthanof3-dayorolderadultsMichaud2004,Kobori. 105

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Transmissiontoushshoots.h(t1,t2)isthenumberofnewinfectedushofaget2ondayt1.OnDay31,transmissionofCLastoushshootscanonlybefrominfectedpsyllidswitht216.Thus,eachage-appropriateinfectedpsyllidisscannedandhasprobabilitypinf=.3oftransmittingCLasduringaninfectivefeeding.Thenumberofinfectivefeedings,B,andthetotalnumberofushpresent,600,isobtainedandtheexpectednumberofinfectedushonDay31,h(31,),iscalculated.Foreachoftheh(31,)infectedush,wethendecidewhatage(t2=30,...,2,1)isassignedtoeachush.Noteherethattheushofwitht2=30willberemovedfromthesystemonthenextday,howevertheyareassumedtostillbelocationsforfeedingonthecurrentdayandthereforeareapartoftheushpopulationthatcanbecomeinfected.Theushwitht2=30onDay31appearedonDay2,ingeneralforthedaysfollowing,ushofaget2onDayt1appearedonDayt1)]TJ /F4 11.955 Tf 12.04 0 Td[(t2+1.Theprobabilityisequaltothecurrentproportionofhealthyushshootsofthatage.ThenumberofhealthyushshootspriortoinfectiononDay31foraget2=2,...,30is20)]TJ /F14 7.97 Tf 11.95 15.06 Td[(t2)]TJ /F9 7.97 Tf 6.58 0 Td[(1Xj=1h(j+31)]TJ /F4 11.955 Tf 11.95 0 Td[(t2,j).Thereare20newemergingushshootsofaget2=1onDay31.ThetotalnumberofhealthyushonDay31isT=20+30Xt2=2 20)]TJ /F14 7.97 Tf 11.96 15.06 Td[(t2)]TJ /F9 7.97 Tf 6.59 0 Td[(1Xj=1h(j+31)]TJ /F4 11.955 Tf 11.95 0 Td[(t2,j)!.Theprobabilityofinfectionofeachushshootofage2t230is20)]TJ /F20 7.97 Tf 6.59 5.98 Td[(Pt2)]TJ /F21 5.978 Tf 5.76 0 Td[(1j=1h(j+31)]TJ /F14 7.97 Tf 6.59 0 Td[(t2,j) T.Theprobabilityofinfectionofnewlyemergedushshootsis20 T.AninventoryofhealthyandinfectedushpresentonDay31istaken.Summarystatistics.ThesummarystatisticscorrespondingtothecompartmentsinFigure 3-1 atushpatchcarerecorded.Thesearethenumberofpsyllidsthatwere 106

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presentandtakingpartinactivitiesonDay31atushpatchcandincludethenewphysicaldistributionofpsyllids.Theseare: Ia(31)=#ofinfectedandsurvivingadultpsyllidsattheendofDay31 Sa(31)=#ofuninfectedandsurvivingadultpsyllidsattheendofDay31 Sn(31)=#ofuninfectedandsurvivingnymphsattheendofDay31 Se(31)=#ofuninfectedandsurvivingeggsattheendofDay31 Iy(31)=#ofinfectedyoungushattheendofDay31 Sy(31)=#ofuninfectedyoungushattheendofDay31 Io(31)=#ofinfectedoldushattheendofDay31 So(31)=#ofuninfectedoldushattheendofDay31Oncetheushshootsbecomemature(t2>30),theyarenolongerincludedinthecountsIoandSoDay61ActivitiesTheactivitiesthattakeplaceonDay32to60followthesamedescriptionofactivitiesthatoccuronDay31.WecontinuewithadescriptionoftheactivitiesthatoccuronDay61,themaindifferenceinactivitiesonDay61isthatthisrepresentstheendoftheushingperiodandistherstdaywherenonewushshootsemerge.Migration.Eachpsyllidwitht217hasprobability.4ofmigratingtoanewushpatch,callitc0.Fortherecentlyemergedadults,eachpsyllidwitht2=15or16hasprobability.1ofmigratingtoanewushpatch.Thus,wescantheolder(resp.younger)adultsinsequenceassigningeachthevalue1withprobability.4(.1)andthevalue0withprobability.6(.9).Foralltheushpatchesinthegrovewithapopulationofadultpsyllids,themovementtonewushpatchesfollowsthemethodologydescribedinSection A.1 .Whenpsyllidsarriveatanewtree,weassumethattheytakepartintheremainingactivitiesonDay61andcombinethepopulationofarrivingpsyllidswiththosethatremainedminusthosethatmigratedawayfromushpatchc. 107

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Removalofoldushshoots.Thelastdaythatnewushshootsemergedis60,thereforeonDayt1,theageoftheyoungestcohortofushbecomesmt2=1+t1)]TJ /F6 11.955 Tf 12.42 0 Td[(60.ForDay61,theyoungestageisnowmt2=2andtheyoungushshootsfrompreviousdaysaredesignatedby(c,(61,t2),20),wheret2rangesfromtheageoftheyoungestcohort,2,to16.Theoldushshootsfrompreviousdaysaredesignatedby(c,(61,t2),20),wheret2rangesfrom17to30.OnDay61,weremovefromthesystemanyushshootsthathavet231astheynolongerimpactanyofthedailyactivitiesatthisage.StartingonDay61,20ushshootsareremovedandnonewushshootsemerge,sothetotalnumberofushshootspresentdecreasesby20ushshootseachday.Ithasbeen1daysincethelastushshootsemerged,thusthetotalnumberofushpresentis600)]TJ /F6 11.955 Tf 12.25 0 Td[(20(1).Sothereare580ushshootsateachpatchonDay61.Psyllidaging.EachnymphhasprobabilitysnofsurvivingfromDay60.Eachnymphisscanned,applyingprobabilitysntoeachofthemtodeterminetheirsurvivorstatus.Thenumberofsurvivingnymphs,sn()=sn(P(c,(60,t2),n)),isrecordedintoP(c,(61,t2+1),sn())foractivitiesonDay61.Inparticular,thenymphswitht2=14thatsurvive,emergeasadults,denotedbyP(c,(61,15),n)),thesearetheemergingadultsthatwillbeconsideredinthetransmissionofCLastopsyllids.Afterthemigrationofpsyllidstotheirnewushpatchesiscompleted,eachadultpsyllidhasprobabilitysaofsurvivingfromDay60.Thus,wescantheadultpsyllids,applyingprobabilitysatoeachofthemtodeterminetheirsurvivorstatus.Forvaluesoft215,thenumberofsurvivingadults,sa()=sa(P(c,(60,t2),n)),isrecordedintoP(c,(61,t2+1),sa())foractivitiesonDay61.Egglayingandsurvivalofpreviouslylaideggs.ThenumberofeggslaidonDay61,E61,ateachushpatchislimitedeitherbyasmallnumberoffemaleadultsorbythenumberofyoungushshoots.Femaleadultswhoemergeonthisdayandthepreviousdaydonotimpactoviposition.Toaccountforthefactthatemergingadultsdonotdeposit 108

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eggsduringtheirrst24to48hours,allfemalepsyllidswitht217thatsurvivedDay60layanother10eggs,thuswehave10sa(Fa(60))eggs.Again,thelastdaythatnewushshootsemergedisonDay60,thereforeonDayt1,theageoftheyoungestcohortofushismt2=1+t1)]TJ /F6 11.955 Tf 11.74 0 Td[(60.ThusthecapacityforeggsonDay61whichdependsonthenumberofyoungushshootsavailableis 4016Xt2=1+t1)]TJ /F9 7.97 Tf 6.59 0 Td[(60(c,(61,t2),20)=4016Xt2=2(c,(61,t2),20)=12000Thenumberofeggslaidistheminimumofthesetwovalues.TheneweggsarenowenteredintoalistofeggsasP(c,(61,1),E61)andtheeggsfromthepreviousdaysareP(c,(61,2),se(E60))andP(c,(61,3),s2e(E59)).EachoftheeggshasprobabilityseofsurvivingDay61.ThesurvivalprocessismodeledasasetofindependenteventsacrosseggsandthenumberofsurvivingeggsareretainedinthepopulationmatrixP(c,(62,3),s2e(E60))andP(c,(62,2),se(E61))forDay62activities.Transmissiontoemergingadults.StartingfromDay61,theminimumoftheushagesisgivenbymt2=1+t1)]TJ /F6 11.955 Tf 12.11 0 Td[(60.Toaccountforthefactthattheushshootareagingandthattherewillbeadaywheretheremainingushshootsareofagegreaterthan15,letMt2=maxf15,mt2g.ThepopulationofushshootswithemergingadultsisE=30Xt2=Mt2(c,(61,t2),20).WeconsideronlytheushshootsthathavebeeninfectedonorbeforeDay60.OnDay61thenumberofinfectedushshootsofaget2ist2)]TJ /F9 7.97 Tf 6.58 0 Td[(1Xj=1h(j+61)]TJ /F4 11.955 Tf 11.96 0 Td[(t2,j).Inorderfortheushshoottohaveemergingadults,wehavetherestrictionMt2t230.ThusonDay61thereare EI=30Xt2=Mt2t2)]TJ /F9 7.97 Tf 6.59 0 Td[(1Xj=1h(j+61)]TJ /F4 11.955 Tf 11.96 0 Td[(t2,j) 109

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infectedushshootswithemergingadults.Theprobability,p,thatanemergingadultbecomesinfectedisobtainedbymultiplyingtheproportionofinfectedushshootswithemergingadults,=EI Eandtheprobabilityoftransmissionfromushshoottoemergingadult,1,sop=1=1EI E.Eachemergingindividualisassigneda1(success)withprobabilitypanda0withprobability1)]TJ /F4 11.955 Tf 12.18 0 Td[(p.Itisimportanttonotethatthenewlyemergedadultsdonotcontributetoegglayingortotransmissiontoushshootsonthedaytheyemerge.Also,themovementprobabilitiesforemergingadultsislowerthanof3-dayorolderadultsMichaud2004,Kobori.Transmissiontoushshoots.StartingfromDay61,theminimumoftheushagesisgivenbymt2=1+t1)]TJ /F6 11.955 Tf 12.22 0 Td[(60.h(t1,t2)isthenumberofnewinfectedushofaget2ondayt1.OnDay61,transmissionofCLastoushshootscanonlybefrominfectedpsyllidswitht216.Thus,eachage-appropriateinfectedpsyllidisscannedandhasprobabilitypinf=.3oftransmittingCLasduringaninfectivefeeding.Thenumberofinfectivefeedings,B,andthetotalnumberofushpresent,580,isobtainedandtheexpectednumberofinfectedushonDay61,h(61,),iscalculated.Foreachoftheh(61,)infectedush,wethendecidewhatage(t2=30,...,mt2)isassignedtoeachush.Onthisday,theyoungestcohortofushisofagemt2=2.Alsonoteherethattheushwitht2=30willberemovedfromthesystemonthenextday,howevertheyareassumedtostillbelocationsforfeedingonthecurrentdayandthereforeareapartoftheushpopulationthatcanbecomeinfected.Theprobabilityisequaltothecurrentproportionofhealthyushshootsofthatage.ThenumberofhealthyushshootspriortoinfectiononDay61foraget2=mt2,...,30is20)]TJ /F14 7.97 Tf 11.95 15.06 Td[(t2)]TJ /F9 7.97 Tf 6.58 0 Td[(1Xj=1h(j+61)]TJ /F4 11.955 Tf 11.95 0 Td[(t2,j). 110

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TherearenoemergingushshootsonDay61.ThetotalnumberofhealthyushonDay61isT=30Xt2=mt2 20)]TJ /F14 7.97 Tf 11.96 15.06 Td[(t2)]TJ /F9 7.97 Tf 6.59 0 Td[(1Xj=1h(j+61)]TJ /F4 11.955 Tf 11.95 0 Td[(t2,j)!.Theprobabilityofinfectionofeachushshootofagemt2t230is20)]TJ /F20 7.97 Tf 6.59 5.98 Td[(Pt2)]TJ /F21 5.978 Tf 5.76 0 Td[(1j=1h(j+61)]TJ /F14 7.97 Tf 6.59 0 Td[(t2,j) T.AninventoryofhealthyandinfectedushpresentonDay61istaken.Summarystatistics.ThesummarystatisticscorrespondingtothecompartmentsinFigure 3-1 atushpatchcarerecorded.ThesearethenumberofpsyllidsthatwerepresentandtakingpartinactivitiesonDay61atushpatchcandincludethenewphysicaldistributionofpsyllids.Theseare: Ia(61)=#ofinfectedandsurvivingadultpsyllidsattheendofDay61 Sa(61)=#ofuninfectedandsurvivingadultpsyllidsattheendofDay61 Sn(61)=#ofuninfectedandsurvivingnymphsattheendofDay61 Se(61)=#ofuninfectedandsurvivingeggsattheendofDay61 Iy(61)=#ofinfectedyoungushattheendofDay61 Sy(61)=#ofuninfectedyoungushattheendofDay61 Io(61)=#ofinfectedoldushattheendofDay61 So(61)=#ofuninfectedoldushattheendofDay61Oncetheushshootsbecomemature(t2>30),theyarenolongerincludedinthecountsIoandSo.Nonewushemergesforthisrstperiodofspringush.Day74ActivitiesTheactivitiesthattakeplaceonDay62to74followthesamedescriptionofactivitiesthatoccuronDay61.WecontinuewithadescriptionoftheactivitiesthatoccuronDay74,thenewadditioninactivitiesonDay74istheremovalofadultpsyllidswitht2>89.Thisistoimposeamaximumlifespanfortheadultpsyllidsof75days. 111

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Migration.Eachpsyllidwitht217hasprobability.4ofmigratingtoanewushpatch,callitc0.Fortherecentlyemergedadults,eachpsyllidwitht2=15or16hasprobability.1ofmigratingtoanewushpatch.Thus,wescantheolder(resp.younger)adultsinsequenceassigningeachthevalue1withprobability.4(.1)andthevalue0withprobability.6(.9).Foralltheushpatchesinthegrovewithapopulationofadultpsyllids,themovementtonewushpatchesfollowsthemethodologydescribedinSection A.1 .Whenpsyllidsarriveatanewtree,weassumethattheytakepartintheremainingactivitiesonDay74andcombinethepopulationofarrivingpsyllidswiththosethatremainedminusthosethatmigratedawayfromushpatchc.Removalofoldushshoots.ThelastdaythatnewushshootsemergedisDay60,thereforeonDay74,theageoftheyoungestcohortofushbecomesmt2=1+74)]TJ /F6 11.955 Tf 12.55 0 Td[(60=15.Theyoungushshootsfrompreviousdaysaredesignatedby(c,(74,t2),20),wheret2rangesfrom15to16.Theoldushshootsfrompreviousdaysaredesignatedby(c,(74,t2),20),wheret2rangesfrom17to30.OnDay74,weremovefromthesystemanyushshootsthathavet231astheynolongerimpactanyofthedailyactivitiesatthisage.Thetotalnumberofushshootspresentdecreasesby20ushshootseachday.Ithasbeen14dayssincethelastushshootsemerged,thusthetotalnumberofushpresentis600)]TJ /F6 11.955 Tf 12.51 0 Td[(20(14).Sothereare320ushshootsateachpatchonDay74.Psyllidagingandremovalofinitialpsyllids.EachnymphhasprobabilitysnofsurvivingfromDay73.Eachnymphisscanned,applyingprobabilitysntoeachofthemtodeterminetheirsurvivorstatus.Thenumberofsurvivingnymphs,sn()=sn(P(c,(73,t2),n)),isrecordedintoP(c,(74,t2+1),sn())foractivitiesonDay74.Inparticular,thenymphswitht2=14thatsurvive,emergeasadults,denotedbyP(c,(74,15),n)),thesearetheemergingadultsthatwillbeconsideredinthetransmissionofCLastopsyllids.Afterthemigrationofpsyllidstotheirnewushpatchesiscompleted,eachadultpsyllidhasprobabilitysaofsurvivingfromDay73.Thus,we 112

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scantheadultpsyllids,applyingprobabilitysatoeachofthemtodeterminetheirsurvivorstatus.Forvaluesoft215,thenumberofsurvivingadults,sa()=sa(P(c,(73,t2),n)),isrecordedintoP(c,(74,t2+1),sa())foractivitiesonDay74.TheinitialpsyllidsenteredthesystemonDay1withage17,thusonDay74thesepsyllidsareofage90andthereforeareremovedfromthecompartmentofadults.WedonotremoveanymoreadultpsyllidsfromthesystemuntilDay90whentheeggslaidonDay1becomeadultsofage75.Egglayingandsurvivalofpreviouslylaideggs.ThenumberofeggslaidonDay61,E61,ateachushpatchislimitedeitherbyasmallnumberoffemaleadultsorbythenumberofyoungushshoots.Femaleadultswhoemergeonthisdayandthepreviousdaydonotimpactoviposition.Toaccountforthefactthatemergingadultsdonotdepositeggsduringtheirrst24to48hours,allfemalepsyllidswitht217thatsurvivedDay60layanother10eggs,thuswehave10sa(Fa(60))eggs.Again,thelastdaythatnewushshootsemergedisonDay60,thereforeonDayt1,theageoftheyoungestcohortofushismt2=1+t1)]TJ /F6 11.955 Tf 11.74 0 Td[(60.ThusthecapacityforeggsonDay61whichdependsonthenumberofyoungushshootsavailableis 4016Xt2=1+t1)]TJ /F9 7.97 Tf 6.59 0 Td[(60(c,(61,t2),20)=4016Xt2=2(c,(61,t2),20)=12000Thenumberofeggslaidistheminimumofthesetwovalues.TheneweggsarenowenteredintoalistofeggsasP(c,(61,1),E61)andtheeggsfromthepreviousdaysareP(c,(61,2),se(E60))andP(c,(61,3),s2e(E59)).EachoftheeggshasprobabilityseofsurvivingDay61.ThesurvivalprocessismodeledasasetofindependenteventsacrosseggsandthenumberofsurvivingeggsareretainedinthepopulationmatrixP(c,(62,3),s2e(E60))andP(c,(62,2),se(E61))forDay62activities. 113

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Transmissiontoemergingadults.StartingfromDay61,theminimumoftheushagesisgivenbymt2=1+t1)]TJ /F6 11.955 Tf 12.12 0 Td[(60.Toaccountforthefactthattheushshootareagingandthattherewillbeadaywheretheremainingushshootsareofagegreaterthan15,letMt2=maxf15,mt2g.ThepopulationofushshootswithemergingadultsisE=30Xt2=Mt2(c,(61,t2),20).WeconsideronlytheushshootsthathavebeeninfectedonorbeforeDay60.OnDay61thenumberofinfectedushshootsofaget2ist2)]TJ /F9 7.97 Tf 6.58 0 Td[(1Xj=1h(j+61)]TJ /F4 11.955 Tf 11.96 0 Td[(t2,j).Inorderfortheushshoottohaveemergingadults,wehavetherestrictionMt2t230.ThusonDay61thereare EI=30Xt2=Mt2t2)]TJ /F9 7.97 Tf 6.59 0 Td[(1Xj=1h(j+61)]TJ /F4 11.955 Tf 11.96 0 Td[(t2,j)infectedushshootswithemergingadults.Theprobability,p,thatanemergingadultbecomesinfectedisobtainedbymultiplyingtheproportionofinfectedushshootswithemergingadults,=EI Eandtheprobabilityoftransmissionfromushshoottoemergingadult,1,sop=1=1EI E.Eachemergingindividualisassigneda1(success)withprobabilitypanda0withprobability1)]TJ /F4 11.955 Tf 12.18 0 Td[(p.Itisimportanttonotethatthenewlyemergedadultsdonotcontributetoegglayingortotransmissiontoushshootsonthedaytheyemerge.Also,themovementprobabilitiesforemergingadultsislowerthanof3-dayorolderadultsMichaud2004,Kobori.Transmissiontoushshoots.StartingfromDay61,theminimumoftheushagesisgivenbymt2=1+t1)]TJ /F6 11.955 Tf 12.22 0 Td[(60.h(t1,t2)isthenumberofnewinfectedushofaget2ondayt1.OnDay61,transmissionofCLastoushshootscanonlybefrominfectedpsyllidswitht216.Thus,eachage-appropriateinfectedpsyllidisscannedandhas 114

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probabilitypinf=.3oftransmittingCLasduringaninfectivefeeding.Thenumberofinfectivefeedings,B,andthetotalnumberofushpresent,580,isobtainedandtheexpectednumberofinfectedushonDay61,h(61,),iscalculated.Foreachoftheh(61,)infectedush,wethendecidewhatage(t2=30,...,mt2)isassignedtoeachush.Onthisday,theyoungestcohortofushisofagemt2=2.Alsonoteherethattheushwitht2=30willberemovedfromthesystemonthenextday,howevertheyareassumedtostillbelocationsforfeedingonthecurrentdayandthereforeareapartoftheushpopulationthatcanbecomeinfected.Theprobabilityisequaltothecurrentproportionofhealthyushshootsofthatage.ThenumberofhealthyushshootspriortoinfectiononDay61foraget2=mt2,...,30is20)]TJ /F14 7.97 Tf 11.95 15.06 Td[(t2)]TJ /F9 7.97 Tf 6.58 0 Td[(1Xj=1h(j+61)]TJ /F4 11.955 Tf 11.95 0 Td[(t2,j).TherearenoemergingushshootsonDay61.ThetotalnumberofhealthyushonDay61isT=30Xt2=mt2 20)]TJ /F14 7.97 Tf 11.96 15.06 Td[(t2)]TJ /F9 7.97 Tf 6.59 0 Td[(1Xj=1h(j+61)]TJ /F4 11.955 Tf 11.95 0 Td[(t2,j)!.Theprobabilityofinfectionofeachushshootofagemt2t230is20)]TJ /F20 7.97 Tf 6.59 5.98 Td[(Pt2)]TJ /F21 5.978 Tf 5.76 0 Td[(1j=1h(j+61)]TJ /F14 7.97 Tf 6.59 0 Td[(t2,j) T.AninventoryofhealthyandinfectedushpresentonDay61istaken.Summarystatistics.ThesummarystatisticscorrespondingtothecompartmentsinFigure 3-1 atushpatchcarerecorded.ThesearethenumberofpsyllidsthatwerepresentandtakingpartinactivitiesonDay61atushpatchcandincludethenewphysicaldistributionofpsyllids.Theseare: Ia(61)=#ofinfectedandsurvivingadultpsyllidsattheendofDay61 Sa(61)=#ofuninfectedandsurvivingadultpsyllidsattheendofDay61 Sn(61)=#ofuninfectedandsurvivingnymphsattheendofDay61 Se(61)=#ofuninfectedandsurvivingeggsattheendofDay61 Iy(61)=#ofinfectedyoungushattheendofDay61 115

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Sy(61)=#ofuninfectedyoungushattheendofDay61 Io(61)=#ofinfectedoldushattheendofDay61 So(61)=#ofuninfectedoldushattheendofDay61Oncetheushshootsbecomemature(t2>30),theyarenolongerincludedinthecountsIoandSo.Nonewushemergesforthisrstperiodofspringush.Day76ActivitiesTheactivitiesthattakeplaceonDay62to75followthesamedescriptionofactivitiesthatoccuronDay61.WecontinuewithadescriptionoftheactivitiesthatoccuronDay76,themaindifferenceinactivitiesonDay76isthattheminimumageofallushshootsismt2=1+76)]TJ /F6 11.955 Tf 12.06 0 Td[(60=17whichmeansthatthisistherstdaywheretherearenoyoungushshootsandthereforethisistherstdaywherenoeggswillbelaid.Migration.Eachpsyllidwitht217hasprobability.4ofmigratingtoanewushpatch,callitc0.Fortherecentlyemergedadults,eachpsyllidwitht2=15or16hasprobability.1ofmigratingtoanewushpatch.Thus,wescantheolder(resp.younger)adultsinsequenceassigningeachthevalue1withprobability.4(.1)andthevalue0withprobability.6(.9).Foralltheushpatchesinthegrovewithapopulationofadultpsyllids,themovementtonewushpatchesfollowsthemethodologydescribedinSection A.1 .Whenpsyllidsarriveatanewtree,weassumethattheytakepartintheremainingactivitiesonDay76andcombinethepopulationofarrivingpsyllidswiththosethatremainedminusthosethatmigratedawayfromushpatchc.Removalofoldushshoots.Thelastdaythatnewushshootsemergedis60,thereforeonDay76,theageoftheyoungestcohortofushbecomesmt2=1+t1)]TJ /F6 11.955 Tf 12.05 0 Td[(60.ForDay76,theyoungestageisnowmt2=17andthereisnopopulationofyoungushshoots.Theoldushshootsfrompreviousdaysaredesignatedby(c,(76,t2),20),wheret2rangesfrom17to30.OnDay76,weremovefromthesystemanyushshootsthathavet231astheynolongerimpactanyofthedailyactivitiesatthisage.Ithas 116

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been16daysincethelastushshootsemerged,sothereare280ushshootsateachpatchonDay76.Psyllidagingandremovalofpsyllids.EachnymphhasprobabilitysnofsurvivingfromDay75.Eachnymphisscanned,applyingprobabilitysntoeachofthemtodeterminetheirsurvivorstatus.Thenumberofsurvivingnymphs,sn()=sn(P(c,(75,t2),n)),isrecordedintoP(c,(76,t2+1),sn())foractivitiesonDay76.Inparticular,thenymphswitht2=14thatsurvive,emergeasadults,denotedbyP(c,(76,15),n)),thesearetheemergingadultsthatwillbeconsideredinthetransmissionofCLastopsyllids.Afterthemigrationofpsyllidstotheirnewushpatchesiscompleted,eachadultpsyllidhasprobabilitysaofsurvivingfromDay75.Thus,wescantheadultpsyllids,applyingprobabilitysatoeachofthemtodeterminetheirsurvivorstatus.Forvaluesoft215,thenumberofsurvivingadults,sa()=sa(P(c,(75,t2),n)),isrecordedintoP(c,(76,t2+1),sa())foractivitiesonDay76.SurvivalofpreviouslylaideggsTherearenoeggslaidonDay76.TheeggsfrompreviousdaysareP(c,(76,2),se(E75))andP(c,(76,3),s2e(E74)).EachoftheeggshasprobabilityseofsurvivingDay76.ThesurvivalprocessismodeledasasetofindependenteventsacrosseggsandthenumberofsurvivingeggsareretainedinthepopulationmatrixP(c,(76,3),s2e(E75))forDay77activities.ThisisthelastcohortofeggsthatwillsurvivetonymphsonDay78withprobabilityse.Transmissiontoemergingadults.OnDay76,theminimumoftheushagesisgivenbymt2=1+t1)]TJ /F6 11.955 Tf 12.29 0 Td[(60.Fromthisdayforward,theushshootsareofaget217.ThepopulationofushshootswithemergingadultsisE=30Xt2=mt2(c,(76,t2),20).WeconsideronlytheushshootsthathavebeeninfectedonorbeforeDay75.OnDay76thenumberofinfectedushshootsofaget2ist2)]TJ /F9 7.97 Tf 6.58 0 Td[(1Xj=1h(j+76)]TJ /F4 11.955 Tf 11.96 0 Td[(t2,j). 117

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Inorderfortheushshoottohaveemergingadults,wehavetherestrictionmt2t230.ThusonDay76thereare EI=30Xt2=mt2t2)]TJ /F9 7.97 Tf 6.59 0 Td[(1Xj=1h(j+76)]TJ /F4 11.955 Tf 11.95 0 Td[(t2,j)infectedushshootswithemergingadults.Theprobability,p,thatanemergingadultbecomesinfectedisobtainedbymultiplyingtheproportionofinfectedushshootswithemergingadults,=EI Eandtheprobabilityoftransmissionfromushshoottoemergingadult,1,sop=1=1EI E.Eachemergingindividualisassigneda1(success)withprobabilitypanda0withprobability1)]TJ /F4 11.955 Tf 12.18 0 Td[(p.Itisimportanttonotethatthenewlyemergedadultsdonotcontributetoegglayingortotransmissiontoushshootsonthedaytheyemerge.Also,themovementprobabilitiesforemergingadultsislowerthanof3-dayorolderadultsMichaud2004,Kobori.Transmissiontoushshoots.OnDay76,theminimumoftheushagesisgivenbymt2=17.h(t1,t2)isthenumberofnewinfectedushofaget2ondayt1.OnDay76,transmissionofCLastoushshootscanonlybefrominfectedpsyllidswitht216.Thus,eachage-appropriateinfectedpsyllidisscannedandhasprobabilitypinf=.3oftransmittingCLasduringaninfectivefeeding.Thenumberofinfectivefeedings,B,andthetotalnumberofushpresent,280,isobtainedandtheexpectednumberofinfectedushonDay76,h(76,),iscalculated.Foreachoftheh(76,)infectedush,wethendecidewhatage(t2=30,...,mt2)isassignedtoeachush.Onthisday,theyoungestcohortofushisofagemt2=17.Alsonoteherethattheushwitht2=30willberemovedfromthesystemonthenextday,howevertheyareassumedtostillbelocationsforfeedingonthecurrentdayandthereforeareapartoftheushpopulationthatcanbecomeinfected.Theprobabilityisequaltothecurrentproportionofhealthyushshootsofthatage.ThenumberofhealthyushshootspriortoinfectiononDay76 118

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foraget2=mt2,...,30is20)]TJ /F14 7.97 Tf 11.95 15.06 Td[(t2)]TJ /F9 7.97 Tf 6.58 0 Td[(1Xj=1h(j+76)]TJ /F4 11.955 Tf 11.95 0 Td[(t2,j).TherearenoemergingushshootsonDay76.ThetotalnumberofhealthyushonDay76isT=30Xt2=mt2 20)]TJ /F14 7.97 Tf 11.96 15.06 Td[(t2)]TJ /F9 7.97 Tf 6.59 0 Td[(1Xj=1h(j+76)]TJ /F4 11.955 Tf 11.95 0 Td[(t2,j)!.Theprobabilityofinfectionofeachushshootofagemt2t230is20)]TJ /F20 7.97 Tf 6.59 5.97 Td[(Pt2)]TJ /F21 5.978 Tf 5.76 0 Td[(1j=1h(j+76)]TJ /F14 7.97 Tf 6.59 0 Td[(t2,j) T.AninventoryofhealthyandinfectedushpresentonDay76istaken.Summarystatistics.ThesummarystatisticscorrespondingtothecompartmentsinFigure 3-1 atushpatchcarerecorded.ThesearethenumberofpsyllidsthatwerepresentandtakingpartinactivitiesonDay76atushpatchcandincludethenewphysicaldistributionofpsyllids.Theseare: Ia(76)=#ofinfectedandsurvivingadultpsyllidsattheendofDay76 Sa(76)=#ofuninfectedandsurvivingadultpsyllidsattheendofDay76 Sn(76)=#ofuninfectedandsurvivingnymphsattheendofDay76 Se(76)=#ofuninfectedandsurvivingeggsattheendofDay76 Iy(76)=0infectedyoungushattheendofDay76 Sy(76)=0uninfectedyoungushattheendofDay76 Io(76)=#ofinfectedoldushattheendofDay76 So(76)=#ofuninfectedoldushattheendofDay76Oncetheushshootsbecomemature(t2>30),theyarenolongerincludedinthecountsIoandSo.NonewushhasemergedsinceDay60,thusonthisdaytherearenoyoungushpresent.ThelastdayeggswerelaidwasonDay75,forthisreason,onDay78,thecompartmentSe(78)=0.Day89Activities 119

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TheactivitiesthattakeplaceonDay77to88followthesamedescriptionofactivitiesthatoccuronDay76.Theagingofpsyllidstakesplaceinthesamemannerduringthesedayswithsomeofthepopulationvaluesequalto0duetothenoegglayingsinceDay75.WecontinuewithadescriptionoftheactivitiesthatoccuronDay89.Onthisday,theminimumageofallushshootsismt2=30whichmeansthatthisisthelastdaywherethereareanyushshootspresent.Migration.Eachpsyllidwitht217hasprobability.4ofmigratingtoanewushpatch,callitc0.Fortherecentlyemergedadults,eachpsyllidwitht2=15or16hasprobability.1ofmigratingtoanewushpatch.Thus,wescantheolder(resp.younger)adultsinsequenceassigningeachthevalue1withprobability.4(.1)andthevalue0withprobability.6(.9).Foralltheushpatchesinthegrovewithapopulationofadultpsyllids,themovementtonewushpatchesfollowsthemethodologydescribedinSection A.1 .Whenpsyllidsarriveatanewtree,weassumethattheytakepartintheremainingactivitiesonDay89andcombinethepopulationofarrivingpsyllidswiththosethatremainedminusthosethatmigratedawayfromushpatchc.Removalofoldushshoots.Thelastdaythatnewushshootsemergedis60,thereforeonDay89,theageoftheyoungestcohortofushis30.Thereisnopopulationofyoungushshootsandtheoldushshootsfrompreviousdaysaredesignatedby(c,(89,30),20).OnDay89,weremovefromthesystemanyushshootsthathavet231astheynolongerimpactanyofthedailyactivitiesatthisage.Thereare20ushshootsremainingateachpatchonDay89.PsyllidagingandremovalofpsyllidsEachnymphhasprobabilitysnofsurvivingfromDay88.Eachnymphisscanned,applyingprobabilitysntoeachofthemtodeterminetheirsurvivorstatus.Thenumberofsurvivingnymphs,sn()=sn(P(c,(88,t2),n)),isrecordedintoP(c,(89,t2+1),sn())foractivitiesonDay89.Inparticular,onDay89,allthenymphshavet2=14andtheonesthatsurviveemergeasadults,denotedbyP(c,(89,15),n)).Thesearetheemergingadultsthatwillbe 120

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consideredinthetransmissionofCLastopsyllids.Afterthemigrationofpsyllidstotheirnewushpatchesiscompleted,eachadultpsyllidhasprobabilitysaofsurvivingfromDay88.Thus,wescantheadultpsyllids,applyingprobabilitysatoeachofthemtodeterminetheirsurvivorstatus.Forvaluesoft215,thenumberofsurvivingadults,sa()=sa(P(c,(88,t2),n)),isrecordedintoP(c,(89,t2+1),sa())foractivitiesonDay89.Transmissiontoemergingadults.OnDay89,theminimumushageis30.ThepopulationofushshootswithemergingadultsisE=(c,(89,30),20).WeconsideronlytheushshootsthathavebeeninfectedonorbeforeDay88.OnDay89thereare EI=29Xj=1h(j+59,j)infectedushshootswithemergingadults.Theprobability,p,thatanemergingadultbecomesinfectedisobtainedbymultiplyingtheproportionofinfectedushshootswithemergingadults,=EI Eandtheprobabilityoftransmissionfromushshoottoemergingadult,1,sop=1=1EI E.Eachemergingindividualisassigneda1(success)withprobabilitypanda0withprobability1)]TJ /F4 11.955 Tf 12.18 0 Td[(p.Itisimportanttonotethatthenewlyemergedadultsdonotcontributetoegglayingortotransmissiontoushshootsonthedaytheyemerge.Also,themovementprobabilitiesforemergingadultsislowerthanof3-dayorolderadultsMichaud2004,Kobori.Transmissiontoushshoots.OnDay89,theminimumushageis30.h(t1,t2)isthenumberofnewinfectedushofaget2ondayt1.OnDay89,transmissionofCLastoushshootscanonlybefrominfectedpsyllidswitht216.Thus,eachage-appropriateinfectedpsyllidisscannedandhasprobabilitypinf=.3oftransmittingCLasduringaninfectivefeeding.Thenumberofinfectivefeedings,B,andthetotalnumberofushpresent,20,isobtainedandthenumberofinfectedushonDay89, 121

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h(89,30),iscalculated.Noteherethattheushwitht2=30willberemovedfromthesystemonthenextday,howevertheyareassumedtostillbelocationsforfeedingonthecurrentdayandthereforeareapartoftheushpopulationthatcanbecomeinfected.AninventoryofhealthyandinfectedushpresentonDay89istaken.Summarystatistics.ThesummarystatisticscorrespondingtothecompartmentsinFigure 3-1 atushpatchcarerecorded.ThesearethenumberofpsyllidsthatwerepresentandtakingpartinactivitiesonDay89atushpatchcandincludethenewphysicaldistributionofpsyllids.Theseare: Ia(89)=#ofinfectedandsurvivingadultpsyllidsattheendofDay89 Sa(89)=#ofuninfectedandsurvivingadultpsyllidsattheendofDay89 Sn(89)=0ofuninfectedandsurvivingnymphsattheendofDay89 Se(89)=0ofuninfectedandsurvivingeggsattheendofDay89 Iy(89)=0infectedyoungushattheendofDay89 Sy(89)=0uninfectedyoungushattheendofDay89 Io(89)=20ofinfectedoldushattheendofDay89 So(89)=20ofuninfectedoldushattheendofDay89Oncetheushshootsbecomemature(t2>30),theyarenolongerincludedinthecountsIoandSo.NonewushhasemergedsinceDay60,thusonthisdaytherearenoyoungushpresent.ThelastdayeggswerelaidwasonDay75,forthisreason,thecompartmentSe(89)=0.ThelastnymphsemergedintoadultsonthisdayandthusthecompartmentSn(89)=0.Day90to120ActivitiesNoushshootsarepresentatanyoftheushpatches,thisbeginsaperiodof31dayswherethereisnomovementortransmissionactivities.Theonlythingthatoccursisthesurvivalandagingofadultpsyllids,aswellastheremovalofpsyllidswitht2>89. 122

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Weassumethatduringthisrstperiodofnoush,theprobabilityforanadultsurvivingchangesfromsa=.9847tos0a=.9659duetoanincreaseintemperature. A.2SummerFlushPeriodThedifferencesbetweenthesummerandspringushperiodsconsistofthesurvivalprobabilityfortheadultdayss0a=.9659,themaximumageforadultsis51,andtheperiodoftimewhereushshootsemerge.Forthesummerperiod,ushshootsemergebeginningonDay121andthelastushshootsemergeonDay151.Day121Activities,Day1SummerThepsyllidadultsthatsurvivetothisdayparticipateinactivitiesonDay121,therstdayofthesummerushperiod.Withthenewsurvivalprobabilities,thedailyactivitiesthatoccuronDays121to150arethesameasthoseonDay1to30.Themaximumlifespanforadultpsyllidsremainsat75days,sotheremovalofadultswitht2>89alsooccurs.Nomigration.OnDay121,therstdayofthissummerushperiod,survivingpsyllidsremainonthesameushpatches.TheywillcontinuemovementbetweenushpatchesbeginningonDay122.Emergenceofnewushshoots.Anewsetof20ushshootsemergesateachlocation,c.Theseusharedesignatedas(c,(121,1),20).Witht1=121andt2=1,thismeanswearerepresentingtheseushshootsatlocationconthe121thdaysincethepsyllidinvasionandontherstdaythatthisushemerged.Thereareatotalof20ushshootspresentateachpatchonDay121.Psyllidagingandremovalofpsyllids.Eachadultpsyllidhasprobabilitys0aofsurvivingfromDay120.Thus,wescantheadultpsyllids,applyingprobabilitys0atoeachofthemtodeterminetheirsurvivorstatus.Forvaluesof15t264,thenumberofsurvivingadults,s0a()=s0a(P(c,(1,t2),n)),isrecordedintoP(c,(2,t2+1),s0a())foractivitiesonDay121.Theadultswitht265areremovedfromconsideration. 123

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Egglaying.ThenumberofeggslaidonDay121,E121,ateachushpatchislimitedeitherbyasmallnumberoffemaleadultsorbythenumberofyoungushshoots.Wehave10s0a(Fa(120))eggs.ThecapacityforeggsonDay121,whichisdependentonthenumberofyoungushshootsavailable,isgivenby 40(c,(121,1),20)=800Thenumberofeggslaidistheminimumofthesetwovalues.TheneweggsarenowenteredintoalistofeggsasP(c,(121,1),E121).EachoftheeggshasprobabilityseofsurvivingDay121.ThesurvivalprocessismodeledasasetofindependenteventsacrosseggsandthenumberofsurvivingeggsareretainedinthepopulationmatrixP(c,(122,2),se(E121))forDay122activities.Transmissiontoushshoots.OnDay121,transmissionofCLascanonlybefrominfectedpsyllidtoushshoot,astherearenoinfectedandinfectiousushpresentintheinitialbatchof20shoots.Thus,eachinfectedpsyllidisscannedandhasprobabilitypinf=.3oftransmittingCLasduringaninfectivefeeding.Thusthenumberofinfectivefeedings,B,andthetotalnumberofushpresent,20,isobtainedandthentheexpectednumberofinfectedushofaget2(t2daysafteremergence)onthet1thdayafterinitialinfection,h(t1,t2)iscalculated.Allushonthisdayareinthesamecohort,thereforeh(121,1)oftheushmovetotheinfectedushand20-h(121,1)remaininthehealthyushpopulation.AninventoryofushispreparedforDay122accordingtotheformalismS(c,(122,2),20)]TJ /F7 11.955 Tf 11.95 0 Td[(h(121,1)))andI(c,(122,2),h(121,1)))forhealthyandinfectedushrespectively.Summarystatistics.ThesummarystatisticscorrespondingtothecompartmentsinFigure 3-1 atushpatchcarerecorded.ThesearethenumberofpsyllidsthatwerepresentandtakingpartinactivitiesonDay121atushpatchc: Ia(121)=#ofinfectedadultpsyllidsattheendofDay121. Sa(121)=#ofuninfectedadultpsyllidsattheendofDay121 124

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Se(121)=#ofuninfectedeggsattheendofDay121 Iy(121)=#ofinfectedyoungushattheendofDay121 Sy(121)=#ofuninfectedyoungushattheendofDay121TheremainingthreecompartmentsinFigure 3-1 pertaintooldushandnymphs,neitherofwhichexistonDay121,therstdayofthesummerush.Eggsdonotbecomerstinstarnymphsuntil3daysaftertheyarelaid.Day122Activities,Day2SummerMigration.Eachpsyllidwitht217hasprobability.4ofmigratingtoanewushpatch,callitc0.Wescantheadultsinsequenceassigningeachthevalue1withprobability.4andthevalue0withprobability.6.Foralltheushpatchesinthegrovewithapopulationofadultpsyllids,themovementtonewushpatchesfollowsthemethodologydescribedinSection A.1 .Whenpsyllidsarriveatanewtree,weassumethattheytakepartintheremainingactivitiesonDay122andcombinethepopulationofarrivingpsyllidswiththosethatremainedminusthosethatmigratedawayfromushpatchc.Emergenceofnewushshoots.Inadditiontothe20ushavailableattheendofDay121,anewsetof20ushshootsemergesateachlocation,c.Theseusharedesignatedas(c,(122,1),20).Witht1=122andt2=1,thismeanswearerepresentingtheseushshootsatlocationconthe122nddaysincethepsyllidinvasionandontherstdaythatthisushemerged.The20ushfromDay121arenowdesignatedby(c,(122,2),20),meaningthatthisisthe122nddaysinceinvasionandthe2nddaysincetheseushemerged.Thereareatotalof40ushpresentateachpatchonDay122.Psyllidagingandremovalofpsyllids.Eachadultpsyllidhasprobabilitys0aofsurvivingfromDay121.Thus,wescantheadultpsyllids,applyingprobabilitys0atoeachofthemtodeterminetheirsurvivorstatus.Forvaluesof15t264,thenumberof 125

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survivingadults,s0a()=s0a(P(c,(1,t2),n)),isrecordedintoP(c,(2,t2+1),s0a())foractivitiesonDay121.Theadultswitht265areremovedfromconsideration.Egglayingandsurvivalofpreviouslylaideggs.ThenumberofeggslaidonDay122,E122,ateachushpatchislimitedeitherbyasmallnumberoffemaleadultsorbythenumberofyoungushshoots.Allfemalepsyllidswitht217thatsurvivedDay121layanother10eggs,thuswehave10s0a(Fa(121))eggs.ThecapacityforeggsonDay122isdependentonthenumberofyoungushshootsavailablewhichisgivenby 402Xi=1(c,(122,i),20)=1600Thenumberofeggslaidistheminimumofthesetwovalues.TheneweggsarenowenteredintoalistofeggsasP(c,(122,1),E122)andtheeggsfromthepreviousdayareP(c,(121,2),se(E121)).EachoftheeggshasprobabilityseofsurvivingDay122.ThesurvivalprocessismodeledasasetofindependenteventsacrosseggsandthenumberofsurvivingeggsareretainedinthepopulationmatrixP(c,(123,3),s2e(E121))andP(c,(122,2),se(E122))forDay123activities.Transmissiontoushshoots.h(t1,t2)isthenumberofnewinfectedushofaget2ondayt1.OnDay122,transmissionofCLastoushshootscanonlybefrominfectedpsyllidswitht216.Thus,eachage-appropriateinfectedpsyllidisscannedandhasprobabilitypinf=.3oftransmittingCLasduringaninfectivefeeding.Thenumberofinfectivefeedings,B,andthetotalnumberofushpresent,40,isobtainedandtheexpectednumberofinfectedushonDay122,h(122,),iscalculated.Foreachoftheh(122,)infectedush,wethendecidewhatage(t2=2,1)isassignedtoeachush.Theprobabilityisequaltothecurrentproportionofhealthyushshootsofthatage.ThenumberofhealthyushshootspriortoinfectiononDay122foraget2=2is20)]TJ /F14 7.97 Tf 11.96 15.06 Td[(t2)]TJ /F9 7.97 Tf 6.59 0 Td[(1Xj=1h(j+t1)]TJ /F6 11.955 Tf 11.96 0 Td[(120)]TJ /F4 11.955 Tf 11.96 0 Td[(t2,j). 126

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Thereare20newemergingushshootsofaget2=1onDay122,sothetotalnumberofhealthyushonDay122isT=20+20)]TJ /F7 11.955 Tf 11.96 0 Td[(h(1,1).Theprobabilityofinfectionofforagest2>1is20)]TJ /F20 7.97 Tf 6.59 5.98 Td[(Pt2)]TJ /F21 5.978 Tf 5.76 0 Td[(1j=1h(j+17)]TJ /F14 7.97 Tf 6.58 0 Td[(t2,j) T.Theprobabilityofinfectionofforaget2=1is20 T.AninventoryofhealthyandinfectedushpresentonDay122istaken.Summarystatistics.ThesummarystatisticscorrespondingtothecompartmentsinFigure 3-1 atushpatchcarerecorded.ThesearethenumberofpsyllidsthatwerepresentandtakingpartinactivitiesonDay122atushpatchcandincludethenewphysicaldistributionofpsyllids. Ia(122)=#ofinfectedadultpsyllidsattheendofDay122 Sa(122)=#ofuninfectedadultpsyllidsattheendofDay122 Se(122)=#ofuninfectedeggsattheendofDay122 Iy(122)=#ofinfectedyoungushattheendofDay122 Sy(122)=#ofuninfectedyoungushattheendofDay122TheremainingthreecompartmentsinFigure 3-1 pertaintooldushandnymphs,neitherofwhichexistonDay122.Eggsdonotbecomerstinstarnymphsuntil3daysaftertheyarelaid.Day151ActivitiesTheactivitiesonDay123,includingmigration,emergenceofnewushshoots,psyllidegglaying,aging,survival,andtransmissiontoushshoots,areidenticaltothoseonDay122.TheactivitiesonDay123toDay150followtheactivitiesfromDay3toDay30,withthestartofactivitiessuchasemergenceofnymphsandadultsoccurringonDay4and15respectively.Wecontinuetoremoveadultswitht2>65fromthesystem.Duetotheshorterushingperiod,newactivitiesonDay151areacombination 127

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ofthenewactivitiesthatbeginonDays31and61.OnDay151,weremoveoldushshootswitht231andnomorenewushshootsemerge.Migration.Eachpsyllidwitht217hasprobability.4ofmigratingtoanewushpatch,callitc0.Fortherecentlyemergedadults,eachpsyllidwitht2=15or16hasprobability.1ofmigratingtoanewushpatch.Thus,wescantheolder(resp.younger)adultsinsequenceassigningeachthevalue1withprobability.4(.1)andthevalue0withprobability.6(.9).Foralltheushpatchesinthegrovewithapopulationofadultpsyllids,themovementtonewushpatchesfollowsthemethodologydescribedinSection A.1 .Whenpsyllidsarriveatanewtree,weassumethattheytakepartintheremainingactivitiesonDay151andcombinethepopulationofarrivingpsyllidswiththosethatremainedminusthosethatmigratedawayfromushpatchc.Removalofoldushshoots.Thelastdaythatnewsummerushshootsemergedis150,thereforeonDayt1150,theageoftheyoungestcohortofushbecomesmt2=1+t1)]TJ /F6 11.955 Tf 12.81 0 Td[(150.ForDay151,theyoungestageisnowmt2=2andtheyoungushshootsfrompreviousdaysaredesignatedby(c,(151,t2),20),wheret2rangesfromtheageoftheyoungestcohort,2,to16.Theoldushshootsfrompreviousdaysaredesignatedby(c,(151,t2),20),wheret2rangesfrom17to30.OnDay151,weremovefromthesystemanyushshootsthathavet231astheynolongerimpactanyofthedailyactivitiesatthisage.StartingonDay151,20ushshootsareremovedandnonewushshootsemerge,sothetotalnumberofushshootspresentdecreasesby20ushshootseachday.Ithasbeen1daysincethelastushshootsemerged,thusthetotalnumberofushpresentis600)]TJ /F6 11.955 Tf 11.61 0 Td[(20(1).Sothereare580ushshootsateachpatchonDay151.Psyllidagingandremovalofpsyllids.EachnymphhasprobabilitysnofsurvivingfromDay150.Eachnymphisscanned,applyingprobabilitysntoeachofthemtodeterminetheirsurvivorstatus.Thenumberofsurvivingnymphs,sn()=sn(P(c,(150,t2),n)),isrecordedintoP(c,(151,t2+1),sn())foractivitiesonDay 128

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151.Inparticular,thenymphswitht2=14thatsurvive,emergeasadults,denotedbyP(c,(151,15),n)),thesearetheemergingadultsthatwillbeconsideredinthetransmissionofCLastopsyllids.Afterthemigrationofpsyllidstotheirnewushpatchesiscompleted,eachadultpsyllidhasprobabilitys0aofsurvivingfromDay150.Thus,wescantheadultpsyllids,applyingprobabilitys0atoeachofthemtodeterminetheirsurvivorstatus.Forvaluesoft215,thenumberofsurvivingadults,s0a()=s0a(P(c,(150,t2),n)),isrecordedintoP(c,(151,t2+1),sa())foractivitiesonDay151.Theadultswitht265areremovedfromconsideration.Egglayingandsurvivalofpreviouslylaideggs.ThenumberofeggslaidonDay151,E151,ateachushpatchislimitedeitherbyasmallnumberoffemaleadultsorbythenumberofyoungushshoots.Femaleadultswhoemergeonthisdayandthepreviousdaydonotimpactoviposition.Toaccountforthefactthatemergingadultsdonotdepositeggsduringtheirrst24to48hours,allfemalepsyllidswitht217thatsurvivedDay150layanother10eggs,thuswehave10s0a(Fa(150))eggs.Again,thelastdaythatnewushshootsemergedisonDay150,thereforeonDayt1,theageoftheyoungestcohortofushismt2=1+t1)]TJ /F6 11.955 Tf 11.28 0 Td[(150.ThusthecapacityforeggsonDay151whichdependsonthenumberofyoungushshootsavailableis 4016Xt2=mt2(c,(151,t2),20)=4016Xt2=2(c,(151,t2),20)=12000Thenumberofeggslaidistheminimumofthesetwovalues.TheneweggsarenowenteredintoalistofeggsasP(c,(151,1),E151)andtheeggsfromthepreviousdaysareP(c,(151,2),se(E150))andP(c,(151,3),s2e(E149)).EachoftheeggshasprobabilityseofsurvivingDay151.ThesurvivalprocessismodeledasasetofindependenteventsacrosseggsandthenumberofsurvivingeggsareretainedinthepopulationmatrixP(c,(152,3),s2e(E150))andP(c,(152,2),se(E151))forDay152activities. 129

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Transmissiontoemergingadults.StartingfromDay151,theminimumoftheushagesisgivenbymt2=1+t1)]TJ /F6 11.955 Tf 12.07 0 Td[(150.Toaccountforthefactthattheushshootareagingandthattherewillbeadaywheretheremainingushshootsareofagegreaterthan15,letMt2=maxf15,mt2g.ThepopulationofushshootswithemergingadultsisE=30Xt2=Mt2(c,(151,t2),20).WeconsideronlytheushshootsthathavebeeninfectedonorbeforeDay150.OnDay151thenumberofinfectedushshootsofaget2ist2)]TJ /F9 7.97 Tf 6.58 0 Td[(1Xj=1h(j+151)]TJ /F4 11.955 Tf 11.96 0 Td[(t2,j).Inorderfortheushshoottohaveemergingadults,wehavetherestrictionMt2t230.ThusonDay151thereare EI=30Xt2=Mt2t2)]TJ /F9 7.97 Tf 6.59 0 Td[(1Xj=1h(j+61)]TJ /F4 11.955 Tf 11.96 0 Td[(t2,j)infectedushshootswithemergingadults.Theprobability,p,thatanemergingadultbecomesinfectedisobtainedbymultiplyingtheproportionofinfectedushshootswithemergingadults,=EI Eandtheprobabilityoftransmissionfromushshoottoemergingadult,1,sop=1=1EI E.Eachemergingindividualisassigneda1(success)withprobabilitypanda0withprobability1)]TJ /F4 11.955 Tf 12.18 0 Td[(p.Itisimportanttonotethatthenewlyemergedadultsdonotcontributetoegglayingortotransmissiontoushshootsonthedaytheyemerge.Also,themovementprobabilitiesforemergingadultsislowerthanof3-dayorolderadultsMichaud2004,Kobori.Transmissiontoushshoots.OnDay151,theminimumoftheushagesisgivenbymt2=1+t1)]TJ /F6 11.955 Tf 12.71 0 Td[(150.h(t1,t2)isthenumberofnewinfectedushofaget2ondayt1.OnDay151,transmissionofCLastoushshootscanonlybefrominfectedpsyllidswitht216.Thus,eachage-appropriateinfectedpsyllidisscannedandhas 130

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probabilitypinf=.3oftransmittingCLasduringaninfectivefeeding.Thenumberofinfectivefeedings,B,andthetotalnumberofushpresent,580,isobtained;thentheexpectednumberofinfectedushonDay151,h(151,),iscalculated.Foreachoftheh(151,)infectedush,wethendecidewhatage(t2=30,...,mt2)isassignedtoeachush.Onthisday,theyoungestcohortofushisofagemt2=2.Alsonoteherethattheushwitht2=30willberemovedfromthesystemonthenextday,howevertheyareassumedtostillbelocationsforfeedingonthecurrentdayandthereforeareapartoftheushpopulationthatcanbecomeinfected.Theprobabilityisequaltothecurrentproportionofhealthyushshootsofthatage.ThenumberofhealthyushshootspriortoinfectiononDay151foraget2=mt2,...,30is20)]TJ /F14 7.97 Tf 11.95 15.06 Td[(t2)]TJ /F9 7.97 Tf 6.59 0 Td[(1Xj=1h(j+151)]TJ /F4 11.955 Tf 11.96 0 Td[(t2,j).TherearenoemergingushshootsonDay151.ThetotalnumberofhealthyushonDay151isT=30Xt2=mt2 20)]TJ /F14 7.97 Tf 11.96 15.06 Td[(t2)]TJ /F9 7.97 Tf 6.59 0 Td[(1Xj=1h(j+151)]TJ /F4 11.955 Tf 11.95 0 Td[(t2,j)!.Theprobabilityofinfectionofeachushshootofagemt2t230is20)]TJ /F20 7.97 Tf 6.59 5.98 Td[(Pt2)]TJ /F21 5.978 Tf 5.76 0 Td[(1j=1h(j+151)]TJ /F14 7.97 Tf 6.58 0 Td[(t2,j) T.AninventoryofhealthyandinfectedushpresentonDay151istaken.Summarystatistics.ThesummarystatisticscorrespondingtothecompartmentsinFigure 3-1 atushpatchcarerecorded.ThesearethenumberofpsyllidsthatwerepresentandtakingpartinactivitiesonDay151atushpatchcandincludethenewphysicaldistributionofpsyllids. Ia(151)=#ofinfectedadultpsyllidsattheendofDay151 Sa(151)=#ofuninfectedadultpsyllidsattheendofDay151 Sn(151)=#ofuninfectedandsurvivingnymphsattheendofDay151 Se(151)=#ofuninfectedeggsattheendofDay151 Iy(151)=#ofinfectedyoungushattheendofDay151 131

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Sy(151)=#ofuninfectedyoungushattheendofDay151 Io(151)=#ofinfectedoldushattheendofDay151 So(151)=#ofuninfectedoldushattheendofDay151Oncetheushshootsbecomemature(t2>30),theyarenolongerincludedinthecountsIoandSo.Nonewushemergesforthisperiodofsummerush.Day166ActivitiesTheactivitiesthattakeplaceonDay152to165followthesamedescriptionofactivitiesthatoccuronDay151.WecontinuewithadescriptionoftheactivitiesthatoccuronDay166,themaindifferenceinactivitiesonDay166isthattheminimumageofallushshootsismt2=1+166)]TJ /F6 11.955 Tf 12.58 0 Td[(150=17whichmeansthatthisistherstdaywheretherearenoyoungushshootsandthereforethisistherstdaywherenoeggswillbelaid.ThisisarepetitionoftheactivitiesthatoccurduringthespringushperiodonDay76.Migration.Eachpsyllidwitht217hasprobability.4ofmigratingtoanewushpatch,callitc0.Fortherecentlyemergedadults,eachpsyllidwitht2=15or16hasprobability.1ofmigratingtoanewushpatch.Thus,wescantheolder(resp.younger)adultsinsequenceassigningeachthevalue1withprobability.4(.1)andthevalue0withprobability.6(.9).Foralltheushpatchesinthegrovewithapopulationofadultpsyllids,themovementtonewushpatchesfollowsthemethodologydescribedinSection A.1 .Whenpsyllidsarriveatanewtree,weassumethattheytakepartintheremainingactivitiesonDay166andcombinethepopulationofarrivingpsyllidswiththosethatremainedminusthosethatmigratedawayfromushpatchc.Removalofoldushshoots.Thelastdaythatnewushshootsemergedis150,thereforeonDay166,theageoftheyoungestcohortofushbecomesmt2=1+t1)]TJ /F6 11.955 Tf 9.93 0 Td[(150.ForDay166,theyoungestageisnowmt2=17andthereisnopopulationofyoungushshoots.Theoldushshootsfrompreviousdaysaredesignatedby(c,(166,t2),20),wheret2rangesfrom17to30.OnDay166,weremovefromthesystemanyush 132

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shootsthathavet231astheynolongerimpactanyofthedailyactivitiesatthisage.Ithasbeen16daysincethelastushshootsemerged,sothereare280ushshootsateachpatchonDay166.Psyllidagingandremovalofpsyllids.EachnymphhasprobabilitysnofsurvivingfromDay165.Eachnymphisscanned,applyingprobabilitysntoeachofthemtodeterminetheirsurvivorstatus.Thenumberofsurvivingnymphs,sn()=sn(P(c,(165,t2),n)),isrecordedintoP(c,(166,t2+1),sn())foractivitiesonDay166.Inparticular,thenymphswitht2=14thatsurvive,emergeasadults,denotedbyP(c,(166,15),n)),thesearetheemergingadultsthatwillbeconsideredinthetransmissionofCLastopsyllids.Afterthemigrationofpsyllidstotheirnewushpatchesiscompleted,eachadultpsyllidhasprobabilitys0aofsurvivingfromDay165.Thus,wescantheadultpsyllids,applyingprobabilitys0atoeachofthemtodeterminetheirsurvivorstatus.Forvaluesof15t264,thenumberofsurvivingadults,s0a()=s0a(P(c,(165,t2),n)),isrecordedintoP(c,(166,t2+1),s0a())foractivitiesonDay166.Survivalofpreviouslylaideggs.TherearenoeggslaidonDay166.TheeggsfrompreviousdaysareP(c,(166,2),se(E165))andP(c,(166,3),s2e(E164)).EachoftheeggshasprobabilityseofsurvivingDay166.ThesurvivalprocessismodeledasasetofindependenteventsacrosseggsandthenumberofsurvivingeggsareretainedinthepopulationmatrixP(c,(166,3),s2e(E165))forDay167activities.ThisisthelastcohortofeggsthatwillsurvivetonymphsonDay168withprobabilityse.Transmissiontoemergingadults.OnDay166,theminimumoftheushagesisgivenbymt2=1+t1)]TJ /F6 11.955 Tf 12.16 0 Td[(150.Fromthisdayforward,theushshootsareofaget217.ThepopulationofushshootswithemergingadultsisE=30Xt2=mt2(c,(166,t2),20).WeconsideronlytheushshootsthathavebeeninfectedonorbeforeDay165.On 133

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Day166thenumberofinfectedushshootsofaget2ist2)]TJ /F9 7.97 Tf 6.58 0 Td[(1Xj=1h(j+166)]TJ /F4 11.955 Tf 11.96 0 Td[(t2,j).Inorderfortheushshoottohaveemergingadults,wehavetherestrictionmt2t230.ThusonDay166thereare EI=30Xt2=mt2t2)]TJ /F9 7.97 Tf 6.59 0 Td[(1Xj=1h(j+76)]TJ /F4 11.955 Tf 11.95 0 Td[(t2,j)infectedushshootswithemergingadults.Theprobability,p,thatanemergingadultbecomesinfectedisobtainedbymultiplyingtheproportionofinfectedushshootswithemergingadults,=EI Eandtheprobabilityoftransmissionfromushshoottoemergingadult,1,sop=1=1EI E.Eachemergingindividualisassigneda1(success)withprobabilitypanda0withprobability1)]TJ /F4 11.955 Tf 12.18 0 Td[(p.Itisimportanttonotethatthenewlyemergedadultsdonotcontributetoegglayingortotransmissiontoushshootsonthedaytheyemerge.Transmissiontoushshoots.OnDay166,theminimumoftheushagesisgivenbymt2=17.h(t1,t2)isthenumberofnewinfectedushofaget2ondayt1.TransmissionofCLastoushshootscanonlybefrominfectedpsyllidswitht216.Thus,eachage-appropriateinfectedpsyllidisscannedandhasprobabilitypinf=.3oftransmittingCLasduringaninfectivefeeding.Thenumberofinfectivefeedings,B,andthetotalnumberofushpresent,280,isobtainedandtheexpectednumberofinfectedushonDay166,h(166,),iscalculated.Foreachoftheh(166,)infectedush,wethendecidewhatage(t2=30,...,mt2)isassignedtoeachush.Onthisday,theyoungestcohortofushisofagemt2=17.Alsonoteherethattheushwitht2=30willberemovedfromthesystemonthenextday,howevertheyareassumedtostillbelocationsforfeedingonthecurrentdayandthereforeareapartoftheush 134

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populationthatcanbecomeinfected.Theprobabilityisequaltothecurrentproportionofhealthyushshootsofthatage.ThenumberofhealthyushshootspriortoinfectiononDay166foraget2=mt2,...,30is20)]TJ /F14 7.97 Tf 11.95 15.06 Td[(t2)]TJ /F9 7.97 Tf 6.59 0 Td[(1Xj=1h(j+166)]TJ /F4 11.955 Tf 11.96 0 Td[(t2,j).TherearenoemergingushshootsonDay166.ThetotalnumberofhealthyushonDay166isT=30Xt2=mt2 20)]TJ /F14 7.97 Tf 11.96 15.06 Td[(t2)]TJ /F9 7.97 Tf 6.59 0 Td[(1Xj=1h(j+166)]TJ /F4 11.955 Tf 11.95 0 Td[(t2,j)!.Theprobabilityofinfectionofeachushshootofagemt2t230is20)]TJ /F20 7.97 Tf 6.59 5.97 Td[(Pt2)]TJ /F21 5.978 Tf 5.76 0 Td[(1j=1h(j+166)]TJ /F14 7.97 Tf 6.58 0 Td[(t2,j) T.AninventoryofhealthyandinfectedushpresentonDay166istaken.Summarystatistics.ThesummarystatisticscorrespondingtothecompartmentsinFigure 3-1 atushpatchcarerecorded.ThesearethenumberofpsyllidsthatwerepresentandtakingpartinactivitiesonDay166atushpatchcandincludethenewphysicaldistributionofpsyllids.Theseare: Ia(166)=#ofinfectedandsurvivingadultpsyllidsattheendofDay166 Sa(166)=#ofuninfectedandsurvivingadultpsyllidsattheendofDay166 Sn(166)=#ofuninfectedandsurvivingnymphsattheendofDay166 Se(166)=#ofuninfectedandsurvivingeggsattheendofDay166 Iy(166)=0infectedyoungushattheendofDay166 Sy(166)=0uninfectedyoungushattheendofDay166 Io(166)=#ofinfectedoldushattheendofDay166 So(166)=#ofuninfectedoldushattheendofDay166Oncetheushshootsbecomemature(t2>30),theyarenolongerincludedinthecountsIoandSo.NonewushhasemergedsinceDay150,thusonthisdaythereare 135

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noyoungushpresent.ThelastdayeggswerelaidwasonDay165,forthisreason,onDay168,thecompartmentSe(168)=0.Day179ActivitiesTheactivitiesthattakeplaceonDay167to178followthesamedescriptionofactivitiesthatoccuronDay166.Theagingofpsyllidstakesplaceinthesamemannerduringthesedayswithsomeofthepopulationvaluesequalto0duetothenoegglayingsinceDay165.WecontinuewithadescriptionoftheactivitiesthatoccuronDay179.Onthisday,theminimumageofallushshootsismt2=30whichmeansthatthisisthelastdaywherethereareanyushshootspresent.Migration.Eachpsyllidwitht217hasprobability.4ofmigratingtoanewushpatch,callitc0.Fortherecentlyemergedadults,eachpsyllidwitht2=15or16hasprobability.1ofmigratingtoanewushpatch.Thus,wescantheolder(resp.younger)adultsinsequenceassigningeachthevalue1withprobability.4(.1)andthevalue0withprobability.6(.9).Foralltheushpatchesinthegrovewithapopulationofadultpsyllids,themovementtonewushpatchesfollowsthemethodologydescribedinSection A.1 .Whenpsyllidsarriveatanewtree,weassumethattheytakepartintheremainingactivitiesonDay179andcombinethepopulationofarrivingpsyllidswiththosethatremainedminusthosethatmigratedawayfromushpatchc.Removalofoldushshoots.Thelastdaythatnewushshootsemergedis150,thereforeonDay179,theageoftheyoungestcohortofushis30.Thereisnopopulationofyoungushshootsandtheoldushshootsfrompreviousdaysaredesignatedby(c,(179,30),20).OnDay179,weremovefromthesystemanyushshootsthathavet231astheynolongerimpactanyofthedailyactivitiesatthisage.Thereare20ushshootsremainingateachpatchonDay179.Psyllidagingandremovalofpsyllids.EachnymphhasprobabilitysnofsurvivingfromDay178.Eachnymphisscanned,applyingprobabilitysntoeachofthemtodeterminetheirsurvivorstatus.Thenumberofsurvivingnymphs,sn()= 136

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sn(P(c,(178,t2),n)),isrecordedintoP(c,(179,t2+1),sn())foractivitiesonDay179.Inparticular,onDay179,allthenymphshavet2=14andtheonesthatsurviveemergeasadults,denotedbyP(c,(179,15),n)).ThesearetheemergingadultsthatwillbeconsideredinthetransmissionofCLastopsyllids.Afterthemigrationofpsyllidstotheirnewushpatchesiscompleted,eachadultpsyllidhasprobabilitys0aofsurvivingfromDay178.Thus,wescantheadultpsyllids,applyingprobabilitys0atoeachofthemtodeterminetheirsurvivorstatus.Forvaluesof15264,thenumberofsurvivingadults,s0a()=s0a(P(c,(178,t2),n)),isrecordedintoP(c,(179,t2+1),s0a())foractivitiesonDay179.Transmissiontoemergingadults.OnDay179,theminimumushageis30.ThepopulationofushshootswithemergingadultsisE=(c,(179,30),20).WeconsideronlytheushshootsthathavebeeninfectedonorbeforeDay178.OnDay179thereare EI=29Xj=1h(j+149,j)infectedushshootswithemergingadults.Theprobability,p,thatanemergingadultbecomesinfectedisobtainedbymultiplyingtheproportionofinfectedushshootswithemergingadults,=EI Eandtheprobabilityoftransmissionfromushshoottoemergingadult,1,sop=1=1EI E.Eachemergingindividualisassigneda1(success)withprobabilitypanda0withprobability1)]TJ /F4 11.955 Tf 11.95 0 Td[(p.Transmissiontoushshoots.OnDay179,theminimumushageis30.h(t1,t2)isthenumberofnewinfectedushofaget2ondayt1.TransmissionofCLastoushshootscanonlybefrominfectedpsyllidswitht216.Thus,eachage-appropriateinfectedpsyllidisscannedandhasprobabilitypinf=.3oftransmittingCLasduringaninfectivefeeding.Thenumberofinfectivefeedings,B,andthetotal 137

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numberofushpresent,20,isobtainedandthenumberofinfectedushonDay179,h(179,30),iscalculated.AninventoryofhealthyandinfectedushpresentonDay179istaken.Summarystatistics.ThesummarystatisticscorrespondingtothecompartmentsinFigure 3-1 atushpatchcarerecorded.ThesearethenumberofpsyllidsthatwerepresentandtakingpartinactivitiesonDay179atushpatchcandincludethenewphysicaldistributionofpsyllids.Theseare: Ia(179)=#ofinfectedandsurvivingadultpsyllidsattheendofDay179 Sa(179)=#ofuninfectedandsurvivingadultpsyllidsattheendofDay179 Sn(179)=0ofuninfectedandsurvivingnymphsattheendofDay179 Se(179)=0ofuninfectedandsurvivingeggsattheendofDay89 Iy(179)=0infectedyoungushattheendofDay179 Sy(179)=0uninfectedyoungushattheendofDay179 Io(179)=20ofinfectedoldushattheendofDay179 So(179)=20ofuninfectedoldushattheendofDay179Oncetheushshootsbecomemature(t2>30),theyarenolongerincludedinthecountsIoandSo.NonewushhasemergedsinceDay150,thusonthisdaytherearenoyoungushpresent.ThelastdayeggswerelaidwasonDay165,forthisreason,thecompartmentSe(179)=0.ThelastnymphsemergedintoadultsonthisdayandthusthecompartmentSn(179)=0.Day180to220ActivitiesNoushshootsarepresentatanyoftheushpatches,thisbeginsaperiodof41dayswherethereisnomovementortransmissionactivities.Theonlythingthatoccursisthesurvivalandagingofadultpsyllids,aswellastheremovalofpsyllidswitht2>65.Weassumethatduringthissecondperiodofnoush,theprobabilityforanadultsurvivingisstills0a=.9659duetoanincreaseintemperature. 138

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A.3FallFlushPeriodThedifferencesbetweenthesummerandfallushperiodsconsistofthesurvivalprobabilityfortheadultdayssa=.9847,themaximumadultageis75,andtheperiodoftimewhereushshootsemerge.Forthefallperiod,ushshootsemergebeginningonDay221andthelastushshootsemergeonDay235.Day221Activities,Day1FallThepsyllidadultsthatsurvivetothisdayparticipateinactivitiesonDay221,therstdayofthefallushperiod.Withthefallsurvivalprobabilities,thedailyactivitiesthatoccuronDays221to236arethesameasthoseonDay1to15.Themaximumlifespanforadultpsyllidsreturnsto75days,sotheremovalofadultswitht2>89alsooccurs.Nomigration.OnDay221,therstdayofthisfallushperiod,survivingpsyllidsremainonthesameushpatches.TheywillcontinuemovementbetweenushpatchesbeginningonDay222.Emergenceofnewushshoots.Anewsetof20ushshootsemergesateachlocation,c.Theseusharedesignatedas(c,(221,1),20).Witht1=221andt2=1,thismeanswearerepresentingtheseushshootsatlocationconthe221thdaysincethepsyllidinvasionandontherstdaythatthisushemerged.Thereareatotalof20ushshootspresentateachpatchonDay221.Psyllidagingandremovalofpsyllids.EachadultpsyllidhasprobabilitysaofsurvivingfromDay220.Thus,wescantheadultpsyllids,applyingprobabilitysatoeachofthemtodeterminetheirsurvivorstatus.Forvaluesof15t288,thenumberofsurvivingadults,sa()=sa(P(c,(1,t2),n)),isrecordedintoP(c,(2,t2+1),sa())foractivitiesonDay221.Theadultswitht2=89areremovedfromconsideration.Egglaying.ThenumberofeggslaidonDay221,E221,ateachushpatchislimitedeitherbyasmallnumberoffemaleadultsorbythenumberofyoungushshoots.Wehave10sa(Fa(220))eggs.ThecapacityforeggsonDay221,whichis 139

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dependentonthenumberofyoungushshootsavailable,isgivenby 40(c,(221,1),20)=800Thenumberofeggslaidistheminimumofthesetwovalues.TheneweggsarenowenteredintoalistofeggsasP(c,(221,1),E221).EachoftheeggshasprobabilityseofsurvivingDay221.ThesurvivalprocessismodeledasasetofindependenteventsacrosseggsandthenumberofsurvivingeggsareretainedinthepopulationmatrixP(c,(222,2),se(E221))forDay222activities.Transmissiontoushshoots.OnDay221,transmissionofCLascanonlybefrominfectedpsyllidtoushshoot,astherearenoinfectedandinfectiousushpresentintheinitialbatchof20shoots.Thus,eachinfectedpsyllidisscannedandhasprobabilitypinf=.3oftransmittingCLasduringaninfectivefeeding.Thusthenumberofinfectivefeedings,B,andthetotalnumberofushpresent,20,isobtainedandthentheexpectednumberofinfectedushofaget2(t2daysafteremergence)onthet1thdayafterinitialinfection,h(t1,t2)iscalculated.Allushonthisdayareinthesamecohort,thereforeh(221,1)oftheushmovetotheinfectedushand20-h(221,1)remaininthehealthyushpopulation.AninventoryofushispreparedforDay222accordingtotheformalismS(c,(222,2),20)]TJ /F7 11.955 Tf 11.95 0 Td[(h(221,1)))andI(c,(222,2),h(221,1)))forhealthyandinfectedushrespectively.Summarystatistics.ThesummarystatisticscorrespondingtothecompartmentsinFigure 3-1 atushpatchcarerecorded.ThesearethenumberofpsyllidsthatwerepresentandtakingpartinactivitiesonDay221atushpatchc: Ia(221)=#ofinfectedadultpsyllidsattheendofDay221. Sa(221)=#ofuninfectedadultpsyllidsattheendofDay221 Se(221)=#ofuninfectedeggsattheendofDay221 Iy(221)=#ofinfectedyoungushattheendofDay221 140

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Sy(221)=#ofuninfectedyoungushattheendofDay221TheremainingthreecompartmentsinFigure 3-1 pertaintooldushandnymphs,neitherofwhichexistonDay221,therstdayofthesummerush.Eggsdonotbecomerstinstarnymphsuntil3daysaftertheyarelaid.Day222to264ActivitiesTheactivitiesinthefallcontinueinthesamepatternasthoseinthespringandsummerushperiods.Duetotheshorterushperiodof15days,thefollowingactivitiestakeplaceasdescribedbeforeonthespecieddays.MigrationbetweenushpatchesoccurfromDay222toDay264.Flushshootsemergeingroupsof20ateachushpatchandthiscontinuesfromDay222tothelastdayushshootsemergewhichisDay235.Flushshootsareremovedfromthesystemoncet2>30,sotheremovalofushshootsbeginsonDay251andcontinuesthroughDay264.EgglayingbeginsonDay221andthelastdaythateggsarelaidarewhentheyoungestushshootshavet217whichisonDay251.Eggssurvivewiththesameprobabilityasbeforeandsurvivingeggsbecomenymphsfort2=4.Psyllidagingcontinues,withthemaximumlifespanofanadultat75daysandthesamerestrictionsonmovementandtransmissionareappliedtoemergingadults.ThesurvivingeggsfromthelastgrouplaidonDay251emergeasadultsonDay264.Transmissiontoushshootsoccursoneverydayinthefallushperiod,whiletransmissiontoemergingadultsbeginsonlywhenthereareemergingadultspresent,whichisonDay235.AfteralloftheactivitiestakeplaceoneachdayinthisfallushperiodwecollectthesummarystatisticsthatrecordsthepopulationcorrespondingtoeachcompartmentfromFigure 3-1 .Day265to365ActvitiesNoushshootsarepresentatanyoftheushpatches,thisbeginsaperiodof101dayswherethereisnomovementortransmissionactivities.Theonlythingthatoccurs 141

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isthesurvivalandagingofadultpsyllids.Thisrepresentstheoverwinteringperiodforthepsyllids,soweassumethatadultpsyllidssurvivelongerinthecolderweatherandtheremovalofpsyllidswitht2>117.Weassumethatduringthisthirdperiodofnoush,theprobabilityforanadultsurvivingiss00a=.9885duetoadecreaseintemperature.OnDay366,thespringushshootsemergetheyearlycycledescribedaboveoccurs. 142

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APPENDIXBSUPPORTINGEVIDENCEInthisappendix,wewillpresentsupportingevidenceforourmodel.Wewillstartwiththeexperimentthatvalidatestheassumedmechanisminourmodel,thetransmissionofCLasbetweengenerationsofpsyllidslocallythroughtheushshoots.Thenwewillproceedtopresentcaseswhereourmodelisabletotodescribethespreadofpsyllidsintoainitiallypsyllid-freegrove,thespreadofHLBinapsyllid-infestedgrovethatresemblesthespreadofinfection,andthepresenceofinfectedD.citrinymphsandadultsbeforedetectionofinfectedtrees.Todifferentiatefromtheeldstudyandthesimulation,wewillrefertothetreesintheeldstudyastheorchardandtreesinthesimulationasthegrove. B.1MechanismforFlushtoPsyllidTransmissionWewillrstpresentthepsyllidexperimentdescribedbyLeeetal.[ 46 ]thatdemonstratetheinfectionofpsyllidprogenythroughushshoots.PsyllidswereobtainedfromanexistingpsyllidcolonythatoriginatedinFloridacitrusgroves.ThecolonyiscurrentlymaintainedingrowthroomsonsourceplantscontainingLasoriginallyinoculatedfromsymptomaticeldtreeslocatedinHendryCounty,Florida,in2007[ 46 ].Generally,6-10previouslyuninfectedplantswereusedforeachexperiment,with50or100adultpsyllidscollectedrandomlyfromthepsyllidpopulationonHLB-infectedplantswithinaplantcontainmentroomandaddedtocages[ 46 ].After15days,adultpsyllids(input)wereremovedandstoredinafreezerforsubsequentDNAextraction.After30days,thosepsyllidprogeny(output)thatemergedfromeggsdepositedbytheinputpsyllidswerealsoremovedandstoredinafreezer.DNAwasextractedfromindividualpsyllidsusingaQiagenDNeasyBloodTissueKitaccordingtomanufacturer'srecommendations.Plantsamples(300mg)wereextractedin3mlofextractionbuffer(100mMTris-HCL,pH8.0;50mMEDTA;500mMNaCl;and10mMdithiothreitol)[ 46 ]. 143

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TableB-1. InfectionofpsyllidprogenybyLasacquiredfromCitrusmacrophyllainfectedbythepriorgenerationofadultpsyllids Exp.##psyllids#plants %Laspositive%LaspositivePlantsaintroduced#w.nymphsinputpsyllidspsyllidprogeny#positive/totalg 1506/225d38@30d2/2@30d2506/221e(33d)0@30d0/2@31d3506/233d33@30d1/2@31d4506/624d5@30d2/4@10d3/4@15d5506/255e0@30d2/2@33d6506/269e13@31d1/2@33d7506/129e8@31d1/1@31d8506/138e13@31d1/1@31d9504/254e22@31d1/2@30d10504/2100e54@30d2/2@30d1110010/651e16@30d4/6@32d12506/nd5e(21d)12@30dndb13506/nd17e(21d)0@22dnd14506/nd8e(21d)17@22dnd15fnd6/2nd83@30d2/2@38d16fnd3/3nd29@40d3/3@58d aRT-qPCRandPCRdetectionofLasininfectedareasofplants.Resultsareshownasthenumberofpositiveplantsoverthetotalnumberofplantstested.bnd=notdetermined.cd=days.dPercentageofLas-positivepsyllidsinthepopulationonHLB-infectedplantsfromwhichtheinputpsyllidswereremovedePercentageofLas-positiveinputpsyllidswithdrawnfromthecagesat15days.fSuccessiveexperimentswerecarriedoutusingpsyllidprogenyfromearlierexperiments.gTotalnumberofplantsthatpriorto30daysharboredeggsandnymphs 144

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TheresultsinTable B-1 showthatpsyllidprogenyacquiredLasfromplantsthat30daysearlierwerenotinfectedwithHLB.Thistimeframewassufcientfortheplantstobecomeefcientdonorhosts.ItisknownthatpsyllidshavehighertitersofLaswhenthebacteriumisacquiredbynymphs[ 37 ].Thus,theassayedadultprogenypsyllidswouldhavehadtoacquirethebacteriumearlierthan30days,likelywithin10to20days.Psyllidsdidnotlayeggsonallplantsineachcage.Usuallyabout1/3to1/2oftheplantswereinfestedwithnymphs.Nymphsgenerallystayinthesameareawhereeggsarelaid.ThustheplantsamplesthatwereanalyzedbyPCRforLassequencesweretakenfromlocationswherenymphsdeveloped.Plantsamplesthatweretakenat10to15daysalreadytestedpositiveforLas.TheresultsconrmthattheareasoftheplantswherepsyllidprogenydevelopedcouldbecomeinfectedwithLaswithinthisshorttimeperiodandtherebyserveasasourceofinoculumforthedevelopingnymphs. B.2LocalDispersalofPsyllidsinSarawak,MalaysiaTheSamarahanDivisionofSouthwestSarawakInMalaysiawascompletelydestroyedbyHLBin1992[ 47 ].Thisisthelocationoftheorchardwheretheeldexperimentwasconductedin1999-2000.Leongetal.[ 47 ]determinedthatthecompleteinfestationofapreviouslypsyllid-freecitrusorchardconsistingof200newly-plantedtreesoccurredwithin1yearand9monthsfromtherstdetectionofpsyllids.ThecitrustreeswereplantedinJanuary1999andtherstdetectionofpsyllidsoccurredinMarch1999,with1.5%ofthetreesinfested.InMay2000,infestationintheorchardwas46%andbyOctober2000,theinfestationwas90%.Theorcharddidnotreceiveanyinsecticidetreatmentsandintheperiodof1yearand9months,theorchardwas100%infestedbythepsyllids.Tosetupthesimulatedgrove,weplace200treesinto10rowswith20treeswithineachrow.Thespacingoftheorchardisnotspecied,soweassumethatthebetween-rowtreedistanceistwicethewithin-rowdistances.Inthesimulation,weassumethattherearenopsyllidspresentonanyofthetreesinitiallyplantedinJanuary 145

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of1999.Inourmodel,theushcyclebeginsinFebruaryandthatiswhentheinitialmigrationofpsyllidsintothegroveoccurs.Weassumethatthreetreesonthesouthwestedgewereinfestedwith10psyllidseach.Wewilllookatthetotalnumberofpsyllidspresentateachtreethroughthenexttwoyearsandthepercentageoftreesinfestedovertime.Usingathresholdof100adultpsyllids,detectioninthegroveisconductedonthe15thofeachmonth.Figure B-3 showsthattherstdetectionoccursinApril1999whentheinfestationis7%andrisesto43%inMarch2000.ByOctober2000,theinfestationisat91.5%andbyDecember2000isat96%.AlthoughfromFigure B-2 theentiregroveisinfestedwith50adultpsyllidsbyOctober2000,sodependingondetectionlevels,theentiregrovecouldbefoundtobeinfestedinaperiodof1yearand7months.ThecorrespondingincreasesininfestationinFigures B-2 and B-3 inthesimulationcorrespondtotheushingperiods.TheinformationonthefrequencyandavailabilityofushinMalaysiaduringthistimewasnotgivenexplicitly,sotheushingperiodsremainthesameasdescribedinSection 2.1.3 .Fromtheagent-basedsimulationmodel,weknowtheexactdaywhentherstpsyllidappearedonatree,thebluelineinFigure B-2 correspondstotheunderlyinginfestationofthetreesbyasinglepsyllid.Detectionintheeldrequiresathresholdofpsyllids,althoughthisthresholdisunknown.Figure B-2 showshowtheinfestationofagrovechangeswhenwerequirevaryingthresholdlevelsfordetection.Todeterminethedailyinfestationweassumethatoneachday,everytreeischeckedforthenumberofadultpsyllidsandifthepopulationisabovethethresholdthenthetreeisinfested.Thisleadstoarticialdecreasesinthenumberoftreesinfestedasthepopulationdeclinesandgoesbelowthethresholdlevel.Withtheassumptionthatonceatreeisfoundtobeinfestedwithpsyllids,itwillremaininfesteddespitebeingundetectable,thenumberofinfestedtreesismonotoneincreasingasshownbytheblacklineinFigure B-3 146

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FigureB-1. SpreadofD.citripopulationinacitrusorchardatJemukan,SamarahanDivision,SouthwestSarawak,Malaysia.(AdaptedfromLeong,S.C.T.etal.2012.ImpactsofhorticulturalmineraloilsandtwoinsecticidepracticesonpopulationuctuationofDiaphorinacitriandspreadofHuanglongbinginacitrusorchardinSarawak.ScienticWorldJournal(Page3,Figure1)) FigureB-2. DailyinfestationofD.citriinasimulatedgrove.Infestationoneachdayisshownforpopulationofadultpsyllidsgreaterthanthethresholdsof0(blue),50(cyan),100(red),200(green),and500(black). 147

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FigureB-3. Surveyofinfestationinthesimulatedgroveatmonthlyintervalsonthefteenthofeachmonth.Monthlyinfestationisshownforpopulationofadultpsyllidsgreaterthanthethresholdsof0(blue)and100(red).Themonotoneincreasingline(black)representstheassumptionthatonceatreeisinfested,itremainsinfestedevenwhenadultpopulationlevelsfallbelowthethresholdlevel. B.3AppearanceofHLBinaCitrusOrchardinColima,MexicoInMexico,therstdetectionofHLBoccurredinJuly2009inurbanareasofthestateofYucatan[ 63 ].ThiseldstudywasconductedinTecoman,Colimastatewheretheclimateissemi-drywith26Caverageannualtemperature[ 63 ]andtherstreportofHLBinthestateofColimawasinApril2010.Thecitrusorchardconsistsoffour-yearoldlimetreeswheretherewasnopsyllidcontrol.AcontrolprogramforthepsyllidwasimplementedoncesymptomsweredetectedstartinginApril2010.Previouslybiologicalcontrolwasbeingconductedintheorchardandthereforetherewasahighpopulationofadultsandnymphs[ 63 ].TheorchardwasscoutedmonthlyforHLBsymptomsoverthecourseofayearandthelocationofthesymptomatictreesat4differenttimesisshowninFigure B-6 148

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Theorchardsize,consistingof2909treesspaced8x4mapart,islargerthanthatofouroriginalsimulation.Weextendthesimulatedgroveto529treesplacedin23rowswith23treeswithineachrowtoapproximateasubsectionofthesoutheastMexicanlimeorchardhighlightedinFigure B-6 .Hereweplottheappearanceofsymptomsinoursimulatedgroveandshowtheunderlyingrateofinfection. AApril2009 BJuly2009 CDec2009 FigureB-4. ProportionofinfectedadultpsyllidsateachushpatchinA)April,B)July,andC)December.Theinitialinfectionislocatedat3treesinthewesterninteriorofthegrove,startingwith30%ofthepsyllidsareinfected.Oneachinfectedpatch,all5thinstarnymphsareinfected. B.4PresenceofInfectedPsyllidsPriortoInfectedTreesinFlorida,USAThedatafromthetransmissionexperimentprovidethemechanismtoexplainobservationsmadebyManjunathetal.[ 50 ],Halbertetal.[ 30 ],andShenetal.[ 67 ].Ineverycase,positivepsyllidswerefoundmonthstoasmuchas6yearspriortodiseasedevelopment.Manjunathetal.[ 50 ]statePsyllidspositivefor`Ca.L.asiaticus'werefoundinsamplescollectedfromcommercialgroves,nurseries,retailstores,residences,andinsecttraps.Whilesomeofthe`Ca.L.asiaticus'-positivesampleswerecollectedfromsymptomatictreesandcountieswithrecordsofplantinfections,alargenumberof`Ca.L.asiaticus'positivesamplescamefromunexpectedsources,includingasymptomatictrees,countieswithnopriorpositiverecordsforHLB,discountgardencenters,smallnurseries,andretailoutlets[...]Whiletheresultsofpsyllid 149

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AApril2010 BJuly2010 CDec2010 DMay2011FigureB-5. ProportionofinfectedadultpsyllidsateachushpatchinA)April,B)July,C)December,andD)May2011.Theinitialinfectionislocatedat3treesinthewesterninteriorofthegrove,startingwith30%ofthepsyllidsareinfected.Oneachinfectedpatch,all5thinstarnymphsareinfected. analysiscamefromthepresentstudy,theinformationon`Ca.L.asiaticus'analysisofplantswasobtainedfromtheDPIrecords.Insixofthirteencounties,Brevard,Highlands,Marion,Nassau,Pasco,andPolk,positivepsyllidswerefoundpriortondingHLBpositiveplantsinsurveys.Halbertetal.[ 30 ]discoveredthatinsomecounties,thereweremorepositiveD.citrisamplesfromplantsforsalethanwouldbeexpectedbasedonnumbersofinfectedplantsinthelandscape.[...]ThediscoveryofLas-positivepsyllidsusuallydidnotcoincidewiththediscoveryofsymptomatic,positiveplants.Infact[...]symptomatic 150

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FigureB-6. DistributionoftreeswithHLBsymptomsobservedoverayearaftertherstdetectioninaMexicanlimeorchard.ThedistributionofsymptomatictreesisshownasofA)April2010,B)July2010,C)December2010,andD)May2011.Thehighlightedareaistheportionthatthesimulatedgroverepresents.(AdaptedfromRobles-Gonzalez,M.M.etal.2013.Huanglongbing(HLB)diseaseinMexicanlimetrees[Citrusaurantifolia(Christm)Swingle]anditsdispersioninColimaState,Mexico.RevistaChapingoSerieHorticultura(Page28,Figure4)) positiveplantswerefoundanaverageof267[days]afterthecollectionofthepositivepsyllids.Shenetal.[ 67 ]foundthatLas-positiveD.citrisamplesweredetectedinGainesvillein2007,andLaspositivetreeswererstfoundontheUFcampusinOctober2011.LaspositiveD.citrisampleswereobtainedfromsymptomlesstreesintheorganicgrovein2006and2007,andLaswasrstdetectedinleafsamplesfromasymptomaticorangetreesinspring2012. 151

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BIOGRAPHICALSKETCH JoAnnLeewasborninAmes,Iowa.ShelivedthereuntilherfathergothisPh.D.degreeatIowaStateUniversityandmovedtoFloridawithherfamily.Forundergraduatestudies,sheattendedtheUniversityofFloridaandearnedaB.S.degreeinmathematicsandapathwaystoteachingminorin2006.ShecontinuedhereducationattheUniversityofFloridatopursueaPh.D.inMathematicswithafocusinmathematicalbiology.Afterhergraduationin2013,sheplanstocontinuewithherresearchasapostdoctoralassociateattheUniversityofFlorida. 159