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Mathematical Modeling of Citrus Greening

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Title:
Mathematical Modeling of Citrus Greening
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1 online resource (78 p.)
Language:
english
Creator:
Stupiansky, Jillian C
Publisher:
University of Florida
Place of Publication:
Gainesville, Fla.
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Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Mathematics
Committee Chair:
Pilyugin, Sergei S
Committee Members:
Keesling, James E
De Leenheer, Patrick
Hager, William Ward
Gillooly, James F

Subjects

Subjects / Keywords:
citrus -- greening -- huanglongbing -- modeling -- psyllid -- roguing -- vector
Mathematics -- Dissertations, Academic -- UF
Genre:
Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

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Abstract:
Huanglongbing (citrus greening) is a bacterial disease that is significantly impacting the citrus industry in Florida and poses a risk to the remaining citrus-producing regions of the United States.  The work presented here centers around the development of a deterministic mathematical model that represents the spread of citrus greening disease within a single grove of trees.  The system models both the tree and insect populations and incorporates a control strategy of roguing infected trees and replanting healthy trees.  The basic reproductive number, R sub one, and its relation to the stability of the equilibria is discussed.  In particular, theorems regarding extinction of the disease when the R sub one 1 are proved.  Variations on the rates of roguing are examined to provide insight into the effect of the control strategy on the level of healthy trees that can be maintained.  A modification to the model is presented which allows for the possibility of the occurrence of a backward bifurcation.  The original model is then made more complex by incorporating a periodic parameter to account for recent biological findings in field studies, and results for this new system are proved.  Numerical studies are performed throughout to illustrate the theoretical findings.  In addition, simulations for a stochastic model are explored, as this type of model may allow for the future incorporation of the economic impact of the disease.
General Note:
In the series University of Florida Digital Collections.
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Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
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This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility:
by Jillian C Stupiansky.
Thesis:
Thesis (Ph.D.)--University of Florida, 2013.
Local:
Adviser: Pilyugin, Sergei S.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2015-08-31

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Applicable rights reserved.
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lcc - LD1780 2013
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UFE0045402:00001

MISSING IMAGE

Material Information

Title:
Mathematical Modeling of Citrus Greening
Physical Description:
1 online resource (78 p.)
Language:
english
Creator:
Stupiansky, Jillian C
Publisher:
University of Florida
Place of Publication:
Gainesville, Fla.
Publication Date:

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Mathematics
Committee Chair:
Pilyugin, Sergei S
Committee Members:
Keesling, James E
De Leenheer, Patrick
Hager, William Ward
Gillooly, James F

Subjects

Subjects / Keywords:
citrus -- greening -- huanglongbing -- modeling -- psyllid -- roguing -- vector
Mathematics -- Dissertations, Academic -- UF
Genre:
Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract:
Huanglongbing (citrus greening) is a bacterial disease that is significantly impacting the citrus industry in Florida and poses a risk to the remaining citrus-producing regions of the United States.  The work presented here centers around the development of a deterministic mathematical model that represents the spread of citrus greening disease within a single grove of trees.  The system models both the tree and insect populations and incorporates a control strategy of roguing infected trees and replanting healthy trees.  The basic reproductive number, R sub one, and its relation to the stability of the equilibria is discussed.  In particular, theorems regarding extinction of the disease when the R sub one 1 are proved.  Variations on the rates of roguing are examined to provide insight into the effect of the control strategy on the level of healthy trees that can be maintained.  A modification to the model is presented which allows for the possibility of the occurrence of a backward bifurcation.  The original model is then made more complex by incorporating a periodic parameter to account for recent biological findings in field studies, and results for this new system are proved.  Numerical studies are performed throughout to illustrate the theoretical findings.  In addition, simulations for a stochastic model are explored, as this type of model may allow for the future incorporation of the economic impact of the disease.
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility:
by Jillian C Stupiansky.
Thesis:
Thesis (Ph.D.)--University of Florida, 2013.
Local:
Adviser: Pilyugin, Sergei S.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2015-08-31

Record Information

Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Classification:
lcc - LD1780 2013
System ID:
UFE0045402:00001


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MATHEMATICALMODELINGOFCITRUSGREENING By JILLIANCLAIRESTUPIANSKY ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2013

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c 2013JillianClaireStupiansky 2

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Tomyamazingparents 3

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ACKNOWLEDGMENTS Iamsogratefultothemanypeoplewhohavemadethisdissertationpossible. Iwouldliketothankmyadvisor,Dr.SergeiS.Pilyugin,forhisconstantguidance, insight,andpatience.Ialsowouldliketoacknowledgemyothercommitteemembers, myprofessors,andtherestofthemathematicsdepartmentfacultyandstaffatthe UniversityofFloridafortheirvaluabletime,input,andhelpthroughoutmygraduate schoolcareer.Inaddition,IwouldliketoextendmythankstoMargaretSomersandmy recommendersfortheirencouragementanddedicationtomyjobsearchprocess. Iwouldalsoliketoacknowledgemytremendoussupportsystemoffriends andfamily.IamsothankfultoKarly,Jessica,Erica,Becky,andCatherinefortheir mathematicalmotivation,butmoreimportantlytheirfriendship.Iwouldliketothank Cheyne,whoalwaysseemedtoknowwhenIcoulduseadistraction,andSebastian,for hiscomputerprowess,advice,andthecountlesshoursspentproductivelyhangingout. Finally,IamsogratefultoLeeforbeingcalmwhenIamcrazyandforsupporting meduringthisprocessandalways.Andlastly,Iwouldliketothankmyparentsfor theirlove,encouragement,andwisdom.Icannotbegintothankthemenoughforthe wonderfullifetheyhavegivenmeandthecondencetheyhaveinstilledinmetoachieve mydreams. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFFIGURES ..................................... 7 ABSTRACT ......................................... 8 CHAPTER 1BACKGROUND ................................... 9 1.1Introduction ................................... 9 1.2Preliminaries .................................. 12 1.2.1EquilibriaandSolutionsofDifferentialEquations .......... 12 1.2.2Matrices ................................. 14 1.2.3FloquetTheoryforPeriodicSystems ................. 15 2MATHEMATICALMODELWITHCONSTANTPARAMETERS .......... 18 2.1DevelopmentoftheModel ........................... 18 2.2Equilibria .................................... 19 2.3TheBasicReproductiveNumber R 0 ..................... 21 2.3.1Calculationof R 0 ............................ 21 2.3.2LocalStabilityoftheDisease-FreeEquilibrium ........... 23 2.4ExtinctionoftheDiseasewhen R 0 1 .................... 25 2.5PersistenceoftheDiseasewhen R 0 > 1 ................... 28 2.5.1PersistenceoftheDiseaseforFullSystem .............. 29 2.5.2StabilityoftheEndemicEquilibriuminaSpecialCase ....... 33 3ROGUINGASACONTROLSTRATEGY ..................... 37 3.1TransientBehavior ............................... 37 3.2DifferentRoguingRates ............................ 40 4MODIFICATIONOFTHEMODEL ......................... 44 5PERIODICITY .................................... 48 5.1PeriodicModel ................................. 48 5.2ExtinctionoftheDiseaseinPeriodicModel ................. 49 5.3IncorporatingthePeriodicFunctionasaSmallPerturbation ........ 53 5.4StabilityoftheDisease-FreeSolution .................... 60 5.5StabilityoftheEndemicSolutioninaSpecialCase ............. 64 6NUMERICALEXPLORATIONS ........................... 70 REFERENCES ....................................... 75 5

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BIOGRAPHICALSKETCH ................................ 78 6

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LISTOFFIGURES Figure page 2-1Flowdiagramforthegrove-scalecitrusgreeningmodel(2 # 1)-(2 # 6) ...... 18 2-2Simulationofsystem(2 # 1)-(2 # 6)with R 0 < 1 ................... 28 2-3Simulationofsystem(2 # 1)-(2 # 6)with R 0 > 1 ................... 29 3-1Simulationsofsystem(3 # 1)-(3 # 6)with R 0 > 1 anddifferentroguingrates ... 40 4-1Simulationsofsystem(4 # 6)-(4 # 6)with R 0 < 1 anddifferentinitialconditions 47 5-1Simulationofsystem(5 # 1)-(5 # 6)satisfyingCondition(5 # 7) .......... 53 5-2Simulationofsystem(5 # 1)-(5 # 6)satisfyingCondition(5 # 9) .......... 64 5-3Simulationofsystem(5 # 10)-(5 # 15)with S (0) < N 0 ............... 69 6-1Simulationofsingle-grovestochasticmodel .................... 71 6-2Timeaveragesofsimulationofsingle-grovestochasticmodel .......... 72 6-3Timeaveragesofsimulationoftwo-grovestochasticmodel ........... 73 6-4Simulationoftwo-grovedeterministicmodel .................... 74 7

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy MATHEMATICALMODELINGOFCITRUSGREENING By JillianClaireStupiansky August2013 Chair:SergeiS.Pilyugin Major:Mathematics Huanglongbing(citrusgreening)isabacterialdiseasethathasasignicantimpact onthecitrusindustryinFloridaandposesarisktotheremainingcitrus-producing regionsoftheUnitedStates.Theworkpresentedherecentersaroundthedevelopment ofadeterministicmathematicalmodelthatrepresentsthespreadofcitrusgreening diseasewithinasinglegroveoftrees.Thesystemmodelsboththetreeandinsect populationsandincorporatesacontrolstrategyofroguinginfectedtreesandreplanting healthytrees.Initially,wewillexamineasysteminwhichallinteractionsoccurata constantrate.Inthiscase,thebasicreproductivenumber, R 0 ,anditsrelationtothe stabilityoftheequilibriaisdiscussed.Inparticular,theoremsregardingextinctionofthe diseasewhen R 0 1 andpersistenceofthediseasewhen R 0 > 1 areproved.Variations ontheratesofroguingareexaminedtoprovideinsightintotheeffectofthecontrol strategyonthelevelofhealthytreesthatcanbemaintained.Amodicationtothemodel ispresentedwhichallowsforthepossibilityoftheoccurrenceofabackwardbifurcation. Theoriginalmodelisthenmodiedtoaccountforseasonalvariationintherateof transmissionofthedisease,andresultspertainingtothestabilityofthisnewsystemare proved.Numericalstudiesareperformedthroughouttoillustratethetheoreticalndings. Inaddition,simulationsofastochasticmodelareexplored,asthistypeofmodelmay allowforthefutureincorporationoftheeconomicimpactofthedisease. 8

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CHAPTER1 BACKGROUND 1.1Introduction Huanglongbing (HLB)isavector-transmittedbacterialinfectionthatiscurrently affectingalltypesofcitrusinthestateofFlorida.InChinese,thenamemeans"yellow dragondisease"[ 20 ],butintheUnitedStatesitisfrequentlyreferredtoas citrus greening disease.Symptomsofthediseaseincludestuntedgrowthandpoorowering ofcitrustreesaswellasblotchymottlingandyellowingoftheirleaves.Inaddition,the fruitthatisproducedbyinfectedtreesismisshapenandsmallerthannormal,witha greentintandabittertaste[ 20 ],makingthefruitinedible.Thediseasehasseverely affectedthecitrusindustryinFlorida,thenation'slargestcitrusproducerandthesecond largestproduceroforangejuiceintheworld.ThepresenceofHLBhasalsobeen detectedinothersoutheasternstatesincludingTexas[ 26 ],andallcitrusproducing regionsoftheUnitedStatesareconsideredtobeatrisk.Becauseofitsimpactonmany sectorsoftheFloridaeconomyandtheimplicationsforthecitrusindustrynationwide, citrusgreeninghasbecomeanimportantissuetostudy. HLBwasrstdiscoveredinChinainthelate1800s[ 17 ].Differentstrainsofthe diseasehavealsocreatedproblemsinthepastinbothAfrica,around1930,andin Brazil,beginningin2004[ 17 ].Itisbelievedthatthevectorsthattransmitthedisease arrivedinsouthernFloridain1998[ 20 ],andcitrusgreeningbeganspreadingnorth throughoutthestatein2000[ 18 ].However,itwasnotuntil2005thatsymptomatic treeswererstdetected[ 17 ].Sincethen,thediseasehasbeenspottedinalmost everycountyasfarnorthasAlachua.Thediseasehasevenaffectedsomecitrusinthe Floridapanhandle,althoughsofaronlyatlargediscountstoresthathandleandship citrustrees[ 20 ]. AninsectknownastheAsiancitruspsyllid, Diaphorinacitri ,carriestheorganism thatcausescitrusgreening, CandidatusLiberibacterasiaticus (Las)[ 17 29 ].The 9

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psyllidshavealifecyclethatlasts45daysonaverage,includingvenymphalstages thattogetherlast15days.Thebacteriaarecarriedinthesalivaofapsyllid;whenan infectedpsyllidfeedsontheleavesofatree,itpassesthediseasetothephloemwithin theveinsofthattree.Similarly,ahealthypsyllidcanacquiretheinfectionbyfeeding onaninfectedtree.Thesetransmissionsaremostlikelytooccurduringtheush, whichhappensinthestateofFloridainthespringandfall[ 20 ].Nymphsarenotableto transmitthediseasewhiletheyareintherstthreelifestages,butafterthispointthey canpassalongthediseaseinaslittleasonedayafterbecominginfected[ 37 ].Itisalso believedthatthediseasecanbespreadamongthepsyllidpopulationtransovarially;a singlepsyllidwilllayaround800eggsduringitslifetime[ 20 ]. Thepsyllidsaretheprimarysourceofthespreadofthedisease.Althoughpsyllids aregenerallysedentaryasnymphs[ 18 ],amaturepsyllidcanyoveramile[ 20 ], andhighwindscanincreasethedistanceapsyllidcantravel[ 35 ].Althoughlarge commercialgrovesgenerallyhavemorethanafewmilesbetweenthem,therearemany abandonedgrovesandresidentialcitrustreesbetweenthemthatfacilitatethespreadof thedisease[ 17 ].Ornamentaltrees,suchas Murrayapaniculata (orangejasmine)are verypopularinthestateofFloridaandserveasanotherhabitatfromwhichthepsyllids canspreadout[ 20 ].Inaddition,manydiscountstoresshipbothinfectedplantsand fruitthroughoutthestate,enablingthediseasetoproliferatequickly.Forthesereasons thediseasehastraveledabout30milesayearinFloridabutonly12milesayearin Brazil[ 18 ].Graftingtogetherhealthyandinfectedtreesalsoaddstothespread[ 20 ]. Withallofthesefactorscontributingtotheadvancementofcitrusgreening,itis verydifculttokeepthediseaseundercontrol.However,becauseoftheimportance ofcitruscropstothestate'seconomy,variousattemptshavebeenmadetoimplement controlstrategies.Oneoptionistorogue(orremove)infectedtrees.Thishelpsto preventmorepsyllidsfromacquiringthedisease.Unfortunately,becauseatreemay beinfectedforacoupleyearsbeforeanysymptomsaredetected,itisdifculttoknow 10

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whichtreesmustberogued[ 17 ].Especiallyincommercialgroves,farmersarereluctant toeliminateahealthy-lookingtreethatisstillcapableofproducinggoodfruit.Oncea treeisrogued,anothercanbereplantedinitsplace.However,itispossiblethatthe remnantsoftreerootsinthesoilmayserveasareservoirforthedisease,asithasbeen observedthatrecentlyplantedtreesaremoresusceptible,creatinganothersetbackto thiscontrolstrategy[ 35 ].Anothermethodistotreatunhealthytreeswithantibiotics.This canbeveryeffective,butitisverycostly.Itisalsonotapermanentx;symptomswill returnaboutayearafterbeingtreated[ 20 ].Witheithercontrolstrategy,itisimportant toknowwhichasymptomatictreesareactuallyinfectedwiththedisease.Onewayis totestplanttissuewiththepolymerasechainreactiontechnique(PCR).However,it isnotfeasibleforeverysingletreeinagrovetobetestedthisway.Also,thedisease isnotevenlydistributedwithinagiventree,andthusthetestmaynotalwaysproduce accurateresults.Manyfarmersusescoutstovisuallyinspecttreesforpsyllids,butitis generallydifculttoseetheminisculeinsectsthatareonlythreetofourmillimetersin diameter[ 20 ]. Thereareafewcontrolstrategiesthataremorerealisticandeasiertoimplement, thoughlesseffective.Theseincludesprayinginsecticidesoverentiregrovesaswell aseliminatinggrafting[ 20 ].Itisalsoimportantforcitrusstocktobeinspectedand screenedbeforeitistransportedorsold,anditisverybenecialfornurseriesthatgrow citrustobescreened-in[ 17 ].AquarantinethatbeganJune17,2010,isdesignedto preventthetransportationofcitrusthroughoutthestateofFlorida[ 3 ]. Citrusgreeninghasbecomeverywide-spreadandisdifculttoeradicate.Because citrusissuchanimportantpartofFlorida'seconomy,thisdiseaseisanissueofgreat concern.ArecentstudybytheUniversityofFlorida'sInstituteofFoodandAgricultural Sciencesestimatesthatfrom2006to2011citrusgreeninghascaused$3.63billion inlostrevenueandover6,000lostjobsinthestate.Theve-yearproductionlevelfor orangejuiceisestimatedtobe1.7billiongallonslessthanprojected[ 21 ].Ifthedisease 11

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continues,eventuallytherewillnotbeenoughcitrusproducedforthejuiceplantsto operate;manyplantshavealreadybeguntoimportextracitrusfromBrazilinorderto keepuptheirproductionlevels[ 35 ].Unfortunately,manybelievethatthedamagemay beirreversibleinthestateofFlorida.However,HLBhasalsobeendetectedinGeorgia, SouthCarolina,andLouisiana.InJanuary2012,theTexasDepartmentofAgriculture andUSDAconrmedthedetectionofthediseaseinacommercialgroveinTexas[ 26 ]. TherstincidencesofHLB-infectedtreeswerealsodiscoveredinCaliforniaandArizona in2012,andasiancitruspsyllidsalonehavebeendiscoveredinHawaii,Mississippiand Alabama.ItisclearthatresearchontheprogressionofthediseaseinFloridacouldbe instrumentalinpreventingthesameoutbreakinotherstates. 1.2Preliminaries Thefollowingdenitionsandtheoremswillbeusedthroughoutthisdissertation. 1.2.1EquilibriaandSolutionsofDifferentialEquations Webeginwiththebasicdenitionofstabilityofanequilibriumpoint,andalso provideresultspertainingtosolutionsofdifferentialequations.Thesefollowfrom Chicone[ 9 ]andSmithandWaltman[ 32 ]unlessotherwisenoted. Denition1. Let t ( x ) beasolutiontothedifferentialequation x = f ( x ) x $ R with 0 ( x )= x .Anequilibriumpoint x 0 ofthedifferentialequationis stable ifforeach > 0 thereexists # > 0 suchthat | t ( x ) # x 0 | < forall t % 0 whenever | x # x 0 | < # .An equilibriumpoint x 0 is asymptoticallystable ifitisstableandthereexists # > 0 such that lim t !" | t ( x ) # x 0 | =0 whenever | x # x 0 | < # .Anequilibriumpoint x 0 is unstable ifit isnotstable. Denition2. Supposethat JxU isanopensubsetof R x R n .Afunction : JxU & R n givenby ( t x ) '& t ( x ) iscalleda ow ofthedifferentialequation x = f ( x ) ifforeach xed x 0 ( x ) ( x and t + s ( x )= t ( s ( x )) Denition3. Aset S ) R n iscalledan invariantset forthedifferentialequation x = f ( x ) x $ R n ,if,foreach x in S ,thesolution t & t ( x ) ,denedonitsmaximal 12

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intervalofexistence,hasitsimagein S S isa maximalinvariantset if,foranyinvariant set T suchthat S T + = , S T Denition4. ([ 31 ])Let ( X d ) beametricspace,andlet ,+ = M X M iscalleda uniformrepeller under ifthereexistssome > 0 suchthat liminf t !" d ( t ( x ), M ) % forall x $ X \ M Denition5. The stablemanifold ofanequilibriumpoint x 0 foranautonomousdifferentialequation x = f ( x ) withow t isthesetofallpoints x suchthat lim t !" t ( x )= x 0 Theorem1.1. ([ 14 ])Let S beacompactsubsetof X suchthat X \ S ispositively invariant.Anecessaryandsufcientconditionfor S tobeauniformrepelleristhatthere existaneighborhood U of S andacontinuousfunction P : X & R + satisfyingthe followingconditions: (1) P ( x )=0 ifandonlyif x $ S (2)forall x $ U \ S thereexists T ( x ) > 0 suchthat P ( T ( x ) ( x )) > P ( x ) Denition6. Let f : R xD & R n ,where D isanopensubsetof R n ,beavector-valued function, f =( f 1 f 2 ,..., f n ) .Thefunction f issaidtobeof typeK in D if,foreach i and all t f i ( t a ) f i ( t b ) foranytwopoints a and b in D satisfying a i b i forall i Theorem1.2. (Kamke'sTheorem)Let f beLipschitzcontinuouson R xD andoftype K Let x ( t ) beasolutionof x = f ( t x ) denedon [ a b ] .If z ( t ) isacontinuousdifferentiable functionon [ a b ] satisfying z f ( t z ) on ( a b ) with z ( a ) x ( a ) ,then z ( t ) x ( t ) forall t in [ a b ] .If y ( t ) iscontinuousanddifferentiableon [ a b ] satisfying y % f ( t y ) on ( a b ) with y ( a ) % x ( a ) ,then y ( t ) % x ( t ) forall t in [ a b ] Theorem1.3. (Gronwall'sInequality)Supposethat a < b andlet $ and % be nonnegativecontinuousfunctionsdenedontheinterval [ a b ] .Moreover,supposethat $ isdifferentiableon ( a b ) withnonnegativecontinuousderivative $ .If,forall t $ [ a b ] ( t ) $ ( t )+ t a % ( s ) ( s ) ds 13

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then ( t ) $ ( t ) e t a ( s ) ds forall t $ [ a b ] 1.2.2Matrices Inthissectionwedenethespectralradiusofamatrixandgiveseveralresultsthat willbeusedtoanalyzestability.Inparticular,thePerron-FrobeniusTheorem,whichis ofgreatimportance,canbefoundhere.Wealsoprovideaspecialresultpertainingto Hurwitzmatrices.Unlessotherwisenoted,theseresultscanbefoundinBermanand Plemmons[ 5 ]andThieme[ 34 ]. Denition7. Let & 1 ,..., & n betheeigenvaluesofan nxn matrix A .Thenthe spectral radius is ( A )=max i ( | & i | ) Denition8. Let & 1 ,..., & n betheeigenvaluesofan nxn matrix A .Thenthe spectral bound is s ( A )= max i ( / ( & i )) Denition9. Asquarematrix A iscalled quasi-positive ifalloff-diagonalentriesare nonnegative. Denition10. Amatrix A is reducible ifthereexistsapermutationmatrix P suchthat P # AP isblocktriangular. A is irreducible ifitisnotreducible. Denition11. Amatrix A % 0 is primitive ifthereexists k $ N suchthat A k > 0 Theorem1.4. (Perron-FrobeniusTheorem)(1)If A ispositive,then ( A ) isasimple eigenvalue,greaterthanthemagnitudeofanyothereigenvalue. (2)If A % 0 isirreducible,then ( A ) isasimpleeigenvalue,anyeigenvalueof A ofthe samemodulusisalsosimple, A hasapositiveeigenvector x correspondingto ( A ) ,and anynonnegativeeigenvectorof A isamultipleof x (3)If A isquasi-positiveandirreducible,then s ( A ) isaneigenvalueof A associatedboth withapositiveeigenvectorof A andapositiveeigenvectorofthetransposedmatrix A # Moreover,if x > 0 isavectorand u $ R suchthat Ax % ux ,thereexistssomevector z > 0 andsomescalar & % u suchthat Az = & z andinparticular s ( A ) % u 14

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Theorem1.5. If A B arematricessuchthat 0 A B ,then ( A ) ( B ) Denition12. Asquarematrix A is Hurwitz ifalleigenvaluesof A havestrictlynegative realpart. Theorem1.6. ([ 9 ])Let A bean nxn matrixthatisHurwitzandlet 0 0 beanormon R n Thenthereareconstants k a > 0 suchthatforall v $ R n andall t % 0 0 e tA v 0" ke $ at 0 v 0 1.2.3FloquetTheoryforPeriodicSystems Toanalyzetheperiodicsystem( 51 )-( 56 ),wewillneedsomebackground informationonFloquettheory.ThesedenitionsandresultsarefromChicone[ 9 ] unlessotherwisespecied. Considerthelinearsystem x = A ( t ) x x $ R n (11) where t & A ( t ) isacontinuous T -periodicmatrix-valuedfunction. Therstpropositionandtheorem,whilenotactuallyapartofFloquettheory,willbe usedintheanalysisofourperiodicsystem. Proposition1.1. (Liouville'sFormula)Supposethat t & ( t ) isamatrixsolutionofthe homogeneouslinearsystem 11 ontheopeninterval J .Thenforany t t 0 $ J det ( t )=det ( t 0 ) e t t 0 tr A ( s ) ds Theorem1.7. ([ 6 ])Let X R beaconnectedintervaland C beasimplyconnected openregion.If f ( x & ) iscontinuousin ( x & ) andanalyticin & throughouttheregion ( X ) ,thesamewillbetrueof ( f / (& Denition13. An nxn matrix-valuedfunction X ( t ) on R iscalleda fundamentalmatrix solution ofthelinearsystem 11 ifitscolumnsformalinearlyindependentsetofvector 15

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solutions.Thefundamentalmatrixsolution X ( t ) isthe principalfundamentalmatrix solution if X (0)= I Theorem1.8. (Floquet'sTheorem)If X ( t ) isafundamentalmatrixsolutionofthe T -periodicsystem 11 ,then,forall t $ R X ( t + T )= X ( t ) X $ 1 (0) X ( T ). Inaddition,thereisamatrix B (whichmaybecomplex)suchthat e TB = X $ 1 (0) X ( T ) anda T -periodicmatrixfunction t & P ( t ) (whichmaybecomplex-valued)suchthat X ( t )= P ( t ) e tB forall t $ R .Also,thereisarealmatrix R andareal 2 T -periodicmatrix function t & Q ( t ) suchthat X ( t )= Q ( t ) e tR forall t $ R Denition14. The Floquetmultipliers ofa T -periodiclineardifferentialequation,such as 11 ,aretheeigenvaluesoftheprincipalfundamentalsolutionevaluatedat t = T Thatis,theFloquetmultipliersaretheeigenvaluesof X ( T ) Theorem1.9. (1)IftheFloquetmultipliersoftheperiodicsystem 11 allhavemodulus lessthanone,thenthezerosolutionisasymptoticallystable. (2)IfatleastoneFloquetmultiplieroftheperiodicsystem 11 hasmodulusgreaterthan one,thenthezerosolutionisunstable. Nowconsiderthenonlinearsystem x = F ( t x ), x $ R n (12) where F : R x R n & R n isa T -periodicsmoothfunctionsuchthat F ( t + T x )= F ( t x ) forall x $ R n andall t $ R .Let t ( x ) bethesolutionto 12 with 0 ( x )= x Denition15. If t ( x ) isdenedonthetimeinterval [0, T ] ,the Poincar ` emap isdened as P ( x )= T ( x ) 16

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Thesetof T -periodicsolutionsofthesystem 12 isinone-to-onecorrespondence withthexedpointsofthePoincar ` emap. Denition16. A T -periodicsolution y ( t ) ofthesystem 12 is locallyorbitallystable if foreachopenset V ) R n thatcontains { y ( t ):0 t T } ,thereisanopenset W ) V suchthateverysolution,startingatapointin W at t =0 ,staysin V forall t % 0 .The periodicsolutioniscalled locallyasymptoticallystablewithaphase if,inaddition, thereexists # > 0 suchthatforeach x 0 with | x 0 # y ( t ) | < # ,thereexists t ( x 0 ) $ [0, T ) withthepropertythat ( t x 0 ) & y ( t + t ( x 0 )) as t &1 ThederivativeofthePoincar ` emapatapoint p is DP ( p )= " x T ( p ) .Thisderivative istheprincipalfundamentalsolutionat t = T ofthevariationalproblem W = F x ( t t ( p )) W W (0)= I (13) Theorem1.10. Thestabilityofa T -periodicsolution y ( t ) ofthesystem 12 isdeterminedbytheFloquetmultipliersoftheperiodicvariationalsystem 13 .Thatis,when ( DP ( p )) < 1 y ( t ) islocallyasymptoticallystable.When ( DP ( p )) > 1 y ( t ) is unstable. 17

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CHAPTER2 MATHEMATICALMODELWITHCONSTANTPARAMETERS 2.1DevelopmentoftheModel Wedevelopamodelthatrepresentsthespreadofcitrusgreeningdiseasewithina singlegroveoftrees.Thecitrustreepopulationisdividedintofourstages. S denotes susceptibletreesand R representsdeadtrees.Duetotheobserveddelayinthe appearanceofsymptomsofcitrusgreening,wesplittheinfectedtrees I intoan asymptomatic(latent)stage, I 1 ,andasymptomaticstage, I 2 .Forthepsyllidvector populationwelet V $ and V + representtheuninfected/susceptibleandinfectedpsyllids, respectively.Weassumethatthegroveissubjecttoarogueandreplantdisease managementstrategy.Inthemodel,symptomaticanddeadtreesareroguedatarate andthecorrespondingspotsarereplantedwithnewtrees.Becausethesoilmay serveasareservoirforthedisease,weassumethataproportion f $ [0,1] ofthenewly plantedtreeswillbehealthyandaproportion 1 # f willbecomeinfectedimmediately.A schematicforthemodelisprovidedinFigure 2-1 Figure2-1. Flowdiagramforthegrove-scalecitrusgreeningmodel(2 # 1)-(2 # 6) Weassumethatinfectedtreesprogressfromthelatentstatetothesymptomatic stateattherate ) .Theparameter $ representsthedisease-associatedmortalityofthe symptomatictrees;thenaturaldeathrateofthetreesisneglected.Themortalityrateof thevectorsisgivenby ,whichisassumedtobethesameforuninfectedandinfected 18

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psyllids.Weassumethatthereisaxednumberofcontactsperunittimebetween treesandpsyllids.Thebitingrate ,givenasunits/time,accountsfortheproportion ofbitesthatresultinsuccessfultransmission.Thetotalnumberoftreesisgivenby N = S + I 1 + I 2 + R while N 0 representsthenumberofspacesinthegrove.Wewill assumethroughoutthat N N 0 .Thetreesareassumedtobereplantedataratethat isproportional,withconstant b ,tothenumberofemptyspacesinthegrove.Thetotal psyllidpopulationisrepresentedby V = V $ + V + withlogisticgrowthrate r andcarrying capacity K .Asanexample,theinfectionrateoftreesisgivenbytheproductofthe bitingrateandtheprobabilitythatabiteinvolvesahealthytreeandaninfectedpsyllid. Withtheaboveassumptions,themodeltakesthefollowingform: I 1 = S N V + V # ) I 1 + (1 # f )( I 2 + R ), (21) I 2 = ) I 1 # ( $ + ) I 2 (22) R = $ I 2 # R (23) V + = V $ V I 1 + I 2 N # V + (24) S = b 1 # N N 0 # # S N V + V + f ( I 2 + R ), (25) V $ = rV 1 # V K # # V $ V I 1 + I 2 N # V $ (26) 2.2Equilibria Webeginouranalysisofthemodelbymakingthefollowingimmediateobservations. First,duetothespecicformofEquations( 21 )-( 26 ),themodeliswell-posed;thatis, allpositivesolutionsarewell-denedandremainpositiveinforwardtime.Second, addingEquations( 21 )-( 23 )and( 25 ),wendthatthetotaltreepopulationis governedby N = b 1 # N N 0 # 19

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Similarly,whenweaddEquations( 24 )and( 26 ),thetotalvectorpopulationsatises V = rV 1 # V K # # V Consequently,allpositivesolutionsof( 21 )-( 26 )areboundedandcanbeextendedto all t $ (0,+ 1 ) .Additionally, N ( t ) & N 0 and V ( t ) & V # =(1 # r ) K (weassumethat < r sothat V # > 0 )exponentiallyfastas t & + 1 Nowwelookforsteadystates ( I # 1 I # 2 R # V # + S # V # $ ) ofsystem( 21 )-( 26 ).Clearly, N = N 0 and V = V # atequilibrium.Trivially, x 0 =(0,0,0,0, N 0 V # ) isanequilibrium whichwedesignateasthedisease-freeequilibrium(DFE).Tondendemicequilibria, weset( 21 )-( 26 )equaltozeroandobtain I 2 = ) $ + I 1 R = $ I 2 V + = V # ( I 1 + I 2 ) N 0 V # + ( I 1 + I 2 ) Thuseachinfectedstatecanbeexpressedintermsof I 1 .If I 1 =0 weseethat I 2 = R = V + =0 andweareinthecaseofthedisease-freeequilibrium.Hencethereexistsan endemicequilibriumifandonlyif I 1 > 0 .If I 1 + =0 ,( 25 )yields S = N 0 f ) I 1 + N 2 0 f V # 2 $ 1 # + 1 $ + % % Recallthat N 0 = S + I 1 + I 2 + R ,sowecancompute N 0 = N 0 f ) I 1 + N 2 0 f V # 2 ( 1 # + 1 $ + % ) + 1+ ) # I 1 Thereforethenecessaryconditionforexistenceofapositive(endemic)equilibriumis: I 1 > 0 2 N 2 0 f V # 2 ( 1 # + 1 $ + % ) < N 0 2 2 ( 1 # + 1 $ + % ) N 0 V # f > 1. (27) Thelastinequalityiscloselyrelatedtothebasicreproductivenumber R 0 whichwe discussinthefollowingsection.Infact, I 1 > 0 ifandonlyif R 0 > 1 ;thatis,theendemic equilibriumexistsifandonlyifthebasicreproductivenumberisgreaterthanone. 20

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2.3TheBasicReproductiveNumber R 0 Thebasicreproductivenumber R 0 representstheaveragenumberofsecondary infectionsthatresultfromtheintroductionofasingleinfectedindividual(atreeora vector)intoasusceptiblepopulation.Therearevariousapproachesforcalculating R 0 whichleadstosomeambiguityinitsdenition.Forinstance,vandenDreisscheand Watmoughdene R 0 tobethespectralradiusofthenextgenerationmatrix FV $ 1 [ 36 ].In Subsection 2.3.1 wecalculate FV $ 1 foroursystem( 21 )-( 26 )andnd R 0 = 1 # f 2 + & (1 # f ) 2 4 + 2 V # N 0 1 ) + 1 $ + # (28) WeprovebelowinSubsection 2.3.2 theexistenceofanequivalentthresholdcondition involvingthequantity T 0 = 2 V # N 0 1 ) + 1 $ + # +1 # f (29) Thequantity T 0 providesthefollowingbiologicalinterpretation.Supposethatasingle infectedtreeinstage I 1 isintroducedintoacompletelysusceptiblegrove.Theaverage numberofsecondaryinfectionsresultingfrompsyllidcontactduringthe I 1 and I 2 stages ofthetreeare,respectively, 2 V # N 0 ) and 2 V # N 0 ( $ + ) Thetreewillnecessarilyberoguedineitherthe I 2 or R stageandwillonaverage produce 1 # f newlyinfected I 1 trees.Thustheexpectednumberofsecondaryinfections isexactly T 0 .InTheorem 2.1 ,weprovetheequivalenceof T 0 and R 0 aswellasa conditionforlocalstabilityoftheDFE. 2.3.1Calculationof R 0 Mathematically, R 0 hasbeendenedtobethespectralradiusofthenextgeneration matrix[ 36 ].Thatis,werewritethevectoreldof( 21 )-( 26 )as x i = F i ( x ) # V i ( x ), i =1,2,3,4,5,6, 21

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where F i ( x ) representstherateofnewinfectionsappearinginstate i and V i = V $ i # V + i where V + i istherateofindividualsenteringstate i byallmeansotherthananew infectionand V $ i istherateofindividualsleavingstate i .Wenowconsideronlythe 4 x 4 matrixcorrespondingtothestates I 1 I 2 R V + ,asthisdenitionfor R 0 dependsonlyon transitionsintoandoutoftheinfectedcompartments.Welet F = ( F i ( x j ( x 0 ) ( V = ( V i ( x j ( x 0 ) ( 1 i j 4, where x 0 istheDFE.Thus F isthematrixofderivativescorrespondingtonewinfections intheinfectedcompartmentswhile V isthematrixofderivativescorrespondingtoall othermodesofenteringorexitinganinfectedcompartment.Thenthebasicreproductive numberisdenedas R 0 = ( FV $ 1 ). Wecomputethenextgenerationmatrix FV $ 1 forsystem( 21 )-( 26 ).Calculationgives F = ) * * * + 0 (1 # f ) (1 # f ) & V 00 00 00 00 & N 0 & N 0 00 , V = ) * * * + ) 000 # )$ + 00 0 # $' 0 000 , and V $ 1 = ) * * * + 1 # 000 1 $ + % 1 $ + % 00 $ % ( $ + % ) $ % ( $ + % ) 1 % 0 000 1 . 22

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Therefore FV $ 1 = ) * * * + 1 # f 1 # f 1 # f & V 0 000 0 000 & N 0 $ 1 # + 1 $ + % % & N 0 ( $ + % ) 00 . Clearly, FV $ 1 hastwozeroeigenvalues.Theremainingeigenvaluesaredeterminedby & 2 # (1 # f ) & # 2 V # N 0 1 ) + 1 $ + # =0. Thus,thespectralradiusof FV $ 1 is: ( FV $ 1 )= 1 # f 2 + & (1 # f ) 2 4 + 2 V # N 0 1 ) + 1 $ + # = R 0 2.3.2LocalStabilityoftheDisease-FreeEquilibrium Theorem2.1. Forthesystem( 21 )-( 26 )and R 0 and T 0 asdenedin( 28 )and( 29 ), respectively: 1. R 0 < 1 ifandonlyif T 0 < 1 2. T 0 < 1 ifandonlyifalleigenvaluesoftheJacobianmatrixofsystem( 21 )-( 26 ) evaluatedattheDFEhavenegativerealparts. Proof. ItisshowninSubsection 2.3.1 that R 0 isthelargestpositiverootof p ( & )= & 2 # (1 # f ) & # 2 V # N 0 1 ) + 1 $ + # Thismeansthat 0= R 2 0 # (1 # f ) R 0 # ( T 0 # (1 # f )), whichrearrangesto T 0 # 1=( R 0 # 1)( R 0 + f ). Since R 0 + f ispositive,thesignsof T 0 # 1 and R 0 # 1 mustbethesameandtherst assertionfollows. 23

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Toprovethesecondassertion,wecomputetheJacobianmatrixofsystem ( 21 )-( 26 )attheDFE: J ( x 0 )= ) * * * * * * * + # )' (1 # f ) (1 # f ) & V 00 ) # ( $ + )0 000 0 $ # 000 & N 0 & N 0 0 # 00 # b N 0 # b N 0 + f # b N 0 + f # & V # & N 0 0 # & N 0 # & N 0 02 # r 0 # r . Weseethat J ( x 0 ) hasablocktriangularformandtherefore # & N 0 and # r are eigenvaluesof J ( x 0 ) .Itfollowsfromourassumption < r thatthesetwoeigenvalues arenegative.Welet J ( x 0 ) betheupperleft 4 3 4 submatrixof J ( x 0 ) .Thentheremaining eigenvaluesaredeterminedbythecharacteristicequationof J ( x 0 ) : p ( & )=( & + )( & + )( & + ) )( & + $ + ) # 2 V # N 0 ( & + )( & + $ + + ) ) # )' (1 # f )( & + )( & + + $ ). Anyroot & of p ( & ) withRe ( & ) % 0 isalsoarootof q ( & ) denedby q ( & )= p ( & ) ( & + )( & + )( & + ) )( & + $ + ) =1 # 2 ( & + $ + + ) ) V # N 0 ( & + )( & + ) )( & + $ + ) # )' (1 # f ) ( & + )( & + ) ) Weobservethat q ( & ) ismonotoneincreasingin & when & > 0 .Thereforesincethe leadingcoefcientof p ( & ) ispositiveweknowthat p ( & ) hasnopositiverealrootsif andonlyif q (0) > 0 .Thereforeallrealeigenvaluesof J ( x 0 ) arenegativeifandonlyif q (0) > 0 ,whichisequivalentto 2 V # N 0 1 ) + 1 $ + # +1 # f < 1; thatis,allrealeigenvaluesof J ( x 0 ) arenegativeifandonlyif T 0 < 1 .Toprovethatall complexeigenvaluesof J ( x 0 ) havenegativerealpartswedene G ( & )=1 # q ( & ) and 24

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suppose p ( & )=0 withRe ( & ) % 0 andIm ( & ) + =0 .Then G ( & )=1 where G ( & )= 2 V # N 0 ( & + )( & + ) ) + 2 ) V # N 0 ( & + )( & + ) )( & + $ + ) + )' (1 # f ) ( & + )( & + ) ) Weclaim | G ( & ) | < G ( Re ( & )) .Indeed, | G ( ) | " 2 V # N 0 | + || + # | + 2 # V # N 0 | + || + # || + $ + % | + #% (1 # f ) | + % || + # | < 2 V # N 0 ( Re ( )+ )( Re ( )+ # ) + 2 # V # N 0 ( Re ( )+ )( Re ( )+ # )( Re ( )+ $ + % ) + #% (1 # f ) ( Re ( )+ % )( Re ( )+ # ) = G ( Re ( )), wherethesecondinequalityisstrictsinceIm ( & ) + =0 .Thereforewehave T 0 < 1 2 q (0) > 0 2 G (0) < 1 ,whichimplies G ( Re ( & )) < 1 since G isdecreasingin & .Then | G ( & ) | < G ( Re ( & )) < 1 givesacontradiction. Inadditiontoprovingtheequivalenceofthethresholdconditions T 0 and R 0 Theorem 2.1 alsostatesthattheDFEislocallyasymptoticallystablewhen R 0 < 1 ,while R 0 > 1 indicatestheDFEisunstable. 2.4ExtinctionoftheDiseasewhen R 0 1 WenowproveaglobalstabilityresultfortheDFEinthecase R 0 1 Theorem2.2. If R 0 1 ,thenallnonnegativesolutionsof( 21 )-( 26 )convergetothe DFE (0,0,0,0, N 0 V # ) Proof. Suppose limsup t !" I 1 ( t )= m > 0 .Thenforevery > 0 thereexists + 1 > 0 such that I 1 ( t ) m + forall t % + 1 .Substituting,wehavethat I 2 ( t ) ) ( m + ) # ( $ + ) I 2 ( t ) forall t % + 1 .Thenthereexists + 2 > + 1 suchthat I 2 ( t ) ) ( m + ) $ + + 25

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forall t % + 2 .Thismeansthat I 1 ( t )+ I 2 ( t ) m +2 + ) ( m + ) $ + forall t % + 2 .Substitutingagain,wegetthat R ( t ) $ ) ( m + ) $ + + # # R ( t ) forall t % + 2 .Thusthereexists + 3 > + 2 suchthat R ( t ) $ ) ( m + ) $ + + # + forall t % + 3 .So I 2 ( t )+ R ( t ) ) ( m + )+ $" +2 forall t % + 3 Wehavethat N ( t )= b (1 # N N 0 ) ,so N ( t ) & N 0 as t &1 .Thusthereexists + 4 > 0 suchthat N 0 # " N ( t ) forall t % + 4 .Wealsohavethat V ( t )= rV (1 # V K ) # V ,and thus V ( t ) & V # as t &1 .Sothereexists + 5 > 0 suchthat V # # " V ( t ) forall t % + 5 Wenowsubstitutethistogetthat V + ( t ) m +2 + # ( m + ) $ + % N 0 # # V + ( t ) forall t % + 2 + 4 .Thenthereexists + 6 > max { + 2 + 4 } suchthat V + ( t ) $ m +2 + # ( m + ) $ + % % ( N 0 # ) + forall t % + 6 .Nowwecanrewrite 26

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I 1 ( t ) S ( t ) N ( t ) & ( m +2 + ( m + ) # + $ ) + ( N 0 $ ) ( N 0 $ )( V $ ) +(1 # f ) $ # ( m + )+ $' % +2 % # ) I 1 ( t ) N ( t ) $ I 1 ( t ) N ( t ) & ( m +2 + ( m + ) # + $ ) + ( N 0 $ ) ( N 0 $ )( V $ ) +(1 # f ) $ # ( m + )+ $' % +2 % # ) I 1 ( t ) & ( m +2 + ( m + ) # + $ ) + ( N 0 $ ) ( N 0 $ )( V $ ) +(1 # f ) $ # ( m + )+ $' % +2 % # ) I 1 ( t ) # & ( m +2 + ( m + ) # + $ ) + ( N 0 $ ) ( N 0 $ )( V $ )( N 0 + ) I 1 ( t ) forall t % + 3 + 5 + 6 Thenthereexists + 7 > max { + 3 + 5 + 6 } suchthat I 1 ( t ) & ( m +2 + ( m + ) # + $ ) + ( N 0 $ ) ( N 0 $ )( V $ ) +(1 # f ) $ # ( m + )+ $' % +2 % ) + & ( m +2 + ( m + ) # + $ ) + ( N 0 $ ) ( N 0 $ )( V $ )( N 0 + ) + forall t % + 7 .RecallfromSection 2.3 thequantity T 0 whosethresholdbehavioris equivalenttothatof R 0 .Nowas & 0 ,theinequalitybecomes I 1 ( t ) % 2 ( m + m # + $ ) N 0 V +(1 $ f ) # m # + % 2 ( m + m # + $ ) N 2 0 V = m # % 2 ( 1 + 1 # + $ ) N 0 V +(1 $ f ) # # + m N 0 % 2 ( 1 + 1 # + $ ) N 0 V # = mT 0 1+ m N 0 ( T 0 $ (1 $ f )) = mT 0 1 $ m N 0 (1 $ f )+ m N 0 T 0 Notethat m =limsup t !" I 1 ( t ) limsup t !" N ( t )= N 0 ,sotheconstant 1 # m N 0 (1 # f ) isalwayspositive.Thismeansthat mT 0 1 $ m N 0 (1 $ f )+ m N 0 T 0 isanincreasingfunctionof T 0 therefore I 1 ( t ) m 1+ f m N 0 < m 27

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since R 0 1 implies T 0 1 .Thus limsup t !" I 1 ( t ) < m ,acontradiction.So m =0 Then limsup t !" I 2 ( t )=limsup t !" R ( t )=limsup t !" V + ( t )=0 followsaswell fromtheinequalitiesobtainedthroughouttheproof.Thismeansthatwemusthave lim t !" S ( t )= N 0 and lim t !" V $ ( t )= V # .Soallnonnegativesolutionsconvergetothe DFE. InFigure 2-2 ,wepresentanumericalsimulationofsystem( 21 )-( 26 )with f = 0.75 r =1.5 b = N 0 = ) = = V # =1 $ = =1.5 =0.25 andinitialcondition ( I 1 (0), I 2 (0), R (0), V + (0), S (0), V $ (0))=(0.5,0.25,0,0.5,0.25,0.5) toshowtheextinction ofthedisease.Withthesehypothesizedparametervalues R 0 4 0.716 ,andweseefrom thesimulationthatthesolutionconvergestothedisease-freeequilibrium. Figure2-2. Simulationofsystem(2 # 1)-(2 # 6)usingMATLABode45solverwith f =0.75 r =1.5 b = N 0 = ) = = V # =1 $ = =1.5 =0.25 ,and initialcondition ( I 1 (0), I 2 (0), R (0), V + (0), S (0), V $ (0))=(0.5,0.25,0,0.5,0.25,0.5) 2.5PersistenceoftheDiseasewhen R 0 > 1 Wenowturntothecasewhen R 0 > 1 .Wearenotabletoanalyticallyestablish thelocalstabilityoftheendemicequilibriumbutsimulationssuggestthatitisstable 28

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wheneveritexists.InFigure 2-3 ,wepresentanumericalsimulationofsystem ( 21 )-( 26 )with f =0.5 r =1.5 b = N 0 = $ = = ) = = = V # =1 andinitialcondition ( I 1 (0), I 2 (0), R (0), V + (0), S (0), V $ (0))=(0.1,0,0,0,0.9,1) Withthesehypothesizedparametervalues R 0 =1.5 andtheendemicequilibriumis ( I # 1 I # 2 R # V # + S # V # $ )=(0.267,0.133,0.133,0.286,0.467,0.714) .Weseefromthe simulationthatthesolutiondoesindeedconvergetotheendemicequilibrium. Figure2-3. Simulationofsystem(2 # 1)-(2 # 6)usingMATLABode45solverwith f =0.5 r =1.5 b = N 0 = $ = = ) = = = V # =1 ,andinitialcondition ( I 1 (0), I 2 (0), R (0), V + (0), S (0), V $ (0))=(0.1,0,0,0,0.9,1) 2.5.1PersistenceoftheDiseaseforFullSystem Thoughstabilityoftheendemicequilibriumisnotproved,weareabletoprove stronguniformpersistenceofthediseaseinthecase R 0 > 1 .Taking N = N 0 and V = V # weconsiderthelimitingsystemoftheinfectedcomponents: I 1 = 1 # I 1 + I 2 + R N 0 # V + V # # ) I 1 + (1 # f )( I 2 + R ), (210) I 2 = ) I 1 # ( $ + ) I 2 (211) R = $ I 2 # R (212) V + = 1 # V + V # # I 1 + I 2 N 0 # V + (213) 29

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withsolutionsrestrictedtothefeasibleregion # = { ( I 1 I 2 R V + ):0 I 1 + I 2 + R N 0 ,0 V + V # } ,apositivelyinvariantsubsetof R 4 + Theorem2.3. Suppose R 0 > 1 .Thenthereexists 0 > 0 suchthat liminf t !" d ( x ( t ), ( R 4 + ) > 0 forallpositivesolutions x ( t ) of( 210 )-( 213 ). Proof. Itsufcestoconsider 0 < f 1 becausetheresultfollowsfromTheorem 2.4 when f =0 .Werstprovethatthereexists 1 > 0 suchthat limsup t !" I 1 ( t )+ I 2 ( t )+ R ( t ) % 1 forallpositivesolutionsof( 210 )-( 213 ).Considerthematrix A ( )= ) * * * + # ) (1 # f ) (1 # f ) & V (1 # N 0 ) ) # ( $ + ) 0 0 0 $ # 0 & N 0 (1 # &' V N 0 ) & N 0 (1 # &' V N 0 )0 # for > 0 .Observethatas & 0 A ( ) & J ( x 0 ) where J ( x 0 ) istheupperleftfourbyfour submatrixcontainedintheJacobianmatrixofthesystem( 21 )-( 26 ).Bycontinuityof eigenvalues,theeigenvaluesof A ( ) convergetotheeigenvaluesof J ( x 0 ) .Since R 0 > 1 byTheorem 2.1 thereexistsaneigenvalueof J ( x 0 ) thathaspositiverealpart.Therefore thereexists 1 > 0 suchthat A ( 1 ) hasaneigenvaluewithpositiverealpart. Bywayofcontradiction,suppose X ( t )=( I 1 I 2 R V + )( t ) isapositivesolutionand limsup t !" I 1 ( t )+ I 2 ( t )+ R ( t ) < 1 .Weseefrom( 213 )that limsup t !" V + ( t ) < &' 1 V N 0 Thusthereexists + > 0 suchthatforall t > + I 1 % V # 1 # 1 N 0 # V + # ) I 1 + (1 # f )( I 2 + R ), I 2 % ) I 1 # ( $ + ) I 2 R % $ I 2 # R V + % N 0 1 # *" 1 N 0 V # # ( I 1 + I 2 ) # V + 30

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Wedene s = t # + andlet Y ( s )=( i 1 i 2 r v + )( s ) $ R 4 + bethesolutionof Y = A ( 1 ) Y withinitialcondition Y (0)= X ( + ) .ApplyingKamke'sTheorem(Theorem 1.2 ),we concludethat X ( s ) % Y ( s ) forall s % 0 Let & betheeigenvalueof A ( 1 ) withthelargestrealpart.Since A ( 1 ) isquasi-positive, byPerron-Frobenius(Theorem 1.4 )thereexistsanonnegativeeigenvector v of A ( 1 ) witheigenvalue & .Choose k sufcientlysmallsothat kv Y (0) entrywise.Thenfor Z ( s )= ke ( s v wehave Z = A ( 1 ) Z and Z (0) Y (0) ,whichimpliesthat Z ( s ) Y ( s ) for all s % 0 .Thenthereexistsacomponent Z i ( s ) suchthat lim s !" Z i ( s )= 1 .Therefore lim s !" Y i ( s )= 1 ,whichimpliesthat lim s !" X i ( s )= 1 ,acontradiction.Therefore limsup t !" I 1 ( t )+ I 2 ( t )+ R ( t ) % 1 forallpositivesolutions x ( t ) of( 210 )-( 213 ). Nowlet M bethemaximalinvariantsetin ( R 4 + # .Suppose x $ M .Let x ( t )= ( I 1 I 2 R V + )( t ) bethesolutionto( 210 )-( 213 )with x (0)= x .Supposethat V + ( t 1 ) > 0 forsome t 1 % 0 .Since x $ ( R 4 + thereexistsan i $ { 1,2,3 } suchthat x i ( t 1 )=0 Suppose I 1 ( t )=0 forall t % t 1 .Then ( I 2 + R )= # ( I 2 + R ) sothereexists t 2 > t 1 suchthat ( I 2 + R )( t 2 ) < N 0 .Then I 1 ( t 2 ) > 0 andcontinuity impliesthatthereexists t 3 > t 2 suchthat I 1 ( t 3 ) > 0 .Thenif I 2 ( t 3 )=0 wehave I 2 ( t 3 )= ) I 1 ( t 3 ) > 0 so I 2 ( t ) becomespositive.Similarly, R ( t ) eventuallybecomes positive.Butthenthereexists t 4 > t 3 suchthat x ( t 4 ) $ R 4 + whichcontradictsthe invarianceof M .So V + ( t )=0 forall t % 0 .Then V + ( t )=0 impliesthat I 1 ( t )= I 2 ( t )=0 forall t % 0 .If f =1 weconcludethat M = { (0,0, R ,0):0 R N 0 } .If f < 1 Equation ( 210 )impliesfurtherthat R ( t )=0 forall t % 0 andhence M istheorigin.Weshow nextthat M isauniformrepellerineithercase. Firstlet f < 1 and < 1 3 .Let x ( t ) beanonnegativenonzerosolutionof ( 210 )-( 213 )with || x (0) || < .Bytheargumentaboveif x (0) $ ( R 4 + thesolution eventuallybecomespositivesoitsufcestoconsiderpositivesolutionsonly.Then 31

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limsup t !" I 1 ( t )+ I 2 ( t )+ R ( t ) % 1 impliesthatthereexists T > 0 suchthat || x ( T ) || > ByTheorem1ofFonda[ 14 ]thereexists > 0 suchthat liminf t !" d ( x ( t ),0) % forall nonnegativenonzerosolutions x ( t ) Nowlet f =1 .Supposethat limsup t !" V + ( t ) " 2 forsomepositivesolution x ( t ) and 2 > 0 .Thenthereexists + > 0 suchthatfor t > + and y =( I 1 ( t ), I 2 ( t ), R ( t )) T y " 2 + Ay where = V # ,0,0 # and A = ) * * + # ) 00 ) # ( $ + )0 0 $ # . Clearlytheprincipaleigenvalueof A isnegative;byPerron-Frobenius(Theorem 1.4 ) thereexists & > 0 andapositivelefteigenvector v suchthat vA = # & v .Dene $ ( y )= v y .Then $ ( y )= v y " 2 ( v )+ vAy = 2 ( v ) # & $ ( x ), fromwhichitfollowsthat limsup t !" $ ( y ) " 2 ( v ) & andhencethereexists C > 0 suchthat limsup t !" I 1 ( t )+ I 2 ( t )+ R ( t ) C 2 .If 2 < 1 C wegetacontradiction.Therefore limsup t !" V + ( t ) % 1 C Toshow M isalsoauniformrepellerinthiscase,suppose < 1 2 C and x ( t ) isa solutionsuchthat x (0) $ # \ M and d ( x (0), M ) < .If x (0) $ ( R 4 + thenthereexists i $ { 1,2,4 } suchthat x i (0) > 0 .Wewillshowthat x ( t ) $ R 4 + forsome t > 0 .Suppose I 1 (0) > 0 .Then I 2 (0), V + (0) > 0 ,sothereexists t > 0 suchthat I 2 ( t ), V + ( t ) > 0 .Then R ( t ) > 0 ,sobycontinuitythereexists t 1 > t suchthat R ( t 1 ) > 0 .Then x ( t 1 ) $ R 4 + Similarly,ifwesupposeeither I 2 (0) > 0 or V + (0) > 0 ,thereexists t > 0 suchthat x ( t ) $ R 4 + .Soitsufcestoconsiderpositivesolutions.Now limsup t !" V + ( t ) % 1 C 32

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impliesthatthereexists T > 0 suchthat d ( x ( T ), M ) % || V + ( T ) || % 1 2 C > .ByTheorem 1.1 thereexists > 0 suchthat liminf t !" d ( x ( t ), M ) % forallsuchsolutions,in particularforpositivesolutions. Thereforefor 0 < f 1 thestablemanifoldofMdoesnotintersect R 4 + .Thusby Theorem4.3in[ 15 ]theowisuniformlystronglypersistent;thatis,thereexists 0 > 0 suchthat liminf t !" d ( x ( t ), ( R 4 + ) > 0 forallpositivesolutions x ( t ) of( 210 )-( 213 ). 2.5.2StabilityoftheEndemicEquilibriuminaSpecialCase Inthecasewhere f =0 thesystemismoretractable.Inthelimitingcase N = N 0 and V = V # ,theequationsofthemodelaregivenby: I 1 = S N 0 V + V # # ) I 1 + ( I 2 + R ), (214) I 2 = ) I 1 # ( $ + ) I 2 (215) R = $ I 2 # R (216) V + = V $ V # I 1 + I 2 N 0 # V + (217) S = # S N 0 V + V # (218) V $ = V # # V $ V # I 1 + I 2 N 0 # V $ (219) Thissystemhastheuniqueendemicequilibrium ( I # 1 I # 2 R # V # + S # V # $ )= N 0 ) + )' N 0 ( $ + )( ) + ) $) N 0 ( $ + )( ) + ) V # + ,0, V # $ # where V # + satises V # + = & N 0 ( I # 1 + I # 2 ) + & N 0 V ( I # 1 + I # 2 ) < V # and V # $ = V # # V # + > 0 .Inthiscaseweareabletoprovethefollowingtheorem concerningglobalasymptoticstabilityoftheendemicequilibrium. Theorem2.4. Allpositivesolutionsofthesystem( 214 )-( 219 )with S (0) < N 0 convergetotheendemicequilibrium ( I # 1 I # 2 R # V # + S # V # $ ) 33

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Proof. Firstweobservethat S ( t ) isdecreasingandboundedbelowby0.So S ( t ) & S as t &1 forsome 0 S < N 0 .Nowlet M ( t )= I 1 ( t )+ I 2 ( t )+ R ( t ) .Sincethetotal populationis N 0 wehavethat M ( t ) & M as t &1 for M = N 0 # S > 0 .Hence I 1 ( t )= M + h ( t ) # I 2 ( t ) # R ( t ) where h ( t ) & 0 as t &1 .Thenweconsiderthe subsystem: I 2 = ) ( M + h ( t ) # I 2 # R ) # ( $ + ) I 2 (220) R = $ I 2 # R (221) Letting I 2 = #% ( $ + % )( # + % ) M and R = $# ( $ + % )( # + % ) M ,weobservethat 0= ) ( M # I 2 # R ) # ( $ + ) I 2 0= $ I 2 # R Wethenperformashiftbydening i 2 = I 2 # I 2 and r = R # R .Then( 220 )-( 221 )can bewrittenas x = Ax + f ( t ) where x = ) + i 2 r , A = ) + # ( ) + $ + ) # ) $ # , f ( t )= ) + ) h ( t ) 0 . Usingvariationofparameters,wendthat x ( t )= t 0 e ( t $ s ) A f ( s ) ds + e tA x 0 Notethatthematrix A isHurwitzsincebotheigenvalueshavenegativerealparts,so e tA & 0 as t &1 .Thisalsomeansthatthereexist C $ > 0 suchthat | e ( t $ s ) A f ( s ) | Ce $ $ ( t $ s ) | f ( s ) | forall t % s % 0 .Let > 0 begiven.Since f ( t ) & 0 ,thereexists + > 0 suchthat | f ( t ) | < '$ 2 C forall t % + .Forthisxed + ) 0 | e ( t $ s ) A f ( s ) | ds e $ t ) 0 Ce $ s | f ( s ) | ds .Sothereexists T % + suchthat e $ $ t ) 0 Ce $ s | f ( s ) | ds < 2 forall 34

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t % T .So / / / / t 0 e ( t $ s ) A f ( s ) ds / / / / t 0 Ce $ $ ( t $ s ) | f ( s ) | ds = ) 0 Ce $ $ ( t $ s ) | f ( s ) | ds + t ) Ce $ $ ( t $ s ) | f ( s ) | ds Ce $ $ t ) 0 e $ s | f ( s ) | ds + "$ 2 t ) e $ $ ( t $ s ) ds < 2 + 2 (1 # e $ $ ( t $ ) ) ) forall t % T < Thus x ( t ) & 0 as t &1 .Hence I 2 & I 2 and R & R as t &1 .Therefore I 1 & % # + % M whichwedeneas I 1 .Now I 1 + I 2 > 0 andwehave I 1 ( t )+ I 2 ( t )= I 1 + I 2 + g ( t ) forsome g ( t ) & 0 as t &1 .Wecanthenexpress( 217 )as V + = N 0 ( I 1 + I 2 + g ( t )) # N 0 V # ( I 1 + I 2 + g ( t ))+ # V + = g 1 ( t ) # g 2 ( t ) V + wherethefunctions g 1 ( t ) and g 2 ( t ) approachthepositivelimits & N 0 ( I 1 + I 2 ) and & N 0 V ( I 1 + I 2 )+ ,respectively,as t &1 .Itthenfollowsthat V + approachesthepositivelimit V + = & N 0 ( I 1 + I 2 ) + & N 0 V ( I 1 + I 2 ) Hence V + > 0 andsothereexists + > 0 and > 0 suchthatforall t > + S "# N 0 V # S andtherefore S ( t ) S ( + ) e %& N 0 V ( t $ ) ) wheretherightsideconvergesto0as t &1 .Therefore S =0 ,whichimpliesthat M = N 0 .Thus I 1 = I # 1 I 2 = I # 2 R = R # ,and V + = V # + 35

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Thuswehaveprovedthatinasimpliedspecialcase,theendemicequilibriumis globallyasymptoticallystable. 36

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CHAPTER3 ROGUINGASACONTROLSTRATEGY Themodelwehavedevelopedincorporatesacontrolstrategyofroguingand replantinginfectedtrees.Modelsforotherplantvirusdiseaseshaveincludedroguing, suchasthoseforbananabunchytop[ 1 ]andcitrustristezavirus[ 13 ].Chanand Jeger[ 8 ]developedamodelincludinghealthy,latentlyinfected,infectiousand post-infectiousplantsandinvestigatedthepopulationdynamicswithandwithout roguing.Analysisoftheirmodelwithroguingshowedthatthebasicreproductivenumber andtheequilibriumhealthypopulationdidnotdependonwhetherroguingwasdone inthepost-infectiouscategory.Howdodifferentroguingmethodsaffectourmodel? Toanswerthisweconsideracontrolmethodwhereroguingofthe I 2 and R treesis doneatrates 1 and 2 ,respectively.Thecasewhere 2 =0 isanalyzedinSection 3.1 whereweconcludethatthediseaseistransientandapositivepopulationofhealthy treeswillremainindenitely.Whilewedon'thaveacompleteanalyticalunderstanding ofthemodelwhen 1 > 0 and 2 =0 ,simulationssuggestthatthelevelofremaining healthytreesislargerwhenroguingisperformedin I 2 thanwhenthereisnoroguingat all.Section 3.2 focusesonthecasewhere 2 > 0 .Inthiscasethediseaseisnolonger transientandanendemicequilibriumwillexistif R 0 > 1 .However,wewillshowthatthe equilibriumhealthypopulationoftreesisincreasingwithrespecttoboth 1 and 2 3.1TransientBehavior Inthemodelwehavebeenconsideringthusfar,wehaveaccountedforroguing ofsymptomaticaswellasdeadtrees.However,itisworthwhiletoexaminethecases whenonlyroguingofsymptomatictreesoccursorwhenthereisnoroguingatall.We showthatineitherofthesesituationsthediseaseistransient;thatis, lim t !" I 1 ( t )=lim t !" I 2 ( t )=lim t !" V + ( t )=0. 37

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Thesystemthatweanalyzenowisthesameasouroriginalsystem,exceptthatitdoes notincludearoguingtermintheequationfor R ,andthusalsodoesnotaccountforthe replantingofarogueddeadtreeineitherthe S or I 1 equations.Weallowforroguingof I 2 treesatrate ,whichcanbezero.Notethatsincedeadtreesarenolongerroguedand replantedpossiblyasinfectedtrees,the R compartmentisnotconsideredtobeinfected. I 1 = S N V + V # ) I 1 + (1 # f ) I 2 (31) I 2 = ) I 1 # ( $ + ) I 2 (32) R = $ I 2 (33) V + = V $ V I 1 + I 2 N # V + (34) S = b 1 # N N 0 # # S N V + V + fI 2 (35) V $ = r ( V $ + V + ) 1 # V $ + V + K # # V $ V I 1 + I 2 N # V $ (36) Theorem3.1. Forallnonnegativesolutionsofthesystem( 31 )-( 36 )andforall % 0 thediseaseistransient. Proof. Forthissystem lim t !" N ( t )= N 0 and lim t !" V ( t )= V # asshownfortheoriginal systemintheproofofTheorem 2.2 .Herewehave R = $ I 2 .Since I 2 % 0 and I 2 ( t ) isbounded,anapplicationofBarbalat'slemma[ 16 ]impliesthat lim t !" I 2 =0 sothat R doesnotincreasewithoutbound.Thissimilarlyforces lim t !" I 1 =0 asaresultof Equation( 32 ).Thus lim t !" V + =0 aswellbecauseofEquation( 34 ).Soallinfected compartmentseventuallybecomeextinct. Theorem3.2. Inthesystem( 31 )-( 36 ), lim t !" S = S > 0 forany % 0 Proof. IntegratingEquation( 33 ),wehavethat R ( t ) # R (0)= $ t 0 I 2 ( + ) d + .Since R (0) and R ( t ) arebothbounded,theirdifferenceisbounded,andthus t 0 I 2 ( + ) d + is bounded.Similarly,Equation( 32 )givesthat I 2 ( t ) # I 2 (0)+( $ + ) t 0 I 2 ( + ) d + = 38

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) t 0 I 1 ( + ) d + ,andthus t 0 I 1 ( + ) d + isboundedaswell.Let > 0 .Since N ( t ) N 0 forall t lim t !" N ( t )= N 0 ,and lim t !" V ( t )= V # ,thereexists T > 0 suchthat N 0 # " N ( t ) N 0 and V # # " V ( t ) V # + forall t % T .Since % 0 byEquation ( 35 )wehavetheinequality S %# S N V + V %# S ( N 0 # ) V + ( V # # ) Therefore S ( t ) % S (0)exp # & ( N 0 $ )( V $ ) t 0 V + ( + ) d + # forall t % T .Bywayof contradiction,assumethat S =0 .Thenbytheaboveinequality, lim t !" t 0 V + ( + ) d + = 1 .Notethat V V I 1 + I 2 N V V I 1 + I 2 N 0 $ forall t % T .Thenforall t % T t 0 V $ ( + ) V ( + ) I 1 ( + )+ I 2 ( + ) N ( + ) d + T 0 V $ ( + ) V ( + ) ( I 1 ( + )+ I 2 ( + )) N ( + ) d + + t T ( I 1 ( + )+ I 2 ( + )) N 0 # d + T + N 0 # t T ( I 1 ( + )+ I 2 ( + ) ) d + T + N 0 # t 0 ( I 1 ( + )+ I 2 ( + ) ) d + whichisbounded.ByEquation( 34 ),wehave V + ( t ) # V + (0)= t 0 V ( ) ) V I 1 ( ) )+ I 2 ( ) ) N d + # t 0 V + ( + ) d + .Sincetherstintegralisboundedbythepreviousinequality,itfollows that lim t !" V + ( t )= #1 ,acontradiction.Thus S > 0 Theorems 3.1 and 3.2 tellusthatwhenroguingofdeadtreesdoesnotoccur, thediseasewilleventuallydieoutleavingonlyhealthyanddeadtrees.Thepresence ofremaininghealthytreesisafeaturesimilartothatofthestandardSIRepidemic model.Thisleadstoaquestionaboutthebenetsofroguingandreplanting.Isit worthwhiletorogueinfected,symptomatictrees,orshouldnotreesberogued?In eithercasethediseaseistransient;thedifferenceisthenumberoftreesthatremain healthy.Weusesimulationstocomparethevaluesof S ,thenumberofunaffected trees,ineachsituation.Considerthehypothesizedparametervalues N 0 = b = = $ = ) = =1 and r =1.5 inbothcases,with =1 and f =0.5 forthesystem 39

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includingroguingand =0 forthesystemexcludingroguing.Webeginwithonly susceptibletreesandasmallnumberofinfectedpsyllids;theinitialconditionsare ( I 1 (0), I 2 (0), R (0), V + (0), S (0), V $ (0))=(0,0,0,0.1,1,0.9) .Thisresultsin S = 0.2750 withroguingand S =0.2192 withoutroguing,somoretreesareunaffected whenroguingoccurs.Withallcombinationsofparametervaluesthatweexplored,the simulationssupportthenotionthatmorehealthytreeswillremainwhensymptomatic treesareroguedthanwhennotreesarerogued. A B Figure3-1. Simulationsofsystem(3 # 1)-(3 # 6)usingMATLABode45solverwith r =1.5 N 0 = b = = $ = ) = = V # =1 ,initialcondition ( I 1 (0), I 2 (0), R (0), V + (0), S (0), V $ (0))=(0,0,0,0.1,1,0.9) ,and(A): =1, f =0.5 ,(B): =0 3.2DifferentRoguingRates Nowweconsider 1 > 0 and 2 > 0 ,inwhichcasethediseaseisnolongertransient andanendemicequilibriumwillexistwhen R 0 > 1 .Substituting 1 and 2 appropriately insystem( 21 )-( 26 ),thereproductivenumberisrecalculatedtobe: R 0 = 1 # f 2 + & (1 # f ) 2 4 + 2 V # N 0 1 ) + 1 $ + 1 # (37) Forourmodel,asforChanandJeger'smodel[ 8 ], R 0 doesnotdependontherateof roguingofthedeadtrees.Althoughthediseasewillbemaintainedinthecasewhen R 0 > 1 ,itmaybethat S # ,theendemicequilibriumhealthytreepopulation,ishighereven 40

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inthepresenceofdiseaseascomparedtothecasewithnoroguingwherethedisease diesout.Thatis,theremaybeatradeoffbetweenallowingthediseasetopersistand maintainingaprotablelevelofhealthytrees.Thereforeitisworthconsideringthe dependenceof S # on 1 and 2 .Indeed, S # isdeterminedtobe: S # = fN 2 0 ) + 1+ V & $ 1+ $# % 2 ( $ + % 1 + # ) % fN 0 & + 1 # + 1 $ + % 1 $ 1+ $ % 2 % . UnliketheresultofChanandJegerweseethattheequilibriumhealthypopulation dependsonwhetherroguingisdoneinboththeinfectious,symptomaticstageandinthe post-infectiousstage.Fromthisexpressionwecandeterminethat S # increasesas 1 and 2 increase,asshownbelow. Proposition3.1. Let R 0 > 1 .Thentheendemicequilibriumlevelofhealthytrees S # = fN 2 0 ) + 1+ V & $ 1+ $# % 2 ( $ + % 1 + # ) % fN 0 & + 1 # + 1 $ + % 1 $ 1+ $ % 2 % isincreasingwithrespectto 1 and 2 Proof. Werstprovethat S # increasesas 1 increases.Wehave S # = fN 2 0 ) + 1+ V & + V $# &% 2 ( $ + % 1 + # ) fN 0 & + 1 # + 1 $ + % 1 $ 1+ $ % 2 % = fN 2 0 0 a + b x + # c + d x 1 = fN 2 0 ax 2 +( a ) + b ) x cx 2 +( c ) + d ) x + d ) ( where x = $ + 1 a =1+ V & b = V $# &% 2 c = fN 0 & + 1 # ,and d =1+ $ % 2 .Notethat x a b c and d arepositivefor 1 % 0 .Afterdifferentiatingandsomesimplicationwedetermine ( S # (' 1 = fN 2 0 ( ad # bc ) x 2 +2 ad ) x + d ) ( a ) + b ) ( cx 2 +( c ) + d ) x + d ) ) 2 ( % fN 2 0 x [( ad # bc ) x +2 ad ) ] ( cx 2 +( c ) + d ) x + d ) ) 2 ( Thereforeitsufcestoshowthat ( ad # bc ) x +2 ad ) > 0 .Recallingtheexpressionfor R 0 (Quantity 37 )weobservethat,similartotheproofofTheorem1, R 0 > 1 ifandonlyif 2 $ 1+ # $ + % 1 % V # N 0 f ) > 1. 41

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Usingthisinequalitywecalculate ( ad # bc ) x +2 ad ) = 1+ V # + $ 2 # V # N 0 f )$ 2 2 ( ( $ + 1 ) +2 ) 1+ V # #" 1+ $ 2 # > 1+ V # + $ 2 # $ 2 1+ ) $ + 1 #( ( $ + 1 ) +2 ) 1+ V # + $ 2 + V # $ *' 2 # > 1+ V # # $) 2 ( $ + 1 ) ( ( $ + 1 )+ 2 $) 2 = 1+ V # # ( $ + 1 )+ $) 2 > 0. Itisprovedsimilarlythat S # isincreasingwithrespectto 2 .Indeed,wehave S # = fN 2 0 0 1+ V & + V $# & ( $ + % 1 + # ) % 2 fN 0 & + 1 # + 1 $ + % 1 + $ ( $ + % 1 ) % 2 1 = fN 2 0 0 g + h % 2 j + k % 2 1 = fN 2 0 g 2 + h j 2 + k ( where g =1+ V # h = V # $) ( $ + 1 + ) ) j = fN 0 + 1 ) + 1 $ + 1 and k = $ $ + 1 Notethat g h j ,and k arepositive.Hence ( S # (' 2 = fN 2 0 gk # hj ( j 2 + k ) 2 # anditsufcestoshowthat gk # hj > 0 .Since R 0 > 1 weusetheinequalityaboveagain toobtain gk # hj = 1+ V # # $ $ + 1 # V # $) ( $ + 1 + ) ) fN 0 + $ + 1 + ) ) ( $ + 1 ) # = $ $ + 1 # V # $ N 0 f ) 2 ( $ + 1 + ) ) = $ $ + 1 2 3 1 # V # N 0 f ) 2 $ 1+ # $ + % 1 % 4 5 > 0. 42

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Theseresultsforvaryingroguingmethodsdependonthenovelfeatureofour modelwhichistheinclusionofthepositiveprobability 1 # f thatareplantedtreewill immediatelybecomeinfected.Thatis,thetreesinthe R stageareapotentialsourceof infectionduetothesoilandremainingrootsystembeingareservoirforthedisease. 43

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CHAPTER4 MODIFICATIONOFTHEMODEL Informulatingtheoriginalmodel,weassumedthatpsyllidsareequallyattracted toalltypesoftrees,includingthedeadones,whichmaybebiologicallyunrealistic.We nowconsideramodiedversionofourmodelinwhichweassumethatthepsyllidsare notattractedtodeadtrees.Thatis,theprobabilitythatatreebittenbyavectorisa susceptibletreeisdenedtobe = S S + I 1 + I 2 Thenewmodelisthengivenby: I 1 = *! V + V # ) I 1 + (1 # f )( I 2 + R ), (41) I 2 = ) I 1 # ( $ + ) I 2 (42) R = $ I 2 # R (43) V + = V $ V (1 # ) # V + (44) S = b 1 # N N 0 # # *! V + V + f ( I 2 + R ), (45) V $ = r ( V $ + V + ) 1 # V $ + V + K # # V $ V (1 # ) # V $ (46) whichwestudyawayfromthesingularity S = I 1 = I 2 =0 .Thesystem( 41 )-( 46 )has thesameDFEasouroriginalsystem( 21 )-( 26 ).Italsoholdsthat N = N 0 and V = V # attheequilibriumof( 41 )-( 46 ).Howeverinthismodiedsystemwehavethepossibility fortwopositiveendemicequilibria.Wedenethefollowingquadraticfunction: g ( )= )$ ( $ + ) 2 # + ) + ) fN 0 # + ) N 0 f ( + V # ) 2 whereeasycalculationshowsthat g ( # )=0 foranyendemicequilibriumvalue # InTheorem 4.1 ,weprovetheexistenceofthetwopositiveequilibriagivencertain parametervalues.Additionally,computationrevealsthatthenextgenerationmatrix of( 41 )-( 46 )isidenticaltothenextgenerationmatrixof( 21 )-( 26 )calculatedin 44

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Subsection 2.3.1 .Thereforeforthemodiedsystemwealsohave R 0 = 1 # f 2 + & (1 # f ) 2 4 + 2 V # N 0 1 ) + 1 $ + # Itiseasytoshowthat R 0 > 1 when f =0 andthereexistsaunique,globallyattractive endemicequilibrium;theproofissimilartothatofTheorem 2.4 .However,inthecase f > 0 ,wehavethefollowingresult.ForuseinTheorem 4.1 wedene crit = $ + 2 $ + ) ) + f N 0 # Theorem4.1. Twodistinctpositiveendemicequilibriaof( 41 )-( 46 )existifandonlyif 1. crit < 1 and 2. 2 f N 0 1 ) + 1 $ + # < V # < crit 1 # 2 + ( + ) ) 2 ') fN 0 # # 1 ( Furthermore, 2 impliesthat R 0 < 1 Proof. For $ (0,1) wehavethat isanequilibriumvalueifandonlyif g ( )=0 .Soit sufcestoshowthat g hastworootsin (0,1) ifandonlyif(i)and(ii)hold.Observethat g (0) > 0 andthatthecriticalvalueof g ( ) isgivenby crit .Then g willhavetworootsin (0,1) ifandonlyif g (1) > 0 0 < crit < 1 ,and g ( crit ) < 0 .Itistrivialthat 0 < crit .We have g (1)= ) N 0 V # f # 1+ ) $ + # Then g (1) > 0 ifandonlyif 2 f N 0 1 ) + 1 $ + # < V # asincondition(ii).Notethatitistrivialfromthisinequalitythat(ii)implies R 0 < 1 .We provenowthat g ( crit ) < 0 ifandonlyiftheright-handinequalityof(ii)holds.Indeed, g ( crit ) < 0 2 ) fN 0 ( V # + ) 2 < ) fN 0 + + ) # crit # )$ ( $ + ) 2 crit 45

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2 V # < 2 ) fN 0 '" ) fN 0 + + ) # crit # )$ ( $ + ) 2 crit ( # 2 V # < 2 ) fN 0 crit ) fN 0 1 # 2 # + + ) 2 ( # 2 V # < crit 1 # 2 + ( + ) ) 2 ') fN 0 # # 1 ( Wenextexaminethestabilityoftheequilibriainthecasethatmultiplepositive endemicequilibriaexist,whichiswhen R 0 < 1 .Considerthehypothesizedparameter values f = =0.1 =0.01 r =1.5 ,and b = N 0 = $ = ) = = V # =1 ,whichresult inthevalue R 0 =0.9199 .WendthattheDFEaswellasoneoftheendemicequilibria willbestable,asalleigenvaluesoftheJacobianevaluatedateachoftheseequilbria havenegativerealparts.However,theotherendemicequilibriumisunstable;forthis particularequilibrium,theJacobianhasapositiverealeigenvalue. Simulationsshowthat,dependingontheinitialconditionsusedincombinationwith theaboveparametervalues,thesolutions ( I 1 ( t ), I 2 ( t ), R ( t ), V + ( t ), S ( t ), V $ ( t )) will convergetooneofthetwostableequilibria(Figure 4-1 ).First,withtheinitialcondition (0,0,0.5,0,0.5,1) ,convergenceistotheDFE (0,0,0,0,1,1) .Ifinsteadweusethe initialcondition (0,0,0.9,0,0.1,1) thesolutionconvergestotheendemicequilibriaat (0.01,0.01,0.977,0.081,0.003,0.919) 46

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A B Figure4-1. Simulationsofsystem(4 # 6)-(4 # 6)usingMATLABode45solverwith f = =0.1 =0.01 r =1.5 b = N 0 = $ = ) = = V # =1 ,andinitial conditions(A): ( I 1 (0), I 2 (0), R (0), V + (0), S (0), V $ (0))=(0,0,0.5,0,0.5,1) (B): ( I 1 (0), I 2 (0), R (0), V + (0), S (0), V $ (0))=(0,0,0.9,0,0.1,1) 47

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CHAPTER5 PERIODICITY 5.1PeriodicModel Recentevidencesuggeststhattransmissionofthediseasemaynotoccurata constantrateasoriginallyassumed.Itisbelievedthatpsyllidsprefertolayeggsand feeduponnewush,whichonlyoccurstwiceayear[ 20 ].Duringtheoffseason,it ispossiblethatthepsyllidsstillfeedontheleavesofcitrusplantsbutperhapsdon't transmitthediseaseaswell,orthattheyfeedandbreedonorangejasmineinstead. Thuswealterouroriginalmodeltoincorporateacontinuousperiodicbitingfunction ( t ) torepresentthisuctuatingtransmission.Thetimeaveragefor ( t ) overtheperiod T is representedby 5 6 = 1 T T 0 ( t ) dt Theincreasedcomplexityofthismodelcanprovideamorerealisticrepresentation ofthedisease,butcanalsobedifculttofullyanalyze.Withthisperiodicity,ourmodel nowbecomes: I 1 = ( t ) S N V + V # ) I 1 + (1 # f )( I 2 + R ), (51) I 2 = ) I 1 # ( $ + ) I 2 (52) R = $ I 2 # R (53) V + = ( t ) V $ V I 1 + I 2 N # V + (54) S = b 1 # N N 0 # # ( t ) S N V + V + f ( I 2 + R ), (55) V $ = rV 1 # V K # # ( t ) V $ V I 1 + I 2 N # V $ (56) Forthenon-periodicsystem( 21 )-( 26 ),wewereabletoperformmuchofour analysisusingthebasicreproductivenumber R 0 .Wewerenotabletodene R 0 explicitlyforthisperiodicsystem,butwearestillabletondsomeconditionsthat willguaranteeextinctionorpersistenceofthediseaseincertaincases,aswellas stabilityandinstabilityofthedisease-freesolution. 48

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5.2ExtinctionoftheDiseaseinPeriodicModel Werstexaminethepossibilitythatthediseasebecomesextinct.Asinthe non-periodicsystem,thetotaltreepopulationagainsatises N = b 1 # N N 0 # whilethetotalvectorpopulationsatises V = rV 1 # V K # # V So N ( t ) & N 0 and V ( t ) & V # =(1 # r ) K (assuming < r )as t & + 1 .The disease-freesolution(DFE)fortheperiodicsystem( 51 )-( 56 )istheconstantsolution x 0 =(0,0,0,0, N 0 V # ) .Weprovethatthediseasecanbecomeextinct. Theorem5.1. Allnonnegativesolutionsofthesystem( 51 )-( 56 )convergetotheDFE (0,0,0,0, N 0 V # ) ifthefollowingconditionholds: ( $ + + ) ) T 2 e 2 T ( + # ) 5 6 2 ( $ + ) N 0 V # f ( e T # 1)( e # T # 1) < 1. Proof. Suppose limsup t !" I 1 ( t )= m > 0 .Thenforevery > 0 thereexists + 1 > 0 such that I 1 ( t ) m + forall t % + 1 .Substituting,wehavethat I 2 ( t ) ) ( m + ) # ( $ + ) I 2 ( t ) forall t % + 1 .Thenthereexists + 2 > + 1 suchthat I 2 ( t ) ) ( m + ) $ + + forall t % + 2 .Thismeansthat I 1 ( t )+ I 2 ( t ) m +2 + ) ( m + ) $ + 49

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forall t % + 2 .Substitutingagain,wegetthat R ( t ) $ ) ( m + ) $ + + # # R ( t ) forall t % + 2 .Thusthereexists + 3 > + 2 suchthat R ( t ) $ ) ( m + ) $ + + # + forall t % + 3 .So I 2 ( t )+ R ( t ) ) ( m + )+ $" +2 forall t % + 3 Wehavethat N ( t )= b (1 # N N 0 ) ,so N ( t ) & N 0 as t &1 .Thusthereexists + 4 > 0 suchthat N 0 # " N ( t ) forall t % + 4 .Wealsohavethat V ( t )= rV (1 # V K ) # V ,and thus V ( t ) & V # as t &1 .Sothereexists + 5 > 0 suchthat V # # " V ( t ) forall t % + 5 Wenowsubstitutethistogetthat,forall t % + 2 + 4 V + ( t ) ( t ) m +2 + # ( m + ) $ + % N 0 # # V + ( t ). Thenbyvariationofparameterswehave V + ( t ) t 0 ( s ) Le ( s $ t ) ds + V + (0) e $ t where,as & 0 L = m +2 + # ( m + ) $ + % N 0 # & ( $ + + ) ) m ( $ + ) N 0 Sinceweareinterestedinthisquantityas t &1 ,wehave V + ( t ) L t 0 ( s ) e ( s $ t ) ds Forany t ,thereexistsaninteger K suchthat t $ ( KT ,( K +1) T ] .Sowecanset t =( K +1) T # / forsome / $ [0, T ) .Then 50

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V + ( t ) Le $ t t 0 ( s ) e s ds Le $ t ( K +1) T 0 ( s ) e s ds = Le $ t 0 T 0 ( s ) e s ds + 2 T T ( s ) e s ds +...+ ( K +1) T KT ( s ) e s ds 1 = Le $ t T 0 ( s ) e s ds + e T T 0 ( s ) e s ds +...+ e KT T 0 ( s ) e s ds ( = Le $ t $ 1+ e T +( e T ) 2 +...+( e T ) K % T 0 ( s ) e s ds = Le $ t e ( K +1) T # 1 e T # 1 # T 0 ( s ) e s ds bysumofageometricseries = Le 1 # e $ ( K +1) T e T # 1 # T 0 ( s ) e s ds since e $ t = e $ ( K +1) T e Le T 1 e T # 1 e T T 0 ( s ) ds = Le 2 T e T # 1 T < ( t ) > as t &1 Turningtotheequationfor I 1 andsubstitutingseveralpreviousinequalitieswehave that,forall t % + 3 + 5 I 1 ( t ) ( t ) V + ( t ) V # # + M # ) I 1 ( t ) where,as & 0 M = ) ( m + )+ $" +2 # (1 # f ) & ) m (1 # f ). Usingvariationofparametersagaingivesusthat I 1 ( t ) t 0 ( s ) V + ( s ) V # # + M # e # ( s $ t ) ds + I 1 (0) e $ # t Sinceweareinterestedinthisquantityas t &1 ,wesubstituteourboundfor V + ( t ) and have 51

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I 1 ( t ) LTe 2 T < ( t ) > ( V # # )( e T # 1) t 0 ( s ) e # ( s $ t ) ds + M t 0 e # ( s $ t ) ds LTe 2 T < ( t ) > ( V # # )( e T # 1) T < ( t ) > e 2 # T e # T # 1 # + M ) (1 # e $ # t ) (similartoabove) LT 2 e 2 T ( + # ) < ( t ) > 2 ( V # # )( e T # 1)( e # T # 1) + M ) as t &1 Nowas & 0 ,thefollowinginequalityholdsas t &1 : I 1 ( t ) ( $ + + ) ) mT 2 e 2 T ( + # ) ( $ + ) N 0 V # ( e T # 1)( e # T # 1) + m (1 # f ). Theaboveinequalitywillyield I 1 ( t ) < m when ( $ + + ) ) T 2 e 2 T ( + # ) 5 6 2 ( $ + ) N 0 V # f ( e T # 1)( e # T # 1) < 1. (57) Thatis,whenCondition 57 holds,thesuppositionthat limsup t !" I 1 ( t )= m > 0 is contradicted.ThustheDFEisgloballystablewhenCondition 57 issatisedandthe diseasebecomesextinct. Notethatas T & 0 ,theleft-handsideofCondition 57 convergestothequantity ( $ + + ) ) 5 6 2 ( $ + ) N 0 V # f ) Thisquantityislessthanoneifandonlyif T 0 (Quantity 29 ofthenon-periodicsystem) islessthanone.Weprovedasimilarresultinthatcasethatextinctionofthedisease occurswhen T 0 1 InFigure 5-1 ,wepresentanumericalsimulationofsystem( 51 )-( 56 )with f = 0.75 r =1.5 b = N 0 = ) = = V # =1 $ = =1.5 T =0.25 ( t )= + sin( 2 + t T ) with =0.25 (so 5 6 =0.25 )andinitialcondition ( I 1 (0), I 2 (0), R (0), V + (0), S (0), V $ (0))= (0.5,0.25,0,0.5,0.25,0.5) toshowtheextinctionofthedisease.Withthesehypothesized parametervaluesCondition 57 issatised,andweseefromthesimulationthatthe solutionindeedconvergestothedisease-freestate.Theparametervaluesusedhere 52

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arethesameasusedinthesimulationinFigure 2-2 ,whichshowstheextinctionofthe diseaseforthenon-periodicsystem( 21 )-( 26 ). Figure5-1. Simulationofsystem(5 # 1)-(5 # 6)usingMATLABode45solverwith f =0.75 r =1.5 b = N 0 = ) = = V # =1 $ = =1.5 T =0.25 ( t )= + sin( 2 + t T ) with =0.25 ,andinitialcondition ( I 1 (0), I 2 (0), R (0), V + (0), S (0), V $ (0))=(0.5,0.25,0,0.5,0.25,0.5) 5.3IncorporatingthePeriodicFunctionasaSmallPerturbation Werstrepresenttheseasonaluctuationofthediseasetransmissionwiththe continuousperiodicperturbation ( t )= + f ( t ) ,where > 0 issmalland f ( t ) isa T -periodicfunction.Sincethisfunctionrepresentsthetransmissionrate,weassume that ( t ) % 0 andthatitstillincorporatestheprobabilitythatabitesuccessfullyresultsin transmissionofthedisease. Asinthenon-periodiccase,weconsidertheinfectedcompartments I 1 I 2 R V + andexaminethe4x4JacobianattheDFE x 0 =(0,0,0,0, N 0 V # ) ,whichisgivenby: J ( x 0 t )= ) * * * + # )' (1 # f ) (1 # f ) & ( t ) V ) # ( $ + )00 0 $ # 0 & ( t ) N 0 & ( t ) N 0 0 # . 53

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Werstconsiderthecasewhen f ( t ) hasmean0overperiod T andexaminethe system X ( t )= J ( x 0 t ) X ( t ) .WerewritetheJacobianmatrixas J ( x 0 t )= A 0 + A 1 ( t ) ,where A 0 = ) * * * + # )' (1 # f ) (1 # f ) & V ) # ( $ + )00 0 $ # 0 & N 0 & N 0 0 # , A 1 ( t )= ) * * * + 000 f ( t ) V 0000 0000 f ( t ) N 0 f ( t ) N 0 00 . Thematrix e A 0 t willbeusedthroughoutouranalysis.Foranyquasi-positivematrix A ,the exponentialmatrix e A isalwaysnonnegative.Forourparticular A 0 ,wehavethat e A 0 t is strictlypositive. Lemma1. Forthematrix A 0 above, e A 0 t isstrictlypositivefor t > 0 Proof. Since A 0 isquasi-positive,thereexists r > 0 suchthat D = rI + A 0 % 0 .Then D k % 0 forallintegers k % 1 .Bydenition, e Dt = 6 k =0 1 k ( Dt ) k .Forourparticular A 0 ,by calculatingtherstfewpowersof D ,wendthat D 3 > 0 (andthus D k > 0 forall k > 3 aswell).So e Dt > 0 .But e Dt = e ( rI + A 0 ) t = e rt e A 0 t forall t > 0 .Since e rt > 0 ,itfollows that e A 0 t > 0 forall t > 0 Substitutingfor J ( x 0 t ) ,weanalyzethesystem X ( t )=( A 0 + A 1 ( t )) X ( t ) where A 0 isquasi-positiveand A 1 ( t ) hasmean0overperiod T .Letting X ( t ) bethe principalfundamentalmatrixsolution,wehave X (0, )= I .TheFloquetmultipliersof thissystemarerepresentedby & i ( ) ,withdominantFloquetmultiplier & ( ) (whichisthe 54

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dominanteigenvalueofthematrixsolution X ( T ) ).Weneedboth X ( T ) and & i ( ) to beanalyticin inordertoexpressthemusingaTaylorseriesexpansionforouranalysis. Lemma2. Theprincipalfundamentalsolution X ( t ) ofthesystem X ( t )=( A 0 + A 1 ( t )) X ( t ) anditssimpleeigenvalues & i ( ) areanalyticin Proof. Forasystem X ( t )= f ( t X ( t ), ) ,byTheorem1.3inChicone[ 9 ], X ( t ) isanalytic ifthefunction f isrealanalytic.Inourcase,thefunction f isleftmultiplicationbythe matrix J ( x 0 t )= A 0 + A 1 ( t ) ,whereeachcomponentisanalyticin .Thus X ( T ) is analyticin Thismeansthatthecharacteristicpolynomial p ( & ) of X ( T ) isalsoanalytic in .Any & i thatisaneigenvalueof X ( T ,0) satises p ( & i ,0)=0 .Thusbyimplicit differentiation,wehave ( p (& ( & i ,0) d & i d + ( p (" ( & i )=0. Notethat p "( ( & i ,0) + =0 foranysimpleeigenvalue & i .Sothereexistsananalyticfunction & i ( ) suchthat & i (0)= & i and p ( & i ( ), )=0 forsufcientlysmall As X ( t ) isanalyticin byLemma 2 ,wecanrewrite X ( t )= X 0 ( t )+ X 1 ( t )+ 2 X 2 ( t )+... asaTaylorserieswithrespectto about0.Wealsohavethatanysimple eigenvalue & i ( ) isanalyticin byLemma 2 ,sowecanwrite & i ( )= & 0 + "& 1 + 2 & 2 +... Recallthat X (0, )= I forall .Inparticular,taking =0 ,wehavethat X 0 (0,0)= I ,and X i (0,0)=0 forall i % 1 .Thisexpansionyieldsthesystem: X 0 = A 0 X 0 ( t ), X 0 (0, )= I X 1 = A 0 X 1 ( t )+ A 1 ( t ) X 0 ( t ), X 1 (0, )=0 X 2 = A 0 X 2 ( t )+ A 1 ( t ) X 1 ( t ), X 2 (0, )=0 . Attime T thissystemhasthesolution: 55

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X 0 ( T )= e A 0 T X 1 ( T )= e A 0 T 7 T 0 e $ A 0 t A 1 ( t ) e A 0 t dt . X k ( T )= e A 0 T 7 T 0 e $ A 0 t A 1 ( t ) e A 0 t X k $ 1 ( t ) dt k =2,3,... As & ( ) representsthedominantFloquetmultiplierofthesystem,weareinterested inhowthisquantitychangeswhentheperturbationisintroduced.Thatis,wewantto nd & % (0) .Notethatat =0 ,oursystemreducesto X ( t )= A 0 X ( t ) ,whichjust hasthesolution X ( T )= X 0 ( T )= e A 0 T .ByLemma 1 e A 0 T ispositive,soby Perron-Frobenius(Theorem 1.4 ),theprincipaleigenvalueissimple.Weareableto showthatforsmallperturbationswithmean0,theprincipaleigenvalueofthesystem hasderivativezero.Thefollowingresultholdsforallsimpleeigenvalues,notjustthe dominantone. Theorem5.2. Let & 0 beasimpleeigenvalueof X ( T ,0) thatsatisesthesystem X ( t )=( A 0 + A 1 ( t )) X ( t ) ,where A 0 isquasi-positiveand A 1 ( t ) hasmean0over period T .Thenthereexistsananalyticfunction & ( ) denedon ( # 0 0 ) suchthat & ( ) isaneigenvalueof X ( T ) forall & (0)= & 0 ,and & % (0)=0 Proof. Considerthe nxn matrix X ( T ) witheigenvalues & i ( ) i =1,2,..., n .The characteristicpolynomial p ( & ) ofthismatrixcanbewrittenas p ( & )= n 8 k =0 ( # 1) k a k & n $ k wherethecoefcients a k aretheelementarysymmetricpolynomials.Newton'sidentities canbestatedintermsoftheseelementarysymmetricpolynomials[ 24 ]: ka k ( & 1 & 2 ,..., & n )= k 8 i =1 ( # 1) i $ 1 a k $ i ( & 1 & 2 ,..., & n ) p i ( & 1 & 2 ,..., & n ), 56

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where p k ( & 1 & 2 ,..., & n ) isthe k -thpowersumdenedby p k ( & 1 & 2 ,..., & n )= n 8 i =1 & k i k % 1. Notethat p k ( & 1 & 2 ,..., & n )=tr( X ( T ) k ) .BysolvingNewton'sidentitiesrecursively, eachcoefcient a k canbewrittenexplicitlyintermsofthetraceofpowersofthematrix X ( T ) .Thusthecharacteristicpolynomialforan n x n matrixcanbewrittensothateach term(otherthantheconstant)isoftheform C tr( X ( T ) i )(tr( X ( T )) j & k ,where C is arealnumberand 0 i j k n areintegers.When k = n ,theconstanttermisjust a k = a n = & 1 & 2 & n =det( X ( T )) Byimplicitdifferentiation, 9 & % ( ) ( p (& + ( p (" : / / / / =0 =0 foranyeigenvalue & .Letting & representanysimpleeigenvalue, p "( ( & 0 ,0) + =0 ,sowe mustshowthat p "' ( & 0 ,0)=0 .Wewillexamineeachtermofthecharacteristicequation individually. Firstnotethat det X ( T )=det X (0, ) e T 0 tr J ( x 0 s ) ds byLiouville'sFormula (Proposition 1.1 ).Butthen det X ( T )=det $ Ie tr( T 0 A 0 + A 1 ( s ) ds ) % sincethetraceisalinearfunction = e tr T 0 A 0 ds +tr T 0 A 1 ( s ) ds = e T tr A 0 since A 1 ( t ) hasmean0overperiod T and A 0 isconstant. Thustheconstantterm det X ( T ) doesnotdependon ,sothederivativeofthisterm is0. Nowconsideranytermoftheform tr( X ( T ) i ) foranyinteger i % 1 .Forease,let X 0 = X 0 ( T ,0) and X 1 = X 1 ( T ,0) .Thederivativeofthistermat =0 is tr( X i $ 1 0 X 1 )+tr( X i $ 2 0 X 1 X 0 )+...+tr( X 1 X i $ 1 0 ) sincethetraceisalinearfunction = i tr( X i $ 1 0 X 1 ) because tr( AB )=tr( BA ) 57

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= i tr $ e T ( i $ 1) A 0 T 0 e ( T $ s ) A 0 A 1 ( s ) e sA 0 ds % = i T 0 tr $ e ( Ti $ s ) A 0 A 1 ( s ) e sA 0 % ds becausethetraceisalinearfunction = i T 0 tr $ A 1 ( s ) e sA 0 e ( Ti $ s ) A 0 % ds because tr( AB )=tr( BA ) = i tr $ T 0 A 1 ( s ) e TiA 0 ds % = i tr $ T 0 A 1 ( s ) dse TiA 0 % =0 since A 1 ( s ) hasmean0overperiod T Thusanytermoftheform tr( X ( T ) i ) & k or (tr X ( T )) i & k willhavezeroderivative. Finally,thederivativeofanyterm tr( X ( T ) i )(tr X ( T )) j & k is0bytheproductrule. Thus & % (0)=0 foranysimpleeigenvalue & Nowconsiderthesystem X ( t )=( A 0 + A 1 + A 2 ( t )) X ( t ) ,where A 0 is quasi-positiveasinthepreviouscase, A 1 > 0 ,and A 2 ( t ) hasmean0overperiod T Thisintroducesaperturbationthathasanon-zeroaverageoveritsperiod T .Weare againinterestedinhowthedominantFloquetmultiplier & ( ) (thedominanteigenvalueof thematrixsolution X ( T ) )changeswithrespectto at =0 .Inthiscaseofintroducing smallperturbationswithpositiveaverage,weshowthatthederivativeoftheprincipal eigenvalueofthesystemispositive. Foruseinouranalysis,wewillwritethissystemas X ( t 1 2 )=( A 0 + 1 A 1 + 2 A 2 ( t )) X ( t 1 2 ) andthenexaminethecasewhere 1 = 2 Theorem5.3. Let & betheprincipaleigenvalueof X ( T ,0) thatsatisesthesystem X ( t )=( A 0 + A 1 + A 2 ( t )) X ( t ) ,where A 0 isquasi-positive, A 1 > 0 ,and A 2 ( t ) has mean0overperiod T .Thenthereexistsananalyticfunction & ( ) denedon ( # 0 0 ) suchthat & ( ) isaneigenvalueof X ( T ) forall & (0)= & ,and & % (0) > 0 Proof. Wewanttond d d & ( " ) / / =0 .Webeginbyexaminingthepartialderivativesof & withrespecttoboth 1 2 individually. 58

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Firstconsider "( "' 1 (0,0) .Since 2 iskeptconstantatzero,thesystemreducesto X ( t 1 )=( A 0 + 1 A 1 ) X ( t 1 ) .Similartotheoriginalperiodiccase,both X ( T 1 ) & ( 1 ) areanalyticin 1 byLemma 2 .Wecanthenusetheexpansion & ( 1 )= & 0 + 1 & 1 + 2 1 & 2 + ... ,so "( "' 1 = & 1 .Since & ( 1 ) isaneigenvalueof X ( T 1 ) ,thereexistsaneigenvector y ( 1 )= y 0 + 1 y 1 + 2 1 y 2 +... suchthat ( X 0 ( T 1 )+ 1 X 1 ( T 1 )+...)( y 0 + 1 y 1 +...)=( & 0 + 1 & 1 +...)( y 0 + 1 y 1 +...). Notethat & 0 isaneigenvalueof X 0 ( T 1 ) ,sothereexistsavector w T 0 suchthat w T 0 ( X 0 ( T 1 ) # & 0 I )=0 .Usingthisandsolvingfor & 1 ,wehave & 1 = w T 0 X 1 ( T 1 ) y 0 w T 0 y 0 Recallthat X 1 ( T 1 )= e A 0 T 7 T 0 e $ A 0 t A 1 e A 0 t dt .Since A 1 > 0 inthiscase,wehave X 1 ( T 1 ) > 0 ,andthus & 1 > 0 .So "( "' 1 (0,0) > 0 Nowconsider "( "' 2 (0,0) .Since 1 iskeptconstantatzero,thesystemreducesto X ( t 2 )=( A 0 + 2 A 2 ( t )) X ( t ) .Wehavethat A 2 ( t ) is T -periodicwithmean0,andthus byTheorem 5.2 wehavethat "( "' 2 (0,0)=0 .NotethatbyLemma 2 & ( 2 ) isanalyticin 2 Bycontinuityofeigenvalues, & ( 1 2 ) iscontinuousinaneighborhoodabout (0,0) Sinceboth & ( 1 ), & ( 2 ) areanalyticintheirrespectivevariables,byTheorem 1.7 both (& / (" 1 and (& / (" 2 arecontinuousinthesameneighborhoodabout (0,0) .Thus & ( 1 2 ) istotallydifferentiableat (0,0) .Bythechainrule,wehave d d & ( " )= (& (" 1 (" 1 (" + (& (" 2 (" 2 (" Since 1 = 2 = ,wehavethat "' 1 "' = "' 2 "' =1 .Thus d d & ( " )= (& (" 1 (" 1 (" + (& (" 2 (" 2 (" = & 1 > 0. 59

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5.4StabilityoftheDisease-FreeSolution Weagainconsiderthesystem( 51 )-( 56 ),whichhastheconstantperiodic solution x 0 =(0,0,0,0, N 0 V # ) asitsdisease-freesolution.Now,however,instead of ( t ) representingasmallperiodicperturbation,wejustrequire ( t ) > 0 tobe continuouswithperiod T .Thesystemhasthesame4x4JacobianmatrixattheDFE x 0 =(0,0,0,0, N 0 V # ) : J ( x 0 t )= ) * * * + # )' (1 # f ) (1 # f ) & ( t ) V ) # ( $ + )00 0 $ # 0 & ( t ) N 0 & ( t ) N 0 0 # . Nowwerewritethismatrixas J ( x 0 t )= A + B ( t ) ,where A = ) * * * + # )' (1 # f ) (1 # f )0 ) # ( $ + )00 0 $ # 0 00 0 # , B ( t )= ) * * * + 000 & ( t ) V 0000 0000 & ( t ) N 0 & ( t ) N 0 00 . Notethatthematrix B ( t ) isnonnegativeandeachtermhasperiod T Because x 0 isaperiodicsolution,itisaxedpointofthePoincar ` emap P ( x ) .Thus thestabilityoftheDFE x 0 canbeanalyzedbydeterminingthespectralradiusof DP ( x 0 ) thederivativeofthePoincar ` emap. DP ( x 0 ) istheprincipalfundamentalsolutionofthe T -periodiclinearvariationalsystem X = F x ( x 0 t ) X (58) 60

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where F x ( x 0 t )= J ( x 0 t )= A + B ( t ) .If X ( t ) denotestheprincipalfundamentalsolution ofsystem 58 ,then ( DP ( x 0 ))= ( X ( T )) .Sowewillexaminetheeigenvaluesof X ( T ) whicharetheFloquetmultipliersofthesystem X =( A + B ( t )) X (equivalenttosystem 58 ).Asprovedbelow,thesystemisstablewhentheaverageofthenormof B ( t ) is sufcientlysmallandunstablewhentheaverageof ( t ) issufcientlylarge. Theorem5.4. Let 0 0 denoteamatrixnorm.Thenthereexists > 0 suchthatthe disease-freesolutionofsystem( 51 )-( 56 )islocallyasymptoticallystablewhen T 0 0 B ( s ) 0 ds < Proof. ItfollowsfromTheorem 2.1 thatthematrix A isHurwitzsinceallofitseigenvalues havenegativerealparts.Because A isHurwitz,forthegivenmatrixnorm 0 0 thereexist a k > 0 suchthat 0 e At 0" ke $ at forall t > 0 byTheorem 1.6 .Forthesystem X =( A + B ( t )) X ,theprincipalfundamentalmatrixsolutionsatises X ( t )= e At + t 0 e A ( t $ s ) B ( s ) X ( s ) ds Then 0 X ( t ) 0 = / / / / / / / / e At + t 0 e A ( t $ s ) B ( s ) X ( s ) ds / / / / / / / / "0 e At 0 + t 0 / / / / e A ( t $ s ) / / / / 0 B ( s ) 00 X ( s ) 0 ds k + k T 0 0 B ( s ) 00 X ( s ) 0 ds since 0 e A ( t $ s ) 0" ke $ a ( t $ s ) k ke 7 t 0 k 0 B ( s ) 0 ds byGronwall'sinequality(Theorem 1.3 ) Rearrangingourprincipalfundamentalsolution,wehave 61

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0 X ( T ) # e AT 0" T 0 0 e A ( T $ s ) 00 B ( s ) 00 X ( s ) 0 ds k T 0 0 B ( s ) 0 ke k s 0 & B ( z ) & dz ds byinequalityfor 0 X ( t ) 0 k 2 T 0 0 B ( s ) 0 e k T 0 & B ( z ) & dz ds since s T Nowas T 0 0 B ( z ) 0 dz & 0 ,wehave 0 X ( T ) # e AT 0& 0 aswell.Thusastheaverage ofthenormof B ( s ) approacheszero,sodoes 0 X ( T ) # e AT 0 .Since A isHurwitz,all oftheeigenvaluesof e AT areinsidetheunitcircle.Bycontinuityofeigenvalues,the eigenvaluesof X ( T ) areinsidetheunitcircleaswell.Soas T 0 0 B ( s ) 0 ds & 0 ,the disease-freesolutionisstablebyTheorem 1.10 Toanalyzethecasewhentheaverageof ( t ) islarge,wewillneedthefollowing lemma. Lemma3. If C and D arequasi-positivematricessuchthat C D ,then e C e D Proof. Since C and D arequasi-positive,thereexists r > 0 suchthat 0 rI + C rI + D Then e rI + C e rI + D .Thisgives e r e C e r e D ,so e C e D Theorem5.5. Thedisease-freesolutionofsystem( 51 )-( 56 )isunstablewhen T 0 ( s ) ds > ; N 0 V # ( e $ T # 1), where 0 =min 1 i 4 { a ii } with a ii representingthediagonalentriesofthematrix A Proof. Forthesystem X =( A + B ( t )) X ,theprincipalfundamentalsolutionsatises X ( t )= e At + t 0 e A ( t $ s ) B ( s ) X ( s ) ds Let r besufcientlylargesothat rI + A + B ( t ) % 0 forall t ,andlet Y ( t )= e rt X ( t ) Then Y ( t )= e rt X ( t )+ rY ( t )=( A + B ( t )+ rI ) Y ( t ) .For Y ( t ) % 0 Y ( t ) % 0 .Sofor Y (0) % 0 Y ( t ) % 0 forall t .Thus X ( t )= e $ rt Y ( t ) % 0 forall t 62

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Since X ( t ) isnonnegative,wehave X ( t ) % e At forall t > 0 .Sotheprincipal fundamentalmatrixsolutionsatises X ( T ) % e AT + T 0 e A ( T $ s ) B ( s ) e As ds Itisgiventhat 0 =min 1 i 4 { a ii } ,where a ii representthediagonalentriesofthematrix A Note 0 0 sincethediagonalentriesof A arenonpositive,and A % 0 I .Then e At % e t I forall t > 0 byLemma 3 .So X ( T ) % e T + T 0 e ( T $ s ) B ( s ) e s ds = e T I + T 0 B ( s ) ds ( Then ( X ( T )) % e T I + T 0 B ( s ) ds (# ,where representsthespectralradiusof thematrix,byTheorem 1.5 .Forthegivensystem,letting b = T 5 6 = T 0 ( s ) ds ,we have T 0 B ( s ) ds = ) * * * + 000 b V 0000 0000 b N 0 b N 0 00 . Theeigenvaluesofthismatrixare 0,0, b ( N 0 V ,so < e T 0 I + T 0 B ( s ) ds 1= = e T 1+ T 5 6 7 N 0 V # # Then X ( T ) willhaveatleastoneeigenvaluethatislargerthanoneaslongas T 0 ( s ) ds > ; N 0 V # ( e $ T # 1). (59) SobyTheorem 1.10 ,thedisease-freesolutionisunstablewhenCondition 59 is satised. 63

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InFigure 5-2 ,wepresentanumericalsimulationofsystem( 51 )-( 56 )with f =0.5 r =1.5 b = N 0 = V # = T =1 $ = = ) = =0.5 ( t )= +sin( 2 + t T ) with =2 (so 5 6 =2 )andinitialcondition ( I 1 (0), I 2 (0), R (0), V + (0), S (0), V $ (0))= (0.05,0,0,0,0.95,1) toshowtheinstabilityofthedisease-freesolution.Withthese hypothesizedparametervaluesCondition 59 issatised,andweseefromthe simulationthat,evenwhenstartingclosetotheDFE,thesolutiondoesnotconverge toit. Figure5-2. Simulationofsystem(5 # 1)-(5 # 6)usingMATLABode45solverwith f =0.5 r =1.5 b = N 0 = V # = T =1 $ = = ) = =0.5 ( t )= +sin( 2 + t T ) with =2 ,andinitialcondition ( I 1 (0), I 2 (0), R (0), V + (0), S (0), V $ (0))=(0.05,0,0,0,0.95,1) 5.5StabilityoftheEndemicSolutioninaSpecialCase Weareunabletoanalyticallydeterminethestabilityofanendemicsolutionofthe periodicsystem,butweagainturntothecasewhere f =0 togainsomeinsight.Inthe limitingcase N = N 0 and V = V # ,theequationsofthemodelaregivenby: I 1 = ( t ) S N 0 V + V # # ) I 1 + ( I 2 + R ), (510) I 2 = ) I 1 # ( $ + ) I 2 (511) 64

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R = $ I 2 # R (512) V + = ( t ) V $ V # I 1 + I 2 N 0 # V + (513) S = # ( t ) S N 0 V + V # (514) V $ = V # # ( t ) V $ V # I 1 + I 2 N 0 # V $ (515) Thissystemhastheuniqueendemicperiodicsolution ( I # 1 I # 2 R # V # + S # V # $ )= N 0 ) + )' N 0 ( $ + )( ) + ) $) N 0 ( $ + )( ) + ) V # + ,0, V # $ # where V # + ( t ) istheperiodicfunction V # + ( t )= e $ t $ $ ( # + $ + ) V ( # + $ )( + $ ) t 0 & ( s ) ds t 0 ( s ) ( $ + + ) ) ( $ + )( ) + ) e s + $ ( # + $ + ) V ( # + $ )( + $ ) s 0 & ( ) ) d ) ds + V + (0) ( withinitialconditionon V + ( t ) givenby V + (0)= % ( $ + % + # ) ( $ + % )( # + % ) 7 T 0 ( s ) e ( s $ T ) e $ ( # + $ + ) V ( # + $ )( + $ ) [ s 0 & ( ) ) d ) $ < & > T ] ds 1 # e $ ( + $ ( # + $ + ) V ( # + $ )( + $ ) < & > ) T (516) and V # $ ( t )= V # # V # + ( t ) > 0 .Noticethat I # 1 I # 2 R # ,and S # havethesameendemic equilibriavaluesasinthecasewhere ( t )= isconstant.Weareabletoprove thefollowingtheorem,similartoTheorem 2.4 ,concerningstabilityoftheendemic equilibrium.Notethatthebeginningoftheproof,whichinvolvesexaminingthelimits oftheendemicequilibriavaluesofthetrees,remainsthesameasintheproofforthe constantcase. WerststatealemmawhichwillbeusedintheproofofTheorem 5.6 Lemma4. Let x ( t )= f 1 ( t ) # f 2 ( t ) x ( t ) and y ( t )= h 1 ( t ) # h 2 ( t ) y ( t ) with m f i h i M for i =1,2 ,all t % 0 ,and m M > 0 .Assumethat f i ( t ) # h i ( t ) & 0 as t &1 for i =1,2 Then x ( t ) # y ( t ) & 0 as t &1 Proof. Dene z ( t )= f 1 ( t ) # h 2 ( t ) z ( t ) forall t .Wewillshowthatboth z ( t ) # x ( t ) & 0 z ( t ) # y ( t ) & 0 as t &1 .Firstconsider d dt ( z # y )( t )= f 1 ( t ) # h 1 ( t ) # h 2 ( t )( z # y )( t ) Since f 1 ( t ) # h 1 ( t ) & 0 as t &1 and h 2 ( t ) % m > 0 ,wehavethat ( z # y )( t ) & 0 as 65

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t &1 .Nowconsider d dt ( z # x )( t )= # f 2 ( t )( z # x )( t )+( f 2 ( t ) # h 2 ( t )) z ( t ) .Notethat z ( t ) isboundedaboveandbelowduetotheequationof z ( t ) ,so ( h 2 ( t ) # f 2 ( t )) z ( t ) & 0 as t &1 .Usingthisandthefactthat f 2 ( t ) % m > 0 ,wehave ( z # x )( t ) & 0 as t &1 Thus ( z # y )( t ) # ( z # x )( t )= x ( t ) # y ( t ) & 0 as t &1 ,asdesired. Theorem5.6. If 5 6 > 0 ,thenallpositivesolutionsofthesystem( 510 )-( 515 )with S (0) < N 0 convergetotheendemicperiodicsolution ( I # 1 I # 2 R # V # + S # V # $ ) Proof. Firstweobservethat S ( t ) isnonincreasingandboundedbelowby0.So S ( t ) & S as t &1 forsome 0 S < N 0 .Nowlet M ( t )= I 1 ( t )+ I 2 ( t )+ R ( t ) .Sincethetotal populationis N 0 wehavethat M ( t ) & M as t &1 for M = N 0 # S > 0 .Hence I 1 ( t )= M + h ( t ) # I 2 # R where h ( t ) & 0 as t &1 .Thenweconsiderthesubsystem: I 2 = ) ( M + h ( t ) # I 2 # R ) # ( $ + ) I 2 (517) R = $ I 2 # R (518) Letting I 2 = #% ( $ + % )( # + % ) M and R = $# ( $ + % )( # + % ) M ,weobservethat 0= ) ( M # I 2 # R ) # ( $ + ) I 2 0= $ I 2 # R Wethenperformashiftbydening i 2 = I 2 # I 2 and r = R # R .Then( 517 )-( 518 )can bewrittenas x = Ax + f ( t ) where x = ) + i 2 r , A = ) + # ( ) + $ + ) # ) $ # , f ( t )= ) + ) h ( t ) 0 . Usingvariationofparameters,wendthat x ( t )= t 0 e ( t $ s ) A f ( s ) ds + e tA x 0 66

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Notethatthematrix A isHurwitzsincebotheigenvalueshavenegativerealparts, so e tA & 0 as t &1 byTheorem 1.6 .Thisalsomeansthatthereexist C $ > 0 suchthat | e ( t $ s ) A f ( s ) | Ce $ $ ( t $ s ) | f ( s ) | forall t % s % 0 .Let > 0 begiven. Since f ( t ) & 0 ,thereexists + > 0 suchthat | f ( t ) | < '$ 2 C forall t % + .Forthis xed + ) 0 | e ( t $ s ) A f ( s ) | ds e $ t ) 0 Ce $ s | f ( s ) | ds .Sothereexists + 1 % + suchthat e $ $ t ) 0 Ce $ s | f ( s ) | ds < 2 forall t % + 1 .So / / / / t 0 e ( t $ s ) A f ( s ) ds / / / / t 0 Ce $ $ ( t $ s ) | f ( s ) | ds = ) 0 Ce $ $ ( t $ s ) | f ( s ) | ds + t ) Ce $ $ ( t $ s ) | f ( s ) | ds e $ $ t ) 0 Ce $ s | f ( s ) | ds + "$ 2 t ) e $ $ ( t $ s ) ds < 2 + 2 (1 # e $ $ ( t $ ) ) ) forall t % + 1 < Thus x ( t ) & 0 as t &1 .Hence I 2 & I 2 and R & R as t &1 .Therefore I 1 & % # + % M whichwedeneas I 1 .Now I 1 + I 2 > 0 andwehave I 1 ( t )+ I 2 ( t )= I 1 + I 2 + g ( t ) forsome g ( t ) & 0 as t &1 .Wecanthenexpress( 513 )as V + = ( t ) N 0 ( I 1 + I 2 + g ( t )) # ( t ) N 0 V # ( I 1 + I 2 + g ( t ))+ # V + = g 1 ( t ) # g 2 ( t ) V + where g 1 ( t ) and g 2 ( t ) convergetothenonnegativeperiodicfunctions & ( t ) N 0 ( I 1 + I 2 ) and & ( t ) N 0 V ( I 1 + I 2 )+ ,respectively,as t &1 .UsingvariationofparametersandLemma 4 V + ( t ) approachestheuniqueperiodicfunction V + ( t )= e $ t $ I 1 + I 2 V N 0 t 0 & ( s ) ds 0 t 0 ( s ) I 1 + I 2 N 0 e s + I 1 + I 2 V N 0 s 0 & ( ) ) d ) ds + V + (0) 1 67

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where V + (0) isgivenbyEquation( 516 ).Hence V + ( t ) > 0 andsothereexists + > 0 and = 1 2 min V + ( t ) > 0 suchthatforall t > + S "# ( t ) N 0 V # S . Thus S ( t ) S ( + ) e & N 0 V ! t & ( s ) ds S ( + ) e & N 0 V K + T & ( s ) ds where K = > t # + T ? = S ( + ) e & N 0 V KT < & > wheretherightsideconvergesto0as t &1 .Therefore S =0 ,whichimpliesthat M = N 0 .Thus I 1 = I # 1 I 2 = I # 2 R = R # ,and V + = V # + InFigure 5-3 ,wepresentanumericalsimulationofsystem( 510 )-( 515 )with r =1.5 b = N 0 = $ = = ) = = V # = T =1 ( t )= +sin( 2 + t T ) with =1 (so 5 6 =1 )andinitialcondition ( I 1 (0), I 2 (0), R (0), V + (0), S (0), V $ (0))= (0.05,0,0,0,0.95,1) .Withthesehypothesizedparametervaluestheendemicsolution is ( I # 1 I # 2 R # V # + S # V # $ )=(0.5,0.25,0.25, V # + ,0, V # $ ) ,where V # + and V # $ arethedened periodicfunctions.Weseefromthesimulationthatthesolutiondoesindeedconvergeto theendemicsolution. 68

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Figure5-3. Simulationofsystem(5 # 10)-(5 # 15)usingMATLABode45solverwith r =1.5 b = N 0 = $ = = ) = = V # = T =1 ( t )= +sin( 2 + t T ) with =1 ,andinitialcondition ( I 1 (0), I 2 (0), R (0), V + (0), S (0), V $ (0))=(0.05,0,0,0,0.95,1) 69

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CHAPTER6 NUMERICALEXPLORATIONS Themotivationforthefollowingnumericalexplorationsistoattempttoincorporate theeconomicaspectofcitrusgreeningdiseaseintoourmodel.AsinBriggsand Sculpher[ 7 ],weconsiderastochasticmodel.Mostoftheliteratureanalyzesthe benetsagainstthecostsoftreatmentforsickhumans,butthesameprinciplescanbe appliedtocitrusgreeningandtheroguingcontrolstrategy.Ideally,thiscostanalysis mightbeusedtoconvincecommercialgrowersthatroguingmorepartially-infected treesnow,whilereducingthecurrentfruityield,couldactuallysavemoremoneyin severalyearswhenfewerdiseasedtreesarepresent.Unfortunately,atthispointthere isnotmuchdataavailablefromgrowersandtoomanybiologicalunknownstoprovide accurateeconomicinsight. Evenwithouttheabilitytodepicttheeconomicimpact,itisstillworthwhileto considerastochasticmodelandcompareitwiththedeterministicmodelwehave usedsofar.Thestochasticmodelisbasedontheoriginalowdiagram,withthesame statesforboththetreesandvectorsaswellasthesamepossibletransitionsbetween states.WeusethisowdiagramtosetuptheGillespiealgorithm.Eachsteponthe graphinFigure 6-1 representsatransition;weonlyallowonetransitiontooccurata time.Thetimebetweentransitionsisdecidedbyanexponentialmeanwaittime.One differencebetweenthissimulationandouroriginalmodelisthatthevectorpopulation isnotconstant(althoughthetreepopulationisconstant).Forthisreason,weplotthe treepopulationsandthevectorpopulationsondifferentaxes,astheirscalesarevastly different. Werstsimulateourstochasticmodelforasinglegroveoftreeswiththeinitial condition ( I 1 (0), I 2 (0), R (0), V + (0), S (0), V $ (0))=(5,0,0,50,15,50) andparameter values =100, ) =.5, $ =.25, =1, f =.9, =.1, r =.5 ,asshowninFigure 70

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6-1 .Usingthesamemodelandvalues,wealsoconsiderthetimeaverages,asshownin Figure 6-2 Figure6-1. Simulationofstochasticsystemwith =100, ) =.5, $ =.25, =1, f =.9, =.1, r =.5 ,andinitialcondition ( I 1 (0), I 2 (0), R (0), V + (0), S (0), V $ (0))=(5,0,0,50,15,50) Wenowexploretheincorporationofasecondgroveintoourmodel,allowingfor psyllidmigration.Eachgroveseparatelyfollowstheoriginalowdiagram,butthereis nowanadditionaltransitionthatrepresentsthepossiblemovementofapsyllidfromone grovetoanother.Weconsiderthesameparametervaluesineachgrove: =100, ) = .5, $ =.1, =1, f =.9, =.1, r =.25, m =.05 .However,therstgrovehasinitial condition ( I 1 (0), I 2 (0), R (0), V + (0), S (0), V $ (0))=(5,0,0,50,15,50) whilethesecond grovehasinitialcondition ( I 1 (0), I 2 (0), R (0), V + (0), S (0), V $ (0))=(0,0,0,0,20,100) ThetimeaveragesofthisstochasticmodelaredepictedinFigure 6-3 71

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Figure6-2. Timeaveragesofsimulationofstochasticsystemwith =100, ) =.5, $ =.25, =1, f =.9, =.1, r =.5 ,andinitialcondition ( I 1 (0), I 2 (0), R (0), V + (0), S (0), V $ (0))=(5,0,0,50,15,50) Nowwecomparethisstochasticmodeltothedeterministicmodelwithtwogroves. ForthesystempicturedinFigure 6-3 ,wesimulatethecorrespondingdeterministic modelwiththesameparametersandinitialconditions.ThisisdepictedinFigure 6-4 Notethatthelimitingbehaviorofeachstatevariableineachgroveisthesameforboth typesofmodels.However,theactuallimitingvaluesvaryslightlybetweenthestochastic anddeterministicsimulations.Thisisduetothenonlinearityofthemodel. 72

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Figure6-3. Timeaveragesofsimulationoftwo-grovestochasticsystemwith =100, ) =.5, $ =.1, =1, f =.9, =.1, r =.25, m =.05 ,grove1(top) initialcondition ( I 1 (0), I 2 (0), R (0), V + (0), S (0), V $ (0))=(5,0,0,50,15,50) andgrove2(bottom)initialcondition ( I 1 (0), I 2 (0), R (0), V + (0), S (0), V $ (0))=(0,0,0,0,20,100) 73

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Figure6-4. Simulationoftwo-grovedeterministicsystemwith =100, ) =.5, $ =.1, =1, f =.9, =.1, r =.25, m =.05 ,grove1(top) initialcondition ( I 1 (0), I 2 (0), R (0), V + (0), S (0), V $ (0))=(5,0,0,50,15,50) andgrove2(bottom)initialcondition ( I 1 (0), I 2 (0), R (0), V + (0), S (0), V $ (0))=(0,0,0,0,20,100) 74

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REFERENCES [1] R.N.A LLEN Epidemiologicalfactorsinuencingthesuccessofroguingforthe controlofbunchytopdiseaseofbananasinNewSouthWales ,AustralianJournal ofAgriculturalResearch,29(1978),pp.535544. [2] Furtherstudiesonepidemiologicalfactorsinuencingcontrolofbanana bunchytopdisease,andevaluationofcontrolmeasuresbycomputersimulation AustralianJournalofAgriculturalResearch,38(1987),pp.373382. [3] USDAA NIMALAND P LANT H EALTH I NSPECTION S ERVICE Aphisnewsrelease: Usdaannouncesquarantinetopreventthespreadofcitrusdisease http://www. freshfromflorida.com/pi/chrp/greening/citrus_disease-june-2010.pdf 2010. [4] R.B.B ASSANEZI ,L.H.M ONTESINO ,M.C.G.G ASPAROTO ,A.B ERGAMIN F ILHO AND L.A MORIM Yieldlosscausedbyhuanglongbingindifferentsweetorange cultivarsinS aoPaulo,Brazil ,EuropeanJournalofPlantPathology,130(2011), pp.577586. [5] A.B ERMANAND R.J.P LEMMONS NonnegativeMatrices ,SIAM,1979. [6] M.B OCHER Onsemi-analyticfunctionsoftwovariables ,TheAnnalsof Mathematics,12(1910),pp.1826. [7] A.B RIGGSAND M.S CULPHER AnintroductiontoMarkovmodellingforeconomic evaluation ,Pharmacoeconomics,13(1998),pp.397409. [8] M.S.C HANAND M.J.J EGER Ananalyticalmodelofplantvirusdiseasedynamics withroguingandreplanting ,JournalofAppliedEcology,31(1994),pp.413427. [9] C.C HICONE OrdinaryDifferentialEquationswithApplications ,vol.34,Springer, 2006. [10] E.J.D ASBACH ,E.H.E LBASHA AND R.P.I NSINGA Mathematicalmodelsfor predictingtheepidemiologicandeconomicimpactofvaccinationagainsthumanpapillomavirusinfectionanddisease ,EpidemiologicReviews,28(2006), pp.88100. [11] E.B.D YNKINAND A.A.Y USHKEVICH MarkovProcesses:TheoremsandProblems PlenumPress,1969. [12] S.F ISHMANAND R.M ARCUS Amodelforspreadofplantdiseasewithperiodic removals ,JournalofMathematicalBiology,21(1984),pp.149158. [13] S.F ISHMAN ,R.M ARCUS ,H.T ALPAZ ,M.B AR -J OSEPH ,Y.O REN ,R.S A LOMON AND M.Z OHAR Epidemiologicalandeconomicmodelsforspreadand controlofcitrustristezavirusdisease ,Phytoparasitica,11(1983),pp.3949. 10.1007/BF02980710. 75

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[14] A.F ONDA Uniformlypersistentsemidynamicalsystems ,Proceedingsofthe AmericanMathematicalSociety,104(1988),pp.111116. [15] H.F REEDMAN ,S.R UAN AND M.T ANG Uniformpersistenceandowsneara closedpositivelyinvariantset ,JournalofDynamicsandDifferentialEquations,6 (1994),pp.583600. [16] K.G OPALSAMY StabilityandOscillationsinDelayDifferentialEquationsofPopulationDynamics ,KluwerAcademicPublishers,Dordrecht,1992. [17] T.G OTTWALD CurrentEpidemiologicalUnderstandingofCitrusHuanglongbing AnnualReviewofPhytopathology,48(2010),pp.119139. [18] S.E.H ALBERT .Personalcommunication,2010. [19] .Personalcommunication,2012. [20] S.E.H ALBERTAND K.L.M ANJUNATH Asiancitruspsyllids(Sternorrhyncha: Psyllidae)andgreeningdiseaseofcitrus:aliteraturereviewandassessmentofrisk inFlorida ,FloridaEntomologist,87(2004),pp.330353. [21] A.H ODGESAND T.S PREEN EconomicImpactsofCitrusGreening(HLB)in Florida,2006/07-2010/11 ,ProceedingsoftheAmericanMathematicalSociety, (2012). [22] K.J ACOBSEN ,J.S TUPIANSKY AND S.S.P ILYUGIN Mathematicalmodelingofcitrusgrovesinfectedbyhuanglongbing ,MathematicalBiosciencesandEngineering, 10(2013),pp.705728. [23] J.E.K EESLING .Personalcommunication,October2012. [24] D.G.M EAD Newton'sidentities ,AmericanMathematicalMonthly,(1992), pp.749751. [25] R.M.N ISBETAND W.S.C.G URNEY ModellingFluctuatingPopulations ,JohnWiley &Sons,Inc.,605ThirdAve.,NewYork,NY10158,416,1982. [26] T EXAS D EPARTMENTOF A GRICULTURE Texasdepartmentofagricultureandusdaconrmdetectionofplantdiseasethatdamagescitrus trees http://www.texasagriculture.gov/tabid/76/Article/1802/ texas-department-of-agriculture-and-usda-confirm-detection-of-plant\ -disease-tha.aspx ,2012. [27] U NIVERSITYOF C ALIFORNIA I NTEGRATED P EST M ANAGEMENT P ROGRAM A newpestincalifornia,diaphorinacitri(asiancitruspsyllid):Provisionaltreatment guidelinesforcitrusinquarantineareas http://www.ipm.ucdavis.edu/EXOTIC/ diaphorinacitri.html ,2011. 76

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[28] S.L.O TT ,S.J.W ELLS AND B.A.W AGNER Herd-leveleconomiclossesassociated withJohne'sdiseaseonUSdairyoperations ,PreventiveVeterinaryMedicine,40 (1999),pp.179192. [29] R.W.H.P LUKE ,J.A.Q URESHI AND P.A.S TANSLY Citrusushingpatterns, Diaphorinacitri(Hemiptera:Psyllidae)populationsandparasitismbyTamarixia radiata(Hymenoptera:Eulophidae)inPuertoRico ,TheFloridaEntomologist,91 (2008),pp.3642. [30] K.M.R ICH Discosem:Anintegratedepidemiological-economicanalysisoffoot andmouthdiseaseinsoutherncone ,tech.report,Workingpaper,TheRegional EconomicsApplicationsLaboratory,UniversityofIllinois,2004. [31] H.L.S MITHAND H.R.T HIEME Dynamicalsystemsandpopulationpersistence vol.118,AmericanMathematicalSociety,2011. [32] H.L.S MITHAND P.W ALTMAN TheTheoryoftheChemostat ,CambridgeUniversity Press,Cambridge,1995. [33] H.M.T AYLORAND S.K ARLIN AnIntroductiontoStochasticModeling ,vol.3, AcademicPressNewYork,1984. [34] H.R.T HIEME MathematicsinPopulationBiology ,PrincetonUniversityPress, 2003. [35] A.V AN B RUGGEN .Personalcommunication,2010. [36] P.V ANDEN D RIESSCHEAND J.W ATMOUGH Reproductionnumbersandsubthresholdendemicequilibriaforcompartmentalmodelsofdiseasetransmission MathematicalBiosciences,180(2002),pp.2948. [37] C.F.X U ,Y.H.X IA ,K.B.L I AND C.K E Furtherstudyofthetransmissionofcitrus huanglongbingbyapsyllid,DiaphorinacitriKuwayama ,inProceedingsofthe10th Conference,InternationalOrganizationofCitrusVirologists.Riverside,California, 1988,pp.243248. 77

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BIOGRAPHICALSKETCH JilliangrewupinLakewood,Ohio,asuburbofClevelandsituatedonLakeErie. AfterplayinggolfandbasketballandgraduatingasvaledictorianfromLakewood HighSchool,sheattendedBirmingham-SouthernCollegeonagolfandacademic scholarship.ShegraduatedfromBSCin2007withabachelorofscienceinmathematics andaminorincomputerscience.JillianthenattendedtheUniversityofFlorida,where sheservedasateachingassistantwhileinthegraduatemathematicsprogram.Whilein Gainesville,shealsohadtheopportunitytocoachtheOakHallgirls'golfteam,leading theEaglestothestatetournamentthreeoutoffouryears.Jillianearnedhermaster's degreein2009whileonherwaytocompletingherPhDinmathematicsin2013inthe areaofmathematicalbiology. 78