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Two Extensions of a Classical Within-Host Virus Model

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Title:
Two Extensions of a Classical Within-Host Virus Model
Creator:
Browne, Cameron Jeffrey
Place of Publication:
[Gainesville, Fla.]
Florida
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University of Florida
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english
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1 online resource (132 p.)

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Mathematics
Committee Chair:
Pilyugin, Sergei
Committee Co-Chair:
Salemi, Marco Maria
Committee Members:
Zhang, Lei
Keesling, James E
De Leenheer, Patrick
Wayne, Marta L
Graduation Date:
8/11/2012

Subjects

Subjects / Keywords:
Antivirals ( jstor )
Drug design ( jstor )
Eigenvalues ( jstor )
Gene therapy ( jstor )
HIV ( jstor )
Infections ( jstor )
Mathematics ( jstor )
Modeling ( jstor )
Semigroups ( jstor )
Sine function ( jstor )
Mathematics -- Dissertations, Academic -- UF
hiv -- mathematical -- model -- optimization -- treatment -- viruses
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bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Mathematics thesis, Ph.D.

Notes

Abstract:
Two extensions of a classical within-host virus model are analyzed.  First, time-periodic combination antiviral drug therapy is incorporated into the model. Floquet theory is applied to find a threshold condition which determines whether the virus persists. Perturbation techniques are then employed to give sharper global dynamical results in the case of small amplitude periodic variations in drug efficacy. It is also found that the timing between dosages of the different drugs in the therapy can critically affect the treatment outcome. Numerical simulations are provided to show potential applications of the results to optimizing HIV treatment.  The second model adds age since infection structure to the infected cell compartment of the original model in order to account for heterogeneity in the infected cell life cycle. The resulting system, a PDE coupled nonlinearly with two ODEs, is analyzed. The basic reproduction number, R0, of this model is calculated. For the case of R0 1, there is a unique positive steady state and global stability is proved. ( en )
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In the series University of Florida Digital Collections.
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Includes vita.
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Includes bibliographical references.
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Description based on online resource; title from PDF title page.
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This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (Ph.D.)--University of Florida, 2012.
Local:
Adviser: Pilyugin, Sergei.
Local:
Co-adviser: Salemi, Marco Maria.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2014-08-31
Statement of Responsibility:
by Cameron Jeffrey Browne.

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UFRGP
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Copyright Browne, Cameron Jeffrey. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Embargo Date:
8/31/2014
Resource Identifier:
857767123 ( OCLC )
Classification:
LD1780 2012 ( lcc )

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TWOEXTENSIONSOFACLASSICALWITHIN-HOSTVIRUSMODELByCAMERONJEFFREYBROWNEADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2012

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c2012CameronJeffreyBrowne 2

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Tomyparents 3

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ACKNOWLEDGMENTS Iamgratefultoallthepeoplewhohavemadethisdissertationpossibleforme.First,Iwouldliketothankmyadvisor,Dr.SergeiPilyugin,forhissupportandguidance,andforteachingmehowtothinkcriticallyaboutmathematicsanditsapplicationtobiology.IalsowanttoespeciallyacknowledgeDr.DeLeenheerforallofhisassistanceandmotivationinmywork.Next,IamveryappreciativeofthetimethatDr.Salemispentineducatingmeaboutcertainaspectsofbiologyandevolution.Iwouldliketothanktherestofmycommittee,theotherprofessorsinthemathematicsdepartment,andthemathdepartmentstaff,formakingmyexperienceattheUniversityofFloridasovaluable.Finally,Iamdeeplygratefulformyparents,DonaandJeffreyBrowne,fortheirunwaveringsupportinallfacetsofmylife.IalsowanttothankmybrothersNickandJordan,andmysisterAlly,fortheirencouragementandalwaysbeingavailabletotalk.Last,butcertainlynotleast,IwanttoacknowledgeandthankHayriyeGulbudak. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFFIGURES ..................................... 7 ABSTRACT ......................................... 8 CHAPTER 1BACKGROUND,STANDARDVIRUSMODEL,MATHPRELIMINARIES .... 9 1.1Introduction ................................... 9 1.2BackgroundonHIV .............................. 11 1.3StandardVirusModel ............................. 13 1.3.1ModelFormulation ........................... 13 1.3.2DynamicsoftheStandardModel ................... 16 1.4MathematicalPreliminaries .......................... 17 1.4.1Matrices ................................. 17 1.4.2FloquetTheory ............................. 18 1.4.3SemigroupsI:BasicDenitions .................... 19 1.4.4SemigroupsII:UniformPersistence .................. 23 1.4.5PerturbationofaGloballyStableSteadyState ............ 26 2PERIODICDRUGTREATMENTINVIRUSMODEL ............... 27 2.1Introduction ................................... 27 2.2FormulationofModelandBoundednessofSolutions ............ 28 2.3ReproductionNumber ............................. 31 2.4PeriodicPerturbations ............................. 37 2.5GlobalStabilityinNon-CriticalCase ..................... 39 2.6CriticalCase,Part1:TranscriticalBifurcation ................ 45 2.7CriticalCase,Part2:BifurcationsofLinearizedSystem .......... 58 2.8Antagonism ................................... 69 2.9PhaseShifts .................................. 76 3NUMERICALSIMULATIONSOFPERIODICHIVMODEL ............ 83 3.1SinusoidalDrugEfcacies ........................... 83 3.2Bang-BangEfcacies ............................. 89 3.3StructuredTreatmentInterruptions ...................... 92 3.4Discussion ................................... 94 4AGE-STRUCTUREDWITHIN-HOSTVIRUSMODEL ............... 97 4.1Introduction ................................... 97 4.2ModelFormulation ............................... 100 5

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4.3ExistenceofSolutions ............................. 104 4.4ReproductionNumber ............................. 110 4.5GlobalExtinctionwhenR0<1 ........................ 111 4.6AsymptoticSmoothness ............................ 112 4.7UniformPersistence .............................. 115 4.8LyapunovFunctionalandGlobalStability .................. 119 5FUTUREWORK ................................... 126 REFERENCES ....................................... 128 BIOGRAPHICALSKETCH ................................ 132 6

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LISTOFFIGURES Figure page 2-1Thethresholdstabilitycurveforconstantdrugefcaciesandasmallamplitudeperiodicperturbation ................................. 37 2-2Phasedifferencebetweendrug-efcacyfunctions ................ 70 2-3DominantFloquetmultiplierasafunctionofphasedifference,( ),forsinusoidaldrugefcacieswithdifferentdosingperiods. .................... 72 2-4( )forsinusoidaldrugefcacieswithsamedosingperiod. ........... 73 2-5Amplitudeof( )vs ................................ 74 2-6( )fortoyexample ................................ 80 3-1Simulationsofstandardvirusmodelforin-phaseandout-of-phaseantiviraltreatments ...................................... 84 3-2Sensitivityofviralsteadystatetodrugefcacyinstandardmodelandmodelwithdensitydependentinfectedcelldeathrate .................. 86 3-3Simulationsofin-phaseandout-of-phasetreatmentsinlowlevelviralpersistencemodels ........................................ 88 3-4( )inthecaseofbang-bangefcacy ....................... 90 3-5( )inthecaseofpiecewiseconstantefcacy .................. 91 3-6( )intheSTIcase ................................. 93 7

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyTWOEXTENSIONSOFACLASSICALWITHIN-HOSTVIRUSMODELByCameronJeffreyBrowneAugust2012Chair:SergeiS.PilyuginCochair:MarcoSalemiMajor:MathematicsTwoextensionsofaclassicalwithin-hostvirusmodelareanalyzed.First,time-periodiccombinationantiviraldrugtherapyisincorporatedintothemodel.Floquettheoryisappliedtondathresholdconditionwhichdetermineswhethertheviruspersists.Perturbationtechniquesarethenemployedtogivesharperglobaldynamicalresultsinthecaseofsmallamplitudeperiodicvariationsindrugefcacy.Itisalsofoundthatthetimingbetweendosagesofthedifferentdrugsinthetherapycancriticallyaffectthetreatmentoutcome.NumericalsimulationsareprovidedtoshowpotentialapplicationsoftheresultstooptimizingHIVtreatment.Thesecondmodeladdsagesinceinfectionstructuretotheinfectedcellcompartmentoftheoriginalmodelinordertoaccountforheterogeneityintheinfectedcelllifecycle.Theresultingsystem,aPDEcouplednonlinearlywithtwoODEs,isanalyzed.Thebasicreproductionnumber,R0,ofthismodeliscalculated.ForthecaseofR0<1,thevirusiscleared.WhenR0>1,thereisauniquepositivesteadystateandglobalstabilityisproved. 8

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CHAPTER1BACKGROUND,STANDARDVIRUSMODEL,MATHPRELIMINARIES 1.1IntroductionMathematicalmodelingofwithin-hostvirusdynamicshasbeenanextensivesubjectofresearchoverthepasttwodecades.Thefateofaviruspopulationwithinaninfectedindividualdependsuponitsabilitytoreplicateandsurviveinsidethebody.Inordertoreplicate,avirusparticlemustenterahosttargetcellandusethecellmachinerytoproducemorevirionsinsidethecell,whicharesubsequentlyreleasedoutsidethecellmembraneandgoontoinfectmoretargetcells.Viruseselicitanimmuneresponse,andthesurvivaloftheviruswithinthebodyistiedtoitsabilitytoevadethehostimmuneresponse.Alloftheseprocessesoccursimultaneouslywithinthebody,creatingacomplexsystemofinteractingpopulations.Amathematicalmodelcanserveasbothadescriptionofthevirus-hostsystemandawaytoassessthedynamicalinteractionsbetweenrelevantvariables.Chronicviralinfections,suchasHIV,HepatitisBandC,areabletoestablishaset-pointlevelafterprimaryinfectionandapproximatelyremainatthispopulationsizeforyears.Inorderforthistohappen,therateofviralproductionandclearanceshouldequilibrate.In1996,Perelsonetal.introducedadifferentialequationsystemwhichmodelsthebasicviralreplicationcycleandcapturesthissteadystateproperty[ 34 ].Amoregeneralversionofthissystemlooksasfollows:. T=f(T))]TJ /F3 11.955 Tf 11.96 0 Td[(kVT,. T=kVT)]TJ /F5 11.955 Tf 11.95 0 Td[(T, (1-1). V=NT)]TJ /F5 11.955 Tf 11.95 0 Td[(V)]TJ /F3 11.955 Tf 11.95 0 Td[(kVT,whereT,T,andVdenotetheconcentrationsofhealthyandinfectedcells,andfreevirusparticles,respectively,andf(T)isthenetgrowthrateofthehealthycells.This 9

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modelhassubsequentlyservedasafundamentaldescriptionofvirus-hostdynamicsandwillbereferredtoasthestandardvirusmodel.DeLeenheerandSmith(2003)rigorouslycharacterizedthedynamicalpropertiesofthestandardvirusmodel[ 12 ].Theyfoundthataquantityknownasthebasicreproductionnumber,R0(denedinSection 1.3 ),largelydeterminestheglobaldynamicsofthesystem.WhenR0>1,aviralsteadystateisestablishedundercertainassumptions,butoscillatorybehaviorcannotberuledoutingeneral.IfR01,theviruswillbecleared.Wewillprovideamoredetaileddescriptionofthestandardmodel(Equation 1-1 )inSection 1.3 .Fromamathematicalpointofview,itisalwaysimportanttoprovepropertiesofadynamicalsystem.Numericalsimulationscanprovideanideaofthedynamicalbehavior,butcannotprovideageneralresultaboutthesystem.Inaddition,mathematicalanalysiscanelucidatesomeofthesubtletiesinvolvedinthemodel,whichcanproducenon-obviousinsightsintothebiologicalsystemandinformonthelimitationsofthemodel.Inthisdissertation,twoextensionsofthestandardvirusmodelareanalyzed.First,weincorporatetime-periodiccombinationantiviraldrugtherapyintothestandardmodel.Combinationantiviraltherapyisacommontreatmentstrategyforviralinfections,especiallyHIV,henceitisimportanttounderstandtheoveralleffectthatthedrugshaveonthedynamicsofthesystem.Applyingatreatmentwithconstantefcacyisequivalenttore-scalingcertainparametersinEquation 1-1 .Hence,inthiscase,thequalitativebehaviorofsolutionsarenotalteredandcanbedeterminedbyre-scalingthereproductionnumber,R0.Inreality,administrationofantiviraldrugsisperiodic,whichresultsintime-periodicefcacy.Therefore,weassumethatthedrugshavetime-periodicefcacy,whichaddssignicantdifcultiestothemathematicalanalysis.Weareabletocharacterizethedynamicsincertaincases,andndaninterestingconsequenceof 10

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theperiodicity,namely:thetimingbetweendosagesofthedifferentdrugscanaffecttreatmentoutcome.Fortheothermodelconsideredinthisdissertation,weaddage-since-infectionstructuretotheinfectedcellstatevariable,T.Thisprovidesageneralwayofincorporatingheterogeneityintotheinfectedcelllifecycle.Age-since-infectionstructuredvariantsofthestandardvirusmodelhaveappearedintheliteratureoverthepastfewyears[ 4 16 30 35 ],butproofsconcerningtheglobaldynamicsremainedanopenproblem.Wewillprovideaglobalanalysiswhichaddressesthisproblem.Althoughthemodelscanapplytovariousviralinfections,wefocusourdiscussionandnumericalsimulationsonthecaseofHIV.WeprovideabasicbackgroundofHIVinSection 1.2 .InSection 1.3 ,wecarefullydescribetheassumptionsofthestandardvirusmodelandstatepreviousresultsregardingthemodel.Chapter 2 and 3 aredevotedtoanalysisandsimulationsoftheperiodiccombinationantiviraltherapymodel.Chapter 4 isconcernedwiththeage-structuredvirusmodel. 1.2BackgroundonHIVHumanimmunodeciencyvirus(HIV)istheretroviruswhichcausesacquiredimmunodeciencysyndrome(AIDS),aconditioninhumansthatleadstothefailureoftheimmunesystemand,subsequently,death.Thereareapproximately34millionpeoplelivingwithHIV/AIDSworldwide[ 43 ].Fromitsdiscoveryin1981to2010,AIDShaskillednearly30millionpeople[ 43 ].HIVisspreadbythetransferofblood,semen,vaginaluid,pre-ejaculate,orbreastmilk.Onceinsidethebody,themaintargetforHIVisCD4+Tcells,butotherimmunecells,suchasmacrophagesanddendriticcells,areinfectedaswell.HIVparticlesenterCD4+TcellsbyinteractingwitheithertheCCR5orCXCR4co-receptoronthesurfaceofthecell.UponentryintoahealthyTcell,theviralRNAisconvertedintoDNAusingtheenzymecalledreversetranscriptase.Oncetheviralcopyhasbeenmade,thedoublestrandedviralDNAisintegratedintocell'snucleusasa 11

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provirus.Subsequently,viralproteinsareproducedaccordingtothegeneticinformationencodedintheprovirus.Theseproteinsareconstructed,thenewvirionsareassembled,thentheybudofffromtheinfectedcell'ssurfaceandcangoontoinfectotherhealthyTcells.Duringthematurationstageofthenewvirion,theproteaseenzymecleaveslongproteinchains,anecessarystepforproducingafunctionalvirus.ForuntreatedHIV-infectedindividuals,theprogressionfrominitialHIVinfectiontoAIDStakes,onaverage,about10years.Therearethreedistinctphasesinthediseaseprogression:acuteinfection,chronicinfection,andAIDS.ShortlyafterrstexposuretoHIV,theconcentrationofvirionsrisesrapidlyinthebloodtoashighasseveralmillioncopies/ml,triggeringastrongimmuneresponseand,often,u-likesymptomsintheindividual.Theviralloadreachesamaximumandabruptlydeclinesbecauseoftheimmuneresponseandtargetcelllimitation.Thevirusisnotcleared,though,andfallstoalevelknownastheset-pointlevel,markingtheendoftheacutestage.Theconcentrationofvirionsremainsrelativelystableattheset-pointlevelformanyyearsandtheindividualisasymptomaticduringthisperiod.TheCD4+Tcellcountslowlydecreasesduringthischronicstage.Oncethiscountfallsbelowacriticallevelof200cellsperL,theindividualissaidtohaveAIDS,inwhichcase,opportunisticinfectionscanbefatal.Aninuentialcollaborationbetweenmodelersandexperimentalistscameaboutintheyearsof1995-1996withpublicationspertainingtotherateofviralproductionduringtheasymptomaticstageofHIV.InanexperimentconductedbyHoetal.[ 20 ],antiviralmedicationswereadministeredtochronicallyinfectedpatientsinordertoperturbtheviralsteadystate,andtheviralloadwascloselymonitored.Arapiddeclineinviralloadwasobservedinthepatients.Perelsonetal.incorporatedperfectinhibitiondrugtreatmentintothestandardvirusmodel(Equation 1-1 )andttheresultingequationstothisviraldeclinedata.TheyfoundthattheHIVturnoverratewasextremelyfastandcalculatedthat,onaverage,atleast1010virionshadtobeproduceddailyinorder 12

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forHIVtomaintainitssteadystatelevelduringtheasymptomaticstage[ 34 ].SincereplicationofHIViserror-prone,resistancetoantiviraldrugscanrapidlyappearduringdrugtreatment.ThisprovidedquantitativejusticationforthestrategyoftreatingHIVinfectedindividualswithcombinationsofantiviralmedications,knownascombinationtherapy.CombinationantiviraltherapyhassignicantlyreducedthemortalityrateofHIVsinceitstartedtobeimplementedonalargescalein1996.However,drugtreatmentcannotcureHIV,asalatentreservoirofvirusremainsintreatedpatients.Also,drugresistance,residualviralreplication,patientnon-adherence,anddrugtoxicitycontinuetobeproblemswithantiviraltherapy.Accesstoantiviralmedicationisimproving,butisstillamajorissueindevelopingcountries,suchasnationsinsub-SaharanAfricawhere68%ofHIVcasesoccur[ 43 ].Thesearchforoptimaltreatments,andnovelpreventativemeasuressuchasanHIVvaccine,continuetobethesubjectofintenseresearch. 1.3StandardVirusModel 1.3.1ModelFormulationThestandardmodelcontainsthreestatevariables:T,theconcentrationofuninfectedtargetcells;T,theconcentrationofproductivelyinfectedtargetcells;andV,theconcentrationoffreevirusparticles.Themodel,anonlinearsystemofthreedifferentialequations,describesthecoupledchangesinthesevariablesthroughtimeinasinglecompartmentofaninfectedindividual.InthecaseofHIV,TrepresentsCD4+Tcells,theprimarytargetofthevirus.ThepopulationdynamicsofCD4+Tcellsisnotwellunderstood.Hence,weassumeageneralmodelforTcellregulationasinDeLeenheerandSmith[ 12 ].ContinualcompetitionamonglymphocytesforsurvivalsignalsandthepresenceofgrowthpromotingcytokinesarebelievedtocauseTcelllevelstoremainrelativelyconstantinhealthyindividuals.Ingeneral,thepropertyofbiologicalsystemstomaintainastablestateisknownashomeostasis.WeassumeT-cellhomeostasisinahealthyindividualis 13

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governedbythefollowingdifferentialequation:. T=f(T),wherefissmoothandthereexistsT0>0suchthat:f(T)>0forall0TT0. (1-2)Bycontinuityoff,f(T0)=0.Thus,T0istheequilibriumconcentrationofTcellsinanuninfectedindividual.Twocommonlyusedfunctionalformsforf(T)are: 1. f(T)=f1(T)=s)]TJ /F3 11.955 Tf 11.95 0 Td[(cT(NowakandMay)[ 32 ] 2. f(T)=f2(T)=s)]TJ /F3 11.955 Tf 11.95 0 Td[(cT+rT(1)]TJ /F3 11.955 Tf 11.95 0 Td[(T=Tmax)(PerelsonandNelson)[ 33 ]Bothf1(T)andf2(T)satisfyCondition 1-2 .Therstform,f1(T),isasimplelinearfunction,whichassumesthatcellsaresuppliedataconstantratesfromasourcesuchasthethymus,anddieatthe(per-capita)ratec.f2(T)addsalogisticproliferationtermtotheequation.TheinfectionofTcellsbythefreevirusisassumedtofollowmassactionkineticswithrateconstantk.Thus,theTcellsaretransformedintoinfectedTcellsataratekVT.Thistypeoftermisreasonable,astheprobabilityoffreevirusparticlesencounteringTcellsshouldbeproportionaltotheproductoftheconcentrationsofTandV.AthigherconcentrationsofV,onemightexpecttherateofnewinfectionstosaturate,buttheconcentrationoffreevirusdoesnotbecomehighrelativetotheconcentrationofTcells(typically1:1ratiointhebloodorlymphoidtissue)[ 33 ].Thus,saturationeffectsareignored.TheaveragenumberofvirusparticlesproducedbyaninfectedcellduringitslifespanisassumedtobeN.Hence,thevirus-celldynamicscanberepresentedinchemicalreactionnotationasfollows:T+V!T!NV. 14

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Weassumethatthelossoffreevirusduringtargetcellinfection(reectingviralentryintotheTcell)canbeignored.Explicitly,theterm)]TJ /F3 11.955 Tf 9.3 0 Td[(kVTisremovedfromthe. Vequation.PerelsonandNelson,NowakandMayarguethatthistermisrelativelysmallandcanbeabsorbedintotheper-capitaclearancerateofthefreevirusparticles[ 32 33 ].Also,forparametersrepresentativeofmostviruses,thedynamicsofthemodelarenotaffected.Thus,thestandardwithin-hostvirusmodelisgivenbythefollowingsystemofdifferentialequations:. T=f(T))]TJ /F3 11.955 Tf 11.96 0 Td[(kVT,. T=kVT)]TJ /F5 11.955 Tf 11.96 0 Td[(T, (1-3). V=NT)]TJ /F5 11.955 Tf 11.96 0 Td[(VInfectedTcells,T,dieattheper-capitarate.Equivalently,thelifespanofinfectedcellsisassumedtobeexponentiallydistributedwithparameter.Hence,theexpectedlifespanofainfectedcellis1=.Wenotethatallinfectedcellsareassumedtobeproductivelyinfectedcells,i.e.latentlyinfectedcellsarenotfactoredintothemodel.Theper-capitarateofviralproductionforaninfectedcellisgivenbyN.Freevirusisclearedattheper-capitarate.Thestandardmodeldescribesthebasicviralreplicationcycle,namely:virusinfectingcellsandinfectedcellsproducingmorevirus.ManydynamicalpropertiesofHIVareignoredinthismodel.Firstoff,thetimelagbetweenviralentryintoatargetcellandsubsequentviralproductionisabsentinEquation 1-3 .InChapter 4 ,wewillstudyamodicationofthestandardmodelwhichallowsforsuchadelay.Inaddition,Equation 1-3 doesnotexplicitlyincludeahostimmuneresponse.OnecanarguethatconstantimmunepressurecanbereectedintheparametersandN,buttheimmuneresponse,inreality,dependsupontheconcentrationofinfectedcells.Also,viralmutationsand 15

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latentlyinfectedcellsaretwofactorsignoredinthestandardmodel,whichgreatlyaffecttheresultsoftreatmentofHIV. 1.3.2DynamicsoftheStandardModelThequalitativebehaviorofsolutionstoEquation 1-3 islargelydeterminedbythebasicreproductionnumber,R0,whereR0=NkT0 (1-4)R0canbeinterpretedastheaveragenumberofsecondaryinfectedcellsarisingfromasingleinfectedcellinahealthyindividual.Indeed,Nistheexpectednumberofvirusproducedbytheinfectedcell,1=istheaveragelifespanofafreevirusparticle,andkT0istheexpectednumberofinfectedcellsproducedbyeachofthesevirusparticlesinthehealthyindividual.WenoticethatE0:=(T0,0,0)isalwaysasteadystatesolutionofEquation 1-3 .E0isreferredtoastheinfection-freeequilibrium.WhenR0>1,thereexistsauniquepositiveequilibriumcorrespondingtoinfection,E1=( T, T, V),where T= kN, T=f( T) V=f( T) k T. (1-5)WhenR01,thereisnopositiveequilibrium.ItcanbeshownthattheeigenvaluesofJ(E0)areallnegativewhenR0<1,whereJ(E0)istheJacobianofEquation 1-3 evaluatedatE0.WhenR0>1,J(E0)hasapositiveeigenvalue.Therefore,E0islocallyasymptoticallystablewhenR0<1andunstablewhenR0>1.DeLeenheerandSmithprovedglobalstabilityofE0whenR01,i.e.thevirusisclearedwhenR01.WhenR0>1,asufcientconditionforLAS(localasymptoticstability)ofE1is:f0( T)0.Ifthisconditionisnotsatised,DeLeenheerandSmithfoundthatforcertainparameters,inthecasef=f2,E1isunstable[ 12 ].Inthisunstablecase,J(E1)hasapairofcomplexconjugateeigenvalueswithpositiverealpart,andonenegative 16

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eigenvalue.FortheglobaldynamicsofEquation 1-3 ,DeLeenheerandSmithprovedthatsolutionswithpositiveinitialconditionseitherconvergetoE1orconvergetoanon-trivialperiodicorbit,andalsostatedsufcientconditionsforeachscenario.DeLeenheerandPilyuginfoundasimplersufcientconditionfortheglobalstabilityofE1whenR0>1[ 11 ],referringtothefollowingasthesectorcondition:)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(f(T))]TJ /F3 11.955 Tf 11.96 0 Td[(f( T)1)]TJ ET q .478 w 266.74 -125.39 m 277 -125.39 l S Q BT /F3 11.955 Tf 266.74 -135.36 Td[(T T0. (1-6)Notethatthisconditionissatisedwhenf(T)isadecreasingfunction,independentlyofthevalueof T,forexamplef(T)=f1(T)=s)]TJ /F3 11.955 Tf 12.29 0 Td[(cT.Inthecaseoff(T)=f2(T)=s)]TJ /F3 11.955 Tf 11.95 0 Td[(cT+rT(1)]TJ /F3 11.955 Tf 11.96 0 Td[(T=Tmax),Condition 1-6 issatisedwhensf( T). Theorem1.1(DeLeenheerandPilyugin,[ 11 ]). Supposef(T)satisesthesectorcondition(Condition 1-6 )andR0>1forthestandardmodel(Equation 1-3 ).ThenE1isGAS(globallyasymptoticallystable)inInt(R3+).Hence,ifR0>1andthesectorconditionissatised,theviralloadwillconvergetoaconstant(positive)level. 1.4MathematicalPreliminaries 1.4.1MatricesLet1,2,....,nbetheeigenvaluesofthennmatrixA.WedenethespectralradiusofA.Denition.ThespectralradiusofA,denotedby(A),isgivenby:(A):=maxijij.Denition.AisprimitiveifthereexistsapositiveintegerksuchAkispositive,i.e.alltheentriesofAkarepositive. Theorem1.2(Perron-FrobeniusTheorem). LetAbeaprimitivennmatrix.ThenAhasapositiveeigenvaluewiththefollowingproperties: (i) isasimplerootofthecharacteristicpolynomialofA, (ii) hasapositiveeigenvectorv, (iii) anyothereigenvalueofAhasmodulusstrictlylessthan, 17

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(iv) anynon-negativeeigenvectorofAisapositivemultipleofv. 1.4.2FloquetTheoryWestatesomebasicdenitionsandresultsconcerningperiodicdifferentialequations.First,weconsiderthelinearsystem:. x=A(t)x,x2Rn (1-7)whereA(t)isa-periodiccontinuousnnmatrix-valuedfunctiondenedonR.Denition.AfundamentalmatrixsolutionofEquation 1-7 ,(t),isannmatrix-valuedfunctiononRsuchthatthecolumnsof(t)arelinearlyindependent(vector)solutionsofEquation 1-7 .Denition.Theprincipalfundamentalsolution(PFS)ofEquation 1-7 ,(t),istheuniquefundamentalmatrixsolutionwith(0)=I. Theorem1.3(Floquet'sTheorem,[ 9 ]). If(t)isafundamentalmatrixsolutionofthe-periodicsystem,Equation 1-7 ,then,forallt2R,(t+)=(t))]TJ /F4 7.97 Tf 6.58 0 Td[(1(0)().Inaddition,thereexistsa(possiblycomplex)matrixBsuchthateB=)]TJ /F4 7.97 Tf 6.59 0 Td[(1(0)(),anda(possiblycomplex)-periodicmatrixfunctiont7!P(t)suchthat(t)=P(t)etBforallt2R.Also,thereisarealmatrixRandareal2-periodicmatrixt7!Q(t)suchthat(t)=Q(t)etRforallt2R.Denition.TheFloquetmultipliersofa-periodiclineardifferentialequationaretheeigenvaluesofthePFSevaluatedatt=.Inotherwords,theFloquetmultipliersofEquation 1-7 aretheeigenvaluesof().Wenextconsiderthegeneral(non-linear)-periodicdifferentialequation:. x=F(x,t),x2Rn (1-8) 18

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whereF:RnR!RnisasmoothfunctionwhichsatisesF(x,t+)=F(x,t)forallx2Rnandt2R.Let'(t,x)denotetheuniquesolutiontoEquation 1-8 suchthat'(0,x)=x.Denition.Assumingthat'(t,x)existsonthetimeinterval[0,],wecandenethePoincaremapasfollows:P(x)='(,x).Hence,wetakeaninitialpointxandintegrateEquation 1-8 overitsperiodtoobtainP(x).NotethatPissmoothanddenoteits(Frechet)derivativeatxasDP(x).Becausethevectoreldis-periodic,xedpointsofPcorrespondto-periodicsolutionsofEquation 1-8 ,i.e.asolution,(t),withthepropertythat(t+)=(t)forallt.Denition.Anyperiodicsolution,(t),ofEquation 1-8 islocallystableifforeachopensetVRnthatcontains)-388(:=f(t):0t0suchthatforeachx0withd(x0,\<,9(x0)2[0,)withthepropertythat'(t,x0)!(t+(x0))ast!1.ThefollowingtheoremrelatesstabilityofaperiodicsolutionofEquation 1-8 tostabilityofthecorrespondingxedpointinthediscretedynamicalsystemgeneratedbythePoincaremap[ 9 ]. Theorem1.4. Let(t)bea-periodicsolutionofEquation 1-8 .ConsiderthespectralradiusofDP(x),(DP(x)).If(DP(x))<1,then(t)islocallyasymptoticallystable(withanasymptoticphase).If(DP(x))>1,then(t)isunstable. 1.4.3SemigroupsI:BasicDenitionsAlloftheresultsandmostofthedenitionsinthissectionarefromHale[ 17 ].AslightdifferencebetweenthecontentofthissectionandthatofHale[ 17 ]isthatweemployaweakerdenitionofglobalattractorthanHaledoes,whichwewillneedforthesystemsstudiedinthisdissertation.Thedenitionweuseforglobalattractoristaken 19

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fromMagalandZhao[ 27 ],andthetermstrongglobalattractorwillbeusedtodenotethetraditionalglobalattractordescribedbyHale[ 17 ].LetXbeacompletemetricspaceandletR+=[0,1).Denition.AfamilyofmappingsT(t):X!X,t0,issaidtobeaC0-semigroup(orsemiow),providedthat (i) T(0)=I, (ii) T(t+s)=T(t)T(s),t0,s0, (iii) T(t)xiscontinuousint,xfor(t,x)2R+X.ThisfamilyofmappingswillsometimesbedenotedasT(t),whichmaybeconfusingasthisalsocanrepresentthespecicmappingattimet.ThereadershouldbeabletotellthecorrectinterpretationofT(t)dependingonthecontext.Denition.Foranypointx2X,thepositiveorbit(orforwardorbit)+(x)isdenedas+=[t0fT(t)xg.Theomegalimitsetofapointx,!(x),isdenedas!(x):=fy2X:9tn"1suchthatT(tn)x!yg.Anegativeorbitorbackwardorbitthroughxisafunction:(,0]!Xsuchthat(0)=x,andforanys0,T(t)(s)=(t+s)for0t)]TJ /F3 11.955 Tf 24.57 0 Td[(s.Acompleteorbitthroughxisafunction:R!Xsuchthat(0)=xand,foranys2R,T(t)(s)=(t+s)fort0.SinceT(t)isnotrequiredtobeonto,anegativeorbitneednotexistforallx.Furthermore,sinceT(t)doesnothavetobeone-to-one,ifanegativeorbitthroughxexists,itisnotnecessarilyunique.Denition.Letthenegativeorbit,)]TJ /F6 11.955 Tf 7.08 -4.34 Td[((x)throughxbedenedastheunionofallnegativeorbitsthroughx.Then)]TJ /F6 11.955 Tf 7.09 -4.94 Td[((x)=[t0H()]TJ /F3 11.955 Tf 9.3 0 Td[(t,x)whereH()]TJ /F3 11.955 Tf 9.3 0 Td[(t,x)=fy2X:thereisanegativeorbit,,throughxwith(0)=x,()]TJ /F3 11.955 Tf 9.3 0 Td[(t)=yg. 20

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Then,thecompleteorbitthroughxisdenedas(x)=)]TJ /F6 11.955 Tf 7.08 -4.34 Td[((x)[+(x).Thealphalimitset,(x),isdenedas(x):=fy2X:9tn#suchthatd(H(tn,x),y)!0g.Itwillbeimportantsometimestoconsiderthealphalimitsetrestrictedtoaspecicnegativeorcompleteorbit(t)throughx.Thiswillbedenotedby(x),where(x):=fy2X:9tn#suchthat(tn)!yg.Denition.ForanysubsetB,let+(B)=[x2B+(x),)]TJ /F6 11.955 Tf 7.09 -4.34 Td[((B)=[x2B)]TJ /F6 11.955 Tf 7.08 -4.34 Td[((x),(B)=[x2B(x)be,respectively,thepositive(forward)orbit,negative(backward)orbit,andcompleteorbitthroughB(thelattertwomaynotexist).Denition.AsetSXissaidtobeforwardinvariantifT(t)SSforallt0.AsetSXissaidtobeinvariantifT(t)S=Sforallt0.Thefollowingequivalentdenitionwillbeimportant:Sisinvariantifandonlyif,foranyx2S,acompleteorbitthroughxexistsand(x)S.Denition.Thestableset(ormanifold)ofacompactinvariantsetAisdenotedbyWsandisdenedasWs(A)=fx2X:!(x)6=;and!(x)Ag.Theunstableset(ormanifold)isdenedbyWu(A)=fx2X:thereexistsabackwardorbit'(t)throughx,'(x)6=;and'(x)Ag.Denition.AsetAXattractsasetBXif,dist(S(t)B,A)!0ast!1,wheredist(B,A)isthedistancefromsetBtosetA,i.e.dist(B,A):=supy2Binfx2Aky)]TJ /F3 11.955 Tf 11.96 0 Td[(xk.Denition.AsetAinXisdenedtobeanattractorifAisnon-empty,compactandinvariant,andthereexistssomeopenneighborhoodUofAinXsuchthatAattractsU. 21

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Denition.AglobalattractorisdenedtobeanattractorwhichattractseverypointinX.Denition.AsetAXissaidtobeastrongglobalattractorifitisaglobalattractor,andinaddition,foranyboundedsetBX,AattractsB.Denition.TheC0semigroupT(t):X!X,t0isasymptoticallysmooth,if,foranynonempty,closedboundedsetBXforwhichT(t)BB,thereisacompactsetJBsuchthatJattractsB.Denition.ThesemigroupT(t)isconditionallycompletelycontinuousfortt1ifforeachtt1andeachboundedsetBXforwhichfT(s)B,0stgisbounded,wehaveT(t)Bprecompact.AsemigroupT(t)iscompletelycontinuousifitisconditionallycompletelycontinuousand,foreacht0,thesetfT(s)B,0stgisbounded.Denition.ThesemigroupT(t)ispointdissipativeinXifthereisaboundednon-emptysetBXsuchthat,foranyx2X,thereisat0=t0(x,B)suchthatT(t)x2Bfortt0.ThefollowingtheoremprovidesasufcientconditionforthesemigroupTtobeasymptoticallysmooth. Theorem1.5([ 17 ]). Foreacht0,supposeT(t)=U(t)+C(t):X!X,t0hasthepropertythatC(t)iscompletelycontinuousandthereisacontinuousfunctionk:R+R+!R+suchthatk(t,r)!0ast!1andkU(t)xkk(t,r)ifkxkr.ThenT(t),t0,isasymptoticallysmooth.Thefollowingtheoremgivesasufcientconditionforexistenceofastrongglobalattractor. Theorem1.6([ 17 ]). IfT(t)isasymptoticallysmoothandpointdissipativeinX,andiftheforwardorbitofboundedsetsisboundedinX,thenthereisastrongglobalattractorAinX. 22

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Asymptoticsmoothnesswillbeanimportantconditionforuniformpersistence,aconceptdiscussedinthenextsection(Section 1.4.4 ).WeintroducethefollowingdenitionforaLyapunovfunction(orfunctional)fortheC0-semigroupT(t),t0.Denition.Supposethat x2XisanequilibriumpointforT(t),i.e.T(t) x= x,8t0.AcontinuousfunctionV:X!RissaidtobeaLyapunovfunctionat xif (i) V(x)>0forallx2Xnf xgandV( x)=0. (ii) V(T(t)x)isnon-increasingintforeachxinX. 1.4.4SemigroupsII:UniformPersistenceThissectionisdevotedtotheconceptofpersistence,alongwithatheoremrelatingpersistencetoexistenceofaglobalattractor.Persistenceprovidesamathematicalformalismfordeterminingwhetheraspecieswillultimatelygoextinctorpersistinadynamicalmodel.Therstpartofthissectioncontainsnecessarydenitionsandatheoremprovidinganequivalentconditionforuniformpersistence,andistakenfromHaleandWaltman[ 19 ].LetXbeametricspace.AssumethatthemetricspaceXistheclosureofanopensetX0;thatis,X=X0[@X0,where@X0(assumedtobenon-empty)istheboundaryofX0.Also,supposethattheC0-semigroupT(t),t0onXsatisesT(t):X0!X0,T(t):@X0!@X0. (1-9)IfT(t)satisestheconditionsofTheorem 1.6 ,thenT@:=T(t)j@X0willsatisfythesameconditionsin@X0.ThereforetherewillbeastrongglobalattractorA@in@X0.Denition.ThesemigroupTissaidtobeuniformlypersistentifthereisan>0suchthat,foranyx2X0,liminft!1d(T(t)x,@X0).Nowwestatedenitionswhichwillbeimportantinndingausefulequivalentconditionforuniformpersistence. 23

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Denition.AnonemptyinvariantsubsetMofXiscalledanisolatedinvariantsetifitisthemaximalinvariantsetofaneighborhoodofitself.Theneighborhoodiscalledanisolatingneighborhood.Denition.LetM,Nbeisolatedinvariantsets(notnecessarilydistinct).MissaidtobechainedtoN,writtenM,!N,ifthereexistsanelementx,x=2M[N,suchthatx2Wu(M)\Ws(N).AnitesequenceM1,M2,....,MkofisolatedinvariantsetsiscalledachainifM1,!M2,!....,!Mk(M1,!M1ifk=1).ThechainwillbecalledacycleifMk=M1.TheparticularinvariantsetsofinterestarefA@=[x2A@!(x).Denition.fA@isisolatedifthereexistsacoveringM=[ki=1MkoffA@bypairwisedisjoint,compact,isolatedinvariantsetsM1,M2,...,MkforT@suchthatMiisalsoanisolatedinvariantsetforT.Miscalledanisolatedcovering.fA@willbecalledacyclicifthereexistssomeisolatedcoveringM=[ki=1MioffA@suchthatnosubsetoftheMi'sformsacycle.Anisolatedcoveringsatisfyingthisconditionwillbecalledacyclic.Thefollowingtheoremwillprovidethemeanstoproveuniformpersistenceofasemigroup. Theorem1.7(HaleandWaltman,[ 19 ]). SupposeT(t)satisesCondition 1-9 andwehavethefollowing: (i) T(t)isasymptoticallysmooth, (ii) T(t)ispointdissipativeinX, (iii) +(U)isboundedifUinX, (iv) fA@isisolatedandhasanacycliccovering.ThenT(t)isuniformlypersistentifandonlyifforeachMi2MWs(Mi)\X0=;. 24

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Weintroduceamoregeneralnotionofuniformpersistence,alongwithawaytoshowuniformlypersistentsystemscontainaninteriorglobalattractor.ThematerialisfromMagalandZhao[ 27 ].SupposethatXisacompletemetricspace,andtheC0semigroupT(t):X!X,t0satisesCondition 1-9 .Supposethereexistsacontinuousfunction:X![0,1)suchthat(x)>0forallx2X0and(x)=0forallx2@X0.Denition.ThesemigroupT(t),t0issaidtobe-uniformlypersistentifthereexists>0suchthatliminft!1(T(t)x)forallx2X0.Thefollowingtheoremrelates-uniformpersistencetoexistenceofaglobalattractorinX0. Theorem1.8(MagalandZhao,[ 27 ]). AssumethatthesemigroupT(t),t0satisesCondition 1-9 ,isasymptoticallysmoothand-uniformlypersistent,andhasaglobalattractorA.ThentherestrictionofT(t)toX0,T(t)jX0,hasaglobalattractorA0.Nowwestateanotherwayofprovinguniformpersistence,inthespecialcaseofanODEonacompactsetinRn.ThefollowingcontentisfromDeLeenheerandPilyugin[ 11 ].Weswitchtotheirnotation.Considerasystem. x=F(x)onacompactforward-invariantsetKRmwithacontinuousow(t,x).LetK0Kbeaclosedforward-invariantsubsetofK.Denition.K0isauniformstrongrepellerinKifthereexistsa>0suchthatforallsolutions(t,x)2KnK0,liminft!1d((t,x),K0).Clearly,uniformstrongrepellerhasthesamedenitionasuniformpersistence.ThefollowingtheoremwillbeneededinSection 2.5 Theorem1.9(DeLeenheerandPilyugin,[ 11 ]). Let:K![0,1)beacontinuouslydifferentiablefunctionsuchthat(x)=0ifandonlyifx2K0.Supposethereexistsalowersemi-continuousfunction :K!Rsuchthat. = ,8x2KnK0. 25

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Inaddition,supposethatthefollowingconditionholds:(H)8x2K0,9T>0:ZT0 ((t,x))dt>0.Then,K0isauniformstrongrepellorinK. 1.4.5PerturbationofaGloballyStableSteadyStateWewillstatearesultfromSmithandWaltman[ 41 ],whichprovidesaconditionfortheglobalstabilityofaxedpointindiscretedynamicalsystemstobepreservedunderperturbations.LetT:U!UbecontinuouswhereUX,XisaBanachspace,andisametricspace.WewriteT(x)=T(x,)andusethenotationBX(x,r)(B(,r))fortheopenballofradiusraboutthepointx2X(2).ForalinearoperatorAonX,asbefore,wewrite(A)foritsspectralradius. Theorem1.10(SmithandWaltman,[ 41 ]). Let(x0,0)2U,BX(x0,)Uforsome>0andassumethatDx(T(x,)existsandiscontinuousinBX(x0,).SupposethatT(x0,0)=x0,(DxT(x0,0))<1,andTn0!x0foreveryx2U.Inaddition,supposethat: (H1) Foreach2,thereisasetBUsuchthatforeachx2U,thereexistsN=N(x,)suchthatTNx2B. (H2) C= [2T(B)iscompactinU.Thenthereexists0>0andacontinuousmap^x:B(0,0)!Usuchthat^x(0)=x0,T(^x(),)=^x()and:Tnx!^x(),x2U,2B(0,0). 26

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CHAPTER2PERIODICDRUGTREATMENTINVIRUSMODEL 2.1IntroductionAprimaryobjectiveforthemathematicalmodelingofwithin-hostHIVdynamicshasbeentostudytheeffectofdrugtreatmentonthevirus-hostsystemandtoassesspotentialtreatmentstrategies.TheaforementionedapplicationofthestandardvirusmodeltodrugperturbationexperimentsbyPerelsonetal.[ 34 ]producedimportantinsightsintodrugtherapyandHIVdynamics.PerelsonandNelson,andDeLeenheerandSmithextendedthisworkbyconsideringtheeffectofnon-perfect,constantefcacyantiviralmedicationonthestandardvirusmodelandfoundthatthevirusisclearediftheefcacyofthedrugtreatmentisaboveacriticallevel[ 12 33 ].Inreality,antiviralmedicationscannotcompletelyeradicateHIVvirus.ThefactthatviraleradicationistheoreticallypossibleshowsthatsomeaspectsofHIVdynamicsarenotcapturedinthestandardmodelorthatcurrentdrugsarenotpotentenough.Viralmutationtodrugresistantstrains[ 6 ],thepresenceoflatentreservoirsofHIV[ 37 39 ],andresidualviralreplicationduringtreatment[ 37 ]haveallbeenproposedtoexplainthetreatmentfailure.GiventhatantiviralmedicationdoesnotcureHIV,researchershavesoughttherapieswhichcancontroltheviralload,whileminimizingtheburdenofdrugcostsanddrugtoxicity.ManymathematicalstudieshavegravitatedtowardStructuredTreatmentInterruptions(STIs),whichentailperiodicbreaksfromdrugtherapyforpatients.However,STIshaveshownlittlesuccessinclinicaltrials.Animportantroleformathematicalmodelingistondoptimaltreatmentstrategies.ForHIV,drugtherapytypicallyconsistsofacombinationofantiviralmedicationsadministeredperiodically(onaday-to-daybasis,orpossiblyalargertime-scaleforSTIs).d'Onofrio[ 13 ]andDeLeenheer[ 10 ]incorporatedtime-periodicantiviraltreatmentintothestandardvirusmodelinordertoreectthisperiodicityofdrugdosages.BothauthorsusedFloquettheoryintheirstudies:d'Onofrioprovedglobalstabilityofthe 27

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infection-freeequilibriumwhenthedominantFloquetmultiplierofthelinearizedsystemislessthanunity;DeLeenheerderivedanexpressionforthisFloquetmultiplierforthecaseofpiecewiseconstantdrugefcaciesandinvestigatedoptimizationproblemsinthiscase.Inspiredbytheseresearchefforts,thenexttwochaptersareconcernedwithadeeperanalysisoftheperiodically-forcedstandardvirusmodel.SomeofthisworkiscontainedinanarticlebyBrowneandPilyugin,whichappearedintheBulletinofMathematicalBiology[ 8 ]. 2.2FormulationofModelandBoundednessofSolutionsCombinationantiviraltherapyconsistsofatleasttwodifferentantiviraldrugs.Combinationtherapyusuallycontainsareversetranscriptaseinhibitorandaproteaseinhibitor.Wewilldescribethemodesofactionofthesedrugsandsubsequentlyincorporatethemintothestandardvirusmodel.ReverseTranscriptaseinhibitors(RT-inhibitors)blocktheenzymaticfunctionofreversetranscriptase.Recallthatreversetranscriptaseisaviralenzymewhichcopiesthevirus'ssingle-strandedRNAgenomeintoadouble-strandedDNAafterenteringatargetcell.Thisstepisnecessaryforintegrationintothecell'snucleusandsubsequentproductionofmorevirus.TherearethreetypesofRT-inhibitors:NucleosideanalogReverseTranscriptaseInhibitors(NRTIs),NucleotideanalogReverseTranscriptaseInhibitors(NtRTIs),andNon-NucleosideReverseTranscriptaseInhibitors(NNRTIs).ThemodeofactionofNRTIsandNtRTIsistocompetewithnaturallyoccurringdeoxynucleotidesforbindingsitesonthegrowingviralDNAchain.IftheNRTIorNtRTIisincorporatedinthisDNAchain,thenthetranscriptionwillnotbecompleted.NNRTIsbindatadifferentsiteontheRTenzymeandinhibit(non-competitively)theprocessofDNAsynthesis.Proteaseinhibitors(P-inhibitors)targetprotease,anenzymefunctioninglateintheviralproductionstageoftheinfectedcelllifecycle.Recallthattheproteaseenzymecleaveslongproteinchainsinimmaturevirionsaroundthetimeofbudding. 28

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P-inhibitorsarecompetitiveinhibitorsthatbinddirectlytositesontheproteaseenzyme,compromisingthefunctionalityofprotease,whichcausesnon-infectiousvirionstobeproduced.TomodeltheeffectsofacombinationtreatmentconsistingofanRT-inhibitorandP-inhibitor,thestandardvirusmodel(Equation 1-3 )canbemodiedtoincludeparametersdescribingthedrugefcacies.ThestandardmodelconsidersinfectionoftargetcellandviralproductionassimultaneouseventsoccurringatrateskVTandNT,respectively.Giventhedrugmechanismsofactiondescribedabove,theRT-inhibitorshouldreducetheparameterk,whiletheP-inhibitorshouldreducetheparameterN(assumingnootherinteractions).Wedenedrugefcacyofadrugastheproportionoftargetedevents(infectionofcellsforRT-inhibitorandproductionofvirusforP-inhibitor)whichareinhibitedbythatdrug.LetRT(t)andP(t)denotethetimedependentefcaciesoftheRT-inhibitorandP-inhibitor,respectively.Thetime-dependenceinthedrugefcaciesreectsthefactthatdrugconcentrationsvaryintime.WeassumethatRT(t)andP(t)aresmooth.WealsoassumethattheRT-inhibitorandP-inhibitorworkindependently.Byindependence,wemeanthatpresenceofRT-inhibitorhasnoeffectontheviralproductionrate,andsimilarlytheP-inhibitorhasnoeffectontherateofinfectionoftargetcells.Then,itcanbeshownthattocompletelymodeltheactionsofthedrugtreatment,wereplacethetermskVTandNTwithk(1)]TJ /F5 11.955 Tf 11.96 0 Td[(RT(t))VTandN(1)]TJ /F5 11.955 Tf 11.95 0 Td[(P(t))T,respectively.Weassumethedrugefcacies,RT(t)andP(t),tobeperiodicfunctionswhichshareacommonperiod.ThisassumptioncanbejustiedonthegroundsthatdrugadministrationfollowsaperiodicschedulewithAbeingthe(drugAspecic)timebetweenconsecutivedoses.TheconcentrationofdrugAcanthentypicallybeassumedtobeA-periodic.Thedrugefcacycanbemodeledasafunctionofthedrugconcentration,oftenwiththeformofaHillfunction.Then,wecanconcludethatRT(t)andP(t)areRTandPperiodicfunctions,respectively.Furthermore,weassumethat 29

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thereexistsarationalnumberp=qsuchthatp qRT=P.ThenRT(t)andP(t)sharethecommonperiod=pRT.Thefollowing-periodicsystemofdifferentialequationsisobtained:. T=f(T))]TJ /F3 11.955 Tf 11.95 0 Td[(k(1)]TJ /F5 11.955 Tf 11.95 0 Td[(RT(t))VT,. T=k(1)]TJ /F5 11.955 Tf 11.96 0 Td[(RT(t))VT)]TJ /F5 11.955 Tf 11.95 0 Td[(T, (2-1). V=N(1)]TJ /F5 11.955 Tf 11.95 0 Td[(P(t))T)]TJ /F5 11.955 Tf 11.95 0 Td[(V,Forconvenience,wewillsometimesrefertothevectoreldinEquation 2-1 asF(x,t):R3R!R3.Inorderforthesystemtobewell-posed,weneedtocheckthatsolutionsstartinginthenon-negativeorthantofR3,denotedasR3+,remaininR3+forallforwardtimes.First,noticethatF(x,t)issmooth,henceastandardtheoremforODEsimpliesexistenceanduniquenessofsmoothsolutions.ForEquation 2-1 ,wewillsometimesdenotesolutionswithinitialconditionxas'(t,x)andothertimeswriteasolutionasT(t),T(t),V(t). Lemma1. R3+isforwardinvariantforEquation 2-1 .Moreover,non-negativeso-lutionsT(t),T(t),V(t)existforallt0andthereexistsM>0suchthatlimsupt!1(T(t)+T(t)+V(t))Mindependentofinitialconditions. Proof. SupposeT=0,andT=V=0.Then. T=f(0)>0and. T=. V=0.Thisimpliesthat@U:=f(T,T,V)2R3:T>0,T=V=0gisaninvariantsetforEquation 2-1 .So,ifx=(T(0),T(0),V(0))2@U[f(0,0,0)g,thenT(t)>0andT(t)=V(t)=0forallt>0.Ifx2R3+n@U,'(t,x)cannotreach@Uforanyt0asthiswouldviolateuniquenessofsolutions.Also,'(t,x)6=(0,0,0)foranyt0becausef(0)>0.Therefore,thecontinuityofsolutionsimpliesthatR3+n@Uisforwardinvariant.Then,certainlyR3+isforwardinvariant.Forthesecondpartofthelemma,noticethat. T(t)f(T(t)))T(t)max(T(0),T0)sincef(T)<08T>T0.Hencethereexistsa0,b0>0(possibly 30

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dependentonT(0))suchthatf(T(t))a0)]TJ /F3 11.955 Tf 11.95 0 Td[(b0T(t).Then:. T(t)+. T(t)=f(T))]TJ /F5 11.955 Tf 11.95 0 Td[(T(t)a0)]TJ /F3 11.955 Tf 11.95 0 Td[(b0T(t))]TJ /F5 11.955 Tf 11.96 0 Td[(T(t)Hence,T(t)+T(t)max(T(0)+T(0),S)whereS=a0 min(b0,).Then. V(t)NR)]TJ /F5 11.955 Tf 12.99 0 Td[(V(t),whereR=max(T(0)+T(0),S).HenceV(t)max(V(0),R).Thereforesolutionscannotblowupinnitetimeandhenceexistforallt0.Thensince. T<0wheneverT>T0,wegetthatTT0+1forallsufcientlylargetime,saytt0.Fortt0,wecannda,b>0suchthatd dt(T+T)a)]TJ /F3 11.955 Tf 12.28 0 Td[(bT)]TJ /F5 11.955 Tf 12.28 0 Td[(T.Hencelimsupt!1(T(t)+T(t))C:=a min(b,).SinceT0,thereexistst1>0suchthat. V(t)=N(1)]TJ /F5 11.955 Tf 11.96 0 Td[(P(t))T(t))]TJ /F5 11.955 Tf 11.96 0 Td[(VNC)]TJ /F5 11.955 Tf 11.96 0 Td[(V.Thenlimsupt!1V(t)NC .Theresultfollows. 2.3ReproductionNumberTheinfection-freeequilibrium,E0,ispreservedintheperiodicallyforcedmodel.Hence,E0,canbeconsideredatrivial-periodicsolutiontoEquation 2-1 .ToinvestigatethestabilityofE0,wecanapplytheFloquettheorypresentedinSection 1.4.2 .E0isaxedpointofthePoincaremap,P(x),associatedwithEquation 2-1 .Recall,bydenition,P(x)='(,x).Theorem 1.4 statesthatstabilityofE0canbedeterminedby(DP(E0)),thespectralradiusofDP(E0).Notice,DP(E0)='x(,E0)and'x(t,E0)isthePFS(principalfundamentalsolution)ofthe-periodiclinearvariationalsystem:. W=Fx(E0,t)W (2-2) 31

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Hence,tostudythestabilityofE0,weturnourattentiontoEquation 2-2 .EvaluatingtheJacobianofF(,t),wend:Fx(E0,t)=0BBBB@f0(T0)0)]TJ /F3 11.955 Tf 9.29 0 Td[(k(1)]TJ /F5 11.955 Tf 11.95 0 Td[(RT(t))T00)]TJ /F5 11.955 Tf 9.3 0 Td[(k(1)]TJ /F5 11.955 Tf 11.95 0 Td[(RT(t))T00N(1)]TJ /F5 11.955 Tf 11.95 0 Td[(P(t)))]TJ /F5 11.955 Tf 9.3 0 Td[(1CCCCA.LetW(t)denotethePFSofEquation 2-2 ,i.e.W(t)='x(t,E0).Then,(DP(E0))=(W()).Hence,weinvestigatetheeigenvaluesofW(),whicharetheFloquetmultipliersofEquation 2-2 .BecauseFx(E0,t)isblock-triangular,weconsiderthetwo-dimensionalsubsystem:. X=B(t)X,X(0)=I (2-3)whereB(t)=0B@)]TJ /F5 11.955 Tf 9.3 0 Td[(k(1)]TJ /F5 11.955 Tf 11.95 0 Td[(RT(t))T0N(1)]TJ /F5 11.955 Tf 11.95 0 Td[(P(t)))]TJ /F5 11.955 Tf 9.3 0 Td[(1CA Lemma2. TheFloquetmultipliersofEquation 2-2 areexactlyef0(T0)andthetwoFloquetmultipliersofthesubsystem,Equation 2-3 Proof. Letc(t)=0)]TJ /F3 11.955 Tf 9.29 0 Td[(k(1)]TJ /F5 11.955 Tf 11.95 0 Td[(RT(t))T0.ThenFx(E0,t)=0B@f0(T0)c(t)0B(t)1CA.DeneX(t)astheuniquematrixsolutiontoEquation 2-3 .Then,letu(t)betheuniquesolutiontotheinitialvalueproblem,(whichcanbeobtainedbyvariationofparameters):. u(t)=f0(T0)u(t)+c(t)X(t)u(0)=00. 32

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Then,d dt0B@ef0(T0t)u(t)0X(t)1CA=Fx(E0,t)0B@ef0(T0t)u(t)0X(t)1CATherefore,W(t)=0B@ef0(T0t)u(t)0X(t)1CA.ThentheeigenvaluesofW()areef0(T0),1,and2,where1,2aretheeigenvaluesofX().Theresultfollows. B(t)isaquasi-positivematrix.Therefore,thevectoreldinEquation 2-3 atanypointon@R2+nf(0,0)gpointsintoInt(R2+).ThisforcesX(t)tobeastrictlypositivematrixforallt>0.Inparticular,becauseX()isstrictlypositive,thePerron-Frobeniustheoremimpliesthat(X())istheprincipaleigenvalueofX().For,notationalconvenience,let=(X())whereX(t)isthematrixsolutiontoEquation 2-3 .Puttingtogethertheprecedingstatements,wearriveatthefollowingproposition: Proposition2.1. ConsiderEquation 2-1 .If<1,thenE0islocallyasymptoticallystable.If>1,thenE0isunstable. Proof. (DP(E0))=max(,ef0(T0)).Notethatef0(T0)<1.ThepropositionfollowsfromTheorem 1.4 Hence,wehaveathresholdconditionforthelocalstabilityofE0basedonthevalueof.Actually,certainstrongerglobalresultscanbeinferredbaseduponthequantity.Infact,when<1,d'Onofrioprovedthatthevirusiscleared[ 13 ]. Theorem2.1. (d'Onofrio,[ 13 ])Suppose<1.ThenE0isGASforEquation 2-1 ,hencetheinfectioniscleared. 33

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Ifistobeareproductionnumberfortheperiodicsystem,thevirusshouldpersistwhen>1.Thenexttheoremestablishesthispersistence. Theorem2.2. Suppose>1.ThenthesemiowgeneratedbyEquation 2-1 isuniformlypersistent.Moreover,thereexists>0(independentofinitialconditions)suchthatallsolutionsofEquation 2-1 withT(0)+V(0)>0havethepropertythat:liminft!1T(t)>,liminft!1T(t)>,liminft!1V(t)> Proof. ThesystemcanberepresentedbythefollowingautonomoussystemonthecylinderX:=R3+R=(Z):. T=f(T))]TJ /F3 11.955 Tf 11.96 0 Td[(kVT(1)]TJ /F5 11.955 Tf 11.95 0 Td[(RT()),. T=kVT(1)]TJ /F5 11.955 Tf 11.96 0 Td[(RT()))]TJ /F5 11.955 Tf 11.96 0 Td[(T,. V=NT(1)]TJ /F5 11.955 Tf 11.96 0 Td[(P()))]TJ /F5 11.955 Tf 11.96 0 Td[(V, (2-4). =12R=(Z)WewillapplyTheorem 1.7 .Xisametricspacewithmetricd((x,1),(y,2)):=kx)]TJ /F3 11.955 Tf 11.95 0 Td[(yk+j1)]TJ /F5 11.955 Tf 12.06 0 Td[(2j.LetS(t)x='(t,(x,))(theowofEquation 2-4 ).ThenS(t):X!X,t0isaC0-semigroup.Let@X0=R+f0gf0gR=(Z)andX0=Xn@X0.NotethatX0and@X0arebothforwardinvariantbytheargumentsintheproofofLemma 1 .Hence,therestrictedsemigroupsS(t)jX0andS(t)j@X0arewell-dened,i.e.S(t)satisesCondition 1-9 .SinceXisnite-dimensional,clearlyS(t)isasymptoticallysmooth.Also,byLemma 1 ,S(t)ispointdissipativeinXandforwardorbitsofboundedsetsareboundedinX.ConsiderthestrongglobalattractorA@forS(t)j@X0.ThenA@=fE0gR=(Z).HencefA@=Sy2A@!(y)=fE0gR=(Z).fA@isisolated(sinceE0ishyperbolic)andfA@isacyclic(becauseorbitscontainedin@X0nfA@cannothaveasubsequencewhichconvergestofA@inbackwardtime).Hence,toapplyTheorem 1.7 ,itsufcestoshowthatWs(fA@)\X0=;.Sinceorbitsareprecompact,thisisequivalentto 34

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thefollowing:(C)limsupt!1d('(t,(x,)),fA@)>0,8(x,)2X0.Supposethat(C)doesnotholdforsomesolution'(t,(x,))toEquation 2-4 with(x,)2X0.Thenforany>0,9t0suchthatd('(t,(x,),fA@)<8tt0.Inparticular,jT(t))]TJ /F3 11.955 Tf 11.96 0 Td[(T0j<8tt0 (2-5)For>0considerthelinearsystem:d dteX=0B@)]TJ /F5 11.955 Tf 9.3 0 Td[(k(T0)]TJ /F5 11.955 Tf 11.95 0 Td[()(1)]TJ /F5 11.955 Tf 11.95 0 Td[(RT(t))NT(1)]TJ /F5 11.955 Tf 11.95 0 Td[(P(t)))]TJ /F5 11.955 Tf 9.3 0 Td[(1CAeX,eX(0)=I (2-6)When=0,thisisjustEquation 2-3 .Recallthat>1isthedominanteigenvalueofX(),whereX(t)isthefundamentalmatrixsolutiontoEquation 2-3 .LeteX(t,)denotethematrixsolutiontoEquation 2-6 asafunctionofbothtimetandtheparameter.NotethateX(,0)=X().ThecontinuityofeX(t,)respecttotheparameterandthecontinuityofeigenvalueswithrespecttomatrixentriesimpliesthatforsufcientlysmall>0,thedominanteigenvalueofeX(,),e(),islargerthan1.Fix>0suchthate()>1.LeteX(t):=eX(t,)ande:=e().BythesemigrouppropertyandInequality 2-5 ,wecanndasolutiontoEquation 2-4 containedinX1suchthatT(t)T0)]TJ /F5 11.955 Tf 11.47 0 Td[(forallt0.Thend dt0B@TV1CA0B@)]TJ /F5 11.955 Tf 9.3 0 Td[(k(T0)]TJ /F5 11.955 Tf 11.95 0 Td[()(1)]TJ /F5 11.955 Tf 11.95 0 Td[(RT(t))NT(1)]TJ /F5 11.955 Tf 11.95 0 Td[(P(t)))]TJ /F5 11.955 Tf 9.3 0 Td[(1CA0B@TV1CA,T(0)>0,V(0)>0. (2-7)ByKamke'scomparisonprincipleforcooperativesystems[ 24 ],itfollowsthat0B@T(t)V(t)1CAeX(t)v8t0,withv0B@T(0)V(0)1CA 35

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BythePerron-Frobeniustheorem,aneigenvectorcorrespondingtoe,u,ispositivedenite.Thenwecannd>0suchthat00onX0.Clearly,-uniformpersistenceexactlythesamedenitionofuniformpersistenceinthiscase.ThenbyTheorem 1.8 ,thereexistsaglobalattractor,A,containedinX0.Noticethatifx='(0,(x,))2X0,i.e.T(0)+V(0)>0,then'(t,(x,))2Int(R3+)R=(Z)forallt>0.HenceAInt(R3+)R=(Z).TheresultfollowsfromthisinclusionandcompactnessofA. Inthenextsection,wewillproveexistenceofagloballystable-periodicsolutionassumingf(T)satisesthesectorcondition,Condition 1-6 ,and>1,inthecasewhereRT(t)andP(t)aresmallamplitudeperiodicperturbations.Inthissection,wehaveshownthedominantFloquetmultiplierofthelinearizedsubsystem,,isathresholdquantityforpersistenceofthevirus,i.e.isabasicReproductionNumberforEquation 2-1 .Theimportantquestionnowis:Howdoesthevalueofdependonthesystemparameters?ForgeneralperiodicfunctionsRT(t)andP(t),wecannotanswerthisquestionbecausethereisnoclosedformsolutionofEquation 2-3 .Inthenextsection,wewillemployperturbationtechniquesinordertoapproximateandprovidecertainqualitativedetailsaboutthesystemdynamicsforthecaseofperiodicdrugefcacieswithsufcientlysmallamplitude. 36

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A BFigure2-1. Thethresholdstabilitycurveforconstantdrugefcaciesandasmallamplitudeperiodicperturbation.A)Thestabilitydiagramforconstantefcacytreatmentsisdisplayed.Thecurverepresentsthethresholdefcacies(i.e.where(1)]TJ /F3 11.955 Tf 11.96 0 Td[(eRT)(1)]TJ /F3 11.955 Tf 11.96 0 Td[(eP)=1 R0).OutsidethecurveE0isstable,insidethecurveE0isunstable.B)Asmallamplitudesinusoidalperturbationisshown(here,weletRT(t)=)]TJ /F6 11.955 Tf 11.29 0 Td[(sintinEquation 2-9 ). 2.4PeriodicPerturbationsInthissection,wewillconsidersmallamplitudeperiodicperturbationsfromconstantdrugefcacies.Assumingthedrugefcaciesareconstantovertime(RT(t)eRTandP(t)eP),thenEquation 2-1 isanautonomoussystemequivalenttoEquation 1-3 .ThestabilityofE0isdeterminedbywhetherornotthemodiedbasicreproductionnumberfR0(eRT,eP)islessthan1,wherefR0(eRT,eP)=T0 k(1)]TJ /F3 11.955 Tf 11.96 0 Td[(eRT)N(1)]TJ /F3 11.955 Tf 11.95 0 Td[(eP)=R0(1)]TJ /F3 11.955 Tf 11.96 0 Td[(eRT)(1)]TJ /F3 11.955 Tf 11.96 0 Td[(eP). (2-8)Theinfection-freeequilibriumE0isGASwhen(1)]TJ /F3 11.955 Tf 11.95 0 Td[(eRT)(1)]TJ /F3 11.955 Tf 11.96 0 Td[(eP)<1 R0,andisunstable(alsotheinfectionpersists),whenthereversedstrictinequalityholds.Thestabilitythresholdcurve(1)]TJ /F3 11.955 Tf 12.47 0 Td[(eRT)(1)]TJ /F3 11.955 Tf 12.46 0 Td[(eP)=1 R0hasintercepts(e,0)and(0,e)wheree=1)]TJ /F4 7.97 Tf 15.74 4.7 Td[(1 R0andisshowninFigure 2-1A .FixeRT2[0,1]andeP2[0,1].Consider 37

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theperiodicperturbations:RT(t,)=eRT)]TJ /F5 11.955 Tf 11.96 0 Td[(RT(t),P(t,)=eP)]TJ /F5 11.955 Tf 11.96 0 Td[(P(t), (2-9)where>0issmallandRT(t),P(t):R!Rare-periodicfunctions.InsertingtheperiodicperturbationsintoEquation 2-1 ,weobtainthefollowingsystem:. T=f(T))]TJ /F3 11.955 Tf 11.96 0 Td[(k(1)]TJ /F3 11.955 Tf 11.96 0 Td[(eRT+RT(t))VT,. T=k(1)]TJ /F3 11.955 Tf 11.95 0 Td[(eRT+RT(t))VT)]TJ /F5 11.955 Tf 11.96 0 Td[(T, (2-10). V=N(1)]TJ /F3 11.955 Tf 11.96 0 Td[(eP+P(t))T)]TJ /F5 11.955 Tf 11.96 0 Td[(V,Recall,thedominantFloquetmultiplierofEquation 2-3 determineslocalstability.WiththedrugefcaciesasinEquation 2-9 ,thelinearsubsystem,Equation 2-3 ,parameterizedbyis:d dtX(t,)=B(t,)X(t,),X(0,)=I (2-11)whereB(t,)=0B@)]TJ /F5 11.955 Tf 9.3 0 Td[(kT0(1)]TJ /F3 11.955 Tf 11.95 0 Td[(eRT+RT(t))N(1)]TJ /F3 11.955 Tf 11.95 0 Td[(eP+P(t)))]TJ /F5 11.955 Tf 9.3 0 Td[(1CA. Lemma3. ThedominantFloquetmultiplierofEquation 2-11 ,denotedby()isananalyticfunctionforsufcientlysmall. Proof. X(,),thematrixsolutionofEquation 2-11 evaluatedatt=,isanalyticintheparameterbecausethevectoreldisanalyticin[ 9 ].ConsiderthecharacteristicpolynomialofX(,)(whichalsoisanalytic):G(s,)=s2)]TJ /F6 11.955 Tf 11.95 0 Td[(trX(,)s+detX(,).ThenG(,0)=0where:=(0)isthedominanteigenvalueofX(,0).BecauseX(,0)isstrictlypositive,thePerron-Frobeniustheoremimpliesthatthenon-dominant 38

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eigenvalue,)]TJ /F1 11.955 Tf 7.09 1.79 Td[(,isstrictlylessthat.Hence@G @s(,0)=2)]TJ /F6 11.955 Tf 11.96 0 Td[(trX(,0)=2)]TJ /F6 11.955 Tf 11.95 -.16 Td[((+)]TJ /F6 11.955 Tf 7.09 1.63 Td[()=)]TJ /F5 11.955 Tf 11.95 0 Td[()]TJ /F5 11.955 Tf 10.4 1.79 Td[(>0,Thereforetheimplicitfunctiontheoremimpliestheexistenceofananalyticfunction,(),suchthat(0)=andG((),)=0forallsufcientlysmall.Moreover,theimplicitfunctiontheoremimpliestheexistenceofananalyticfunction)]TJ /F6 11.955 Tf 7.09 1.79 Td[(()suchthatsuchthat)]TJ /F6 11.955 Tf 7.08 1.8 Td[((0)=)]TJ /F1 11.955 Tf 10.41 1.8 Td[(andG()]TJ /F6 11.955 Tf 7.08 1.8 Td[((),)=0forallsufcientlysmallsince@G @s()]TJ /F6 11.955 Tf 7.08 1.79 Td[(,0)=2)]TJ /F2 11.955 Tf 9.75 1.79 Td[()]TJ /F6 11.955 Tf 11.95 0 Td[(trX(,0)=2)]TJ /F6 11.955 Tf 11.96 -.16 Td[((+)]TJ /F6 11.955 Tf 7.08 1.63 Td[()=)]TJ /F2 11.955 Tf 9.74 1.79 Td[()]TJ /F5 11.955 Tf 11.96 0 Td[(<0,Hence,()>)]TJ /F6 11.955 Tf 7.09 1.79 Td[(()forallsufcientlysmall. Proposition2.2(Non-criticalcase). ConsiderEquation 2-10 .SupposethatfR0(eRT,eP)6=1. (i) IffR0(eRT,eP)>1,thenforallsufcientlysmall,E0isunstable. (ii) IffR0(eRT,eP)<1,thenforallsufcientlysmall,E0isGAS. Proof. LocalstabilitydependsuponthedominantFloquetmultiplier,(),oftheperturbedlinearsubsystem(Equation 2-11 ).ByLemma 3 ,()isananalyticfunctionforsufcientlysmall.LetB(0)=B(t,0)(sincethesystemisautonomouswhen=0).ThenX(,0)=eB(0).IffR0(eRT,ep)<1,thenB(0)hasdistinctnegativeeigenvalues.Hence(0)=(eB(0))<1.Therefore,forsufcientlysmall>0,()<1andhenceE0isGASbyTheorem 2.1 .Thisproves(ii).IffR0(eRT,eP)>1,thendetB(0)<0(andtrB(0)<0).Hence,B(0)hasonepositiveeigenvalue.Then(0)>1andforsufcientlysmall>0,()>1.(i)followsfromTheorem 1.4 2.5GlobalStabilityinNon-CriticalCaseInthecaseoffR0(eRT,eP)>1,whenf(T)satisesthesectorcondition,Condition 1-6 ,wecanestablishtheexistenceofagloballyattractingperiodicsolutioninInt(R3+).OurproofwillapplytheresultfromSmithandWaltman[ 41 ]thatprovidesconditionsfor 39

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globalstabilityofahyperbolicxedpointofadiscretedynamicalsystemtobepreservedunderperturbations,Theorem 1.10 .Webeginwithnecessarypreliminaryresultsthatmodifyargumentspresentedin[ 11 ].First,weshowuniformpersistencewithrespecttotheperturbationparameter. Lemma4. SupposethatfR0(eRT,eP)>1andconsiderEquation 2-10 .Then,thereexist0,1>0suchthatliminft!1(T+V)0>0forallsolutionswithT(0)+V(0)>0and2[)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1]. Proof. TheprooffollowsalongthesamelinesastheproofforLemma5inDeLeenheerandPilyugin[ 11 ].Forconvenience,letkd=k(1)]TJ /F3 11.955 Tf 12.61 0 Td[(eRT)andNd=N(1)]TJ /F3 11.955 Tf 12.61 0 Td[(eP).SinceR0(1)]TJ /F3 11.955 Tf 12.21 0 Td[(eRT)(1)]TJ /F3 11.955 Tf 12.21 0 Td[(eP)=k(1)]TJ /F10 7.97 Tf 6.58 0 Td[(eRT)N(1)]TJ /F10 7.97 Tf 6.59 0 Td[(eP)T0 >1,thereexistsapositivenumbereNsuchthat k(1)]TJ /F10 7.97 Tf 6.59 0 Td[(eRT)T00suchthatvM(T0,,t),whereM(T,,t):=0B@)]TJ /F5 11.955 Tf 9.3 0 Td[(Tk(1)]TJ /F3 11.955 Tf 11.96 0 Td[(eRT+RT(t))N(1)]TJ /F3 11.955 Tf 11.96 0 Td[(eP+P(t)))]TJ /F5 11.955 Tf 9.3 0 Td[(1CAisapositivevectorforall2[)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1]WenowinserttwomoreequationsintoEquation 2-10 inordertomakethesystemarticiallyautonomousandalsotoincludeasavariable.Henceweconsiderthesystem:. T=f(T))]TJ /F3 11.955 Tf 11.95 0 Td[(kVT(1)]TJ /F3 11.955 Tf 11.96 0 Td[(eRT+RT()),. T=kVT(1)]TJ /F3 11.955 Tf 11.95 0 Td[(eRT+RT()))]TJ /F5 11.955 Tf 11.95 0 Td[(T,. V=NT(1)]TJ /F3 11.955 Tf 11.95 0 Td[(eP+P()))]TJ /F5 11.955 Tf 11.95 0 Td[(V, (2-12). =02[)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1],. =12R=(Z) 40

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WewillapplyTheorem 1.9 .LetK0= B(0,M)R3+whereM>0wasestablishedinLemma 1 .NotethatK0iscompactandforwardinvariantforEquation 2-1 anddeneK=K0[)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1]R=(Z).KiscompactandforwardinvariantunderEquation 2-12 .K0:=[0,T0]f0gf0g[)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1]R=(Z)Kisacompactforwardinvariantset.Let(T,T,V,,):=v0B@TV1CAissmooth,zeroonK0,andpositiveonKnK0.. =:=vM(T,,)(T,V)0 v(T,V)0wheretheprimenotationdenotestranspose.Ontheboundary,dene(T,0,0,,)=min(vM(T,,))i viThenislowersemi-continuousonK[ 11 ].Thefunction(T,0,0,,)iscontinuousin(T,,).Notethatc0:=inf2[)]TJ /F14 7.97 Tf 6.59 0 Td[(1,1],2R=(Z)(T0,0,0,,)>0AllsolutionsinK0havethepropertythatlimt!1T=T0.Therefore,bycontinuity,(T(t),0,0,,(t))>c0 2>0forallsufcientlylarget.Hence,R0(T(t),0,0,,(t))dt>0forsufcientlylarget.ByTheorem 1.9 ,K0isauniformstrongrepellerinK.Hence,thereexistsa0>0suchthatliminft!1(T+V)0forallsolutionsinKnK0.Therefore,liminft!1(T+V)0forallsolutionsofEquation 2-10 withT(0)+V(0)>0andall2[)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1]. Lemma5. LetU=Int(R3+).ThereexistsacompactsetK1Usuchthatforany2[)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1]andforanysolutionofEquation 2-1 inU,thereexistsat0suchthat(T(t),T(t),V(t))2K1forallt>t0.Here1isthesameasinthestatementofLemma 4 41

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Proof. Firstwenoticethatsincef(0)>0andthefactthat9M>0suchthatV(t)0suchthatT0.ThereforeifT<,then. T>0.HenceT(t)forallsufcientlylarget.FromLemma 4 ,weknowliminft!1(T+V)>0 2forallsufcientlylarget.HenceT(t)>0 2)]TJ /F3 11.955 Tf 11.96 0 Td[(V(t).Thereforesubstitutingthisintothe. Vequation,weget. V(0 2)]TJ /F3 11.955 Tf 11.96 0 Td[(V(t))N(1)]TJ /F3 11.955 Tf 11.96 0 Td[(eP+P(t)))]TJ /F5 11.955 Tf 11.95 0 Td[(V8larget=C0 2)]TJ /F6 11.955 Tf 11.95 0 Td[((C+)V(t)whereC=N(1)]TJ /F3 11.955 Tf 11.95 0 Td[(eP+kPk).Aftersolvingthedifferentialinequality,wegetthatliminft!1V1=0C 2(C+)>0HenceV(t)>1 2forallsufcientlylarget.WealsoknowT(t)forallsufcientlylarget.Substitutingtheseestimatesintothe. Tequation,weobtain. TC1 2)]TJ /F5 11.955 Tf 11.95 0 Td[(T(t)whereC=k(1)]TJ /F3 11.955 Tf 11.96 0 Td[(eRT+kRTk).Henceliminft!1T2=1C 2>0LetK1=f(T,T,V)2R3jT,T2,V1g.ThenK1Uisacompactsetsuchthatforany2[)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1]andforanysolutionofEquation 2-10 inU,thereexistsat0suchthat(T(t),T(t),V(t))2K1forallt>t0. Theorem2.3. Supposethatf(T)satisesCondition 1-6 ,i.e.thesectorconditionholds.AssumefR0(eRT,eP)>1.RefertofamilyofdifferentialequationsparameterizedbytheperturbationinEquation 2-10 asfS()g2()]TJ /F14 7.97 Tf 6.59 0 Td[(0,0)forsome0>0.Also,letE1denotetheuniquepositiveequilibriumwhen=0,i.e.forS(0).Thenthereexists0>0andacontinuousmapG:[)]TJ /F5 11.955 Tf 9.3 0 Td[(0,0]!Usatisfying 42

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(i) G(0)=E1,andforeach2[0,0],)]TJ /F14 7.97 Tf 6.77 -1.79 Td[(:=St2[0,)'(t,G())isa-periodicorbitofsystemS()insideU=Int(R3+).(E1persistsunder-periodicperturbationsasa-periodicsolution). (ii) Foreach2[)]TJ /F5 11.955 Tf 9.29 0 Td[(0,0],'(t,G())isagloballyasymptoticallystable(withaphase)-periodicsolutioninUforthesystemS(). Proof. Forany,denotethePoincaremapassociatedwithsystemS()asP(x,).P(x,):R3+()]TJ /F5 11.955 Tf 9.3 0 Td[(1,1)!R3+isasmoothfunctionbecauseofsmoothnessofsolutionswithrespecttotheinitialconditionandparameters.Considerthedisplacementfunction:(x,):=P(x,))]TJ /F3 11.955 Tf 12.39 0 Td[(x.Notethat(E1,0)=E1andx(E1,0)=Px(E1,0))]TJ /F3 11.955 Tf 12.38 0 Td[(I.NotethatPx(E1,0)isthesolutiontothefollowingvariationalequationevaluatedatt=:. Z=J(E1)ZZ(0)=IwhereJ(E1)istheJacobianoftheautonomoussystem,Equation 1-3 ,atE1.HencePx(E1,0)=eJ(E1).Becausef(T)satisesthesectorconditionandR0(eRT,eP)>1,theeigenvaluesofJ(E1)arecontainedintheopenlefthalfofthecomplexplane.Then,byTheorem2.52in[ 9 ],(eJ(E1))<1.Hencex(E1,0)=eH(E1))]TJ /F3 11.955 Tf 12.21 0 Td[(I,isinvertible.Thenbytheimplicitfunctiontheorem,thereexistscontinuousfunctionG:[)]TJ /F5 11.955 Tf 9.3 0 Td[(0,0]!Uforsome0>0,suchthatG(0)=E1andP(G(),)=G().Foreach2[)]TJ /F5 11.955 Tf 9.3 0 Td[(0,0],G()isaxedpointofthePoincaremap.ThenSt2[0,)'(t,G())isa-periodicorbitofsystemS(),denotedas)]TJ /F14 7.97 Tf 6.77 -1.79 Td[(.AlsonotethatG()iscontinuous.Thisprovespart(i).Toprovepart(ii),weuseTheorem 1.10 .Followingthenotationofthetheorem,weletUbethesetasalreadydened,denetheset:=[)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1],deneTx:=P(x,).ItishasalreadybeenstatedthatE1isagloballyasymptoticallystableequilibriuminUforS(0).HenceP(E1,0)=E1,thespectralradius(DxP(E1,0))<1,andPn(x,0)!E1foreveryx2U.Bytheprevioustheorem,thereisacompactsetK1Usuchthatforeach2[)]TJ /F5 11.955 Tf 9.29 0 Td[(1,1]andeachsolution(T(t),T(t),V(t))2U,thereexistst0>0suchthat(T(t),T(t),V(t))2K1forallt>t0.Then,Condition (H1) issatisedchoosingB=K1foreach2=[)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1].Thenweneedtocheckifthefollowingset 43

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iscompact:C:= [2T(B)=P(K1[)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1])SincePiscontinuousandK1[)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1]iscompact,theabovesetiscompact.ThereforeCondition (H2) issatised.Hence,Theorem 1.10 impliesthatPn(x,)!G0()forallx2Uandsufcientlysmall,say2[)]TJ /F5 11.955 Tf 9.3 0 Td[(0,0].ThisimpliesthatPn(x,)!G()forallx2Uand2[)]TJ /F5 11.955 Tf 9.29 0 Td[(0,0].Therefore'(t,G())isgloballyasymptoticallystable(withaphase)for2[)]TJ /F5 11.955 Tf 9.3 0 Td[(0,0]. Severalremarksconcerningthisresultareinorder.First,wehavenotcharacterizedthelevelofviralloadoccurringinthe-periodicsolution.E1,thepositiveequilibriumintheautonomousmodel,isoftenanattractingspiralpointwhenf(T)satisesthesectorcondition.Inotherwords,solutionsexhibitdampedoscillationsenroutetoconvergingtotheviralsteadystate.Hence,periodicforcingatcertainfrequenciescaninteractwiththenaturalfrequencyofthedampedoscillationstoproduceweakparametricresonanceinthesystem.BrebanandBlowerstudythisphenomenonandsuggestitasapossibleexplanationforthevarianceofresultsinpatientsundergoingSTItherapy[ 7 ].Itshouldalsobenotedthatiff(T)doesnotsatisfythesectorcondition,aperiodicsolutioncanoccurintheautonomousmodel[ 12 ].Inthiscase,theconsequencesofperiodicforcingaredifculttoanalyze.Astrongerresonanceeffectwouldmostlikelybepossible.Inaddition,akeyassumptionforourresultsisthattheperiodicperturbationshavesufcientlysmallamplitude.Thestandardvirusmodel,Equation 1-3 isequivalenttoanSEIRmodelinmathematicalepidemiology,althoughthetwomodelshavedifferentinterpretationsandarelikelytohavedifferentparametervalues.ManyresearchershavestudiedtheSEIRmodelwithaseasonalcontactrate.TheseasonalforcingenterstheequationsexactlywheretheperiodicRT-inhibitortreatmentisinsertedforourvirus-hostsystem.Forcertainparameterregions,increasingtheamplitudeofseasonalitycancauseacascadeofperiod-doublingbifurcations,andchaoticbehaviorintheSEIRmodel[ 40 ].Thishasbeenproposedasamechanismforbiennialepidemics. 44

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Furtherstudyisneededtoseeifthesebifurcationscanoccurforparametersmorerepresentativeofawithin-hostvirussystem.Inaddition,whenfR0(eRT,eP)iscloseto1,theresultsfromSections 2.7 and 2.8 cancomeintoplayinthecaseofsufcientlylargeamplitudeperturbations.InthissectionwehaveshownthatwhenfR0(eRT,eP)6=1inEquation 2-10 ,smallamplitudeperiodicperturbationsoftheconstantdrugefcacies,eRTandeP,donotaffectthepersistenceofthevirus.WhenfR0(eRT,eP)<1,thevirusisstillcleared,andwhenfR0(eRT,eP)>1,theuniqueviralsteadystatebecomesagloballyattractingperiodicorbitcorrespondingtopersistentinfection.AmoreinterestingscenarioforstudyingsmallamplitudeperiodicperturbationsisthecasewhenfR0(eRT,eP)=1,whichwecallthecriticalcaseandinvestigateinthenextsection. 2.6CriticalCase,Part1:TranscriticalBifurcationForEquation 2-1 ,wechooseconstantdrugefcacies,RT(t)eRTandPeP,tobeonthethresholdcurve,fR0(eRT,eP)=1,depictedinFigure 2-1A .Inthiscase,the(autonomous)systemundergoesatranscriticalbifurcation[ 33 ].Inotherwords,ifwetakethebranchesofequilibriaasafunctionofthereproductionnumber,fR0(eRT,eP),thenthebranchesofequilibriaintersectatfR0(eRT,eP)=1.Ifweintroduceperiodicperturbationsintothemodel,bifurcationresultsbecomedifculttoestablishandrigorouslyprove.However,wecanprovethatthePoincaremapundergoesatranscriticalbifurcationforthecasewhentheRT-inhibitorefcacy,eRT,isperiodicallyperturbedbyafunctionwithnon-zeroaverageovertheperiod,whileePremainsconstant.Fortheothercase,wewillbecontentwithinvestigatinghowtheperturbationsaffectthedominantFloquetmultiplier,,ofthelinearizedsystem.Thisanalysiswillproduceaninterestingresultshowingthephasedifferencebetweenthedrugefcacyfunctions,RT(t)andP(t),canbeacriticalparameterindeterminingthefateofthevirus.First,weexplicitlystateandprovetheaforementionedbifurcationinthespecialcaseofasingle(weighted)periodicperturbation. 45

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First,westateandproveasufcientconditionforatranscriticalbifurcation.Wenotethatthisresultholds(withessentiallythesameproof)inthemoregeneralsettingofindexzerononlinearFredholmoperatorsonarbitraryBanachspaces.ThetheoremandproofisbasedonthemethodofpseudoresolventsofErhardSchmidt(1908)[ 46 ]. Theorem2.4. LetXandYbenitedimensionalBanachspacesoverK,whereK=RorC.SupposeF:KX!Yisanonlinearoperator.SupposefurtherthatFisanalyticat(0,0)andalsoF(,0)0forall2K.Also,assumeker(Fx(0,0))=span(x1)wherex16=0andsupposethefollowingconditionshold:Fxx1=2ran(Fx(0,0)) (E1)Fxxx21=2ran(Fx(0,0))whereFxxx21denotesthebilinearform,xT1Fxxx1. (E2)Thenthereisananalyticfunction:K!Xsuchthat(0)=0,0(0)6=0andF(,())=0forallinsomeneighborhoodaround=0.(Wenotethatthisisaspecialcaseofatranscriticalbifurcation). Proof. LetB=Fx(0,0).BisaboundedlinearmapfromXtoY.Letx12ker(B).Nowdim(ker(B))=dim(ker(B))(sinceXandYarenitedimensional)whereBistheadjointoperatorofB.HenceB:Y!X,whereXandYarethedualspacesofXandY,respectively.Forx2X,lethx,idenotetheactionofxonX,i.ehx,i:X!K.LikewiseforelementsofY.Letx12ker(B).Findy12Y,y12Xsuchthathx1,y1i=hy1,x1i=1(thisisacorollaryoftheHahn-BanachTheorem).DeneS:X!YwhereSx=Bx+hy1,xiy1WeclaimSisinvertible.AssumeSx=0.Notehx1,Bxi=hBx1,xi=0.Thisimplies0=hx1,Sxi=hy1,xihx1,y1i=hy1,xi.Hencehy1,xi=0.ThereforeBx=0bythedenitionofS.Sox=ax1forsomenonzeroscalara.ThenSx=a1hy1,x1iy1=a1y1=0)a1=0.Hencex=0.SoSisinvertible. 46

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NoticesinceSx1=y1,wehaveS)]TJ /F4 7.97 Tf 6.59 0 Td[(1y1=x1.WeconsiderxwhichsatisfyF(,x)=0.ThenSx=Bx+sy1)]TJ /F3 11.955 Tf 11.96 0 Td[(F(,x)wheres=hy1,xi,0=)]TJ /F3 11.955 Tf 9.3 0 Td[(x+S)]TJ /F4 7.97 Tf 6.59 0 Td[(1Bx+sx1)]TJ /F3 11.955 Tf 11.95 0 Td[(S)]TJ /F4 7.97 Tf 6.59 0 Td[(1F(,x)) (2-13)ThepartialderivativeoftheRHS(RightHandSide)ofEquation 2-13 withrespecttoxat(=0,s=0,x=0)is)]TJ /F3 11.955 Tf 9.3 0 Td[(I.Hencebytheimplicitfunctiontheoremthereisaneighborhoodof(,s)=(0,0)onwhichEquation 2-13 hasauniqueanalyticsolution:x=Xk,lkslxklAlso,sinceFisanalytic,F(,x)=1Xk,m=0kMkmxmwhereM01=Fx(0,0),M11=Fx(0,0),M02=Fxx(0,0).AlsoMk0=08k(sinceF(,0)0).WepluginbothexpansionstoEquation 2-13 andnotethatxk0=08ksinceMk0=08k,andthereforethereisalwaysanonzerofactorofsontheRHSofEquation 2-13 .Webegintocollectterms:0s1x01+1s1x11+0s2x02+O(3)=0s1x1+S)]TJ /F4 7.97 Tf 6.58 0 Td[(1(Fx(0,0)x01)]TJ /F3 11.955 Tf 11.95 0 Td[(M01x01)+1s1S)]TJ /F4 7.97 Tf 6.58 0 Td[(1(Fx(0,0)x11)]TJ /F3 11.955 Tf 11.96 0 Td[(M11x01)]TJ /F3 11.955 Tf 11.96 0 Td[(M01x11)+0s2S)]TJ /F4 7.97 Tf 6.58 0 Td[(1(Fx(0,0)x02)]TJ /F3 11.955 Tf 11.96 0 Td[(M01x02)]TJ /F3 11.955 Tf 11.96 0 Td[(M02x01x01+O(3)Hencex01=x1,x11=S)]TJ /F4 7.97 Tf 6.59 0 Td[(1Fx(0,0)x1,andx02=S)]TJ /F4 7.97 Tf 6.59 0 Td[(1Fxx(0,0)x1x1.Thereforex=sx1)]TJ /F5 11.955 Tf 11.95 0 Td[(sS)]TJ /F4 7.97 Tf 6.59 0 Td[(1M11x1+s(sS)]TJ /F4 7.97 Tf 6.58 0 Td[(1M02x1x1+O(2)) (2-14) 47

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Hence,s=hy1,xi=shy1,x1i)]TJ /F5 11.955 Tf 19.26 0 Td[(sy1,S)]TJ /F4 7.97 Tf 6.59 0 Td[(1M11x1)]TJ /F3 11.955 Tf 11.96 0 Td[(s2y1,S)]TJ /F4 7.97 Tf 6.59 0 Td[(1M02x1x1+s(O(2))Recallthathy1,x1i=1anddividebys,toobtain0=)]TJ /F5 11.955 Tf 9.3 0 Td[(y1,S)]TJ /F4 7.97 Tf 6.58 0 Td[(1M11x1)]TJ /F3 11.955 Tf 11.96 0 Td[(sy1,S)]TJ /F4 7.97 Tf 6.58 0 Td[(1M02x1x1+O(2) (2-15)Ifwerequirehy1,S)]TJ /F4 7.97 Tf 6.58 0 Td[(1M11x1i6=0,thenthepartialderivativeoftheRHSwithrespecttoisnonzeroandtheimplicitfunctiontheoremapplies.Therefore,thereexistsananalyticfunction(s)inaneighborhoodofs=0wheretheRHSofEquation 2-15 iszerowhen=(s).Wecansubstitute(s)intoEquation 2-14 togetx(s).Thus((s),x(s))isasolutionofF(,x)=0.Weclaimthattherequirementhx1,M11x1i6=0implieshy1,S)]TJ /F4 7.97 Tf 6.58 0 Td[(1M11x1i6=0.Toshowthis,supposebywayofcontradictionhy1,S)]TJ /F4 7.97 Tf 6.59 0 Td[(1M11x1i=0.ThenB(S)]TJ /F4 7.97 Tf 6.59 0 Td[(1M11x1)=S(S)]TJ /F4 7.97 Tf 6.59 0 Td[(1M11x1)=M11x1.ThereforeBx1,S)]TJ /F4 7.97 Tf 6.58 0 Td[(1M11x1=hx1,M11x1i6=0ThisisacontradictiontothefactthatBx1=0.ThenbytheFredholmAlternative,wehavethefollowing:hx1,M11x1i6=0,M11x1=2ran(B)sincex12ker(B).RecallM11=Fx(0,0),soM11x1=2ran(B)isjustcondition(H1).Since(s)isanalyticabouts=0,weget(s)=1Xk=1ksk (2-16)PlugthisintoEquation 2-15 toget:1=)-166(hy1,S)]TJ /F4 7.97 Tf 6.58 0 Td[(1M02x1x1i hy1,S)]TJ /F4 7.97 Tf 6.58 0 Td[(1M11x1i 48

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Suppose16=0.Thenlettings!0inEquation 2-16 ,wegets= 1+O(2)forsufcientlysmall.PluggingthisintoEquation 2-14 ,weget:():=x()= 1x1+O(2)insomeneighborhoodaround=0.Noticethatthisisapowerseriesin.Usingargumentsfromabove,itiseasytoseethatConditions E1 and E2 imply1isniteandnonzero.Thisprovesthetheorem. Theorem2.5. ConsiderEquation 2-10 whenfR0(eRT,eP)=1.Assumethefollowing:RT(t)=+(t)with>0,(t)-periodic,R0RT(t)=0,andP(t)0.Denethe-dependentPoincaremap,P(x,)asbefore.Thenatranscriticalbifurcationoccursat=0forP(x,).Moreprecisely,thereexistsananalyticfunction:()]TJ /F5 11.955 Tf 9.3 0 Td[(0,0)!R3suchthatP((),)=()forasufcientlysmall0>0,where(0)=E0and0(0)>0. Proof. Rescaletheparameterssothesystemiswrittenasfollows:. T=f(T))]TJ /F3 11.955 Tf 11.95 0 Td[(kVT(1+'(t)),. T=kVT(1+'(t)))]TJ /F5 11.955 Tf 11.95 0 Td[(T, (2-17). V=NT)]TJ /F5 11.955 Tf 11.96 0 Td[(V,whereR0=kNT0 =1and'(t)isa-periodicfunction,'(t)=+(t)where>0andR0(t)dt=0.When=0,theJacobianatE0isJ:=0BBBB@f0(T0)0)]TJ /F3 11.955 Tf 9.3 0 Td[(kT00)]TJ /F5 11.955 Tf 9.3 0 Td[(kT00N)]TJ /F5 11.955 Tf 9.3 0 Td[(1CCCCA.WhenR0=1,Jhasazero-eigenvalueandJcanbediagonalizedinthefollowingway:J=ALA)]TJ /F4 7.97 Tf 6.59 0 Td[(1 49

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whereL=0BBBB@)]TJ /F6 11.955 Tf 9.3 0 Td[((+)0000000f0(T0)1CCCCA,A=0BBBB@kT0 ++f0(T0)kT0 f0(T0)1)]TJ /F4 7.97 Tf 6.58 0 Td[(1 N N01101CCCCA,A)]TJ /F4 7.97 Tf 6.58 0 Td[(1=0BBBB@0)]TJ /F10 7.97 Tf 6.59 0 Td[(N + +0N + +1)]TJ /F14 7.97 Tf 6.59 0 Td[( f0(T0)(++f0(T0)))]TJ /F10 7.97 Tf 6.59 0 Td[(kT0 (+)(++f0(T0))+)]TJ /F10 7.97 Tf 6.59 0 Td[(kT0 f0(T0)(+)1CCCCA,Forconvenience,wedenoteEquation 2-17 as:. x=F(x,t,) (2-18)Alsowhen=0,wewriteF(x,t,0)asF(x).Lett7!x(t,,)bethesolutiontoEquation 2-17 suchthatx(0,,)=2R3.Asbefore,denethe-dependentPoincaremapP:R3R!R3asP(,)=x(,,).Denethedisplacementfunction:R3R!R3as(,)=P(,))]TJ /F5 11.955 Tf 12.96 0 Td[(.Clearly(E0,)08.Also(E0,0)=P(E0,0))]TJ /F3 11.955 Tf 11.96 0 Td[(I.AndfromtherstvariationofEquation 2-18 :P(E0,0)=eR0DF(E0)dt=eJSince0isaneigenvalueofJ,1isaneigenvalueofeJ,andeJ)]TJ /F3 11.955 Tf 12.75 0 Td[(Iisnotinvertible.Therefore(E0,0),sowecannotapplytheimplicitfunctiontheorem.Butwenotethatdim(ker((E0,0)))=1,hencewetrytochecktheconditionsoftheTheorem 2.4 .Let 50

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v2ker((E0,0))withv6=0.Weneedtocheckthefollowing:(E0,0)v=2ran((E0,0)) (E1-1)(E0,0)v2=2ran((E0,0)) (E2-1)where(E0,0)v2isshorthandforabilinearform.TherstorderofbusinessischeckingCondition E1-1 .DeneB:R3R!L(R3,R3)asB(,):=A)]TJ /F4 7.97 Tf 6.59 0 Td[(1(,)A.LetB:=B(E0,0),C:=(E0,0),B:=B(E0,0)andC:=(E0,0).Clearlydim(ker(B))=1.Letw2R3beabasisforker(B).WeclaimthatCondition E1-1 isequivalenttothefollowingcondition:Bw=2ran(B) (G1)Indeed,sinceB=A)]TJ /F4 7.97 Tf 6.59 0 Td[(1CA,wehavethatBw=0)Aw2ker(C).NoticeBw=y,A)]TJ /F4 7.97 Tf 6.59 0 Td[(1CAw=y,C(Aw)=AyHenceitsufcestoshowy=2ran(B))Ay=2ran(C).Todothis,supposeAy2ran(C).Thereforethereexistsu6=0suchthatCu=Ay.SoA)]TJ /F4 7.97 Tf 6.58 0 Td[(1Cu=y.Butu=Azforsomez6=0.ThereforeA)]TJ /F4 7.97 Tf 6.59 0 Td[(1CAz=yandhenceBz=y,whichimpliesy2ran(B).Theclaimisproved.Tocomputethederivativesofthedisplacementfunction,weconsidervariationalequationsofEquation 2-18 .Inthefollowing,wewritethesolutionx(t,,)asxsometimes.Equation 2-18 alongaspecicsolutioncanbewrittenas:. x=F(x(t,,),t,),x(0,,)= (2-19)DifferentiatingEquation 2-19 withrespectto,weobtaintherstvariation:. x=DxF(x(t,,),t,)x,x(0,,)=I (2-20) 51

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NowdifferentiatingEquation 2-20 withrespectto:. x=D2xxF(x(t,,),t,)x+D2xF(x(t,,),t,)x+DxF(x(t,,),t,)x,x(0,,)=0 (2-21)WewanttoevaluateEquation 2-21 at=E0,=0.Notethatx(t,E0,)=0forall,t,hencex(t)08t.Henceat(E0,),Equation 2-21 becomes:. x=D2xF(E0,t,0)x+DxF(E0,0)x,x(0)=0 (2-22)Now=x().Letz(t,)=A)]TJ /F4 7.97 Tf 6.59 0 Td[(1x(t,E0,)A.SoB=z()j=0 (2-23)Here(whenevaluatedat=0),. z=A)]TJ /F4 7.97 Tf 6.58 0 Td[(1D2xF(E0,t,0)Az+A)]TJ /F4 7.97 Tf 6.59 0 Td[(1DF(E0)Az,z(0)=0 (2-24)Thisisjustalinearequationandthesolutionatt=is:z()=eLZ0e)]TJ /F10 7.97 Tf 6.58 0 Td[(LtM(t)eLtdt (2-25)whereM(t):=A)]TJ /F4 7.97 Tf 6.58 0 Td[(1D2xF(E0,t,0)A.D2xF(E0,t,0)=d d0BBBB@f0(T0)0)]TJ /F3 11.955 Tf 9.3 0 Td[(kT0(1+'(t))0)]TJ /F5 11.955 Tf 9.3 0 Td[(kT0(1+'(t))0N)]TJ /F5 11.955 Tf 9.3 0 Td[(1CCCCA=0BBBB@00)]TJ /F3 11.955 Tf 9.3 0 Td[(kT0'(t)00kT0'(t)0001CCCCAInthefollowingcalculationssomenon-essentialquantitiesforourpurposesaredenotedwithanasterisk.WeobtainM(t)=0BBBB@0NkT0'(t) +001CCCCA 52

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eLt=0BBBB@e)]TJ /F4 7.97 Tf 6.59 0 Td[((+)t0001000ef0(T0)t1CCCCAe)]TJ /F10 7.97 Tf 6.59 0 Td[(Lt=0BBBB@e(+)t0001000e)]TJ /F10 7.97 Tf 6.58 0 Td[(f0(T0)t1CCCCAHence,e)]TJ /F10 7.97 Tf 6.59 0 Td[(LtM(t)eLt=0BBBB@0NkT0'(t) +001CCCCA (2-26)Therefore,byEquations 2-23 2-41 ,and 2-26 ,B=eL0BBBB@0R0NkT0'(t)dt +001CCCCA=0BBBB@0NkT0 +001CCCCANowB=eL)]TJ /F3 11.955 Tf 12.3 0 Td[(I,henceker(B)=span(v)wherevisaneigenvectorcorrespondingtothe0-eigenvalueofB.Clearlywecanletv=e2:=(0,1,0)T.Also,ran(B)=span(e1,e3)(wherefe1,e2,e3gisthestandardbasisofR3)Hence,Bv=0BBBB@NkT0 +1CCCCA=2ran(B)sinceNkT0 +6=0.ThereforeCondition G1 issatised,andbytheclaim,Condition E1-1 issatised.TocheckCondition E2-1 ,weintroducethelinearcoordinatechange 53

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y=A)]TJ /F4 7.97 Tf 6.59 0 Td[(1x,whichtransformsEquation 2-18 into:. y=A)]TJ /F4 7.97 Tf 6.59 0 Td[(1F(Ay,t,) (2-27)ThentherstvariationofEquation 2-27 alongasolutiony(t,,)is:. y=A)]TJ /F4 7.97 Tf 6.59 0 Td[(1DF(Ay(t,,),t,)Ay(t,,)y(0,,)=I (2-28)At=E0,=0,thesolutiontoEquation 2-28 isy(t)=eLtasbefore.DifferentiatingEquation 2-28 withrespectat=E0,=0,weobtain. y=(Iy(t))T(IA)T(A)]TJ /F4 7.97 Tf 6.59 0 Td[(1I)D2F(E0)Ay(t)+A)]TJ /F4 7.97 Tf 6.59 0 Td[(1DF(E0)AI)y,y(0)=0=(y(t)TI)(ATA)]TJ /F4 7.97 Tf 6.58 0 Td[(1I)D2F(E0)Ay(t)+A)]TJ /F4 7.97 Tf 6.59 0 Td[(1DF(E0)AI)y,y(0)=0 (2-29)HeredenotestheKroneckerproductofmatrices;FortwomatricesR,SwhereRismp,andSisnq,RS=0BBBB@r11S...r1nS.........rm1S...rmnS1CCCCAWenotethatD2F(E0)andyare93matrices.Equation 2-29 islinearandthesolutionatt=is:y()=e(A)]TJ /F15 5.978 Tf 5.76 0 Td[(1DF(E0)AI)Z0e)]TJ /F4 7.97 Tf 6.58 0 Td[((A)]TJ /F15 5.978 Tf 5.76 0 Td[(1DF(E0)AI)t(y(t)I)(ATA)]TJ /F4 7.97 Tf 6.58 0 Td[(1I)D2F(E0)Ay(t)dtItisimportanttonotethaty(t),e(A)]TJ /F15 5.978 Tf 5.76 0 Td[(1DF(E0)AI),e)]TJ /F4 7.97 Tf 6.58 0 Td[((A)]TJ /F15 5.978 Tf 5.76 0 Td[(1DF(E0)AI)t,andy(t)Iarealldiagonalmatrices.Remembery(t)=e(A)]TJ /F15 5.978 Tf 5.75 0 Td[(1DF(E0)A)t=eLt 54

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RecallthatF:R3!R3.WeseethatD2F(E0)=0BBBB@H(F1(E0))H(F2(E0))H(F3(E0))1CCCCAwhereH(Fi(E0))isthehessianoftheithcomponentofF.ThereforeD2F(E0)=0BBBBBBBBBBBBBBBBBBBBBBBB@f00(T0)00000)]TJ /F3 11.955 Tf 9.3 0 Td[(k0000k000k000000000001CCCCCCCCCCCCCCCCCCCCCCCCAHenceD2F(E0)A=0BBBBBBBBBBBBBBBBBBBBBBBB@f00(T0)a11)]TJ /F3 11.955 Tf 11.96 0 Td[(kf00(T0) f0(T0)kT0)]TJ /F3 11.955 Tf 11.96 0 Td[(kf00(T0)000)]TJ /F3 11.955 Tf 9.3 0 Td[(ka11)]TJ /F10 7.97 Tf 6.59 0 Td[(k2T0 f0(T0))]TJ /F3 11.955 Tf 9.3 0 Td[(kkk0000ka11k2T0 f0(T0)k0000000001CCCCCCCCCCCCCCCCCCCCCCCCA 55

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Fora33matrixPanda93matrixQ,observethefollowing:((PI)Q)(3(i)]TJ /F4 7.97 Tf 6.59 0 Td[(1)+j,k)=3Xn=1P(i,n)Q(3(n)]TJ /F4 7.97 Tf 6.59 0 Td[(1)+j,k)i,j,k=1,2,3 (2-30)Forconvenience,letM:=(ATA)]TJ /F4 7.97 Tf 6.58 0 Td[(1I)D2F(E0)A.ByEquation 2-30 andthefactthat(D2F(E0)A)(3(n)]TJ /F4 7.97 Tf 6.59 0 Td[(1)+2,k)=0foralln,k=1,2,3,wegetM(i,j)=0fori=2,5,8andj=1,2,3,i.e.thesecond,fth,andeighthrowsofMarerowsofzeroes.InEquation 2-29 ,Mismultipliedbydiagonalmatrices,hencetherowsofzeroesarepreserved.Thisleadstothefollowingstructurefory():y()=0BBBBBBBBBBBBBBBBBBBBBBBB@0000000001CCCCCCCCCCCCCCCCCCCCCCCCA (2-31)Nowrecallx(t,,)=Ay(t,,).Differentiatingthistwicewithrespecttoat=E0,=0,t=,weobtainx()=(AI)y()HencebyEquation 2-30 ,x()hasthesamematrixstructureasEquation 2-31 .Remember(E0,0)=eJ)]TJ /F3 11.955 Tf 12.39 0 Td[(I.TheeigenvectorcorrespondingthezeroeigenvalueofeJ)]TJ /F3 11.955 Tf 11.95 0 Td[(IissimplytheeigenvectorofthezeroeigenvalueofJ:v:=kT0 f0(T0) N1T 56

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Clearly(E0,0)=x().HencetocheckCondition E2-1 ,wecalculate:x()v2=(vTI)x()v=0BBBB@0000000000000000001CCCCA0BBBBBBBBBBBBBBBBBBBBBBBB@0001CCCCCCCCCCCCCCCCCCCCCCCCA=0BBBB@01CCCCAAlso,eJ)]TJ /F3 11.955 Tf 11.95 0 Td[(I=AeLA)]TJ /F4 7.97 Tf 6.59 0 Td[(1)]TJ /F3 11.955 Tf 11.95 0 Td[(I=0BBBB@ef0(T0))]TJ /F6 11.955 Tf 11.96 0 Td[(10e)]TJ /F15 5.978 Tf 5.75 0 Td[((+)+ +)]TJ /F6 11.955 Tf 11.95 0 Td[(10)]TJ /F10 7.97 Tf 6.59 0 Td[(Ne)]TJ /F15 5.978 Tf 5.75 0 Td[((+) +1CCCCAWeneedtochecktheconditionx()v2=2(eJ)]TJ /F3 11.955 Tf 13.23 0 Td[(I).Nowran(eJ)]TJ /F3 11.955 Tf 13.24 0 Td[(I)isatwodimensionalsubspaceofR3.Clearlye1=(1,0,0)T2ran(eJ)]TJ /F3 11.955 Tf 12.97 0 Td[(I)sincee1isaneigenvectorof(eJ)]TJ /F3 11.955 Tf 12.17 0 Td[(I)(correspondingtoeigenvalueef0(T0))]TJ /F6 11.955 Tf 12.17 0 Td[(1).Also,sincee1ande2arelinearlyindependentfromv,R3=span(e1,e2,v).Thereforeran(eJ)]TJ /F3 11.955 Tf 11.95 0 Td[(I)=span((eJ)]TJ /F3 11.955 Tf 11.95 0 Td[(I)e1,(eJ)]TJ /F3 11.955 Tf 11.96 0 Td[(I)e2)=span(e1,(eJ)]TJ /F3 11.955 Tf 11.96 0 Td[(I)e2)Wehave(eJ)]TJ /F3 11.955 Tf 11.95 0 Td[(I)e2=0BBBB@e)]TJ /F15 5.978 Tf 5.75 0 Td[((+)+ +)]TJ /F6 11.955 Tf 11.95 0 Td[(1+)]TJ /F10 7.97 Tf 6.58 0 Td[(Ne)]TJ /F15 5.978 Tf 5.76 0 Td[((+) +1CCCCA 57

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Nowe)]TJ /F4 7.97 Tf 6.58 0 Td[((+)+ +)]TJ /F6 11.955 Tf 11.96 0 Td[(1+)]TJ /F3 11.955 Tf 9.3 0 Td[(Ne)]TJ /F4 7.97 Tf 6.59 0 Td[((+) += +(e)]TJ /F4 7.97 Tf 6.58 0 Td[((+)(1)]TJ /F3 11.955 Tf 11.96 0 Td[(N))]TJ /F6 11.955 Tf 11.95 0 Td[(1)<0Therefore,e3=2ran(eJ)]TJ /F3 11.955 Tf 13.04 0 Td[(I)anditisdeducedthatx()v2=2ran(eJ)]TJ /F3 11.955 Tf 13.04 0 Td[(I)sincex()v22span(e1,e3).Hence,Condition E2-1 ischecked.ByTheorem 2.4 ,thereisananalyticfunction:R!R3suchthat((),)=0forsufcientlysmall,where(0)=E0and0(0)6=0.ThisimpliesthatP((),)=()forall2()]TJ /F5 11.955 Tf 9.3 0 Td[(0,0)where0>0issufcientlysmall.IthasbeenshownthatwhenR0<1,thenE0isgloballyattractinginthenon-negativeorthantofR3.Thisimpliesthatwhen<0,()2R3nR3+.Hence,when>0issufcientlysmall()2R3+.Inparticular,()=(T0,0,0)T+cv+o()wherec>0andvistheeigenvectorcorrespondingtothezeroeigenvalueofJ,i.e.v=kT0 f0(T0), N,1T. WeremarkthatifR0RT(t)dt=0,i.e.RTisamean-zeroperturbation,thenwewouldhavetocalculateevenhigherorderderivativesintheproofofTheorem 2.4 toobtainanappropriatebifurcationcondition.Forthiscaseandtheotherremainingcases,wewillnotpursuerigorousnonlinearbifurcationresults. 2.7CriticalCase,Part2:BifurcationsofLinearizedSystemWenowinvestigatetheresponseofthedominantFloquetmultiplier,,toperiodicperturbationsinthecriticalcasewherefR0(eRT,eP)=1.Forsmall,recallthelinearizedsubsystem,Equation 2-11 (thistimewrittenwiththevariableY):d dtY(t,)=0B@)]TJ /F5 11.955 Tf 9.3 0 Td[(kT0(1)]TJ /F3 11.955 Tf 11.95 0 Td[(eRT+RT(t))N(1)]TJ /F3 11.955 Tf 11.96 0 Td[(eP+P(t)))]TJ /F5 11.955 Tf 9.3 0 Td[(1CAY(t,),Y(0,)=I. (2-32)ThedominantFloquetmultiplierofEquation 2-32 ,(),isananalyticfunctionofforsufcientlysmallbyLemma 3 .Recallthat()isthedominanteigenvalueofthematrix 58

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solutiontoEquation 2-32 .ObservingthatkT0(1)]TJ /F3 11.955 Tf 11.96 0 Td[(eRT+RT(t))=kT0 R0(1)]TJ /F3 11.955 Tf 11.96 0 Td[(eP)+kT0RT(t)= N(1)]TJ /F3 11.955 Tf 11.96 0 Td[(eP)+kT0RT(t),Equation 2-32 canberewrittenas. Y=(B0+RT(t)BRT+P(t)BP)Y,Y(0)=I, (2-33)whereB0=0B@)]TJ /F5 11.955 Tf 9.3 0 Td[( N(1)]TJ /F10 7.97 Tf 6.59 0 Td[(eP)N(1)]TJ /F3 11.955 Tf 11.96 0 Td[(eP))]TJ /F5 11.955 Tf 9.3 0 Td[(1CA,BRT=0B@0kT0001CA,andBP=0B@00N01CA.TheeigenvaluesofB0are)]TJ /F6 11.955 Tf 9.29 0 Td[((+)and0.Hence,diagonalizingB0,weobtainB0=Q0B@000)]TJ /F6 11.955 Tf 9.3 0 Td[((+)1CAQ)]TJ /F4 7.97 Tf 6.59 0 Td[(1,whereQ=0B@11N(1)]TJ /F10 7.97 Tf 6.59 0 Td[(eP) )]TJ /F3 11.955 Tf 9.3 0 Td[(N(1)]TJ /F3 11.955 Tf 11.96 0 Td[(eP)1CAandQ)]TJ /F4 7.97 Tf 6.59 0 Td[(1=1 1+=0B@11 N(1)]TJ /F10 7.97 Tf 6.59 0 Td[(eP)=)]TJ /F4 7.97 Tf 6.59 0 Td[(1 N(1)]TJ /F10 7.97 Tf 6.59 0 Td[(eP)1CA.Forcomputationalconvenience,weintroducetheauxiliaryquantitiesa:==,b:=(+)andM:=N(1)]TJ /F3 11.955 Tf 11.95 0 Td[(ep).WealsodeneA0:=Q)]TJ /F4 7.97 Tf 6.59 0 Td[(1B0Q=0B@000)]TJ /F3 11.955 Tf 9.3 0 Td[(b1CA,ART:=Q)]TJ /F4 7.97 Tf 6.59 0 Td[(1BRTQ=kT0 1+a0B@Ma)]TJ /F3 11.955 Tf 9.3 0 Td[(MMa2)]TJ /F3 11.955 Tf 9.3 0 Td[(Ma1CA,AP:=Q)]TJ /F4 7.97 Tf 6.59 0 Td[(1BPQ= (1+a)(1)]TJ /F3 11.955 Tf 11.96 0 Td[(eP)0B@11)]TJ /F6 11.955 Tf 9.3 0 Td[(1)]TJ /F6 11.955 Tf 9.29 0 Td[(11CA, 59

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X(t,):=Q)]TJ /F4 7.97 Tf 6.59 0 Td[(1Y(t,)Q.UponintroducingA1(t):=RT(t)ART+P(t)AP,X(t,)istheprincipalfundamentalmatrixsolutionto. X=(A0+A1(t))X .(2-34)TheeigenvaluesofX(,)coincidewiththoseofY(,)sincethematricesaresimilar.ThusthedominanteigenvalueofX(,)is(),whichdeterminesthestabilitypropertiesofE0forEquation 2-10 .TheplanofactionistoexpandX(t,)asaTaylorseries.Then,wewillbeabletocalculatederivativesof()at=0inordertoseewheretheperturbationtakesthedominanteigenvalue.Notethat(0)=1inthiscriticalcaseoffR0(eRT,eP)=1sinceX(,0)=eA0isadiagonalmatrixwith1ande)]TJ /F10 7.97 Tf 6.59 0 Td[(basthediagonalentries.ByLemma 3 ,()canbewrittenasaTaylorseriesforsufcientlysmall:()=1+0(0)+1 200(0)2+O(3)Werstndthelinearapproximationbycalculating0(0).Throughouttheremainderofthischapter,wewilldenotethetimeaverageofa-periodicfunctionas :=1 R0(t)dt. Proposition2.3. Let>0besufcientlysmallandconsiderthenonlinearperturbedvirusmodel,Equation 2-10 ,withfR0(eRT,eP)=1.Recallthat()isthedominantFloquetmultiplierofEquation 2-11 andananalyticfunctionof.Then0(0)= + RT 1)]TJ /F3 11.955 Tf 11.96 0 Td[(eRT+ P 1)]TJ /F3 11.955 Tf 11.95 0 Td[(ePIf0(0)<0,then()<1andtheinfection-freeequilibriumE0forEquation 2-10 isGAS.If0(0)>0,then()>1andhenceE0isunstableforEquation 2-10 60

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Wenotethatthisresultiscertainlyexpectedassign(0(0))=sign( RT)ifj RTj 1)]TJ /F10 7.97 Tf 6.58 0 Td[(eRT>j Pj 1)]TJ /F10 7.97 Tf 6.59 0 Td[(ePandvice-versafortheoppositecase. Proof. WeareinterestedincalculatingtheeigenvaluesofX(,),thematrixsolutionofEquation 2-34 .TherighthandsideofEquation 2-34 isanalyticin,thereforethesolutionX(t,)isanalyticin.SowecanexpandX(t,)asaTaylorserieswithrespecttoaround0.X(t,)=1Xn=0nXn(t)=X0(t)+X1(t)+2X2(t)+...Thenwehave(. X0(t)+. X1(t)+2. X2(t)+...)=. X(t,)=(A0+A1(t)X(t,)=A0X0(t)+(A0X1(t)+A1(t)X0(t))+2(A0X2(t)+A1(t)X1(t))+...SinceX(t,)isaprincipalfundamentalsolution,X(0,)=Iforall.Letting!0weseethatX0(0)=I,sothatXi(0)=0foralli1.Hence,weobtainthefollowingsystemofdifferentialequations:. X0(t)=A0X0(t),X0(0)=I, (2-35). X1(t)=A0X1(t)+A1(t)X0(t),X1(0)=0, (2-36). X2(t)=A0X2(t)+A1(t)X1(t),X2(0)=0. (2-37)Wenotethat:eA0t=0B@100e)]TJ /F10 7.97 Tf 6.58 0 Td[(bt1CA,e)]TJ /F10 7.97 Tf 6.58 0 Td[(A0t=0B@100ebt1CA. 61

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ThesolutionstoEquations 2-35 2-36 ,and 2-37 att=(respectively)are:X0()=eA0,X1()=eA0Z0e)]TJ /F10 7.97 Tf 6.58 0 Td[(A0tA1(t)eA0tdt,X2()=eA0Z0e)]TJ /F10 7.97 Tf 6.58 0 Td[(A0tA1(t)eA0tZt0e)]TJ /F10 7.97 Tf 6.59 0 Td[(A0sA1(s)eA0sdsdt.ToinvestigatetheeigenvaluesofX(,),weconsiderthecharacteristicequationofX(,):G(,)=2)]TJ /F6 11.955 Tf 11.96 0 Td[(trX(,)+detX(,)=0, (2-38)where:=().At=0,=1.Also@G @(1,0)=2)]TJ /F6 11.955 Tf 11.96 0 Td[(trX(,0)=2)]TJ /F6 11.955 Tf 11.95 0 Td[(tr(eA0)=2)]TJ /F11 11.955 Tf 11.95 9.68 Td[()]TJ /F6 11.955 Tf 5.48 -9.68 Td[(1+e)]TJ /F10 7.97 Tf 6.58 0 Td[(b=1)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ /F10 7.97 Tf 6.59 0 Td[(b.Then,takingthetotalderivativeofG(,)at=0,=1,wend:0(0)=)]TJ /F5 11.955 Tf 9.3 0 Td[(@G=@ @G=@(1,0),0(0)=)]TJ /F11 11.955 Tf 10.49 19.03 Td[(d d(detX(,))]TJ /F6 11.955 Tf 11.96 0 Td[(trX(,))=0 1)]TJ /F3 11.955 Tf 11.95 0 Td[(e)]TJ /F10 7.97 Tf 6.58 0 Td[(b.Nowd d(detX(,))]TJ /F6 11.955 Tf 11.96 0 Td[(trX(,))j=0=deteA0d ddet)]TJ /F3 11.955 Tf 5.47 -9.68 Td[(I+C+2E+O(3))=0)]TJ /F6 11.955 Tf 11.96 0 Td[(trX1(),whereC=Z0e)]TJ /F10 7.97 Tf 6.58 0 Td[(A0tA1(t)eA0tdt,E=Z0e)]TJ /F10 7.97 Tf 6.58 0 Td[(A0tA1(t)eA0tZt0e)]TJ /F10 7.97 Tf 6.58 0 Td[(A0sA1(s)eA0sdsdt.trX1()=tr)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(eA0C=tr0B@0B@100e)]TJ /F10 7.97 Tf 6.59 0 Td[(b1CAC1CA=C11+e)]TJ /F10 7.97 Tf 6.58 0 Td[(bC22. 62

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deteA0d ddet)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(I+C+2E+O(3)=0=e)]TJ /F10 7.97 Tf 6.59 0 Td[(bd ddet0B@1+C11+O(2)C12+O(2)C21+O(2)1+C22+O(2)1CA=0=e)]TJ /F10 7.97 Tf 6.59 0 Td[(b(C11+C22).Combiningtheaboveresults,weobtain0(0)=C11+e)]TJ /F10 7.97 Tf 6.58 0 Td[(bC22)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ /F10 7.97 Tf 6.59 0 Td[(b(C11+C22) 1)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ /F10 7.97 Tf 6.59 0 Td[(b=C11(1)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ /F10 7.97 Tf 6.59 0 Td[(b) 1)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ /F10 7.97 Tf 6.59 0 Td[(b=C11. (2-39)Inwhatfollows,wewillneedthefollowingquantitye)]TJ /F10 7.97 Tf 6.58 0 Td[(A0tA1(t)eA0t=RT(t)e)]TJ /F10 7.97 Tf 6.59 0 Td[(A0tARTeA0t+P(t)e)]TJ /F10 7.97 Tf 6.59 0 Td[(A0tAPeA0t=RT(t)0B@100ebt1CAART0B@100e)]TJ /F10 7.97 Tf 6.59 0 Td[(bt1CA+P(t)0B@100ebt1CAAP0B@100e)]TJ /F10 7.97 Tf 6.59 0 Td[(bt1CA=RT(t)MkT0 1+a0B@a)]TJ /F3 11.955 Tf 9.3 0 Td[(e)]TJ /F10 7.97 Tf 6.58 0 Td[(bta2ebt)]TJ /F3 11.955 Tf 9.29 0 Td[(a1CA+P(t) (1+a)(1)]TJ /F3 11.955 Tf 11.95 0 Td[(eP)0B@1e)]TJ /F10 7.97 Tf 6.59 0 Td[(bt)]TJ /F3 11.955 Tf 9.29 0 Td[(ebt)]TJ /F6 11.955 Tf 9.3 0 Td[(11CA (2-40)Hence,thematrixCisgivenbyC=1 1+aZ0RT(t)N(1)]TJ /F3 11.955 Tf 11.95 0 Td[(eP)kT00B@a)]TJ /F3 11.955 Tf 9.3 0 Td[(e)]TJ /F10 7.97 Tf 6.59 0 Td[(bta2ebt)]TJ /F3 11.955 Tf 9.3 0 Td[(a1CA+P(t) 1)]TJ /F3 11.955 Tf 11.96 0 Td[(eP0B@1e)]TJ /F10 7.97 Tf 6.58 0 Td[(bt)]TJ /F3 11.955 Tf 9.3 0 Td[(ebt)]TJ /F6 11.955 Tf 9.3 0 Td[(11CAdtFromwhichitfollowsthat0(0)=1 1+a RTkT0N(1)]TJ /F3 11.955 Tf 11.96 0 Td[(eP)a+ P 1)]TJ /F3 11.955 Tf 11.95 0 Td[(eP= + RT 1)]TJ /F3 11.955 Tf 11.95 0 Td[(eRT+ P 1)]TJ /F3 11.955 Tf 11.96 0 Td[(ePIf0(0)6=0,thenthesignof0(0)determineswhether0()<1or0()>1forsufcientlysmall>0. Wenowinvestigateamoresubtlequestion,namely,whathappenswhenbothperturbationsaremean-zero,i.e. RT=0and P=0.Then0(0)=0,hencealinear 63

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approximationgivesnoinformationandweneedtocalculate00(0).Weintroducethefollowingquantities:C:=Z0e)]TJ /F10 7.97 Tf 6.59 0 Td[(A0tA1(t)eA0tdt,E:=Z0e)]TJ /F10 7.97 Tf 6.59 0 Td[(A0tA1(t)eA0tZt0e)]TJ /F10 7.97 Tf 6.59 0 Td[(A0sA1(s)eA0sdsdt. (2-41) Proposition2.4. ConsiderthesameassumptionsofProposition 2.3 .Also,supposethat RT=0and P=0.Then00(0)=2E11+e)]TJ /F10 7.97 Tf 6.59 0 Td[(b 1)]TJ /F3 11.955 Tf 11.95 0 Td[(e)]TJ /F10 7.97 Tf 6.59 0 Td[(bC12C21If00(0)<0,thentheinfection-freeequilibriumE0forEquation 2-1 isGAS.If00(0)>0,thenE0isunstable. Proof. Let 'RT=0and 'P=0.Weevaluatethesecondderivativeof():00(0):=d2 d2(1,0)=d2 d2(trX(,))]TJ /F6 11.955 Tf 11.96 0 Td[(detX(,))=0 1)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ /F10 7.97 Tf 6.59 0 Td[(b,whered2 d2[trX(,))]TJ /F6 11.955 Tf 11.95 0 Td[(detX(,)]=0=2trX2())]TJ /F6 11.955 Tf 11.96 0 Td[(det)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(eA0d ddet)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(I+C+2E+O(3))=0=2tr)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(eA0E)]TJ /F6 11.955 Tf 11.96 0 Td[(det)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(eA0d2 d2det)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(I+C+2E+O(3))=0=2)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(E11+e)]TJ /F10 7.97 Tf 6.58 0 Td[(bE22)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ /F10 7.97 Tf 6.59 0 Td[(b(2E11+2E22)]TJ /F6 11.955 Tf 11.96 0 Td[(2C12C21)=2)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(E11(1)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ /F10 7.97 Tf 6.59 0 Td[(b)+e)]TJ /F10 7.97 Tf 6.58 0 Td[(bC12C21.Therefore00(0)=2E11+e)]TJ /F10 7.97 Tf 6.58 0 Td[(b 1)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ /F10 7.97 Tf 6.59 0 Td[(bC12C21, (2-42)whichconcludestheproof. 64

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WenowapplyProposition 2.4 tosinusoidalfunctionsinordertocalculatethesecondorderapproximationinaspeciccase. Proposition2.5. SupposethattheperiodicperturbationsinEquation 2-9 areoftheform:RT(t)=1sint,P(t)=1sintwhere1,22R.Thenforsufcientlysmall,00(0)=R022 (+)[(+)2+1])]TJ /F3 11.955 Tf 9.3 0 Td[(R0)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(21(1)]TJ /F3 11.955 Tf 11.95 0 Td[(eRT)2+22(1)]TJ /F3 11.955 Tf 11.95 0 Td[(eP)2+12 + Therefore,if12<0,then00(0)<0andE0isGASforEquation 2-10 .If12>0,thenforcertainparameterregionsE0isunstable.Hence,thephasedifferencebetweendrugefcaciescancriticallyaffectthestabilityofE0. Proof. Weprovethistheoremin3steps.First,letusassumethatRT(t)=sintandP=0.ThenE=Z20kT0sint 1+a0B@Ma)]TJ /F3 11.955 Tf 9.29 0 Td[(Me)]TJ /F10 7.97 Tf 6.59 0 Td[(bt1CAZt0kT0sins 1+a0B@MaMa2ebs1CAdsdt=(kT0)2 (1+a)2Z20sint0B@Ma)]TJ /F3 11.955 Tf 9.3 0 Td[(Me)]TJ /F10 7.97 Tf 6.59 0 Td[(bt1CA0B@Ma(1)]TJ /F6 11.955 Tf 11.95 0 Td[(cost)Ma2Rt0ebssinsds1CAdt,whereasterisksdenotenon-essentialentries.WendthatE11=)]TJ /F6 11.955 Tf 9.3 0 Td[((kT0Ma)2 (1+a)2Z20sint(cost)]TJ /F6 11.955 Tf 11.96 0 Td[(1)+(sint)e)]TJ /F10 7.97 Tf 6.59 0 Td[(btbebtsint)]TJ /F3 11.955 Tf 11.96 0 Td[(ebtcost+1 b2+1dt=)]TJ /F6 11.955 Tf 9.3 0 Td[((kT0Ma)2 (1+a)2(b2+1)Z20bsin2t+e)]TJ /F10 7.97 Tf 6.59 0 Td[(btsintdt=)]TJ /F6 11.955 Tf 9.3 0 Td[((kT0Ma)2 (1+a)2(b2+1)2(b(b2+1)+1)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ /F4 7.97 Tf 6.59 0 Td[(2b),C12C21=)]TJ /F6 11.955 Tf 9.3 0 Td[((kT0Ma)2 (1+a2)Z20e)]TJ /F10 7.97 Tf 6.59 0 Td[(btsintdtZ20ebtsintdt 65

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=)]TJ /F6 11.955 Tf 9.29 0 Td[((kT0Ma)2 (1+a)2(b2+1)2(1)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ /F4 7.97 Tf 6.59 0 Td[(2b)(1)]TJ /F3 11.955 Tf 11.95 0 Td[(e2b).Soe)]TJ /F10 7.97 Tf 6.59 0 Td[(b 1)]TJ /F3 11.955 Tf 11.95 0 Td[(e)]TJ /F10 7.97 Tf 6.58 0 Td[(bC12C21=)]TJ /F6 11.955 Tf 9.3 0 Td[((kT0Ma)2 (1+a)2(b2+1)2(e)]TJ /F4 7.97 Tf 6.59 0 Td[(2b)]TJ /F6 11.955 Tf 11.96 0 Td[(1).ThereforebyProposition 2.4 ,00(0)=)]TJ /F6 11.955 Tf 9.3 0 Td[((kT0Ma)2 (1+a)2(b2+1)2(b(b2+1)). (2-43)Next,supposethatRT=0andP(t)=sint.Itfollowsthate)]TJ /F10 7.97 Tf 6.59 0 Td[(b2 1)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ /F10 7.97 Tf 6.59 0 Td[(b2C12C21=e)]TJ /F10 7.97 Tf 6.59 0 Td[(b2 1)]TJ /F3 11.955 Tf 11.95 0 Td[(e)]TJ /F10 7.97 Tf 6.58 0 Td[(b2)]TJ /F5 11.955 Tf 9.3 0 Td[( (1)]TJ /F3 11.955 Tf 11.96 0 Td[(eP)2(1+a2)Z20e)]TJ /F10 7.97 Tf 6.59 0 Td[(btsintdtZ20ebtsintdt=e)]TJ /F10 7.97 Tf 6.59 0 Td[(b2 1)]TJ /F3 11.955 Tf 11.95 0 Td[(e)]TJ /F10 7.97 Tf 6.58 0 Td[(b2)]TJ /F5 11.955 Tf 9.3 0 Td[( (1)]TJ /F3 11.955 Tf 11.96 0 Td[(eP)2(1+a)2(b2+1)2(1)]TJ /F3 11.955 Tf 11.95 0 Td[(e)]TJ /F4 7.97 Tf 6.59 0 Td[(2b)(1)]TJ /F3 11.955 Tf 11.96 0 Td[(e2b)=)]TJ /F5 11.955 Tf 9.3 0 Td[( ((1)]TJ /F3 11.955 Tf 11.95 0 Td[(eP)(1+a))2(b2+1)2(e)]TJ /F4 7.97 Tf 6.58 0 Td[(2b)]TJ /F6 11.955 Tf 11.96 0 Td[(1).E=Z20sint (1+a)(1)]TJ /F3 11.955 Tf 11.96 0 Td[(eP)0B@1e)]TJ /F10 7.97 Tf 6.59 0 Td[(bt1CAZt0sins (1+a)(1)]TJ /F3 11.955 Tf 11.96 0 Td[(eP)0B@Na)]TJ /F3 11.955 Tf 9.3 0 Td[(ebs1CAdsdt.PerformingasimilarcomputationinvolvedinndingEquation 2-43 ,itcanbeseenthatE11=)]TJ /F5 11.955 Tf 9.3 0 Td[(2 ((1)]TJ /F3 11.955 Tf 11.95 0 Td[(eP)(1+a))2(b2+1)2(b(b2+1)+1)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ /F4 7.97 Tf 6.59 0 Td[(2b),andbyProposition 2.4 weobtain00(0)=)]TJ /F5 11.955 Tf 9.29 0 Td[(2 (1)]TJ /F3 11.955 Tf 11.96 0 Td[(eP)2(1+a)2(b2+1)2(b(b2+1)). (2-44)Finally,weletRT=1sint,P=2sintwhere1,22R.ThenC=Z20)]TJ /F6 11.955 Tf 5.48 -9.68 Td[((1sint)e)]TJ /F10 7.97 Tf 6.59 0 Td[(A0tARTeA0t+(2sint)e)]TJ /F10 7.97 Tf 6.59 0 Td[(A0tAPeA0tdt=1 1+aZ200B@1kT00B@)]TJ /F3 11.955 Tf 9.3 0 Td[(Me)]TJ /F10 7.97 Tf 6.58 0 Td[(btsintMa2ebtsint1CA 66

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+2 1)]TJ /F3 11.955 Tf 11.95 0 Td[(eP0B@e)]TJ /F10 7.97 Tf 6.58 0 Td[(btsint)]TJ /F3 11.955 Tf 9.3 0 Td[(ebtsint1CA1CAdt.E=1 (1+a)2Z208><>:264kT01sint0B@Ma)]TJ /F3 11.955 Tf 9.3 0 Td[(Me)]TJ /F10 7.97 Tf 6.58 0 Td[(bt1CA+2sint 1)]TJ /F3 11.955 Tf 11.96 0 Td[(eP0B@1e)]TJ /F10 7.97 Tf 6.58 0 Td[(bt1CA375264kT010B@Ma(1)]TJ /F6 11.955 Tf 11.96 0 Td[(cost)Ma2Rt0ebssinsds1CA+2 1)]TJ /F3 11.955 Tf 11.95 0 Td[(eP0B@(1)]TJ /F6 11.955 Tf 11.96 0 Td[(cost)Rt0ebssinsds1CA3759>=>;dt.UsingEquations 2-43 and 2-44 ,andProposition 2.4 ,itcanbeseenthat00(0)=b (1+a)2(b2+1))]TJ /F6 11.955 Tf 9.3 0 Td[((1kT0Ma)2)]TJ /F6 11.955 Tf 19.92 8.09 Td[((2)2 (1)]TJ /F3 11.955 Tf 11.96 0 Td[(eP)2+12kT0N(1+a2)=bR0 (1+a)2(b2+1)2)]TJ /F3 11.955 Tf 9.29 0 Td[(R0)]TJ /F5 11.955 Tf 5.48 -9.69 Td[(21(1)]TJ /F3 11.955 Tf 11.95 0 Td[(eP)2+22(1)]TJ /F3 11.955 Tf 11.95 0 Td[(eRT)2+12 + =R022 (+)[(+)2+1])]TJ /F3 11.955 Tf 9.3 0 Td[(R0)]TJ /F5 11.955 Tf 5.48 -9.69 Td[(21(1)]TJ /F3 11.955 Tf 11.95 0 Td[(eRT)2+22(1)]TJ /F3 11.955 Tf 11.95 0 Td[(eP)2+12 + Werstconsiderthesingle-drugperturbationcase,whereeither1=0or2=0.Inthiscase,00(0)<0andtherefore,E0isdriventostability.Nowwefurtherinvestigate00(0)when16=0and26=0.Factorout12andrecallthatR0(1)]TJ /F3 11.955 Tf 12.08 0 Td[(eRT)(1)]TJ /F3 11.955 Tf 12.08 0 Td[(eP)=1,toobtain:00(0)=c12 + )]TJ /F11 11.955 Tf 11.95 16.86 Td[(1 2R0(1)]TJ /F3 11.955 Tf 11.96 0 Td[(eP)2+2 1R0(1)]TJ /F3 11.955 Tf 11.96 0 Td[(eP)2 (2-45)=c12[g(a))]TJ /F3 11.955 Tf 11.95 0 Td[(g(r)]wherec=R022 (+)[(+)2+1],a= ,r=1 2R0(1)]TJ /F3 11.955 Tf 11.96 0 Td[(eP)2andg(z)=z+1 zforz2R 67

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Wenoticethat12)]TJ /F3 11.955 Tf 16.21 0 Td[(g(r)turnsouttobenegativeforall1,22Rnf0g,while12g(z1)ispositivewhensign(1)=sign(2)(in-phaseperturbations),andnegativewhensign(1)=)]TJ /F6 11.955 Tf 9.3 0 Td[(sign(2)(out-of-phaseperturbations).Therefore,sign(00(0))=sign(g(a))-222(jg(r)j)forin-phaseperturbations,andsign(00(0))=sign()]TJ /F3 11.955 Tf 9.3 0 Td[(g(a))-222(jg(r)j)forout-of-phaseperturbations.Thus,inthecaseofout-of-phaseperturbations,00(0)<0andE0isstabilized,asstatedinProposition 2.5 .Forthecaseofin-phaseperturbations,weneedtodeterminesign(g(a))-222(jg(r)j)inordertodetermineifE0isstableornot.g(a))-221(jg(r)j>0,a+1 a)]TJ /F11 11.955 Tf 11.96 16.85 Td[(jrj+1 jrj>0,a)-222(jrj+jrj)]TJ /F3 11.955 Tf 17.94 0 Td[(a ajrj>0,(a)-222(jrj)(ajrj)]TJ /F6 11.955 Tf 17.93 0 Td[(1)>0Thefunctiong(a)attainsaglobalminimumwhen=,inwhichg(1)=2,andincreasestoinnityasagoestoinnityorzero.NotethateP2[,e)]TJ /F5 11.955 Tf 13.18 0 Td[(]wheree=1)]TJ /F4 7.97 Tf 16.51 4.71 Td[(1 R0.Inthecaseofj1j=j2j,g(r)attainstheuniqueminimumof2wheneP=1)]TJ /F4 7.97 Tf 19.79 4.71 Td[(1 p R0(eP=eRT).Ifj1j>j2j,g(r)attainsitsmaximumat,whileitattainsitsmaximumate)]TJ /F5 11.955 Tf 13.06 0 Td[(whenthereverseinequalityistrue.Inthecaseof1=2andeP=eRT,sign(00(0))0andin-phaseperturbationswillalwaysdestabilizeE0whenever6=.InSection 2.9 ,wewillprovethatthedominantFloquetmultiplier,,isafunctionofthephase-differencebetweendrugefcacies.Wewilldenotethisby( ).Hence,ifweconsiderRT(t)=sint,P(t)=sin(t)]TJ /F5 11.955 Tf 12.44 0 Td[( ),then(())( )isaperiodicfunctionof withperiod2.Herewehavecalculatedtheleadingorderapproximationof(),00(0),forthecase =0and =.Thedifferencebetween00(0)inthesetwocases( =0and =)givesanapproximationoftheamplitudeof(())( ).Fix>0andlet( )denote(())( ).Noticethattheamplitudeof( )dependsonthemagnitudeof 68

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cg(a).FixandR0.ConsiderG(a):=cg(a).Let1=1,2=1.Thentheabsolutedifferenceof00(0)inthecase =0and00(0)inthecase =is:G(a)=2R0(a+a3) (a+1)[(a+1)2+1=2]ItcanbeshownG0(a)>0foralla0(withxed).Hence,G(a)isstrictlyincreasing.Also,G(a)!2R0asa!1.Thus,theamplitude( )(whileholdingR0andconstant)shouldapproximatelybeanincreasingfunctionofapproachingalimitas!1.Recallthattheparametermeasuresthedecayrateoftheinfectedcells(equivalentlytheaveragelife-spanoftheinfectedcellsis1=),andNistheviralproductionrateofinfectedcells.Considercg(a)asvaries,butR0isheldconstantbyalteringsomeoftheotherparametersN,k,orT0.Theexpressioncg(a)issymmetricinandwhenR0isheldconstant.Hence,thesameconclusionsfromthepreviousparagraphwillapplytothiscase.Explicitly,theamplitudeof( )shouldincreasewithasR0andareheldconstant.Thus,ifweconsiderG(a)asafunctionofand,H(,),withR0heldconstant,then@H(,) @>0and@H(,) @>0.Therefore,weexpecttheamplitudeof( )toincreaseiforisincreasedandR0isheldconstant.Viralgenerationtimecanbedenedastheaveragetimeforthe Vvirionstobereplacedbyanewpopulationof Vvirions(calculatedattheinfectionequilibrium).Theaverageviralgenerationtimeinthestandardmodelisgivenby1 +1 [ 33 ].Basedupontheaboveanalysis,decreasingtheviralgenerationtimeisexpectedtoproducelargeramplitudein( )whenthedrugefcaciesareatcriticallevels. 2.8AntagonismWithsecondorderapproximationsofthedominantFloquetmultiplier,(),weanalyticallyshowed(forcriticalefcacies,i.e.fR0(eRT,eP)=1)thatin-phaseperturbationsmayresultinviralpersistence,whileout-of-phaseperturbationsalwaysdrivethediseasetoextinction.Eventhoughtheaveragedrugefcaciesremain 69

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Figure2-2. Thephasedifferencebetweendrug-efcacyfunctions(displayedhere)canbeanimportantparameterindeterminingstabilityofE0. constant,thetreatmentoutcomechanges.WhatisthemechanismbehindthisqualitativechangethatcanoccurwhenshiftingthephasedifferencebetweenRT(t)andP(t)?AcloserlookatthemodiedreproductionnumberfR0(eRT,eP)revealsthepossibledrivingforce.RecallfR0(eRT,eP)=R0(1)]TJ /F3 11.955 Tf 12.25 0 Td[(eRT)(1)]TJ /F3 11.955 Tf 12.26 0 Td[(eP).Wecancalculatetheaveragereproductionnumber, R0,overtheperiod,where R0=1 Z0R0(1)]TJ /F5 11.955 Tf 11.95 0 Td[(RT(t))(1)]TJ /F5 11.955 Tf 11.96 0 Td[(P(t))dt (2-46)LeteRT,ePbethresholdefcacies,i.e.R0(1)]TJ /F3 11.955 Tf 11.29 0 Td[(eRT)(1)]TJ /F3 11.955 Tf 11.29 0 Td[(eP)=1andassumethefollowingsinusoidalperturbationswithphaseshift :RT(t)=eRT)]TJ /F5 11.955 Tf 11.96 0 Td[(sin!t,P(t)=eP)]TJ /F5 11.955 Tf 11.96 0 Td[(sin(q p!(t)]TJ /F5 11.955 Tf 11.96 0 Td[( )) (2-47)HeretheperiodoftheRT-inhibitorisRT=2 !andtheperiodoftheP-inhibitorisP=p qRT.Thecommonperiodbetweenthedrugsatises=pRT=qP=p2 !.Let R0( )betheaveragereproductionnumberasafunctionofthephaseshift ,where 2[0,P].Then, R0( )=1 Z0R0(1)]TJ /F3 11.955 Tf 11.95 0 Td[(eRT+sin!t)(1)]TJ /F3 11.955 Tf 11.96 0 Td[(eP+sin(q p!(t)]TJ /F5 11.955 Tf 11.95 0 Td[( )))dt 70

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=1 Z0R0(1)]TJ /F3 11.955 Tf 11.95 0 Td[(eRT)(1)]TJ /F3 11.955 Tf 11.95 0 Td[(eP)dt+Z0R0(1)]TJ /F3 11.955 Tf 11.96 0 Td[(eP)sin(!t)dt+Z0R0(1)]TJ /F3 11.955 Tf 11.95 0 Td[(eRT)sin(q p!(t)]TJ /F5 11.955 Tf 11.96 0 Td[( ))dt+Z0R02sin(!t)sin(q p!(t)]TJ /F5 11.955 Tf 11.96 0 Td[( ))dt=1+R02 Z0sin(!t)sin(q p!(t)]TJ /F5 11.955 Tf 11.95 0 Td[( ))dtMakethesubstitution!t=sp=1+R02 2Z20sin(ps)sinqs)]TJ /F3 11.955 Tf 13.15 8.09 Td[(q p! ds=8>><>>:1ifp6=q1+R02cos(! )ifp=qAnotheraveragingtechniquetoconsideristheaveragedeigenvalue, r( ):=1 R0r(t, )dt,wherer(t, )isthetimedependenteigenvalueofthematrixA(t, )withA(t, )=0B@)]TJ /F5 11.955 Tf 9.3 0 Td[(kT0(1)]TJ /F3 11.955 Tf 11.95 0 Td[(eRT+sin(!t))N(1)]TJ /F3 11.955 Tf 11.95 0 Td[(eP+sin(q p!(t)]TJ /F5 11.955 Tf 11.96 0 Td[( ))))]TJ /F5 11.955 Tf 9.29 0 Td[(1CATheintegralintheformulafor r( )isnotanalyticallytractable,butwedoprovidenumericalcalculations.Toconverttoapotentialreproductionnumberforthesystem,weconsidere r( ).InFigures 2-3 and 2-4 R0( ),e r( ),and( )(thedominantFloquetmultiplierasafunctionof )aredisplayedforthesystem. x=A(t, )xwith=0.2.TocomputethedominantFloquetmultiplier,( ),thePFSiscalculatedbynumericallysolvingthedifferentialequation. x=A(t, )x.Inthecasethatthedrugdosingperiodsaredifferent,i.e.p6=q,then R0( )1.Figure 2-3 displaysthecasewithRT=1dayandP=1=2days,andRT=3daysandP=1day.WeseethattheFloquetmultiplier,( ),alwaysstaysbelow1,andhaslessvariationthanthecaseinwhichRT=P=(displayedinFigure 2-4 ). 71

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A BFigure2-3. ThedominantFloquetmultiplier,( ),(red)isshownasfunctionofphaseshiftfortwocasesofsinusoidalperturbations,Equation 2-47 ,with=0.2,eRT=eP=1)]TJ /F4 7.97 Tf 19.22 4.71 Td[(1 p R0,anddifferentdosingperiods(inA,RT=1andP=1 2;inB,RT=3andP=1).Theaveragedreproductionnumber, R0( ),(green)andexponentiatedaveragedeigenvalue,e r( ),(blue)arealsoshown.TheremainingparametersarechosenaccordingtopublishedestimatesforHIVfromRongandPerelson[ 37 ].ThevaluesfortheparameterswillbeexplicitlystatedinChapter 3 FortherestofthissectionwefocusonthecasewhereRT=P=,i.e.theRT-inhibitorandP-inhibitorhavethesamedosingperiod.Then R0( )=1+1 2R02cos(! ).Hence, R0>1forin-phaseperturbationsand R0<1forout-of-phaseperturbations.Thisdifferenceinaveragereproductionnumber, R0,mayexplainwhyout-of-phasetreatmentsaremoreeffectivethanin-phasetreatment.However, R0isnotastabilitythreshold,andmoreover,predictstheincorrectstabilityincertaincases,asseeninFigure 2-4 .Westressthatinperiodicdifferentialequations,theFloquetmultipliersofaperiodicsolutiondeterminelocalstability.Noticethat R0( )doesnotdependonthecommonperiod.Onthecontrary,theperiodaffectsthegraphof( )(thedominantFloquetmultiplerasafunctionof ).InFigures 2-4A and 2-4B ,thegraphsaredepictedforthecases=1 2daysand=1days.Theamplitudeof( )increasesastheperiodincreases. 72

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A B C DFigure2-4. ( ), R0( ),ande r( )aredisplayedforcertaincaseswhenthedosingperiodsareequal,i.e.RT=P=.A)=1 2daysandalltheotherparametersarechosenasinFigure 2-3 .B)Weincreaseto1dayandleavetheotherparametersthesame.C)andD)=1dayandisincreasedto13(thevalueof)and50respectively. InFigures 2-4B 2-4C ,and 2-4D ,theperiod=1day,buttakesondifferentvalues.Recall R0( )issimplyanunshiftedcosinefunction.Henceontheinterval[0,], R0( )hasauniqueminimumat=2andmaximaat0and.Whereas,asshowninFigure 2-4 ,theminimaandmaximaof( )canbeshiftedtodifferentvalues.Inaddition, R0( )hasamplitudeproportionaltoR02,whileamplitudeof( )stronglydependsupon.Also,noticeinthecase=andeRT=ep,(0)=1eventhough R0(0)=1+2R0 2.Therearecertainlyotherfactorsbeside R0( )whichplayaroleindeterminingstability.Ingeneral,thegraphof( )canexhibitvaryingamplitude,alongwithhorizontalandverticalshifts,asvaries. 73

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A BFigure2-5. Theamplitudeof( )asafunctionofinfectedcelldeathrateisdisplayed.Alltheparameters(except)areasinFigures 2-4B 2-4C ,and 2-4D .Thegraphsconrmanalyticalpredictionsfromtheprevioussection. Overall,e r( )iscloseto( )forthethespecicparameterschosen.Therearedifferences,though,ase r( )alwayshasaminimumat =0.5,whichisnotthecasefor( ).Thisleadstovaluesof inwhiche r( )>1,yetE0isstable,andothervaluesof wheree r( )<1,butE0isunstable.Itiswellknownthatforaperiodiclinearsystem. x=A(t)x,theeigenvaluesofA(t)donot,ingeneral,determinethevalueoftheFloquetmultipliers.Infact,inthenextsectionwewillprovideanexampleofaperiodiclinearsystem. x=A(t)xinwhichalltheeigenvaluesA(t)havenegativerealpart,yetthezerosolutionisunstable.Attheendofthelastsection,wepredictedthattheamplitudeof( )wouldgenerallyincreaseasincreasesandapproachalimitas!1.InFigure 2-5A ,theamplitudeof( )isshownasafunctionofasrangesfrom0.1to50.Forthechosenparameters,theamplitudeof( )isactuallydecreasingasvariesfromaround5to13,butincreasingeverywhereelseinthedomain.Ourpredictionofanincreasingfunctionwasbasedonthefactthattheleadingorderapproximationofj(0))]TJ /F5 11.955 Tf 9.67 0 Td[((0.5)j(theperiodis1)isanincreasingfunction.Thereasonforthediscrepancywhen5<<13seemstobeaconsequenceofneglectinghigherorderterms.InFigure 2-5B ,the 74

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amplitudeof( )isshownforlargevaluesforandappearstobeapproachingalimitas!1.ThelevelsetsofthemodiedreproductionnumberfR0(eRT,eP)=R0(1)]TJ /F3 11.955 Tf 9.96 0 Td[(eRT)(1)]TJ /F3 11.955 Tf 9.97 0 Td[(eP)correspondtodrugcombinationsofequalinhibitorypower.Inpharmaceuticalresearch,druginteractionsareoftenevaluatedbyttingacurvetodataofcombinationsofdrugconcentrationscorrespondingto50%inhibition(thisisoftenreferredtoasanisobolegraph).ThistheoreticallydiffersfromourgraphoflevelssetsoffR0(eRT,eP)sincedrugefcaciesareusuallynonlinearfunctionsofdrugconcentration.Thettedcurveisthencomparedtoanullmodel,i.e.azerointeractionreference,toasseswhetherthedrugsexhibitsynergism,antagonism,orzerointeraction.Twonullmodels,LoeweadditivityandBlissindependence,arecommonlyusedandthereisconsiderabledebateonthecorrectmodel[ 15 ].Loeweadditivityassumesthatthetwodrugsactonatargetthroughasimilarmechanism,whileBlissindependenceassumesnon-interactingdrugsactingindependentlyofeachother.Inourcase,theRT-inhibitorandP-inhibitorareassumedtoexactlysatisfytheconditionsofBlissindependence.Specically,thecriteriaforBlissindependenceisthatFUAB=FUAFUB,whereFUABdenotesthefractionofuninhibitedeventsundercombinationtherapyconsistingofdrugAanddrugB,andFUAandFUBdenotethefractionsofuninhibitedeventsundersingledrugtherapywithdrugAanddrugB,respectively.Inourcase,thereproductionnumberofourmodelmeasuresreplicationevents.VericationofBlissindependencefortheRT-inhibitorandP-inhibitorfollows:fR0(eRT,eP) R0=R0(1)]TJ /F3 11.955 Tf 11.96 0 Td[(eRT)(1)]TJ /F3 11.955 Tf 11.96 0 Td[(eP) R0=R0(1)]TJ /F3 11.955 Tf 11.95 0 Td[(eRT) R0R0(1)]TJ /F3 11.955 Tf 11.95 0 Td[(eP) R0=fR0(eRT,0) R0fR0(0,eP) R0ThisproducesconcavecurvesforthelevelsetsoffR0(eRT,eP).Essentially,ifwehaveaxedleveloftotalefcacy,thenconcentratingasmuchofthisefcacyaspossibleinonedrugtypeisthemosteffectivestrategyinreducingfR0(eRT,eP). 75

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Forourpresenttheoreticalwork,drugconcentrationsdonotneedtobetakenintoaccount.BlissindependenceallowsustoequatethephasedifferencebetweenRT(t)andP(t)asthetimingbetweendosagesoftherespectivedrugtypes.Wehaveshownthatthistimingbetweendosagescanbecriticalfordeterminingthequalitativedynamicsofthevirus.Thisseemsrathercounter-intuitiveasonemightexpectthetimingbetweendosagesofindependent,non-interactingdrugstonotmakeadifference.ItseemsasthoughtheassumptionofBlissindependence,whichdirectlyleadstothecalculationof R0( ),isthedrivingfactorbehindtheadvantageofout-of-phasetreatments.Certainly, R0(Equation 2-46 )doesnotdeterminestability.Averagingatimedependentreproductionnumbergenerallydoesnotprovideathresholdconditionforstabilityinperiodicmodels[ 5 ].Despitethesefailuresofaveraging,thevariationin R0( )doesofferasimpleexplanationforthequalitativeresultofout-of-phasetreatmentsbeingsuperiortoin-phasetreatments.Thisraisesthequestionofwhetherzero-interactiondrugsinacombinationtherapyshouldactuallybeconsideredantagonisticwhenperiodicdosingistakenintoaccount.Moreworkneedstobedoneininvestigatingthisprincipleandassessingpossiblerelevancetopharmaceuticalresearch. 2.9PhaseShiftsWewouldliketondthephasedifference ,whichminimizesthedominantFloquetmultiplier( ).ThiscouldthenbetranslatedintotheoptimaltimingbetweendosagesoftheRT-inhibitorandP-inhibitor.Unfortunately,thiscannotbedoneexplicitly,butwecanndanequivalentminimizationproblemwhichrequireslessnumericalcomputation.ConsiderthelinearizedsubsystemofEquation 2-1 ,parameterizedbyphaseshift, 2[0,):. x=A(t, )x (2-48) 76

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whereA(t, )=0B@)]TJ /F5 11.955 Tf 9.3 0 Td[(k(1)]TJ /F5 11.955 Tf 11.96 0 Td[(RT(t))T0N(1)]TJ /F5 11.955 Tf 11.96 0 Td[(P(t)]TJ /F5 11.955 Tf 11.95 0 Td[( )))]TJ /F5 11.955 Tf 9.29 0 Td[(1CA.Let( )denotethedominantFloquetmultiplierofEquation 2-48 .Wedenetheoptimalphaseshift, ,asthevalueof whichminimizes( ),i.e. :=argmin0 <( ).LetX(t, )denotetheprincipalfundamentalsolutionofEquation 2-48 Proposition2.6. =argmin0 0sinceisstrictlylargerthantheothereigenvalueofX(, )(followsfromthePerron-FrobeniustheoremasX(, )ispositive).Thereforebytheimplicitfunctiontheorem,wecandenethefunction( )for0 <.FromEquation 2-49 ,trX(, )=( )+detX(, ) ( ).WendthatdetX(, )=e)]TJ /F4 7.97 Tf 6.59 0 Td[((+)=e)]TJ /F10 7.97 Tf 6.59 0 Td[(bforall0 <.Hence,( )>p e)]TJ /F10 7.97 Tf 6.59 0 Td[(bandtrX(, )=( )+e)]TJ /F10 7.97 Tf 6.59 0 Td[(b ( ).TheRHSisstrictlyincreasingwith( ).So iswheretrX(, )isminimized. 77

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For 1, 22R,weconsiderthe(linearized)phaseshiftedsystem:. x=A(t, 1, 2)x (2-50)whereA(t, 1, 2)=0B@)]TJ /F5 11.955 Tf 9.29 0 Td[(k(1)]TJ /F5 11.955 Tf 11.95 0 Td[(RT(t)]TJ /F5 11.955 Tf 11.96 0 Td[( 1))T0N(1)]TJ /F5 11.955 Tf 11.96 0 Td[(P(t)]TJ /F5 11.955 Tf 11.96 0 Td[( 2)))]TJ /F5 11.955 Tf 9.3 0 Td[(1CA. Proposition2.7. Let( 1, 2)bethedominantFloquetmultiplierofEquation 2-50 .Then,( 1, 2)=(0,( 2)]TJ /F5 11.955 Tf 12.2 0 Td[( 1)modulo).Inotherwords,thedominantFloquetmulti-plierisa-periodicfunctionofthephasedifferencebetweendrugefcacyfunctions. Proof. Let\(t)beaPFS(principalfundamentalsolution)toEquation 2-50 .Let(t)beaPFSto:. x=A(t+ 1, 1, 2)x=A(t,0,( 2)]TJ /F5 11.955 Tf 12.03 0 Td[( 1)modulo)x.Thene(t):=(t)]TJ /F5 11.955 Tf 12.02 0 Td[( 1)isaFStoEquation 2-50 withe( 1)=I.UsingFloquet'stheorem,weobtain:()=e(+ 1)=\(+ 1)e(0)=\( 1)\()e(0)=e( 1)e)]TJ /F4 7.97 Tf 6.59 0 Td[(1(0)\()e(0)=e)]TJ /F4 7.97 Tf 6.58 0 Td[(1(0)\()e(0).Hence( 1, 2)=(0,( 2)]TJ /F5 11.955 Tf 11.96 0 Td[( 1)modulo). Therefore,itsufcestoconsiderEquation 2-48 whenphaseshiftisaparameterofinterest.WeremarkthatProposition 2.7 holdsforgenerallineardifferentialequationsystemscontainingmultipletime-periodiccoefcients.Webrieydiscussourresultsinthecontextoflinearperiodicdifferentialequations.Considerthesystem:. x=A(t)xwherex2RnandA(t)isannn-periodicmatrix.ItiswellknownthattheeigenvaluesofA(t)donot,ingeneral,determinethestabilitytypeofthezerosolution.Thisdiffersfromanautonomouslinearsystem,. x=Ax.Inthecaseoftheperiodiclinearizedvirussubsystem,Equation 2-48 ,wehavefoundanexampleinwhichthedominanteigenvalueofthe(autonomous)averagedmatrixA:= A(t)iszero,yetthe(non-autonomous) 78

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periodicsystemcaneitherhaveanunstableorstablezerosolution(E0)(dependingonthephasedifference).TodemonstratetheproblemswithinferringstabilitydirectlyfromthematrixA(t)inperiodiclinearsystems,andalsotoshowtheimportanceofthephasedifferenceparameter,weendthischapterwithatoyexample.SeveralresearchershavefoundperiodicsystemsinwhichtheeigenvaluesofA(t)havenegativerealpartforallt,yetthezerosolutionisunstable,oralternatively,caseswhereA(t)hasaneigenvaluewithpositiverealpartforallt,yetthezerosolutionisstable.Onesuchexampleisthefollowing,discoveredbyMarkusandYamabe[ 28 ]:A(t)=0B@)]TJ /F6 11.955 Tf 9.3 0 Td[(1+3 2cos2t1)]TJ /F4 7.97 Tf 13.15 4.7 Td[(3 2sintcost)]TJ /F6 11.955 Tf 9.29 0 Td[(1)]TJ /F4 7.97 Tf 13.15 4.71 Td[(3 2sintcost)]TJ /F6 11.955 Tf 9.3 0 Td[(1+3 2sin2t1CATheeigenvaluesofA(t)are1 4)]TJ /F2 11.955 Tf 5.48 -9.69 Td[()]TJ /F6 11.955 Tf 9.3 0 Td[(1p 7iforallt,hencetherealpartofeacheigenvalueisnegative.Bydirectsubstitution,itcanbeveriedthatthefollowingexponentiallygrowingfunctionisasolutionto. x=A(t)x:x(t)=et=20B@)]TJ /F6 11.955 Tf 11.29 0 Td[(costsint1CAHence,thezerosolutionisunstable.Usingtrigonometricidentities,wendthatA(t)=A0+3 4A1(t)+3 4A2(t),whereA0=0B@)]TJ /F4 7.97 Tf 6.59 0 Td[(1 41)]TJ /F6 11.955 Tf 9.3 0 Td[(1)]TJ /F4 7.97 Tf 10.49 4.71 Td[(1 41CA,A1(t)=0B@cos2t00)]TJ /F6 11.955 Tf 11.29 0 Td[(cos2t1CA,A2(t)=0B@0)]TJ /F6 11.955 Tf 11.29 0 Td[(sin2t)]TJ /F6 11.955 Tf 11.29 0 Td[(sin2t01CA.Then,wecanthinkofthedifferentialequation. x=A(t)xasalinearautonomoussystemwithtwo-periodicforcesindependentlyaffectingthecoefcientsofthesystembytheactionofthematricesA1(t)andA2(t).WhatistheeffectofshiftingthephasedifferencebetweenA1(t)andA2(t)? 79

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Figure2-6. Forthetoyexample,thephasedifferenceisacriticalparameter.When( )>1,solutionsareunbounded.When( )<1,solutionsconvergetozero. Considerthephase-shiftedsystem:. x=A(t, )x=(A0+A1(t)+A2(t)]TJ /F5 11.955 Tf 12.55 0 Td[( ))x,where0 <.WenotethatbothA(t,0)andA)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(t, 2haveeigenvalues1 4)]TJ /F2 11.955 Tf 5.48 -9.68 Td[()]TJ /F6 11.955 Tf 9.3 0 Td[(1p 7iforallt.Let( )denotethemodulusofthedominantFloquetmultiplierasafunctionof .Thegraphof( )isshowninFigure 2.9 .Weseethatthephasedifference, ,isacriticalparameterforthesystem.Thefunction( )intersectsthecriticalvalueof1twice,andattainsitsmaximumandminimumat =0and ==2,respectively.Actually,theprincipalfundamentalsolutioncanbefoundexplicitlywhen =0and ==2,sothereisnoneedtorelyonnumericsforthisexample.FollowingtheworkofJosicandRosenbaum[ 23 ],letG=0B@0)]TJ /F6 11.955 Tf 9.3 0 Td[(1101CA.ThenetG=0B@cost)]TJ /F6 11.955 Tf 11.29 0 Td[(sintsintcost1CA, 80

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i.e.etGisarotationmatrix,whichrotatesvectorsinR2bytheanglet.NoticethatA(t,0)=0B@)]TJ /F6 11.955 Tf 9.3 0 Td[(1+3 2cos2t1)]TJ /F4 7.97 Tf 13.15 4.71 Td[(3 2sintcost)]TJ /F6 11.955 Tf 9.3 0 Td[(1)]TJ /F4 7.97 Tf 13.15 4.71 Td[(3 2sintcost)]TJ /F6 11.955 Tf 9.3 0 Td[(1+3 2sin2t1CA=0B@costsint)]TJ /F6 11.955 Tf 11.3 0 Td[(sintcost1CA0B@1 21)]TJ /F6 11.955 Tf 9.3 0 Td[(1)]TJ /F6 11.955 Tf 9.3 0 Td[(11CA0B@cost)]TJ /F6 11.955 Tf 11.3 0 Td[(sintsintcost1CAandAt, 2=0B@)]TJ /F6 11.955 Tf 9.3 0 Td[(1+3 2cos2t1+3 2sintcost)]TJ /F6 11.955 Tf 9.3 0 Td[(1+3 2sintcost)]TJ /F6 11.955 Tf 9.3 0 Td[(1+3 2sin2t1CA=0B@cost)]TJ /F6 11.955 Tf 11.29 0 Td[(sintsintcost1CA0B@1 21)]TJ /F6 11.955 Tf 9.3 0 Td[(1)]TJ /F6 11.955 Tf 9.3 0 Td[(11CA0B@costsint)]TJ /F6 11.955 Tf 11.29 0 Td[(sintcost1CAHence,A(t,0)=e)]TJ /F10 7.97 Tf 6.58 0 Td[(tGBetGandA)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(t, 2=etGBe)]TJ /F10 7.97 Tf 6.59 0 Td[(tGwhereB=0B@1 21)]TJ /F6 11.955 Tf 9.3 0 Td[(1)]TJ /F6 11.955 Tf 9.3 0 Td[(11CA.LetX(t,0)=e)]TJ /F10 7.97 Tf 6.59 0 Td[(tGet(B+G).Thend dtX(t,0)=)]TJ /F3 11.955 Tf 9.3 0 Td[(e)]TJ /F10 7.97 Tf 6.58 0 Td[(tGGet(B+G)+e)]TJ /F10 7.97 Tf 6.59 0 Td[(tG(B+G)et(B+G)=e)]TJ /F10 7.97 Tf 6.58 0 Td[(tG)]TJ /F3 11.955 Tf 9.3 0 Td[(Get(B+G)+Bet(B+G)+Get(B+G)=e)]TJ /F10 7.97 Tf 6.58 0 Td[(tGBetGe)]TJ /F10 7.97 Tf 6.58 0 Td[(tGet(B+G)=A(t,0)X(t,0)ThereforethePFSto. x=A(t,0)xisX(t,0)=e)]TJ /F10 7.97 Tf 6.58 0 Td[(tGet(B+G).Similarly,thePFSto. x=A(t,)xisX)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(t, 2:=etGet(B)]TJ /F10 7.97 Tf 6.59 0 Td[(G).ByFloquet'sTheorem,thereexistsa(possiblycomplex)-periodicmatrixSandmatrixCsuchthatX(t,0)=S(t)etC,wheretheeigenvaluesofeCaretheFloquetmultipliersof. x=A(t,0)x.Then,X(2,0)=e2(B+G)=)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(e2C2.SotheFloquetexponentsof. x=A(t,0)xaretheeigenvaluesof(B+G).LikewisetheFloquetexponentsof. x=A(t,)xaretheeigenvaluesof(B)]TJ /F3 11.955 Tf 11.58 0 Td[(G). 81

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TheeigenvaluesofB+Gare)]TJ /F6 11.955 Tf 9.3 0 Td[(1and1 2,whiletheeigenvaluesofB)]TJ /F3 11.955 Tf 12.19 0 Td[(Gare)]TJ /F4 7.97 Tf 6.59 0 Td[(1 4p 55 2i.Hencethezerosolutionisunstablewhen =0,butstablewhen = 2.ThisexampleshowsthatstabilitydoesnotingeneraldependontheeigenvaluesofA(t).Italsodemonstratesthepotentialsignicanceofphasedifferenceasaparameterinlinearperiodicdifferentialequationswithmultipleperiodiccoefcients.Incertainnonlinearsystemswithtwoperiodicforces,thisphaseeffecthasbeenstudiedinthecontextofcontrollingchaoticdynamics[ 45 ].Phaseshiftsmaybeaninterestingcontrolparameterinsystemswithmultipleperiodicforces,suchastheviralmodelwithcombinationtherapy,becauseenhancedresultscanbeachievedwithoutadjustingthemagnitudeoftheforces.Sincestabilityofperiodiclinearsystemsisoftenimportantincontroltheory,itmaybevaluabletoinvestigatehowphaseshiftsaffectFloquetmultipliersincertainmodels. 82

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CHAPTER3NUMERICALSIMULATIONSOFPERIODICHIVMODEL 3.1SinusoidalDrugEfcaciesTheperturbationresultsandsubsequentnumericalcalculationsofthedominantFloquetmultiplierofEquation 2-48 ,( ),demonstratedthatthephasedifference, ,betweendrugefcacyfunctionscancriticallyaffectthestabilityoftheinfection-freeequilibriumE0,andhencealtertheglobaldynamicsofthesolutionstoEquation 2-1 .InFigure 3-1 ,numericalsimulationsofsystemEquation 2-1 areshownforin-phaseandout-of-phasedrugtreatments,specicallywhen:RT(t)=0.7373)]TJ /F6 11.955 Tf 11.96 0 Td[(0.2sin(2t)P(t)=0.7373)]TJ /F6 11.955 Tf 11.96 0 Td[(0.2sin(2t)(in-phase)RT(t)=0.7373)]TJ /F6 11.955 Tf 11.96 0 Td[(0.2sin(2t)P(t)=0.7373+0.2sin(2t)(out-of-phase). (3-1)InFigure 3-1B ,RT(t)isshown.AllthenumericalsolutionstoordinarydifferentialequationswerecomputedonMatlabusinganexplicitRunge-Kutta(4,5)formula,theDormand-Princepair[ 14 ].WeuseestimatesofHIVparametersfromRongandPerelson[ 37 ],inwhichR0=18.4651.Notice,thecriticalconstantdrugefcacycombinationwheneRT=ePiseRT=eP=1)]TJ /F6 11.955 Tf 12.5 -.01 Td[(1=p R0=0.7673,hencetheaveragedrugefcacycombinationinEquation 3-1 isbelowthecriticallevelneededtocleartheinfection.Thefunctionalformf(T)=a)]TJ /F3 11.955 Tf 12.76 0 Td[(bTisusedtomodelthe(net)growthrateoftheuninfectedcellpopulation.Thevaluesoftheparametersareasfollows:a=104ml)]TJ /F4 7.97 Tf 6.58 0 Td[(1day)]TJ /F4 7.97 Tf 6.58 0 Td[(1andb=0.01day)]TJ /F4 7.97 Tf 6.59 0 Td[(1(T0=106ml)]TJ /F4 7.97 Tf 6.59 0 Td[(1),k=810)]TJ /F4 7.97 Tf 6.59 0 Td[(7ml)]TJ /F4 7.97 Tf 6.58 0 Td[(1day)]TJ /F4 7.97 Tf 6.59 0 Td[(1,=0.7day)]TJ /F4 7.97 Tf 6.59 0 Td[(1,N=300,=13day)]TJ /F4 7.97 Tf 6.59 0 Td[(1.TheinitialconditionsaretakentobeinfectionequilibriumofEquation 1-3 (T(0)=5.42104ml)]TJ /F4 7.97 Tf 6.59 0 Td[(1,T(0)=1.35104ml)]TJ /F4 7.97 Tf 6.58 0 Td[(1,V(0)=2.18105ml)]TJ /F4 7.97 Tf 6.58 0 Td[(1).InFigure 3-1C ,weseethatthein-phasedrugtreatmentsdonotcleartheinfection.Thevirusdecaysrapidly,butthenreboundsafteraround250daysandthesolutionconvergestoaperiodicorbit(Figure 3-1D ).Observethattherearetwoperiodsof 83

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A B C D E FFigure3-1. ViralLoadvs.timeforEquation 2-1 .A)Theviralload,undertheassumptionofaconstantdrugefcacyofmagnitude0.7373.B)DrugefcacyasinEquation 3-1 usedinsimulationsforperiodictreatment(C,D,E,F).C)Thein-phasetreatmentinitiallybringtheviralloadclosetozero,butthenthereisviralreboundbeforeconvergingtoaperiodicsolutionwithfairlyhighviralload.D)Theasymptoticperiodicsolutionfortheviralloadinthecaseofin-phasetreatments.E)andF)Theout-of-phasetreatmentscausetheviralloadtodecaytozero. 84

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oscillationthatoccur.Therearehighfrequencyoscillations(periodof1day)duetotheperiodicforcingfromthesinusoidaldrugefcacies.Therearealsolowfrequencydampedoscillationsduetothefactthatthepositivesteadystateoftheautonomoussystemisalocallyasymptoticallystablespiralpoint.Onthecontrary,theout-of-phasesinusoidalforcingclearsthevirus,asshowninFigures 3-1E and 3-1F .Acriticismofthestandardwithin-hostvirusmodel,Equation 1-3 ,isthatitcannotrobustlygeneratealowlevelviralsteadystateoftenobservedinpatientsundergoingtherapy[ 37 ].Asthedrugefcaciesapproachthethresholdlevel,themagnitudeoftheviralsteadystateisextremelysensitivetosmallchangesinefcacy,rapidlydecreasingtozero,asshowninFigure 3-2A .Thesefeaturesofthemodelstillexistwithperiodicdrugefcacies,although,atleastforsmallamplitudeperiodicperturbations,theviralsteadystatebecomesaperiodicsolution.Thestandardmodelwithtreatment,Equation 2-1 ,doesnotseemtocapturesomerelevantdynamicswhichcontributetolowlevelviralpersistence.AnexcellentreviewofmodelingHIVpersistenceisprovidedbyRongandPerelson[ 37 ].WebrieydiscussanddisplaysimulationsfortwoslightlymodiedversionsofEquation 2-1 thatpartiallyattempttoaddresstheaforementionedcriticism.First,weconsideramodelintroducedbyHolteetal,whichallowsfortheper-capitadeathrateoftheinfectedcellstobedensity-dependent[ 21 ]:. T=f(T))]TJ /F3 11.955 Tf 11.95 0 Td[(kVT,. T=kVT)]TJ /F5 11.955 Tf 11.96 0 Td[((T)!T, (3-2). V=pT)]TJ /F5 11.955 Tf 11.96 0 Td[(V.Theconstantper-capitadecayrate,,fromthestandardvirusmodelisreplacedbydensitydependentdecayratefollowingthepowerlawfunction:(T)!.Themotivationforthisformisthefollowing:theper-capitadeathrateoftheinfectedcellsdependsupontheconcentrationofimmuneeffectors,E,andEisafunctionoftheconcentrationofinfectedcells,T.Thepreciserelationshipsbetweenthesequantitiesisunknown. 85

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A BFigure3-2. Thenaturallogoftheviralsteadystateisshownasafunctionof(constant)drugefcacyasacriticalefcacyisapproached.A)Theviralload(atsteadystate)isextremelysensitivetoefcacynearthecriticalefcacyinthestandardmodel.B)Theviralsteadystateforthemodelwithdensitydependentinfectedcelldeathratedoesnotshowthesamesensitivitywithrespecttoefcacy.Hence,thedensitydependentmodel,Equation 3-2 ,canrobustlygeneratealowlevelviralload. Thepowerlawfunctionrepresentsasimplewaytoincorporatetheseinteractionsintothedeathrate.Theparameterprepresentstheviralproductionrateofaninfectedcell,asimmuneeffectorscanlyseinfectedcellsbeforeviralproductioniscomplete,henceN(T)!wouldbeaninaccurateexpressionfortheviralproductionrate.ThemajoradvantageofEquation 3-2 isthatalow-levelviralsteadystatecanberobustlygenerated.Thesensitivityoftheviralsteadystatewithrespecttodrugefcacyinasingle-drugtreatmentforEquation 3-2 isdisplayedinFigure 3-2B .Inaddition,Holteetal.haveshownthatthemodel,Equation 3-2 ,performsbetterthanthestandardmodelinttingviraldeclinedatafrompatientsinadrugperturbationexperiment[ 21 ].Figure 3-3 showssimulationsofthisdensitydependentmodelwithbothin-phaseandout-of-phaseforcingwithsinusoidaldrugefcacyfunctions.Itisseenthattheout-of-phasetreatmentscontrolthevirustolevelsjustbelow50copies/ml.Thein-phasetreatmentsinitiallydrivethevirustolowlevels,butthenthevirusreboundstolevelsinexcessof10,000copies/ml.Atthesevirallevels,thein-phasetreatmentswouldcertainlybeafailure,whereastheout-of-phasetreatmentswouldbesuccessful 86

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insuppressingthevirustolevelsbelowdetectionbystandardassays.Thecontrastinresultswhenvaryingthephasedifference(whilekeepingtheaveragedrugstrengthsconstant)isconsistentwithouranalysisoftheperiodicallyforcedstandardmodel.Arathernaivewayofkeepingtheviruslevelawayfromzero,istosimplyperturbmodelEquation 2-1 byaconstant.Explicitly,wejustaddaconstant,A>0,tothe. Tequation.ThemotivationbehindthiscanbeaveryslowreplicatingdrugsanctuarywhichleaksinfectedcellsatarateAday)]TJ /F4 7.97 Tf 6.59 0 Td[(1andalsoproducesinfectedcellsatthatsameconstantrate.Hencethedrugsanctuaryremainsxedinsize.ForsufcientlysmallvaluesofAthequalitativebehaviorofthefamilyofsolutionswillnotchange,however,V(t)willalwaysbestronglypersistentnomatterwhatthedrugefcacy,i.e.9>0(dependentonA)suchthatV(t)>8t>0.AsimulationofthismodelwithourperiodicdrugefcaciesisshowninFigure 3-3 .Theseresultsillustratetheimportanceofunderstandingthepharmacokineticsofthedrugsinacombinationtherapy.Drugsandtheirdosingregimensareoftendesignedtolimitthevariabilityofdrugefcacyoverthecourseoftreatment.Interestingly,wehaveshownthatwheninsertingperiodicdrugefcaciesintoadynamicalmodelofthevirus,thephasedifferencebetweentheRT-inhibitorandP-inhibitorefcacyfunctionscangreatlyinuencethetreatmentoutcome.Thiscanbeviewedasanafrmationofthegoalofminimizingvariabilityindrugefcacyovertime.Alternatively,especiallygiventoxicityofmedications,time-varyingefcacyfunctionscanbeviewedassomethingtoexploitifthephasedifferencebetweentheRT-andP-inhibitorscanbecontrolled.Therearemanyfactorswhichaffecthowthedrugefcacyfunctions,RT(t)andP(t),mightlookinrealityandhowmuchdrugdesigncaneffectivelyshapethesedrugefcacyfunctions.Thecomplexitiesofthisissueconstituteasignicantportionofpharmaceuticalresearch.Inaway,ourresultsaddanextrafactorofcomplexityintotheequation. 87

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A B C D E FFigure3-3. Simulationsofin-phaseandout-of-phasetreatmentsinlowlevelviralpersistencemodels.A,B,C,andDaresimulationsbasedonthesystemwithdensitydependentdecayrateforinfectedcells,Equation 3-2 .EandFaresimulationsbasedonaperturbedstandardmodel:Equation 2-1 with. T=k(1)]TJ /F5 11.955 Tf 11.95 0 Td[(RT(t))VT)]TJ /F5 11.955 Tf 11.96 0 Td[(T+1.Thedrugefcacyfunctionsusedare:RT(t)=0.79)]TJ /F6 11.955 Tf 11.96 0 Td[(0.2sin(2t)andP(t)=0.80)]TJ /F6 11.955 Tf 11.95 0 Td[(0.2sin(2(t+ )).A)Viralloadforin-phasetreatments.B)Asymptoticperiodicsolutionforin-phasetreatments.C)Viralloadforout-of-phasetreatments.D)Asymptoticperiodicsolutionforout-of-phasetreatments.E)Viralloadforin-phasetreatments.F)Viralloadforout-of-phasetreatments. 88

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Aswevarythephasedifferenceofthedrugefcacyfunctionsinthesemodels,awiderangeofviralreboundlevelsarerealized.Inclinicalstudiesofviralreboundduringcombinationtherapy,patientshavedisplayedvariableresults.Forexample,inastudyofvirallevelinpatients1yearaftertheinitiationoftreatment,71%achievedvirallevelsbelow500copies/ml,10%hadvirallevelsat500-5000copies/ml,and19%hadaviralreboundabove5000copies/ml[ 1 ].Thereasonfordifferingviralreboundsinpatientsisnotalwaysknown.Aspatientscertainlymighthavedifferentdosingschedules,levelsofadherence,anddifferentpharmacokinetic/pharmacodynamicparameters,ourresultsofferapossibleexplanationforsomeofthevarianceinresults. 3.2Bang-BangEfcaciesWeconsiderdrugefcacyfunctionsofthebang-bangtypeandmoregenerallypiecewiseconstantfunctionsasin[ 10 ].First,wedeneRT(t),P(t):R![0,1]asperiodicfunctionswithperiod=1suchthat:RT(t)=8>><>>:eRTift2[0,1 2),0ift2[1 2,1).andP(t)=8>><>>:ePift2[0,1 2),0ift2[1 2,1).Wewillconsidervariablephaseshifts, 2[0,1),ofP(t).Hence,ifthephaseshift 2[0,1 2],thenon[0,1):P(t)]TJ /F5 11.955 Tf 11.95 0 Td[( )=8>><>>:ePift2[ ,1 2+ ),0ift2[0, )[[1 2+ ,1].If 2(1 2,1),thenon[0,1):P(t)]TJ /F5 11.955 Tf 11.96 0 Td[( )=8>><>>:ePift2[ ,1)[[0, )]TJ /F4 7.97 Tf 13.15 4.7 Td[(1 2),0ift2[ )]TJ /F4 7.97 Tf 13.15 4.7 Td[(1 2, ).HereeRTandeParexedin[0,1].Hence,theefcacyoftheRT-inhibitorandP-inhibitor 89

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A B CFigure3-4. A)Anexampleofthebang-bangfunctionswithaphasedifference .B)Weassumethedrugefcacyfunctionstobeofthebang-bangtypedepictedwitheRT=eP=e2[0.3,1].Thebluelinegraphs(0)asafunctionofefcacye.Theredlinedepicts(0.5)asafunctionofefcacye.Weseethein-phasetreatmentsdonotcleartheinfection,whiletheoutofphasetreatmentscancleartheinfection.C)( )isplottedasafunctionofthephasedifference, ,intwocases.Theredcurve(thecurvewhichiscloserto0)representsthecasewheneRT=eP=0.85andthebluecurverepresentsthecasewheneRT=0.9,eP=0.5oreRT=0.5,eP=0.9(bothgivethesamegraph). areeRTandeP(respectively)for12hoursinadayand0fortheother12hours.ThegraphofthisfunctionisshowninFigure 3-4A .Thisbang-bangcontroliscertainlynotperfectformodelingtherealdrugefcacyfunctions,butitallowsustohaveanexplicitformula(intermsofmatrixexponentials)forthefundamentalsolutiontothelinearizedsubsystem,Equation 2-3 [ 10 ].Also,ifwelengthentheperiod,wecanobtainapproximationsfordrugefcaciesunderanSTI(StructuredTreatmentInterruption),whichwillbediscussedinthenextsection.ThePFStothelinearizedsubsystem(Equation 2-3 )withbang-bangdrugefcaciesis:(eRT,eP, )=exp[(0.5)]TJ /F5 11.955 Tf 11.96 0 Td[( )E(0,0)]exp[ E(0,eP)]exp[(0.5)]TJ /F5 11.955 Tf 11.95 0 Td[( )E(eRT,eP)]exp[ E(eRT,0)] 90

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A BFigure3-5. A)thedrugefcacyfunctionsRT(t)andP(t)]TJ /F5 11.955 Tf 11.96 0 Td[( )areassumedtohaveatheshapeabove,whichisacloserapproximationtorealdrugefcacies.B)ThedominantFloquetmultiplier,( ),isshownasafunctionofphasedifference, ,whenRT(t)andP(t)]TJ /F5 11.955 Tf 11.96 0 Td[( )areasintheleftpanel(A). when 2[0,1 2],and(eRT,eP, )=exp[( )]TJ /F6 11.955 Tf 11.96 0 Td[(0.5)E(0,eP)]exp[ E(0,0)]exp[( )]TJ /F6 11.955 Tf 11.96 0 Td[(0.5)E(eRT,0)]exp[ E(eRT,eP)]when 2(1 2,1);whereE(eRT,eP)=0B@)]TJ /F5 11.955 Tf 9.3 0 Td[(k(1)]TJ /F3 11.955 Tf 11.96 0 Td[(e1)T0N(1)]TJ /F3 11.955 Tf 11.95 0 Td[(e2))]TJ /F5 11.955 Tf 9.3 0 Td[(1CA.NumericalcalculationsofthedominantFloquetmultiplier( )areshowninFigure 3-4 .AbetterapproximationofanactualdrugefcacyfunctionbyapiecewiseconstantfunctionisconsideredinFigure 3-5 .WeusetheparametersasinRongetal.[ 36 ],andtheyareasfollows:f(T)=a)]TJ /F3 11.955 Tf 12.58 0 Td[(bTwitha=104ml)]TJ /F4 7.97 Tf 6.59 0 Td[(1day)]TJ /F4 7.97 Tf 6.58 0 Td[(1andb=0.01day)]TJ /F4 7.97 Tf 6.58 0 Td[(1(T0=106ml)]TJ /F4 7.97 Tf 6.59 0 Td[(1),k=2.410)]TJ /F4 7.97 Tf 6.59 0 Td[(8ml)]TJ /F4 7.97 Tf 6.59 0 Td[(1day)]TJ /F4 7.97 Tf 6.58 0 Td[(1,=1day)]TJ /F4 7.97 Tf 6.59 0 Td[(1,N=3000,=23day)]TJ /F4 7.97 Tf 6.59 0 Td[(1.Thedifferenceinparameterchoicefromthepriorsimulationsdoesnotchangequalitativeresults,butgivesalowerreproductionnumber,R0=3.1304. 91

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3.3StructuredTreatmentInterruptionsThebang-bangdrugefcacyfunctionsareareasonablewaytomodelthedrugefcacyfunctionsforSTIs.DuringanSTIregimen,apatientwilltakeadrugforacertainperiodoftime(theon-days)andthentakeabreakfromthatmedicationforanotherperiodoftime(theoff-days).Thiscyclemayrepeat.Thusonecanconsiderthedrugefciencyfunctionasbeingnearlyconstant,eRT(oreP),fortheon-daysandroughly0fortheoff-days.TheperiodanddurationofthedrugefcacyfunctionforanSTIdependsonthenatureoftheprogram.DurationhereismeantasanopenintervalI2Rsuchthatthedrugefcacyfunction(t)supportedinI,isperiodicwithperiod.Ingeneral,anSTImighthaveadrugefcacyfunction1(t)withperiod1onanintervaloftime(t1,t2),thenchangethedrugefcacyfunctionto2(t)withperiod2on(t2,t3)andsoon.AdynamicSTIwouldchoose2(t)basedoninformationreectingtheinfectionstatusatt2.Weconsiderthebang-bangfunctionsdescribedabovewiththeperiodincreasedto60days.ThentheFloquetmultipliersaretheeigenvaluesof(eRT,eP, )=exp[(30)]TJ /F5 11.955 Tf 11.96 0 Td[( )E(0,0)]exp[ E(0,eP)]exp[(30)]TJ /F5 11.955 Tf 11.95 0 Td[( )E(eRT,eP)]exp[ E(eRT,0)],when 2[0,30],and(eRT,eP, )=exp[(60)]TJ /F5 11.955 Tf 11.95 0 Td[( )E(0,eP)]exp[( )]TJ /F6 11.955 Tf 11.95 0 Td[(30)E(0,0)]exp[(60)]TJ /F5 11.955 Tf 11.96 0 Td[( )E(eRT,0)]exp[( )]TJ /F6 11.955 Tf 11.95 0 Td[(30)E(eRT,eP)],when 2(30,60),whereE(eRT,eP)isdenedasbefore.TheseareexamplesofstaticSTIswherethedrugefcacyfunctions,RT(t)andP(t)]TJ /F5 11.955 Tf 13.01 0 Td[( ),havetheon-daysinaconsecutiveblockofdays.Theperiodis=60daysandtheon-daysfortheRT-inhibitorare30consecutivedays(theoff-daysare 92

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ASTIdrugefcacyfunction BFloquetMultipliervs.phasedifferenceFigure3-6. A)AnexampleshowingtheSTIdrugefcacyfunctions.B)ThenaturallogarithmofthedominantFloquetmultiplier,log( ),isshownasafunctionofphasedifference, ,whenRT(t)andP(t)]TJ /F5 11.955 Tf 11.96 0 Td[( )areofthetypedepictedintheleftpanel(A).Inthesecalculations,eRT=eP=0.85.Noticethelargerangeofvaluesweobtainfor( ).Increasingthedrugperiodscausesanincreaseintheamplitudeof( ). similarlya30dayblock),likewisefortheP-inhibitor.Thegraphoflog( )forefcacieseRT=eP=0.85isshowninFigure 3-6 .Certainlythereisamuchlargersetofpossibleregimenswhichincludetreatmentschedulesthathavemultipleblocksofon-daysandoff-dayswithinaperiod.NumericaloptimalcontrolmethodshavebeenusedtoaddressoptimizationoveralargersetofpossibleSTIs[ 2 ].Ifwedochoosearegimenwhere,withinaperiod,theon-daysareinaconsecutiveblockandoff-daysareinaconsecutiveblock,ourresultsshowthattheon-daysoftheRT-inhibitorshouldcoincidewiththeoff-daysoftheP-inhibitorandvice-versa(theoptimalphasedifferenceinthisexampleisapproximately30days,asshowninFigure 3-6 ).TheseresultsonSTIsshouldbeviewedwithahealthydoseofcautionasdrugresistanceisnotfactoredinandmostlikelywouldbeasignicantproblem. 93

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3.4DiscussionWebrieysummarizetheresultsfromtheprevioustwochaptersanddiscussimplications.InChapter 2 ,weanalyzedthedynamicalconsequencesofincorporatingperiodiccombinationdrugtreatmentinthestandardvirusmodel,Equation 2-1 .Contrarytotheautonomousmodel,Equation 1-3 ,areproductionnumbercannotbeexpressedasasimplefunctionofsystemparameters.However,thedominantFloquetmultiplierofthelinearizedsystem(Equation 2-11 ),,doesgiveathresholdconditionforlocalstabilityofE0andalsoisprovedtoprovidemoreglobalresults.d'Onofrioprovedglobalextinctionwhen<1[ 13 ]andweproveduniformpersistencewhen>1.GlobalconvergencetoaninfectionequilibriumoccurswhenR0>1andf(T)satisesthesectorconditioninthestandardmodel(Equation 1-3 ).Weprovedananalogousresultfortheperiodicsystem(Equation 2-1 )forsufcientlysmallamplitudeefcacyfunctions.NamelywhenfR0(eRT,ep)>1(themodiedreproductionnumberusingaveragedrugefcacy),globalconvergencetoa-periodicsolutionensues,whereistheperiodoftreatment.Akeyassumptionhereissmallamplitudeefcacyfunctions,astheresultmaynotbetrueforlargeramplitudeperiodicperturbations.Despitethemoderatesuccessindescribingglobaldynamicsbasedupon,itisnotpossibletondaformulaforintermsoftheparametersforgeneraldrugefcacyfunctions.Hence,weexpandedasaTaylorseriesin,theamplitudeofperiodicperturbations,inthecaseofcriticalcombinationdrugefcacies.Thisenabledustocalculatesecond-orderapproximationsforin-phaseandout-of-phasesinusoidalperturbations.Basedupontheapproximation,(0)(in-phaseFloquetmultiplier)isgreaterthan()(out-of-phaseFloquetmultiplier)foranyparameters.Thisanalysisisimportantasparametersareneverknownforcertainandvaryacrossdifferentviruses.Hence,theanalyticalmethodsprovidemoregeneralresultsthannumericalcomputations. 94

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InRongetal.[ 36 ],itisarguedthataverageddrugefcaciesprovidesagoodindicatoroftreatmentoutcomeforEquation 2-1 .Ouranalyticalandnumericalresultsshowthatthisisnotnecessarilythecasewithcombinationtherapy.Forsinusoidaldrugefcacies,weshowedthatout-of-phasetreatmentsoutperformedin-phasetreatmentsdespitethefactthatthetimeaveragedefcacyremainsconstant.Calculationoftheaveragereproductionnumber, R0(Equation 2-46 ),doespredictthisdifferencebetweenin-phaseandout-of-phasetreatmentsandalsoprovidesanintuitiveexplanationfortheseresults.Still, R0doesnotdeterminetreatmentoutcomeasitcanbecriticallydifferentfromthedominantFloquetmultiplier,,forcertainparameters.Addingheterogeneitytothereplicationcyclesuchastimedelaysandnon-constantviralproductionrate,bothofwhichcanoccurinreality,mayaffecttheresultsaboutoptimizingperiodiccombinationtherapy.InChapter 4 ,wewillanalyzeanautonomousage-structuredmodelwhichallowsfortheseheterogeneitiesinthereplicationcycleasinfectedcelldeathrateandviralproductionareallowedtovarywithagesinceinfectionofaninfectedcell.ForHIV,therearecertainlyconfoundingfactorsinadditiontotheinherentdelaysintheinfectedcellcycle.First,HIVisanotoriouslyfastevolvingvirusandmutationtodrugresistantstrainscancausetreatmentfailure.Drugresistantstrainscanbeincorporatedintothemodel,andthisremainsasfuturework.UseofmultipleRT-inhibitorsandmultipleP-inhibitorsmaybeawayofavoidingdrugresistance,butcarefulconsiderationofcross-resistancepatternswouldbenecessary.Anotherimportantfactortoconsideristhemechanismoflowlevelviralpersistenceinpatientsundergoingtreatment.Inthischapterwehaveconsideredtwomodicationsofthestandardmodelwhichcangeneratelowlevelviralpersistence.Inbothofthesesystems,thephasedifferencebetweendrugefcacyfunctionscanbeacrucialparameterindeterminingviralload.Therealityofviralpersistenceintreatedpatientscombinedwiththetoxicityofantiviralmedicationscreatestheneedforoptimaltherapies. 95

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Potentially,eitherdrugsorstructuredregimenscanbedesignedtoreducetoxicityinducedinpatients,whilecontrollingviralload.Optimizationofdrugtherapiesisanimportanttopicformathematicalbiology.Theincreasedcomputingcapabilitiesoftodayallowresearcherstosimulatemodelsinvolvingalargenumberofinteractingvariables.Yet,theanalysisofsimplemathematicalmodelscanprovideinsightsandgeneralitythatmaybelostinmorecomplexmodels.Despiteitsaws,thestandardvirusmodeloffersafundamentaldescriptionofvirus-hostdynamics.Themodelprovidesthesimplestsystemforassessingiftheinherentperiodicityindrugtreatmentsinacombinationtherapycanaffectdynamics.Thesimplicityenabledustoanalyticallyshowthatthephasedifferencebetweenthedrugefcacyfunctions, ,canbeacriticalparameterfordeterminingtreatmentoutcome.Moreworkisneededtoseeifthistheoreticalinsightcanhavepracticalapplications. 96

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CHAPTER4AGE-STRUCTUREDWITHIN-HOSTVIRUSMODEL 4.1IntroductionDuringHIVinfection,thereisarelativelylargetimeperiodbetweenviralentryintoanactiveTcellandsubsequentproductionofvirions.Arecentanalysisofinvivomulti-drugperturbationexperimentsshowedthatreversetranscriptiontakesonaverage24hourstocomplete,whichisabout2/3oftheaveragelife-spanofaninfectedcell[ 29 ].Thereisalsoevidencethatthedeathrateofaninfectedcellvarieswithdifferentstagesofthereplicationcycle[ 38 ].Inaddition,therateofviralproductionmayvaryafterthetranscribedviralDNAhasbeenintegratedintothehostcellDNA[ 30 ].Thestandardvirusmodel(Equation 1-3 )assumessimultaneousinfectionoftargetcellsandviralproduction,andhenceignorestheinfectedcelllifecycle.Toaccountforthetimelagbetweenviralentryofatargetcellandsubsequentviralproductionfromthenewlyinfectedcell,Perelsonetal.includeddiscreteanddistributeddelaysinthestandardmodel[ 31 ].Nelsonetal.consideredamodelwithagestructureintheinfectedcellcomponent,whichgeneralizesthedelaystandardvirusmodelbyallowingforinfectedcelldeathrateandviralproductiontovarywithagesinceinfectionofaninfectedcell[ 30 ].Wewillprovideaglobalanalysisofthismodel.Themodelcanbewrittenasfollows:dT(t) dt=f(T(t)))]TJ /F3 11.955 Tf 11.96 0 Td[(kV(t)T(t),dV(t) dt=Z10p(a)T(t,a)da)]TJ /F5 11.955 Tf 11.95 0 Td[(V(t),@T(t,a) @t+@T(t,a) @a=)]TJ /F5 11.955 Tf 9.3 0 Td[((a)T(t,a), (4-1)T(t,0)=kV(t)T(t)whereT(t,a)denotesthedensityofinfectedcellconcentrationwithrespecttoagesinceinfection.Thefunctions(a)andp(a)aretheage-dependent(per-capita)ratesof 97

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infectedcelldeathandvirionproduction,respectively.IntheSection 4.2 wewillderivethePDEinEquation 4-1 andintroduceamoregeneralformulation.Thismodelhasbeenofrecentinterestintheliterature[ 4 16 22 35 ].Gilchristetal.investigatedviralproductionstrategieswhichoptimizeviraltnessindifferentenvironments[ 4 ].Althausetal.estimatedHIVparametersintheage-structuredmodelusingviraldeclinedatafrompatientsundergoingdrugtherapy[ 16 ].Rongetal.incorporateddrugRT-inhibitorsintotheage-structuredmodelandobtainedlocalstabilityresults[ 35 ].Ingeneral,theadditionoftimedelaystoadifferentialequationsystemcaninduceinstabilitywhichwasabsentfromtheoriginalmodel.Asimpleexampleisthelogisticgrowthmodelfrompopulationdynamics.ThegloballystablepositivesteadystatecanundergoaHopfbifurcationleadingtoinstabilitywhenatimedelayisinsertedintotheequation.Inthestandardvirusmodel,whenR0>1andf(T)satisesthesectorcondition,DeLeenheerandPilyuginprovedthatthepositivesteadystateisgloballyasymptoticallystable[ 11 ].Ourmainresultinthischapterisprovingthatthisglobalstabilityispreservedwhenagesinceinfectionstructureisincorporatedintothestandardmodel.Theanalysisrequiredforthisproofiscomplicatedbythefactthattheunderlyingstatespaceforanage-structuredmodelisinnitedimensional.Variousapproacheshavebeendevelopedforanalyzingagestructuredmodels.Thegeneralideaistostudythenonlinearsemigroupgeneratedbythefamilyofsolutions.Oneapproachistousethetheoryofintegratedsemigroups[ 26 42 ].Weemployanothermethod,namelyintegratingsolutionsalongthecharacteristicstoobtainanequivalentintegralequation.ThisapproachwasdevelopedbyWebbforagedependentmodels[ 44 ],butthesettingthereisslightlydifferentfromourhybridODE-PDEsystem.Hence,weusefundamentalprinciples,resultsfromHale[ 17 ]onasymptoticsmoothness,andacompactnessconditionforLpspacestorigorouslyprove 98

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existence,uniquenessandeventualcompactnessofthenonlinearsemigroupassociatedwiththesolutionstoEquation 4-1 .Inordertostudytheglobalbehaviorwhenthebasicreproductionnumber,R0,exceeds1,wefollowasimilarapproachtoMagaletal.[ 26 ].WeuseresultsfromHaleandWaltman[ 19 ]andMagalandZhao[ 27 ]toproveuniformpersistenceandexistenceofaglobalattractor,A.WesubsequentlydeneaLyapunovfunctionalonAwhichreliesontheuniformpersistenceinordertobewelldened.Then,weproveconvergenceofbackwardorbitsonAto xviatheLyapunovfunction,andsubsequentlyusethisresult,incombinationwiththesameLyapunovfunction,toprovelocalstabilityoftheuniquepositiveequilibrium, x.ThiswillthenimplyA=f xgandglobalstabilityfollows.IncontrasttoMagaletal.[ 26 ],ouranalysisdoesnotuseanylinearizationarguments.Hence,wehavenoneedtoinvestigatespectralpropertiesofthegeneratorofanabstractCauchydifferentialequationwhichtheintegratedsemigroupformulationwouldprovide.Thisallowsamoredirectapproachtotheglobalanalysis.Wewanttobrieydiscussthearticlepublishedveryrecentlycontainingaproofoftheglobalstabilityoftheinfection-equilibriuminaparticularcaseofEquation 4-1 (whenf(T)=s)]TJ /F5 11.955 Tf 12.3 0 Td[(T)[ 22 ].Asthisarticlewaspublishedin2012,wewereunawareoftheseresultsbyHuangetal.untilafterthemathematicalworkcontainedinthispaperwascompleted.Despitethefactthattheresultsareverysimilar,wehaveafewconcernswiththepublicationofHuangetal.Thereisalackofclarityuponthestatespaceoftheinfectedcellcomponent,T(t,a),intheirarticle.TheauthorsclaimtobeconsideringL1(0,1)astheunderlyingspace.Theyreasonthatlima!1T(t,a)=0foranytaimpliesthatthereisanitemaximumage,butthisseemstobeafalseimplication.Infact,ifT(t,a)>0forsomea,t0,thenT(t+a)]TJ /F3 11.955 Tf 12.53 0 Td[(a,a)>08a2[0,1).TheproblemisthattheyuseaLyapunovfunctionwhichisundenedontheentirestatespace.Inparticular,oneofthetermsisundenedwhenever9a2[0,1)andt0suchthatT(t,a)=0.OnewaytoxthisproblemistoconsiderL1(0, a)astheunderlying 99

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spaceforsomexed a<1,i.e.imposeanitemaximumage.Thentherewillexist>0suchthatT(t,)isstrictlypositivefort.Hence,theirLyapunovfunctionwillbedenedforallsufcientlylargetime.Anotherissueisthecompactnessofthesemiow.Iftheunderlyingspaceisinnitedimensional,boundedsetsarenotnecessarilyprecompact.BecausetheuseofaLyapunov-LaSalleargumenttoinferglobalstabilityrequiresprecompactnessoforbits,appropriateconsiderationsshouldbetakenwhenthedynamicstakesplaceinaninnite-dimensionalspace.WithanitemaximumageinEquation 4-1 ,i.e.T(t,)2L1(0, a),onecanusethestatespaceC([0, a])forT(t,)forallt> a.Here,C([a,b])denotestheBanachspaceofreal-valuedcontinuousfunctionsontheclosedinterval[a,b]R.Thisspaceisoftenusedastheunderlyingstatespacefordelaydifferentialequationswithnitedelay.Inthiscase,undertheassumptionthatthevectoreldiscompletelycontinuous,theAscolitheoremwillimplyprecompactnessofboundedorbits,henceaLyapunov-Lasalletypeargumentcanbereadilyinvoked[ 18 ].However,ifthereisnonitemaximumage,thentheunderlyingstatespacewillbeL1(0,1),whichrequiresitsowncompactnesscriteria.SinceweconsiderL1(0,1)asthestatespace,webelievethatourresultsarestrongerthanthecasewithnitemaximumage.Thischapterisorganizedasfollows:InSection 4.2 wederivetheage-structuredvirusmodel.InSection 4.3 weprovetheexistenceanduniquenessofsolutionsofanequivalentintegralformulationofEquation 4-1 usingthecontractionmappingtheoremandalsoproveboundednessofsolutions.InSections 4.4 4.5 4.6 4.7 ,and 4.8 ,westudytheglobaldynamicsofthesemiowassociatedwiththesolutionsofEquation 4-1 .ThemainresultsofthesesectionsaretheglobalextinctionwhenR0<1andtheglobalasymptoticstabilityoftheuniqueinfectionequilibriumwhenR0>1. 4.2ModelFormulationDenoteT(t,a)asthedensity(attimet)ofinfectedcellconcentrationwithrespecttoagesinceinfection,a,wherea2[0,1).Equivalently,theconcentrationattimetof 100

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infectedcellswhichwereinfectedbetweena1anda2timeunitsago,wherea2a10,isdenedtobeZa2a1T(t,a)da.Weassumetheinfectedcellsaresubjecttodeathattheage-dependentrate,(a).Assumethat(a)2L1:=L1(0,1),i.e.(a)ismeasurableandboundedalmosteverywhere(a.e.).Let>0.SupposethatT(t,)isaLebesgueintegrablefunctionon[0,1),i.e.T(t,)2L1:=L1(0,1),forallt2[0,].L1isanaturalstatespaceforT(t,),sincethetotalconcentrationofinfectedcells,R10T(t,a)da,shouldbenite.Furthermore,assumethatt7!T(t,)isacontinuousfunctionfrom[0,]toL1.WesaythatT2LwhereL:=C)]TJ /F6 11.955 Tf 5.48 -9.68 Td[([0,];L1)istheBanachspaceofcontinuousL1functionson[0,].Considerarbitrarya1,a2wherea2>a10.Fixt2[0,]andlet0suchthatt+.Theconcentrationofinfectedcellsofagebetweena1+anda2+attimet+isequaltotheconcentrationofinfectedcellswhichhaveagebetweena1anda2attimetandhavesurvivedthetimestepofunits.Considera2(a1,a2).Theconcentrationofinfectedcellswhichhasagebetweenaanda+daattimetandsubsequentlysurvivesunitsoftimeis:T(t,a)da)]TJ /F11 11.955 Tf 11.96 16.86 Td[(Z0(a+s)T(t+s,a+s)dsdaHence,thefollowingbalanceequationdescribestheagingprocessoftheinfectedcells:Za2a1T(t+,a+)da=Za2a1T(t,a))]TJ /F11 11.955 Tf 11.95 16.28 Td[(Z0(a+s)T(t+s,a+s)dsda (4-2)AssumeforthemomentthatT(t,a)isalsodifferentiable.DifferentiateEquation 4-2 withrespectto:@ @Za2a1T(t+,a+)da=)]TJ /F5 11.955 Tf 13.7 8.09 Td[(@ @Za2a1Z0(a+s)T(t+s,a+s)dsda 101

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WecanbringthederivativesinsidetheintegralsbecauseT2Land2L1:Za2a1@ @Tda=)]TJ /F11 11.955 Tf 11.29 16.27 Td[(Za2a1@ @Z0(a+s)T(t+s,a+s)dsda=Za2a1(a+)T(t+,a+)daEvaluateat=0toobtain:Za2a1@ @T(t+,a+)j=0da=)]TJ /F11 11.955 Tf 11.29 16.27 Td[(Za2a1(a)T(t,a)daZa2a1@ @tT(t,a)+@ @aT(t,a)da=Za2a1)]TJ /F5 11.955 Tf 9.3 0 Td[((a)T(t,a)daSincea1,a2arearbitrary,thefollowingequationholdswhena2(0,1):@T(t,a) @t+@T(t,a) @a=)]TJ /F5 11.955 Tf 9.3 0 Td[((a)T(t,a) (4-3)ThenewlyinfectedcellsattimetisequaltokV(t)T(t).Hence,theboundaryconditionisthefollowing:T(t,0)=kV(t)T(t)Supposethattherateofvirionproductionisagedependentandgivenbyanon-negative,boundedfunctionp(a).WesubsequentlyarriveatEquation 4-1 :dT(t) dt=f(T(t)))]TJ /F3 11.955 Tf 11.96 0 Td[(kV(t)T(t),dV(t) dt=Z10p(a)T(t,a)da)]TJ /F5 11.955 Tf 11.95 0 Td[(V(t),@T(t,a) @t+@T(t,a) @a=)]TJ /F5 11.955 Tf 9.3 0 Td[((a)T(t,a),T(t,0)=kV(t)T(t)ThePDEinEquation 4-1 isalineartransportequationwithdecay.Henceitcanbesolved,atleastformally,byintegratingalongthecharacteristicsandincorporatingtheboundaryandinitialconditions.Indeed,denez(s)=1 (a+s)T(t+s,a+s), 102

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where(a)=e)]TJ /F23 7.97 Tf 7.99 6.42 Td[(Ra0(s)ds. (4-4)d dsz(s)=1 (a+s)@ @tT(t+s,a+s)+@ @aT(t+s,a+s)+1 (a+s)(a+s)T(t+s,a+s)=0(byEquation 4-3 )Usingtheequalitiesz(0)=z()]TJ /F3 11.955 Tf 9.3 0 Td[(a)andz(0)=z()]TJ /F3 11.955 Tf 9.3 0 Td[(t),alongwiththeconstraintsthatbothargumentsofT(,)mustbenon-negative,wend:T(t,a)=8>>>>>><>>>>>>:(a)kV(t)]TJ /F3 11.955 Tf 11.95 0 Td[(a)T(t)]TJ /F3 11.955 Tf 11.96 0 Td[(a),ift>a(a) (a)]TJ /F10 7.97 Tf 6.59 0 Td[(t)T(0,a)]TJ /F3 11.955 Tf 11.96 0 Td[(t),ifta, (4-5)Observethattheproportionofinfectedcellsnewlyinfectedattimet(wheret0)whichsurvivetoagea,isprecisely(a).Hence,(a)istheprobabilityaninfectedcellwillsurvivetoagea.Thesolution(Equation 4-5 )alsosatisesthebalancelaw(Equation 4-2 )forallt2[0,],2[0,)]TJ /F3 11.955 Tf 12.71 0 Td[(t],anda2a10(assumingthesameboundaryconditionT(t,0)=kV(t)T(t)).Indeed,assumethat0a1a2
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Hence,T(t,a)=(a)kV(t)]TJ /F3 11.955 Tf 12.15 0 Td[(a)T(t)]TJ /F3 11.955 Tf 12.15 0 Td[(a)1ft>agsatisesthebalancelaw,Equation 4-2 .Thecaseinwhichta2canbesimilarlyveried.LetTdenotethemapt7!T(t,)whereT(t,)2L1isdenedbyEquation 4-5 .NotethatT2L.ByTheorem2.1inWebb[ 44 ],TistheuniqueLfunctionwhichsatisesthebalancelaw(Equation 4-2 )forallt2[0,],2[0,)]TJ /F3 11.955 Tf 12.54 0 Td[(t],anda2a10.Hence,thereisnoneedtoassumeT(t,a)isdifferentiableorevencontinuous.ItissufcienttoassumethatT2Landconsiderthefollowingsystem:dT(t) dt=f(T(t)))]TJ /F3 11.955 Tf 11.96 0 Td[(kV(t)T(t),dV(t) dt=Z10p(a)T(t,a)da)]TJ /F5 11.955 Tf 11.95 0 Td[(V(t), (4-6)T(t,a)=(a)kV(t)]TJ /F3 11.955 Tf 11.96 0 Td[(a)T(t)]TJ /F3 11.955 Tf 11.95 0 Td[(a)1ft>ag+(a) (a)]TJ /F3 11.955 Tf 11.96 0 Td[(t)T(0,a)]TJ /F3 11.955 Tf 11.95 0 Td[(t)1fa>tgT(t,0)=kV(t)T(t)T(0,a)2L1+(0,1),T(0)=T02R+,V(0)=V02R+.Here,(a)andp(a)areassumedtobeinL1+(bounded,non-negativea.e.andmeasurableon[0,1)).Weimposethefollowingrestrictionon(a):9b>0suchthat(a)ba.e.on[0,1).IfT(t,a)isinsertedintotheVequationinEquation 4-6 ,thenweobtainanODEcoupledwithaVolterraintegro-differentialequation.Equation 4-6 isequivalenttoEquation 4-1 whenT(t,a)isdifferentiable. 4.3ExistenceofSolutionsWenowprovelocalexistenceanduniquenessofsolutionstoEquation 4-6 andhencetoEquation 4-1 Theorem4.1. Letx0=((T(0),V(0),T(0,a))2R2+L1+.Thenthereexists>0andneighborhoodB0R2+L1+(0,1)withx02B0suchthatthereexistsauniquecontinuousfunction, :[0,]B0!R2L1+where (t,x)isthesolutiontoEquation 4-6 with (0,x)=x. 104

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Proof. AnysolutiontoEquation 4-6 mustsatisfythefollowingintegralequation:T(t)=T(0)+Zt0f(T(s)))]TJ /F3 11.955 Tf 11.95 0 Td[(kV(s)T(s)ds,V(t)=V(0)+Zt0Z10p(a)T(s,a)dads)]TJ /F5 11.955 Tf 11.95 0 Td[(Zt0V(s)ds, (4-7)T(t,a)=(a)kV(t)]TJ /F3 11.955 Tf 11.96 0 Td[(a)T(t)]TJ /F3 11.955 Tf 11.95 0 Td[(a)1ft>ag+(a) (a)]TJ /F3 11.955 Tf 11.96 0 Td[(t)T(0,a)]TJ /F3 11.955 Tf 11.96 0 Td[(t)1fa>tg.LetY:=C([0,]B0,R2L1)where>0andB0R2+L1+,aneighborhoodcontainingx0,aretobedetermined.LetBYcontainfunctionswhoserangeiscontainedintheclosedballinBR2L1(0,1)whereB= B((T(0),V(0),T(0,a)),r)forsomer>0.NowdenetheoperatoronBasfollows:Letx=(x1,x2,`(a))2B0andthevectorvaluedfunction=(1,2,3)2B.Dene()(t,x)=0BBBB@x1+Rt0[f(1(s,x)))]TJ /F3 11.955 Tf 11.95 0 Td[(k2(s,x)1(s,x)]dsx2+Rt0R10p(a)3(s,x)(a)da)]TJ /F5 11.955 Tf 11.95 0 Td[(2(s,x)ds(a)k1(t)]TJ /F3 11.955 Tf 11.96 0 Td[(a,x)2(t)]TJ /F3 11.955 Tf 11.95 0 Td[(a,x)1ft>ag+(a) (a)]TJ /F10 7.97 Tf 6.58 0 Td[(t)`(a)]TJ /F3 11.955 Tf 11.95 0 Td[(t)1fa>tg1CCCCA.Wehavethat()2Y,indeedsinceZ10(a)k1(t)]TJ /F3 11.955 Tf 11.96 0 Td[(a,x)2(t)]TJ /F3 11.955 Tf 11.95 0 Td[(a,x)1ft>ag+(a) (a)]TJ /F3 11.955 Tf 11.95 0 Td[(t)`(a)]TJ /F3 11.955 Tf 11.96 0 Td[(t)1fa>tgdaZt0(a)kj1(t)]TJ /F3 11.955 Tf 11.95 0 Td[(a,x)2(t)]TJ /F3 11.955 Tf 11.96 0 Td[(a,x)jda+Z1t(a) (a)]TJ /F3 11.955 Tf 11.96 0 Td[(t)`(a)]TJ /F3 11.955 Tf 11.95 0 Td[(t)dat1 b(1)]TJ /F3 11.955 Tf 11.95 0 Td[(e)]TJ /F10 7.97 Tf 6.58 0 Td[(bt)kjT(0)+rjjV(0)+rj+k`k<1Hereisanupperboundofp(a).TakeB0=B((T(0),V(0),T(0,a)),r=2).Thenk()(t,x))]TJ /F3 11.955 Tf 11.95 0 Td[(x0k=jx1)]TJ /F3 11.955 Tf 11.95 0 Td[(T(0)+Zt0f(1(s,x)))]TJ /F3 11.955 Tf 11.95 0 Td[(k2(s,x)1(s,x)dsj+jx2)]TJ /F3 11.955 Tf 11.95 0 Td[(V(0)+Zt0Z10p(a)3(s,x)(a)dads)]TJ /F5 11.955 Tf 11.96 0 Td[(Zt02(s)dsj+Z10(a)k1(t)]TJ /F3 11.955 Tf 11.95 0 Td[(a,x)2(t)]TJ /F3 11.955 Tf 11.95 0 Td[(a,x)1ft>ag+(a) (a)]TJ /F3 11.955 Tf 11.95 0 Td[(t)`(a)]TJ /F3 11.955 Tf 11.96 0 Td[(t)1fatg)]TJ /F3 11.955 Tf 11.95 0 Td[(T(0,a)dajx1)]TJ /F3 11.955 Tf 11.95 0 Td[(T(0)j+maxz2B(T(0),r)jf(z)j+kjT(0)+rjjV(0)+rj+jx2)]TJ /F3 11.955 Tf 11.96 0 Td[(V(0)j 105

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+(kT(0,a)k+r)+jV(0)+rj+1 b(1)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ /F10 7.97 Tf 6.59 0 Td[(b)kjT(0)+rjjV(0)+rj+Z10(a) (a)]TJ /F3 11.955 Tf 11.96 0 Td[(t)1fatgj`(a)]TJ /F3 11.955 Tf 11.96 0 Td[(t))]TJ /F3 11.955 Tf 11.95 0 Td[(T(0,a)]TJ /F3 11.955 Tf 11.95 0 Td[(t)jda+Z10(a) (a)]TJ /F3 11.955 Tf 11.95 0 Td[(t)1fatgT(0,a)]TJ /F3 11.955 Tf 11.96 0 Td[(t))]TJ /F3 11.955 Tf 11.96 0 Td[(T(0,a)da.NoticethatZ10(a) (a)]TJ /F3 11.955 Tf 11.96 0 Td[(t)1fatgj`(a)]TJ /F3 11.955 Tf 11.96 0 Td[(t))]TJ /F3 11.955 Tf 11.96 0 Td[(T(0,a)]TJ /F3 11.955 Tf 11.95 0 Td[(t)jdak`)]TJ /F3 11.955 Tf 11.96 0 Td[(T(0,)k.Also,Z10(a) (a)]TJ /F3 11.955 Tf 11.96 0 Td[(t)1fatgT(0,a)]TJ /F3 11.955 Tf 11.96 0 Td[(t))]TJ /F3 11.955 Tf 11.95 0 Td[(T(0,a)daZ101fatgT(0,a)]TJ /F3 11.955 Tf 11.95 0 Td[(t)(a) (a)]TJ /F3 11.955 Tf 11.96 0 Td[(t))]TJ /F6 11.955 Tf 11.95 0 Td[(1da (J1)+Z101fatgT(0,a)]TJ /F3 11.955 Tf 11.96 0 Td[(t))]TJ /F3 11.955 Tf 11.96 0 Td[(T(0,a)da. (J2)BytheDominatedConvergenceTheorem,( J1 )!0ast!0.Hence( J1 )
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0sufcientlysmall.whereM1,M2arepositiveconstants.Therefore:B!B.NotethatBisacompletemetricspace.WewillnowshowthatisacontractiononBforsufcientlysmall.Let,2B.Thenk()(t,x))]TJ /F6 11.955 Tf 11.95 0 Td[(()(t,x)kZt0jf(1(s,x)))]TJ /F3 11.955 Tf 11.95 0 Td[(f(1(s,x))jds+kZt0j2(s,x)1(s,x))]TJ /F5 11.955 Tf 11.96 0 Td[(2(s,x)1(s,x)jds+Zt0Z10p(a)j3(s,x)(a))]TJ /F5 11.955 Tf 11.96 0 Td[(3(s,x)(a)jdads+Zt0j2(s,x))]TJ /F5 11.955 Tf 11.95 0 Td[(2(s,x)jds+Zt0e)]TJ /F10 7.97 Tf 6.59 0 Td[(bakj2(t)]TJ /F3 11.955 Tf 11.95 0 Td[(a,x)1(t)]TJ /F3 11.955 Tf 11.96 0 Td[(a,x))]TJ /F5 11.955 Tf 11.95 0 Td[(2(t)]TJ /F3 11.955 Tf 11.96 0 Td[(a,x)1(t)]TJ /F3 11.955 Tf 11.96 0 Td[(a,x)jdamaxz2B(T(0),r)jf0(z)jk1)]TJ /F5 11.955 Tf 11.95 0 Td[(1k+k(T(0)+r)k2)]TJ /F5 11.955 Tf 11.95 0 Td[(2k+k(V(0)+r)k1)]TJ /F5 11.955 Tf 11.95 0 Td[(1k+k3)]TJ /F5 11.955 Tf 11.95 0 Td[(3k+k2)]TJ /F5 11.955 Tf 11.96 0 Td[(2k+k(T(0)+r)k2)]TJ /F5 11.955 Tf 11.95 0 Td[(2k+k(V(0)+r)k1)]TJ /F5 11.955 Tf 11.95 0 Td[(1kMk)]TJ /F5 11.955 Tf 11.95 0 Td[(k.whereM>0isaconstant.ThereforeisacontractionmappingonBforsufcientlysmall.BytheContractionMappingTheoremthereexistsauniquexedpointofinB,denotethisfunctionby .Then (t,x)solvestheinitialvalueproblemandiscontinuouson[0,]B0. Wenowprovethatthesystemiswell-posedinthesensethatsolutionsremainnon-negativeforalmosteveryaandareboundedinforwardtime.NotethatsolutionstoEquation 4-6 aresolutionstoEquation 4-1 iftheyhaveappropriatedifferentiabilityinthevariablea.Ifnot,solutionstoEquation 4-6 areweaksolutionstoEquation 4-1 Lemma6. SolutionstoEquation 4-6 remainnon-negativeforalmosteveryaandboundedinforwardtime. 107

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Proof. WenotebylookingattheintegralequationsthatT(t),V(t),andR10T(t,a)daaredifferentiableintbythefundamentaltheoremofcalculusforT(t),V(t)andforthecaseofR10T(t,a)da,thesmoothingpropertiesofconvolution.LetT(t),T(t,a),andV(t)beaparticularsolutiontoEquation 4-6 andhencethe(possiblyweak)solutiontoEquation 4-1 ontheinterval[0,].Firstweshowsolutionsremainnon-negativeon[0,].WendthatT(t)>08t2(0,]bythecontinuityofT(t)andtheassumptionthatf(0)>0andfissmooth.Supposebywayofcontradictionthattheresultdoesnothold.Nowdene=min)]TJ /F6 11.955 Tf 5.47 -9.68 Td[(infft0:V(t)<0g,inft0:T(t,)=2L1+.Suppose=inft0:T(t,)=2L1+(0,1).Then9(tn)suchthattn#andforalln,T(tn,)=2L1+.Also,T(0,)2L1+andEquation 4-5 implyfa2[0,1):T(tn,a)<0g=fa2[0,tn):(a)kV(tn)]TJ /F3 11.955 Tf 11.96 0 Td[(a)T(tn)]TJ /F3 11.955 Tf 11.96 0 Td[(a)<0g=fa2[0,tn):V(tn)]TJ /F3 11.955 Tf 11.95 0 Td[(a)<0g.Thereforeforalln,9t2[0,tn),suchthatV(t)<0.Sincetn#,wendthatinfft0:V(t)<0g.Hence,itsufcestoconsider=infft0:V(t)<0g.Bythesemigroupproperty,itsufcestoconsider=0.ThenV(0)=0.. V(t)=Z10p(a)T(t,a)da)]TJ /F5 11.955 Tf 11.95 0 Td[(V(t)=Zt0p(a)(a)kV(t)]TJ /F3 11.955 Tf 11.95 0 Td[(a)T(t)]TJ /F3 11.955 Tf 11.95 0 Td[(a)da+Z1tp(a)(a) (a)]TJ /F3 11.955 Tf 11.96 0 Td[(t)T(0,a)]TJ /F3 11.955 Tf 11.95 0 Td[(t)da)]TJ /F5 11.955 Tf 11.96 0 Td[(V(t)(. V(t)+V(t))et=Zt0e(t)]TJ /F10 7.97 Tf 6.59 0 Td[(a)p(t)]TJ /F3 11.955 Tf 11.96 0 Td[(a)(t)]TJ /F3 11.955 Tf 11.95 0 Td[(a)kT(a)eaV(a)da+etZ1tp(a)(a) (a)]TJ /F3 11.955 Tf 11.95 0 Td[(t)T(0,a)]TJ /F3 11.955 Tf 11.96 0 Td[(t)da. 108

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LeteV(t)=V(t)etandg(t,a)=e(t)]TJ /F10 7.97 Tf 6.58 0 Td[(a)p(t)]TJ /F3 11.955 Tf 12.2 0 Td[(a)(t)]TJ /F3 11.955 Tf 12.19 0 Td[(a)kT(a).Noteg(t,a)0.Thenwehavethefollowingdifferentialequation:d dteV(t)=Zt0g(t,a)eV(a)da+etZ1tp(a)(a) (a)]TJ /F3 11.955 Tf 11.96 0 Td[(t)T(0,a)]TJ /F3 11.955 Tf 11.95 0 Td[(t)da,eV(0)=0.Forthisequation,weclaimthatthereexistsanon-negativesolutiononasufcientlysmallintervaloftime.Indeed,leteY=C([0,],R+)andeBeYbethesetoffunctionswhoserangeiscontainedin[0,1],i.e.eB=C([0,],[0,1]).DeneeoneBase()(t)=Zt0Zs0g(s,a)(a)dads+Zt0etZ1sp(a)(a) (a)]TJ /F3 11.955 Tf 11.96 0 Td[(s)T(0,a)]TJ /F3 11.955 Tf 11.95 0 Td[(s)da,ds.For>0sufcientlysmall,wecanshow,similartotheproofofTheorem 4.1 ,thefollowing:e:eB!eBandeisacontractionmappingonthecompletemetricspaceeB.Uponapplicationofthecontractionmappingtheorem,existenceofanon-negativesolutionisobtained.Butthenthiseithercontradictstheassumptionthat=0ortheforwarduniquenessofsolutionstoEquation 4-6 .HencethesolutionstoEquation 4-6 mustremainnon-negativeon[0,].Fromassumptions,weknowthat(a)b>08a2[0,1)and0p(a).FromtheproofofLemma 1 ,noticethatforwardcompletenessandultimateboundedness(assumingforwardcompleteness)canbeprovedwiththesameargument.Hence,wesupposethatthesolutionsareforwardcomplete,i.e.existonthetimeinterval[0,1),andshowultimateboundedness.ThereexistsA>0andB>0suchthatf(T)A)]TJ /F3 11.955 Tf 9.33 0 Td[(BTbythesameargumentintheproofofLemma 1 .ConsiderT+R10Tda+b 2V.Thend dt(T+Z10Tda+b 2V)=f(T))]TJ /F11 11.955 Tf 11.96 16.27 Td[(Z10(a)Tda+b 2(Z10p(a)T(t,a)da)]TJ /F5 11.955 Tf 11.96 0 Td[(V)A)]TJ /F3 11.955 Tf 11.95 0 Td[(BT)]TJ /F3 11.955 Tf 11.95 0 Td[(bZ10Tda+b 2Z10Tda)]TJ /F3 11.955 Tf 16.42 8.08 Td[(b 2V=A)]TJ /F3 11.955 Tf 11.95 0 Td[(BT)]TJ /F3 11.955 Tf 13.15 8.09 Td[(b 2Z10Tda)]TJ /F3 11.955 Tf 16.43 8.09 Td[(b 2VA)]TJ /F5 11.955 Tf 11.95 0 Td[((T+Z10Tda+b 2V) 109

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where=min(B,b 2,).Thisimpliesthatlimsupt!1(T+R10Tda+b 2V)A .Boundednessfollowsfromnonnegativenessofsolutions. NowdeneXasX=R2+L1+.NotethatXisaclosedsubsetofaBanachSpace,andhenceisacompletemetricspace.Letx=(x1,x2,`(a))2X.Let (t,x)bethesolutiontoEquation 4-6 asinTheorem 4.1 ,whichweknowfromLemma 6 isboundedandforwardcomplete.Fort0denetheow,S(t):X!XasS(t)x= (t,x).WeclaimthefamilyoffunctionsfS(t)gt0isaC0semigrouponX.ClearlyS(0)x=x.Nowtoshowthesemigroupproperty,bystandardarguments8t0,s0 (t, (s,x))= (t+s,x).(Dene(t)= (t+s,x),then(t)isasolutiontoEquation 4-6 withinitialcondition (s,x),andtheninvokeforwarduniquenessfromTheorem 4.1 ).AlsothecontinuityfollowsfromTheorem 4.1 .Inthefollowing,wewilloftenletS(t)x=(T(t),V(t),T(t,a))withx=(T(0),V(0),T(0,a)). 4.4ReproductionNumberThesteadystatesolutionstoEquation 4-6 areconstantthroughtime.NoticethatlettingT0,V=0,andT=T0producestheinfection-freeequilibriumx0:=(T0,0,0).Werefertothisequilibriumasx0insteadofE0todifferentiatefromthepreviousmodels.Supposethat( T, V, T(a))isanequilibriumwith V>0.Then T(a)=k V T(a), T(a)=(a) (a)]TJ /F3 11.955 Tf 11.95 0 Td[(t) T(a)]TJ /F3 11.955 Tf 11.95 0 Td[(t). T(a):=k V T(a)istheuniquesolutiontobothequations.Thed dtV=0equationbecomes:0=Z10p(a)k V T(a)da)]TJ /F5 11.955 Tf 11.96 0 Td[( VHence, T= kR10p(a)(a)da 110

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and V=f( T) k TRecallthatf(T)>0ifandonlyifT0ifandonlyif T0ifandonlyifR0>1,wherethereproductionnumber,R0,isdenedas:R0=kT0R10p(a)(a)da (4-8)Wedenetheinfectionequilibrium, x:=( T, V, T(a)),whichexistsifandonlyifR0>1: T=T0 R0, V=f( T) k T, T(a)=k V T(a), (4.4.2)NoticethatR0,Equation 4-8 ,canbeinterpretedinthesamewayasthereproductionnumberoftheautonomousODE(Equation 1-3 ).TheaverageamountofvirionsproducedinthelifetimeofaninfectedcellisnowZ10p(a)(a)da.Thus,R0isintuitivelytheaverageamountofsecondaryinfectedcellsinducedbyasingleinfectedcellinapopulationoftargetcellsatcarryingcapacity. 4.5GlobalExtinctionwhenR0<1 Theorem4.2. IfR0<1,thenx0isgloballyasymptoticallystableinX. Proof. LetT1:=limsupt!1T(t)andV1:=limsupt!1V(t).Noticefromtheequationfor. T,T1T0.Forall>0,thereexists>0suchthat8t,T(t)T0+,V(t)V1+.AlsonotethatZ1tp(a)(a) (a)]TJ /F3 11.955 Tf 11.95 0 Td[(t)T(0,a)]TJ /F3 11.955 Tf 11.95 0 Td[(t)dae)]TJ /F10 7.97 Tf 6.58 0 Td[(btZ10T(0,a)!0ast!1 111

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Hence,wecanpickthe>0,suchthatR1tp(a)(a) (a)]TJ /F10 7.97 Tf 6.59 0 Td[(t)T(0,a)]TJ /F3 11.955 Tf 12.97 0 Td[(t)da<.Bythesemigroupproperty,wecanwithoutlossofgeneralityassume=0.Then. V=Zt0kV(t)]TJ /F3 11.955 Tf 11.96 0 Td[(a)T(t)]TJ /F3 11.955 Tf 11.96 0 Td[(a)p(a)(a)da+Z1tp(a)(a) (a)]TJ /F3 11.955 Tf 11.95 0 Td[(t)T(0,a)]TJ /F3 11.955 Tf 11.95 0 Td[(t)da)]TJ /F5 11.955 Tf 11.96 0 Td[(Vk(T0+)(V1+)Zt0p(a)(a)da+)]TJ /F5 11.955 Tf 11.95 0 Td[(V=(T0+)(V1+)R0 T0+)]TJ /F5 11.955 Tf 11.96 0 Td[(V.Hence,V1(T0+)(V1+)R0 T0+ .BecauseR0<1,fromtheaboveinequality,wendthatfor>0sufcientlysmall,ifV1>0,thenV11,itisimportanttoprovethatthesemigroupS(t)isasymptoticallysmooth.WerefertodenitionsandresultsinSection 1.4.3 .Also,sinceL1+isacomponentofourstatespaceX,weneedanotionofcompactnessinL1+.Beinganinnitedimensionalspace,boundednessdoesnotimplyprecompactness.Weusethefollowingresult. 112

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Lemma7([ 3 ]). LetKLp+(0,1)beclosedandboundedwherep1.ThenKiscompactiffthefollowinghold:limh!0Z10ju(z+h))]TJ /F3 11.955 Tf 11.95 0 Td[(u(z)jpdz=0uniformlyforu2K.(u(z+h)=0ifz+h<0). (i)limh!1Z1hju(z)jpdz=0uniformlyforu2K. (ii) Proposition4.1. Thesemigroup,S(t),isasymptoticallysmooth. Proof. Inourcase,itisclearthatifforanyboundedsetBX,weprojectS(t)BontoR2,1S(t)B,wehavethat1S(t)Bisprecompactbecausesolutionsremainbounded.SupposeweshowthattheprojectionofS(t)ontoL1(0,1),2S(t),canbewrittenas2S(t)=U(t)+C(t),wherethereexistsk(t,r)!0ast!1withkU(t)xkk(t,r)ifkxkr,andforanyBXwhichisclosedandbounded,wehaveC(t)Biscompact.ThenwecanapplyLemma 1.5 forS(t)=eU(t)+eC(t)whereeU(t)=0B@0U(t)1CA,eC(t)=0B@1S(t)C(t)1CAIndeed,ifBXisclosedandbounded,theneC(t)B1S(t)BC(t)Bisaclosedsubsetofacompactset,andhenceiscompact.Also,thedecayingrequirementforeU(t)iscertainlysatisted.Inordertofollowthisplanofaction,let2S(t)=U(t)+C(t)where(U(t)x)(a)=(a) (a)]TJ /F3 11.955 Tf 11.96 0 Td[(t)T(0,a)]TJ /F3 11.955 Tf 11.95 0 Td[(t)1fa>tg,(C(t)x)(a)=(a)kV(t)]TJ /F3 11.955 Tf 11.96 0 Td[(a)T(t)]TJ /F3 11.955 Tf 11.96 0 Td[(a)1fa
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Hence,ifweletk(t,r)=re)]TJ /F10 7.97 Tf 6.59 0 Td[(bt,thencertainlyk(t,r)!0ast!1andkU(t)xkk(t,r)ifkxkr.ToshowthatC(t)satisesthecompactnesscondition,weapplyLemma 7 .LetBXbeclosedandbounded.Supposer>0suchthatkxkrforallx2B.Noticethatforallx2B,R1hj(C(t)x)(a)jda=08ht.ThereforeCondition ii issatisedforthesetC(t)BL1.TocheckCondition i ,observe:Z10j(C(t)x)(a))]TJ /F6 11.955 Tf 11.95 0 Td[((C(t)x)(a+h)jda=Zt0j(a)kV(t)]TJ /F3 11.955 Tf 11.95 0 Td[(a)T(t)]TJ /F3 11.955 Tf 11.96 0 Td[(a))]TJ /F5 11.955 Tf 11.96 0 Td[((a+h)kV(t)]TJ /F3 11.955 Tf 11.95 0 Td[(a)]TJ /F3 11.955 Tf 11.95 0 Td[(h)T(t)]TJ /F3 11.955 Tf 11.96 0 Td[(a)]TJ /F3 11.955 Tf 11.96 0 Td[(h)jda=Zt0(a)kV(t)]TJ /F3 11.955 Tf 11.96 0 Td[(a)T(t)]TJ /F3 11.955 Tf 11.95 0 Td[(a))]TJ /F5 11.955 Tf 13.15 8.09 Td[((a+h) (a)kV(t)]TJ /F3 11.955 Tf 11.95 0 Td[(a)]TJ /F3 11.955 Tf 11.96 0 Td[(h)T(t)]TJ /F3 11.955 Tf 11.96 0 Td[(a)]TJ /F3 11.955 Tf 11.95 0 Td[(h)daZt0e)]TJ /F10 7.97 Tf 6.58 0 Td[(bakV(t)]TJ /F3 11.955 Tf 11.95 0 Td[(a)T(t)]TJ /F3 11.955 Tf 11.96 0 Td[(a))]TJ /F5 11.955 Tf 13.15 8.09 Td[((a+h) (a)kV(t)]TJ /F3 11.955 Tf 11.95 0 Td[(a)T(t)]TJ /F3 11.955 Tf 11.96 0 Td[(a)da+Zt0e)]TJ /F10 7.97 Tf 6.59 0 Td[(ba(a+h) (a)jkV(t)]TJ /F3 11.955 Tf 11.96 0 Td[(a)T(t)]TJ /F3 11.955 Tf 11.95 0 Td[(a))]TJ /F3 11.955 Tf 11.95 0 Td[(kV(t)]TJ /F3 11.955 Tf 11.96 0 Td[(a)]TJ /F3 11.955 Tf 11.96 0 Td[(h)T(t)]TJ /F3 11.955 Tf 11.95 0 Td[(a)]TJ /F3 11.955 Tf 11.95 0 Td[(h)jda (4-9)LetM=max(r,2A b)whereA,aredenedinLemma 6 .NoticethatZt0e)]TJ /F10 7.97 Tf 6.59 0 Td[(bakV(t)]TJ /F3 11.955 Tf 11.96 0 Td[(a)T(t)]TJ /F3 11.955 Tf 11.96 0 Td[(a))]TJ /F5 11.955 Tf 13.15 8.08 Td[((a+h) (a)kV(t)]TJ /F3 11.955 Tf 11.96 0 Td[(a)T(t)]TJ /F3 11.955 Tf 11.96 0 Td[(a)da=Zt0e)]TJ /F10 7.97 Tf 6.58 0 Td[(bakV(t)]TJ /F3 11.955 Tf 11.96 0 Td[(a)T(t)]TJ /F3 11.955 Tf 11.95 0 Td[(a)1)]TJ /F5 11.955 Tf 13.15 8.09 Td[((a+h) (a)daMZ10e)]TJ /F10 7.97 Tf 6.58 0 Td[(ba1)]TJ /F5 11.955 Tf 13.15 8.09 Td[((a+h) (a)da,limh!0Z10e)]TJ /F10 7.97 Tf 6.59 0 Td[(ba1)]TJ /F5 11.955 Tf 13.15 8.09 Td[((a+h) (a)da=Z10e)]TJ /F10 7.97 Tf 6.58 0 Td[(ba1)]TJ /F6 11.955 Tf 13.16 0 Td[(limh!0(a+h) (a)da=0,whereweappliedDominatedConvergenceTheorem.Also,Zt0e)]TJ /F10 7.97 Tf 6.59 0 Td[(ba(a+h) (a)kV(t)]TJ /F3 11.955 Tf 11.96 0 Td[(a)T(t)]TJ /F3 11.955 Tf 11.95 0 Td[(a))]TJ /F5 11.955 Tf 13.15 8.09 Td[((a+h) (a)kV(t)]TJ /F3 11.955 Tf 11.96 0 Td[(a)]TJ /F3 11.955 Tf 11.96 0 Td[(h)T(t)]TJ /F3 11.955 Tf 11.96 0 Td[(a)]TJ /F3 11.955 Tf 11.95 0 Td[(h)daksup2[0,t]jV()T())]TJ /F3 11.955 Tf 11.95 0 Td[(V()]TJ /F3 11.955 Tf 11.95 0 Td[(h)T()]TJ /F3 11.955 Tf 11.96 0 Td[(h)jZ10e)]TJ /F10 7.97 Tf 6.58 0 Td[(bada 114

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ksup2[0,t](jV()jjT())]TJ /F3 11.955 Tf 11.95 0 Td[(T()]TJ /F3 11.955 Tf 11.95 0 Td[(h)j+jT()]TJ /F3 11.955 Tf 11.96 0 Td[(h)jjV())]TJ /F3 11.955 Tf 11.96 0 Td[(V()]TJ /F3 11.955 Tf 11.95 0 Td[(h)j)Z10e)]TJ /F10 7.97 Tf 6.59 0 Td[(bada (4-10)Bytheintegralformulation,wendthatjV())]TJ /F3 11.955 Tf 11.95 0 Td[(V()]TJ /F3 11.955 Tf 11.95 0 Td[(h)j=jZ)]TJ /F10 7.97 Tf 6.58 0 Td[(hZ10p(a)T(s,a)dads)]TJ /F5 11.955 Tf 11.96 0 Td[(Z)]TJ /F10 7.97 Tf 6.59 0 Td[(hV(s)dsjh(kTk+kVk)h(+)rjT())]TJ /F3 11.955 Tf 11.96 0 Td[(T()]TJ /F3 11.955 Tf 11.96 0 Td[(h)jZ)]TJ /F10 7.97 Tf 6.59 0 Td[(hjf(T(s)))]TJ /F3 11.955 Tf 11.95 0 Td[(kV(s)T(s)jdsmaxs2[0,r]jf(s)j+r2hHence,byInequality 4-10 ,Zt0e)]TJ /F10 7.97 Tf 6.58 0 Td[(ba(a+h) (a)kV(t)]TJ /F3 11.955 Tf 11.96 0 Td[(a)T(t)]TJ /F3 11.955 Tf 11.95 0 Td[(a))]TJ /F5 11.955 Tf 13.15 8.08 Td[((a+h) (a)kV(t)]TJ /F3 11.955 Tf 11.96 0 Td[(a)]TJ /F3 11.955 Tf 11.96 0 Td[(h)T(t)]TJ /F3 11.955 Tf 11.95 0 Td[(a)]TJ /F3 11.955 Tf 11.95 0 Td[(h)dahMwhereM=r(++maxs2[0,r]jf(s)j+r2)R10e)]TJ /F10 7.97 Tf 6.59 0 Td[(bada.Thisconvergesuniformlyto0ash!0.ThereforeEquation 4-9 convergesuniformlyto0ash!0andCondition i isshownforthesetC(t)B.Hence,byLemma 7 ,C(t)Biscompact.BytheaforementionedargumentwecanapplyLemma 1.5 andconcludethatS(t)isasymptoticallysmooth. 4.7UniformPersistenceInthissection,weprovetheuniformpersistenceandtheexistenceofaglobalattractorbyusingresultsfromSection 1.4.4 .WeneedtopartitionXasX=X0[@X0,where@X0isgoingtobearepeller.Let a=supfa2(0,1):p(a)>0g.Wenotethat aisallowedtobeinnity.Let@M0=n(a)2L1+:R a0(a)da=0oandM0=L1+n@M0.Let@X0=R+f0g@M0andX0=Xn@X0.ThenX=X0[@X0.AlsodeneX0+=R+(0,1)M0. 115

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Lemma8. X0and@X0areforwardinvariantunderthesemigroupS(t).Also,8x2@X0,wehaveS(t)x!x0ast!1wherex0=(T0,0,0).Inaddition,S(t)X0X0+8t>0. Proof. Firstweshowtheconclusionsfor@X0.Recallthatt7!R a0T(t,a)daisdifferentiable.Wehavethefollowingsystem:. V=Z a0p(a)T(t,a)da)]TJ /F5 11.955 Tf 11.96 0 Td[(Vd dtZ a0T(t,a)da=kVT)]TJ /F3 11.955 Tf 11.96 0 Td[(T(t, a))]TJ /F11 11.955 Tf 11.95 16.27 Td[(Z a0(a)T(t,a)daClearly,. V=0andd dtR a0T(t,a)da0ontheset@Xo.Thenon-negativenessofthesolutionsthenforces@X0tobeforwardinvariant.Theninviewofoursystemandthepropertiesoff(T),itisclearthat8x2@X0,wehaveS(t)x!x0ast!1wherex0=(T0,0,0).NowtoshowX0isforwardinvariant.Noticethat. V)]TJ /F5 11.955 Tf 21.99 0 Td[(V.HenceV(t)V(0)e)]TJ /F14 7.97 Tf 6.58 0 Td[(tforallt0.IfV(0)>0,thentheresultfollows.IfV(0)=0,thenR10p(a)T(0,a)da>0(sincex(0)2X0).Thend dtV(0)>0,sothat9>0suchthat8t2(0,],wehaveV(t)>0.Notethatinthiscase,wecanchoosesuchthatR10p(a)T(t,a)da>0forallt2[0,].Then,thesameargumentapplieswithV(t)V()e)]TJ /F14 7.97 Tf 6.59 0 Td[(tfort.HenceV(t)>08t>0.Then,sinceT(t)>08t>0,wehavethatT(t,a)kV(t)]TJ /F3 11.955 Tf 11.95 0 Td[(a)T(t)]TJ /F3 11.955 Tf 11.96 0 Td[(a)(a)>0forallt>0.Therefore,S(t)X0X0+forallt>0. WewillneedthefollowingresultaboutlinearscalarVoterraintegro-differentialequations. Lemma9. Considerthefollowingscalarintegro-differentialequation:. y(t)=Zt0h(a)y(t)]TJ /F3 11.955 Tf 11.95 0 Td[(a)da)]TJ /F3 11.955 Tf 11.95 0 Td[(cy(t)y(0)>0whereh(a)2L1+(0,1),c>0,andR10h(a)da>c.Thereisauniquesolution,y(t),whichisunbounded. 116

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Proof. FromsimilarargumentstoTheorem 4.1 andLemma 6 ,wecangetexistence,uniqueness,differentiability,andpositivityofasolution,y(t),totheintegro-differentialequation.Positivityofy(t)isclear,sincey(t)y(0)e)]TJ /F10 7.97 Tf 6.59 0 Td[(ct.SupposeBWOC,y(t)isbounded.Then,sincey(t)iscontiuousandbounded,theLaplacetransformofy(t),L[y](s):=R10e)]TJ /F10 7.97 Tf 6.59 0 Td[(sty(t)dtisdenedforalls>0.ThenusingpropertiesoftheLaplacetransform,thesolutionmustsatisfy:L[y](s)=y(0) s+c)-222(L[h](s)whereL[h](s)isdened8s0sinceh2L1+(0,1).Now,L[h](s)!L[h](0)ass!0bythedominatedconvergencetheorem.SinceL[h](0)=R10h(a)da>c,thereexists>0suchthatc)-234(L[h](s)<)]TJ /F5 11.955 Tf 9.3 0 Td[(foralls2[0,).Hence,L[y](s)<0foralls2[0,).Butthiscontradictsthepositivityofy(t). Theorem4.3. SupposethatR0>1.ThenS(t)isuniformlypersistent.Moreover,thereexistsacompactsetAX0+whichisaglobalattractorforfS(t)gt0inX0,and9>0suchthatliminft!1V(t),andliminft!1d(T(t,a),@M0) Proof. WewillapplyTheorem 1.7 .Noticethatfx0gistheglobalattractorforS(t)restrictedto@X0.Weclaimthefollowingforthestablemanifoldoffx0g:Ws(fx0g)=fx2X:S(t)x!x0g.Indeeditisobviousthatfx2X:S(t)x!x0gWs(fx0g).SupposeBWOCtheconverseisnottrue.Then9x2Ws(fx0g)forwhichS(t)xdoesnotconvergetox0.Then9>0suchthat9tn"1withkS(tn))]TJ /F3 11.955 Tf 11.95 0 Td[(x0k.SincethesemigroupS(t)isasymptoticallysmooth,[t0fS(t)xgiscompact.Hence,thereexistsaconvergentsubsequenceofS(tn)x.Butthen!(x)isnotcontainedinfx0g),whichisacontradiction.Observethat@X0Ws(fx0g).Also)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(@X0nfx0g\Wu(fx0g)=;.Indeed,letx2@X0nfx0g.Anybackwardorbitofxmuststayin@X0sinceX0(thecomplementof@X0)isforwardinvariant.Supposex=(T(0),0,`(a)).If`(a)=0(inL1),thenwehaveascalarODEwithauniquepositiveequilibriumandlimt!T(t)=0or1.Suppose 117

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R10`(a)da>0.Sincex2@X0,R a0`(a)da=0.Suppose9>0,x1=(T()]TJ /F5 11.955 Tf 9.29 0 Td[(),0,`1(a))2@X0suchthatS()x1=x.Then,R1 a`(a)da=R1 a+e)]TJ /F23 7.97 Tf 7.99 6.42 Td[(Raa)]TJ /F16 5.978 Tf 5.75 0 Td[((s)ds`(a)]TJ /F5 11.955 Tf 12.14 0 Td[()da0,soWLOGwecantakeVn(0)>0.Weclaimthatfornsufcientlylarge,ynisunbounded.TheassumptionR0>1isequivalentto)]TJ /F5 11.955 Tf 9.3 0 Td[(+kT0R10p(a)(a)da>0.Hence9N2Nsuchthat)]TJ /F5 11.955 Tf 9.3 0 Td[(+k)]TJ /F3 11.955 Tf 5.47 -9.68 Td[(T0)]TJ /F4 7.97 Tf 14.23 4.7 Td[(1 NR10p(a)(a)da>0.ThenbyLemma 9 ,yNisunbounded.SinceVNyN,wegetthatVNisunboundedandhenceS(t)xNisunboundedwhichiscertainlyacontradiction.ThereforeWs(fx0g)\X0=;.ByTheorem 1.7 ,wegetthatS(t)isuniformlypersistent.Forx=(x1,x2,x3(a))2X,let(x)bedenedas:(x)=x2+Z a0x3(a)da. 118

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Then:X![0,1)iscontinuous.Also,(x)>0onX0and(x)=0on@X0.Noticethatforx2X0:d(x,@X0)=infy2@X0jx1)]TJ /F3 11.955 Tf 11.96 0 Td[(y1j+jx2)]TJ /F3 11.955 Tf 11.96 0 Td[(y2j+Z10jx3(a))]TJ /F3 11.955 Tf 11.96 0 Td[(y3(a)jda=infy2@X00+x2+Z a0jx3(a))]TJ /F3 11.955 Tf 11.95 0 Td[(y3(a)jda+Z1 ajx3(a))]TJ /F3 11.955 Tf 11.95 0 Td[(y3(a)jda=x2+Z a0x3(a)daTherefore,-uniformpersistenceisequivalenttouniformpersistenceinthiscase.ThenbyTheorem 1.8 ,wecanconcludethatthereexistsacompactsetAX0whichisaglobalattractorforfS(t)gt0inX0.BecauseS(t)X0X0+,theglobalattractor,A,isactuallycontainedinX0+.Becauseofthis,9>0suchthatliminft!1V(t),andliminft!1d(T(t,a),@M0) BecauseAisinvariant,wecanndacompleteorbitthroughxwhichiscontainedinA.Thismeanswecanndfz(t)gt2RAsuchthat8s2R,S(t)z(s)=z(t+s)fort0andz(0)=x. 4.8LyapunovFunctionalandGlobalStabilityInthefollowing,weassumethatf(T)satisesthesectorcondition,Condition 1-6 .Forx2A,letx=(T(0),V(0),T(0,a))2A.ExtendT(t),V(t),T(t,a)sothatt2R,i.e.z(t)x=(T(t),V(t),T(t,a))8t2Risacompleteorbitthroughx.Herez(t)mustsatisfythesystem:dT(t) dt=f(T(t)))]TJ /F3 11.955 Tf 11.95 0 Td[(kV(t)T(t),dV(t) dt=Z10p(a)T(t,a)da)]TJ /F5 11.955 Tf 11.96 0 Td[(V(t), (4-12)T(t,a)=(a)kV(t)]TJ /F3 11.955 Tf 11.95 0 Td[(a)T(t)]TJ /F3 11.955 Tf 11.96 0 Td[(a)(T(0),V(0),T(0,a))2AR2+L1+ 119

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Deneu(t,a)=1 (a)T(t,a)andq(a)=(a)p(a).Forx=(x1,x2,x3(a))2A,leth(x)=x1,x2,1 (a)x3(a).Underthischangeofvariables,Equation 4-12 becomesthefollowingsystem:dT(t) dt=f(T(t)))]TJ /F3 11.955 Tf 11.95 0 Td[(kV(t)T(t),dV(t) dt=Z10q(a)u(t,a)da)]TJ /F5 11.955 Tf 11.96 0 Td[(V(t), (4-13)u(t,a)=kV(t)]TJ /F3 11.955 Tf 11.95 0 Td[(a)T(t)]TJ /F3 11.955 Tf 11.96 0 Td[(a)(T(0),V(0),(a)u(0,a))2AR2+L1+Noticethereisapositiveequilibriumto,with u(a)=1 (a) T(a)=k V T,whichisconstantina,solet u(a)= u.Thenwehavethefollowingconditions:=k TZ10q(a)daf( T)=k V T= uFromTheorem 4.3 anditsconsequences,wehavethefollowing:9,M>0suchthatforallt2R,T(t)Mk2u(t,a)kM28a2[0,1)V(t)MDeneg(x)=x)]TJ /F6 11.955 Tf 12.25 0 Td[(1)]TJ /F6 11.955 Tf 12.25 0 Td[(log(x).Notethatg(x)isnon-negativeandcontinuouson(0,1)withauniquerootatx=1.Let(a)=R1aq(l)dl.Wedenethefollowingfunctiononh(A):W:(T,V,u)7!WT+WV+WuwhereWT= T ugT T,WV=k T V ugV V,Wu=k T Z10(a)gu(t,a) uda 120

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Notethatsincek2u(t,a)kM2foralla2[0,1)andt2R,wehavethat9M1>0suchthat0gu(t,a) uM1Then,Z10(a)gu(t,a) udaM1Z10Z1a(`)p(`)d`daM1Z10Z1ae)]TJ /F10 7.97 Tf 6.58 0 Td[(b`d`da=M1 b2<1ThereforeitfollowsthatW=WT+Wu+WViswell-denedandbounded.Forconvenience,WT(T(t))isdenotedbyWT,andlikewisefortheothertwocomponents.d dtWu=d dtk T Z10(a)gu(t,a) uda=k T d dtZ10(a)gu(t)]TJ /F3 11.955 Tf 11.96 0 Td[(a,0) uda=k T d dtZt(t)]TJ /F3 11.955 Tf 11.96 0 Td[(s)gu(s,0) uds=k T (0)gu(t,0) u+Zt0(t)]TJ /F3 11.955 Tf 11.95 0 Td[(s)gu(s,0) uds=k T (0)gu(t,0) u+Z100(a)gu(t,a) uda=k T Z10q(a)u(t,0) u)]TJ /F6 11.955 Tf 11.95 0 Td[(1)]TJ /F6 11.955 Tf 11.95 0 Td[(logu(t,0) u)]TJ /F3 11.955 Tf 13.15 8.09 Td[(u(t,a) u+1+logu(t,a) uda=k T Z10q(a)u(t,0) u)]TJ /F3 11.955 Tf 13.15 8.09 Td[(u(t,a) u+logu(t,a) u(t,0)daWeusethefollowingequilibriumconditionsinthenextcalculation:f( T)=k T V= u, V V=T u Tu(t,0), k T=Z10q(a)da.d dt(WT+WV)=d dt T ugT T+k T V ugV V 121

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=1 u" Tg0T T. T T+k T V g0V V. V V#=1 u1)]TJ ET q .478 w 143.01 -40.57 m 153.27 -40.57 l S Q BT /F3 11.955 Tf 143.01 -50.54 Td[(T T(f(T))]TJ /F3 11.955 Tf 11.95 0 Td[(kVT)+k T 1)]TJ ET q .478 w 307.3 -40.57 m 317.31 -40.57 l S Q BT /F3 11.955 Tf 307.3 -50.54 Td[(V VZ10q(a)u(t,a)da)]TJ /F5 11.955 Tf 11.95 0 Td[(V=1 u)]TJ /F3 11.955 Tf 5.47 -9.69 Td[(f(T))]TJ /F3 11.955 Tf 11.96 0 Td[(f( T)1)]TJ ET q .478 w 223.56 -74.69 m 233.82 -74.69 l S Q BT /F3 11.955 Tf 223.56 -84.67 Td[(T T+f( T)1)]TJ ET q .478 w 317.79 -74.69 m 328.05 -74.69 l S Q BT /F3 11.955 Tf 317.79 -84.67 Td[(T T)]TJ /F3 11.955 Tf 11.96 0 Td[(kVT+kV T+k T Z10q(a)u(t,a)1)]TJ ET q .478 w 246.61 -108.82 m 256.62 -108.82 l S Q BT /F3 11.955 Tf 246.61 -118.79 Td[(V Vda)]TJ /F3 11.955 Tf 11.96 0 Td[(kV T+k T V=1 u)]TJ /F3 11.955 Tf 5.47 -9.68 Td[(f(T))]TJ /F3 11.955 Tf 11.96 0 Td[(f( T)1)]TJ ET q .478 w 217.25 -142.94 m 227.51 -142.94 l S Q BT /F3 11.955 Tf 217.25 -152.91 Td[(T T+1 uk T k Tf( T))]TJ /F3 11.955 Tf 11.95 0 Td[(f( T) T T)]TJ /F3 11.955 Tf 11.96 0 Td[(kVT+1 uk T Z10q(a)u(t,a)1)]TJ ET q .478 w 264.2 -177.06 m 274.21 -177.06 l S Q BT /F3 11.955 Tf 264.2 -187.04 Td[(V Vda+ k Tk T V=1 u)]TJ /F3 11.955 Tf 5.47 -9.68 Td[(f(T))]TJ /F3 11.955 Tf 11.96 0 Td[(f( T)1)]TJ ET q .478 w 217.25 -211.19 m 227.51 -211.19 l S Q BT /F3 11.955 Tf 217.25 -221.16 Td[(T T+k T Z10q(a))]TJ /F3 11.955 Tf 9.3 0 Td[(u(t,0) u)]TJ ET q .478 w 254.34 -245.31 m 264.6 -245.31 l S Q BT /F3 11.955 Tf 254.34 -255.29 Td[(T T+u(t,a) u)]TJ /F3 11.955 Tf 14.56 8.08 Td[(Tu(t,a) Tu(t,0)+2daTherefore,d dt(WT+WV+Wu)=1 u)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(f(T))]TJ /F3 11.955 Tf 11.96 0 Td[(f( T)1)]TJ ET q .478 w 276.68 -317.04 m 286.93 -317.04 l S Q BT /F3 11.955 Tf 276.68 -327.01 Td[(T T+k T Z10q(a)2)]TJ ET q .478 w 274.62 -351.16 m 284.88 -351.16 l S Q BT /F3 11.955 Tf 274.62 -361.13 Td[(T T)]TJ /F3 11.955 Tf 14.56 8.09 Td[(Tu(t,a) Tu(t,0)+logu(t,a) u(t,0)dak T Z10q(a) 2)]TJ /F6 11.955 Tf 11.96 0 Td[(2s u(t,a) u(t,0)+logu(t,a) u(t,0)!da=2k T Z10q(a) 1)]TJ /F11 11.955 Tf 11.95 21.67 Td[(s u(t,a) u(t,0)+logs u(t,a) u(t,0)!da=)]TJ /F6 11.955 Tf 9.3 0 Td[(2k T Z10q(a)g s u(t,a) u(t,0)!da0Herewehaveusedthesectorcondition(Condition 1-6 ),thearithmetic/geometricmeaninequality,andthepositivityofg.Hence,wendthatdW dt=0,u(t,a)=u(t,0)and T T=Tu(t,a) Tu(t,0) 122

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,u(t,a)=u(t,0)andT= T,d dtu(t,0)=0andd dtT=0,d dtkVT=0andd dtT=0,d dtV=0andd dtT=0,(T,u,V)=( T, u, V)Nowwearereadytoproveglobalasymptoticstabilityoftheinteriorequilibrium, x,whenR0>1andf(T)satisesthesectorcondition. Theorem4.4. LetR0>1andthesectorcondition,Condition 1-6 ,holds.Then x:=( T, V, T(a))isgloballyasymptoticallystableinX0. Proof. WeprovethatA=f xg.SupposeBWOCthereexistsx2Anf xg.Thereisacompleteorbitthroughx,fz(t)gt2R,containedinA.Considerthealphalimitsetonthisspecicorbit,z(x).SinceAiscompact,weconcludethatz(x)isnon-empty,compact,invariant,andbelongstoA.Weclaimthatz(x)=f xg.Letex=(ex1,ex2,eT(a))2z(x).Letz(t)=(T(t),V(t),T(t,a)).Then9tn#suchthatxn:=z(tn)!ex2A.InparticularT(tn,a)!eT(a)inL1astn#.Then,weclaimWu(1 (a)T(tn,a))!Wu(1 (a)eT(a))inL1astn#.Becausexn,ex2A,weobtainthatk21 (a)T(tn,a)kM2k21 (a)eT(a)kM2Letu(t,a)=1 (a)T(t,a)andeu(a)=1 (a)eT(a)asbefore.ThenjWu(u(tn,a))]TJ /F3 11.955 Tf 11.95 0 Td[(Wu(eu(t,a)j=Z10Z1a(`)p(`)d`gu(tn,a) u)]TJ /F3 11.955 Tf 11.96 0 Td[(geu(a) udaZ10Z1a(`)d`gu(tn,a) u)]TJ /F3 11.955 Tf 11.96 0 Td[(geu(a) udaZ10Z1a(`)d`maxk2skM2jg0(s)ju(tn,a) u)]TJ /F11 11.955 Tf 13.21 8.59 Td[(eu(a) uda 123

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=M1 uZ10Z1a(`)d`1 (a)T(tn,a))]TJ /F11 11.955 Tf 13.76 3.16 Td[(eT(a)da=M1 uZ10Z1ae)]TJ /F23 7.97 Tf 7.99 6.42 Td[(R`a(s)dsd`T(tn,a))]TJ /F11 11.955 Tf 13.76 3.15 Td[(eT(a)daM1 uZ10Z1ae)]TJ /F10 7.97 Tf 6.58 0 Td[(b(`)]TJ /F10 7.97 Tf 6.59 0 Td[(a)d`T(tn,a))]TJ /F11 11.955 Tf 13.77 3.16 Td[(eT(a)da=M1 b uZ10T(tn,a))]TJ /F11 11.955 Tf 13.76 3.15 Td[(eT(a)da!0astn#TheconvergenceoftheothercomponentsofWisaconsequenceofthecontinuityofg.Foranyy=(y1,y2,y3(a))2X,leth(y)=y1,y2,1 (a)y3(a).Then,W(h(z(tn)))!W(h(ex))astn#.SinceW(h(z(t)))isanon-increasingmap,whichisboundedabove,weconcludethatW(h(z(t)))"c<1ast#.Therefore,W(h(^x))=cforall^x2z(x).Combiningthiswiththefactthatz(x)isinvariant,wegetthatW(h((t)))=cforallt2R,where(t)isacompleteorbitthroughex(with(0)=ex).Hence,d dt(W(h((t)))=0forallt2R.Therefore,h((t))=h( x)forallt,inparticularwhent=0.So,ex= x.Thisshowsthatz(x)=f xg.Now,wewillapplyLyapunov'sdirectmethodtondthat xislocallystableonA(inthesenseofthesubspacetopologyonX).WenotethatthecompactnessofAandouraboveresultthatz(x)=f xgarebothvitalforthisargument.Letd=maxy2Aky)]TJ ET q .478 w 261.81 -446.34 m 268.46 -446.34 l S Q BT /F3 11.955 Tf 261.81 -453.66 Td[(xk.Thendkx)]TJ ET q .478 w 81.07 -482.2 m 87.71 -482.2 l S Q BT /F3 11.955 Tf 81.07 -489.52 Td[(xk>0.Forr>0,denethefollowingsets:Ar:=fx2A:kx)]TJ ET q .478 w 107.52 -518.06 m 114.17 -518.06 l S Q BT /F3 11.955 Tf 107.52 -525.38 Td[(xk
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0d.WenotethatCrand ArarecompactsincetheyareclosedsubsetsofthecompactsetA.NoticeWhisastrictLyapunovfunctionat xinthetraditionalsenseonA.Indeed,(fromtheaboveintegralcalculation)WhiscontinuousonAand (i) Wh( x)=0. (ii) Wh>08y2Anf xg. (iii) d dtWh(S(t)y)<08y2Anf xgand8t2R+.Theproofthat xislocallystablefollowsbystandardarguments.Forcompleteness,weincludetheargument.Let0<0suchthatmWh(x)forally2C.BycontinuityofWh,9>0suchthatWh(y)m 2forally2 A.Lety2A,thenWh(S(t)y)0sinced dtWh(S(t)y)<0.Hence,S(t)y2Aforallt0.Otherwise,thecontinuityoft7!kS(t)y)]TJ ET q .478 w 77.22 -325.41 m 83.86 -325.41 l S Q BT /F3 11.955 Tf 77.22 -332.72 Td[(xkandinvarianceofAimplythat9t1>0suchthatS(t1)y2C,whichcontradictsthefactthatWh(S(t1)y)0.Bystability,thereexists>0suchthaty2A)S(t)y2A8t0.Sincez(x)= x,thereexistst)]TJ /F5 11.955 Tf 13.43 1.8 Td[(<0suchthatz(t)]TJ /F6 11.955 Tf 7.09 1.8 Td[()x2A.Thenz(t+t)]TJ /F6 11.955 Tf 7.08 1.79 Td[()=S(t)z(t)]TJ /F6 11.955 Tf 7.08 1.79 Td[()2Aforallt0.Thiscontradictsthefactthatx=z(0)2C.Hence,A=f xg.Thisprovestheresult. 125

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CHAPTER5FUTUREWORKEachtypeofantiviralmedicationforHIVworksatadistinctstageintheinfectedcelllifecycle.Theage-structuredvirusmodelprovidesthemeansofincorporatingdrugsatthesedistinctstagesofthelifecycle,whichgivesamoreaccuraterepresentationofthemodeofactionoftheantiviralmedication.RongandPerelsonexploitedthisfeatureoftheage-structuredmodelin[ 35 ]bystudyingtheeffectofthreedifferenttypesofdrugs(RT-inhibitors,P-inhibitors,andEntryInhibitors)throughincorporatingthemedicationsintheappropriateageclassesoftheinfectedcell.Theyassumedthatthedrugefcacieswereconstantintime.Afuturelineofworkformewillbetoincorporatetime-periodicdrugtreatmentintotheage-structuredmodel,inotherwords,combinethetwomodelsthatIstudiedinthisdissertation.Theinclusionofperiodicityandage-structureaddssubstantialmathematicaldifculty,buttheresultsmaybeinteresting.BacaerhasshownthataSEIRmodelwithseasonalityandatimedelaycanproduceresonanceinthereproductionnumberofthesystem[ 5 ].Addinganextraperiodicforce,asincombinationtherapyfortheequivalentwithin-hostvirusmodel,maybeabletocontrolthisresonance.Ingeneral,theoptimaltimingofthedrugsinacombinationtherapywouldbeofinterestinthismodel,astheadditionalage-structurecouldaltertheresultsfromtheperiodically-forcedODEcase.Theadditionofmultipletypesoftargetcellsandmultiplecompartmentsisanotherpotentialtopicoffutureresearch.TheabilityofHIVtoestablishareservoirofinfectedcellsinthebraincontributestothepersistenceofthevirusduringantiviraltreatmentsincethedrugsoftenhavepoorpenetrationoftheblood-brainbarrier.Theinfectionofmacrophagesplayalargeroleintrafckingthevirustothebrain,hencetheneedtoincludemultipletypesoftargetcells.Viralevolutioninthepresenceofdrugtreatmentisanotherfactortoconsider.Theoptimaladministrationofdrugsinordertominimizedrugresistanceisanimportant 126

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topicofstudyandwillbepursuedinthefuture.Explicitlyincludingimmuneresponseisanotherwaytoimprovethemodels,andagoalcanbetondpotentialtherapieswhichcanaidincreatingasustained,effectiveimmuneresponsetoHIV.ModelingthevariousaspectsofHIVwhichcontributetoitspersistenceunderdrugtreatmentscanbeinsightfulfordesigningtherapies,andthisprovidesmotivationforfutureresearch. 127

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[29] J.M.MURRAY,A.D.KELLEHER,ANDD.A.COOPER,TimingoftheComponentsoftheHIVLifeCycleinProductivelyInfectedCD4+TCellsinaPopulationofHIV-InfectedIndividuals,J.Virol.,85(2011),pp.10798. [30] P.W.NELSON,M.A.GILCHRIST,D.COOMBS,J.M.HYMAN,ANDA.S.PEREL-SON,Anage-structuredmodelofHIVinfectionthatallowsforvariationsintheproductionrateofviralparticlesandthedeathrateofproductivelyinfectedcells,Math.Biosci.Eng.,1(2004),pp.267. [31] P.W.NELSONANDA.S.PERELSON,MathematicalanalysisofdelaydifferentialequationmodelsofHIV-1infectionMath.Biosci.,179(2002),pp.73. [32] M.A.NOWAKANDR.M.MAY,VirusDynamics,OxfordUniversityPress,NewYork,2000. [33] A.S.PERELSONANDP.W.NELSONMathematicalanalysisofHIV-1dynamicsinvivo,SIAMRev.,41(1999),pp.3. [34] A.S.PERELSON,A.U.NEUMANN,M.MARKOWITZ,J.M.LEONARDANDD.D.HO,HIV-1dynamicsinvivo,virionclearancerate,infectedcelllife-span,viralgenerationtimeScience,271(1996),pp.1582. [35] L.RONG,Z.FENG,ANDA.S.PERELSON,Mathematicalanalysisofage-structuredHIV-1dynamicswithcombinationantiviraltheraphy,SIAMJ.Appl.Math.,67(2007),pp.731. [36] L.RONG,Z.FENG,A.S.PERELSON,EmergenceofHIV-1drugresistanceduringantiretroviraltreatment,Bull.Math.Biol.,69(2007)pp.2027. [37] L.RONG,A.S.PERELSON,ModelingHIVpersistence,thelatentreservoir,andviralblips,J.Theor.Biol.,260(2009),pp.308. [38] J.SACHA,C.CHUNG,E.RAKASZ,S.P.SPENCER,A.K.JONAS,A.T.BEAN,W.LEE,B.J.BURWITZ,J.J.STEPHANY,J.T.LOFFREDO,D.B.ALLISON,S.ADNAN,A.HOJI,N.A.WILSON,T.C.FRIEDRICH,J.D.LIFSON,O.O.YANG,ANDD.I.WATKINS,Gag-specicCD8+TlymphocytesrecognizeinfectedcellsbeforeAIDS-virusintegrationandviralproteinexpression,J.Immunol.,178(2007),pp.2746-54. [39] J.D.SILICIANO,J.KAJDAS,D.FINZI,T.C.QUINN,K.CHADWICK,J.B.MAR-GOLICK,C.KOVACS,S.J.GANGE,ANDR.F.SILICIANO,Longtermfollow-upstudiesconrmtheextraordinarystabilityofthelatentreservoirforHIV-1inrestingCD4+Tcells,Nat.Med.,9(2003),pp.727. [40] I.B.SCHWARTZANDH.L.SMITH,InnitesubharmonicbifurcationinanSEIRepidemicmodel,J.Math.Biol.,18(1993),pp.233. [41] H.L.SMITHANDP.WALTMAN,Perturbationofagloballystablesteadystate,Proc.Amer.Math.Soc.,127(1999),pp.447. 130

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[42] H.R.THIEME,SemiowsgeneratedbyLipschitzperturbationsofnon-denselydenedoperators,DifferentialIntegralEquations,3(1990),pp.1035. [43] UNAIDS,UNAIDSWorldAIDSDayReport2011,2011. [44] G.F.WEBB,TheoryofNonlinearAge-DependentPopulationDynamics,MarcelDekker,NewYork,1985. [45] J.YANG,Z.QU,ANDG.HU,Dufngequationwithtwoperiodicforcings:Thephaseeffect,Phys.Rev.,E53(1996),pp.4402. [46] E.ZEIDLER,NonlinearFunctionalAnalysisanditsApplicationsI:Fixed-PointTheorems,Springer-Verlag,NewYork,1986. 131

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BIOGRAPHICALSKETCH CameronBrownereceivedaBachelorofScienceinmathematicswithaminorinphysicsfromtheUniversityofFloridain2007.HewentontograduateschoolatUniversityofFloridabeginningin2007,andcompletedhisPhDinmathematicsin2012,specializinginmathematicalbiology. 132