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# A Mathematical Model for Tumor Therapy with a Fusogenic Oncolytic Virus

## Material Information

Title: A Mathematical Model for Tumor Therapy with a Fusogenic Oncolytic Virus
Physical Description: 1 online resource (83 p.)
Language: english
Creator: Jacobsen, Karly A
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2013

## Subjects

Subjects / Keywords: advection -- boundary -- cancer -- differential -- diffusion -- equations -- mathematical -- model -- moving -- oncolytic -- syncytia -- tumor -- virotherapy -- virus
Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

## Notes

Abstract: Oncolytic virotherapy is a tumor treatment which uses viruses to selectively target and destroy cancer cells. Clinical trials have demonstrated varying degrees of success for the therapy with limitations predominantly due to barriers to viral spread throughout the tumor and the immune response to the virus. Fusogenic viruses, capable of causing cell-to-cell fusion upon infection of a tumor cell, have shown promise in recent experimental studies. The fusion causes the formation of large, multinucleated syncytia which eventually leads to cell death. We formulate a partial differential equations model with a moving boundary to describe the treatment of a spherical tumor with a fusogenic oncolytic virus. Syncytia formation, lysis, budding and interstitial diffusion are incorporated as mechanisms of viral spread. The tumor cells move via advection; the tumor radius is a time-dependent variable. A proof is presented for existence and uniqueness of local solutions to the nonlinear hyperbolic-parabolic system. In a special case, the model is reduced to an ordinary differential equations system for which a global stability analysis is performed. This provides a prediction of success or failure of the treatment in terms of long-term behavior of the tumor radius.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Karly A Jacobsen.
Thesis: Thesis (Ph.D.)--University of Florida, 2013.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2015-05-31

## Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2013
System ID: UFE0045381:00001

## Material Information

Title: A Mathematical Model for Tumor Therapy with a Fusogenic Oncolytic Virus
Physical Description: 1 online resource (83 p.)
Language: english
Creator: Jacobsen, Karly A
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2013

## Subjects

Subjects / Keywords: advection -- boundary -- cancer -- differential -- diffusion -- equations -- mathematical -- model -- moving -- oncolytic -- syncytia -- tumor -- virotherapy -- virus
Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

## Notes

Abstract: Oncolytic virotherapy is a tumor treatment which uses viruses to selectively target and destroy cancer cells. Clinical trials have demonstrated varying degrees of success for the therapy with limitations predominantly due to barriers to viral spread throughout the tumor and the immune response to the virus. Fusogenic viruses, capable of causing cell-to-cell fusion upon infection of a tumor cell, have shown promise in recent experimental studies. The fusion causes the formation of large, multinucleated syncytia which eventually leads to cell death. We formulate a partial differential equations model with a moving boundary to describe the treatment of a spherical tumor with a fusogenic oncolytic virus. Syncytia formation, lysis, budding and interstitial diffusion are incorporated as mechanisms of viral spread. The tumor cells move via advection; the tumor radius is a time-dependent variable. A proof is presented for existence and uniqueness of local solutions to the nonlinear hyperbolic-parabolic system. In a special case, the model is reduced to an ordinary differential equations system for which a global stability analysis is performed. This provides a prediction of success or failure of the treatment in terms of long-term behavior of the tumor radius.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Karly A Jacobsen.
Thesis: Thesis (Ph.D.)--University of Florida, 2013.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2015-05-31

## Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2013
System ID: UFE0045381:00001

Full Text

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c2013KarlyA.Jacobsen 2

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Totheteachersandmentorsthatinspiredandencouragedme 3

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFFIGURES ..................................... 7 ABSTRACT ......................................... 8 CHAPTER 1BACKGROUND ................................... 9 1.1OncolyticVirotherapy ............................. 9 1.1.1HistoryandClinicalTrials ....................... 9 1.1.2MechanismsforTargetingandCellDeath .............. 11 1.2PriorMathematicalModels .......................... 12 1.2.1ModelswithCell-to-CellFusion .................... 13 1.2.2PDEModelsforVirotherapy ...................... 13 1.2.3NumericalMethodsforVirotherapyModels ............. 15 1.3MathematicalPreliminaries .......................... 15 1.3.1OrdinaryDifferentialEquations .................... 16 1.3.2PartialDifferentialEquations ...................... 18 1.3.3NumericalMethods ........................... 20 2FORMULATIONANDWELL-POSEDNESSOFTHEMODEL .......... 24 2.1TheMathematicalModel ............................ 24 2.1.1ModelFormulation ........................... 24 2.1.2BoundaryandInitialConditions .................... 27 2.1.3DerivationofIntegralTerms ...................... 28 2.1.4TumorGrowthintheAbsenceofInfection .............. 32 2.2TransformationoftheSystem ......................... 32 2.3ExistenceandUniquenessofSolutions ................... 35 2.3.1PreliminaryDenitions ......................... 35 2.3.2TheParabolicandHyperbolicSubsystems .............. 38 2.3.3TheHyperbolic-ParabolicSubsystem ................. 42 2.3.4ExistenceandUniquenessfortheFullSystem ........... 46 3SPATIALLYHOMOGENEOUSCASE ........................ 54 3.1SimplicationsandAssumptions ....................... 54 3.2EquilibriaandStability ............................. 55 3.3TheCase=0(NoViralBudding) ...................... 57 3.3.1TheCase>)]TJ /F3 11.955 Tf 11.96 0 Td[( .......................... 57 3.3.2TheCase<)]TJ /F3 11.955 Tf 11.96 0 Td[( .......................... 60 5

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4NUMERICALMETHODS .............................. 63 4.1TheNumericalAlgorithm ........................... 64 4.1.1Then=0Step ............................. 64 4.1.2Then=1Step ............................. 65 4.1.3ThenthStep .............................. 65 4.2AnalysisoftheNumericalMethod ...................... 68 4.2.1ParameterEstimation ......................... 68 4.2.2IntegralTermsandSpatialHomogeneity ............... 69 4.2.3SimulationsofTumorGrowthBehavior ................ 71 4.2.4StabilityandAccuracy ......................... 73 5FUTUREWORK ................................... 77 REFERENCES ....................................... 79 BIOGRAPHICALSKETCH ................................ 83 6

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LISTOFFIGURES Figure page 2-1Localcoordinatesystems .............................. 28 2-2Schematicdrawingsofcellarrangements ..................... 30 3-1Phaseportraitswith=0and>)]TJ /F3 11.955 Tf 11.95 0 Td[( ..................... 59 3-2Phasediagramswith=0 ............................. 60 3-3Phaseportraitswith=0and<)]TJ /F3 11.955 Tf 11.95 0 Td[( ..................... 61 4-1Effectofintegralcalculationondensityplots .................... 70 4-2ConvergencetotheIFE ............................... 72 4-3Exponentialtumorgrowth .............................. 73 4-4Convergencetotheendemicequilibrium ...................... 74 4-5Burstsizeeffectsontumorgrowthbehavior .................... 75 4-6Instabilityatlargetimes ............................... 76 7

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asingletransfectedcellcankillinexcessof150-200bystandercells[ 3 ].Ameaslesvaccinestrain,modiedherpessimplexvirus,andrecombinantvesicularstomatitisvirushavealsobeenshowntocauseanincreasedcytopathiceffectthroughtheformationofsyncytia[ 11 18 27 ].Fusogeniconcolyticvirusesalsohavetheadvantageofskippinganintracellularstepwheretherearebarrierstodiffusionandthepossibilityofbeingneutralizedbyantibodies.Additionally,thepresenceofasecondmechanismisbenecialwhentumorcellsmightbecomeresistanttothevirus[ 18 ].Thereareconictingremarksintheliteratureabouttheparticularmechanismswhicheventuallyleadtosyncytialdeath.Batemanetal.claimthatsyncytiaundergoanautophagicprocesswhichisnonapoptoticinnature[ 4 ]whileothershavefoundevidenceforapoptoticmechanisms[ 26 27 ].However,whatismorerelevantfortumortherapyisthatthedeathofsyncytiahasbeenshowntocauseapotentantitumorimmuneresponse[ 4 11 26 29 ]. 1.2PriorMathematicalModelsTheyear2001markedtheonsetofmathematicalmodelsintheliteraturedescribingvirotherapytreatmentoftumors.Severalordinarydifferentialequations(ODE)modelshavebeenformulatedtoanalyzedifferentfunctionalformsforviralinfectionandcancerproliferation.Someoftheseincludetheviruspopulationexplicitlyorimplicitlythroughincorporationofavirus-specicimmunesystemcomponent[ 12 28 38 43 44 49 ].Othertwo-dimensionalmodelsonlyconsidertheuninfectedandinfectedtumorcellpopulations[ 1 23 30 ].In2009,Paivaetal.presentedamultiscalemodelforthetherapy[ 31 ].Mostrecently,Crivellietal.developedamodelforcellcycle-specicvirotherapywhichincorporatedatime-delay[ 8 ],andWodarzetal.usedanagent-based,stochasticcomputationalmodeltostudythespatialdynamicsforvirusspreadthroughtumorpopulations[ 45 ].ThemodelswhicharemostrelevanttothepresentworkareODEmodelsincorporatingthemechanismofcell-to-cellfusionandpartialdifferentialequations(PDE)modelsforgeneralvirotherapy. 12

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atime-dependentvariableandhencethesystemisformulatedwithamovingboundary.Themodel,for00,thustakestheform[ 46 ]@x(r,t) @t=x(r,t))]TJ /F3 11.955 Tf 13.15 8.09 Td[(x(r,t) 2rcZr+rcr)]TJ /F7 7.97 Tf 6.58 0 Td[(rcv(s,t)ds)]TJ /F4 11.955 Tf 15.24 8.09 Td[(1 r2@ @r)]TJ /F5 11.955 Tf 5.48 -9.69 Td[(r2u(r,t)x(r,t),@y(r,t) @t=x(r,t) 2rcZr+rcr)]TJ /F7 7.97 Tf 6.58 0 Td[(rcv(s,t)ds)]TJ /F3 11.955 Tf 11.96 0 Td[(y(r,t))]TJ /F4 11.955 Tf 15.23 8.08 Td[(1 r2@ @r)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(r2u(r,t)y(r,t),@n(r,t) @t=y(r,t))]TJ /F3 11.955 Tf 11.96 0 Td[(n(r,t))]TJ /F4 11.955 Tf 15.23 8.08 Td[(1 r2@ @r)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(r2u(r,t)n(r,t),@v(r,t) @t=NZr+rcr)]TJ /F7 7.97 Tf 6.59 0 Td[(rc3[r2c)]TJ /F4 11.955 Tf 11.95 0 Td[((r)]TJ /F5 11.955 Tf 11.95 0 Td[(s)2] 4r3cy(s,t)ds)]TJ /F3 11.955 Tf 11.95 0 Td[(v(r,t), r2@ @r(r2u(r,t))=x(r,t))]TJ /F3 11.955 Tf 11.95 0 Td[(n(r,t),dR(t) dt=u(R(t),t).Wuetal.considerthreetypesofinitialconditionscorrespondingtouniform,core,andriminjectionofvirus.Foreachcase,aftersimplicationstothemodel,theyarriveataconditionwhichdetermineswhetherornotthevirusultimatelycontrolsthetumor.Theirmodelpredictsthatavirusdoeshavethecapacitytocontrolafast-growingtumorbuttheinjectionmustbeaggressive,thatiswithahighconcentrationofthevirusoverabroadandveryprecisespatialdistribution.In2003,Wuetal.expandedupontheirpreviousmodelbyincorporatingtheimmuneresponse[ 47 ].Inthatworktheyconcludedthat,withoutthepresenceofanimmunesuppressortomitigatetheimmuneresponsetothevirus,itwasimpossibleforthevirustoeradicatethetumor.Thismodelwasvalidatedwithpreclinicalandclinicaldataandanalysisofsimulationsconrmedthenecessityofanaggressiveadministrationofvirusfromaspatialstandpoint,althoughtheinitialvirusconcentrationwasdeemedlessimportant[ 42 ].FriedmanandTaoalsopublishedavirotherapymodelin2003[ 16 ].TheirmodeldiffersfromthatofWuetal.throughtheinclusionofaviraldiffusionterm,withoutwhichtheyarguethemodelofWuetal.wasnotmathematicallywell-posed.Inaddition, 14

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1.3.1OrdinaryDifferentialEquationsConsidertheautonomousordinarydifferentialequation _x=f(x)(1)denedonURnandletjjdenotethen-dimensionalEuclideannorm.ConsiderJUanopensubsetofRRnandsupposeallsolutionsofthesystemarecomplete,i.e.theyaredenedforallt2R. Denition1. Afunction:JU!Rngivenby(t,x)!(t,x)iscalledaowif(0,x)xand(t+s,x)=(t,(s,x))foralls,t2R.Clearly,ift!(t,x)denesthefamilyofsolutionsofthedifferentialequation( 1 )suchthat(0,x)x,thenisaow. Denition2. x02Rniscalledanequilibriumpointiff(x0)=0.Thatis,anequilibriumisaxedpointoftheow. Denition3. Anequilibriumpointofthedifferentialequation( 1 )isstableifforeach>0,thereisanumber>0suchthatj(t,x))]TJ /F5 11.955 Tf 12.85 0 Td[(x0j0suchthatlimt!1j(t,x))]TJ /F5 11.955 Tf 11.96 0 Td[(x0j=0wheneverjx)]TJ /F5 11.955 Tf 11.96 0 Td[(x0j<. Denition4. If(t,x)isanon-constantsolutionof( 1 )andthereexistsT>0suchthat(T,x)=xthen(t,x)iscalledaperiodicsolution.Theclosedcurvef(t,x)jt2Rgiscalledaperiodicorbit. Denition5. Aperiodicorbit)]TJ /F13 11.955 Tf 10.1 0 Td[(ofthedifferentialequation( 1 )isorbitallystableifforeachopensetVRnthatcontains)]TJ /F13 11.955 Tf 6.78 0 Td[(,thereisanopensetWVsuchthateverysolution,startingatapointinWatt=0,staysinVforallt0.Theperiodicorbitisorbitallyasymptoticallystableif,inaddition,thereisanopensubsetXWsuchthateverysolutionstartinginXisasymptoticto)]TJ /F13 11.955 Tf 10.1 0 Td[(ast!1. 16

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WenowproveapropositionpertainingtotheplanarsystemofgeneralizedLotka-Volterraformgivenby_x=x(a1x+b1y+c1) (1)_y=y(a2x+b2y+c2) (1)whereai,bi,cifori=1,2areconstants.LetJ(x,y)denotetheJacobianmatrixforthetransformation. Proposition1.1. Considerthesystem( 1 )-( 1 )withpositiveendemicequilibrium(x,y)anda1b2)]TJ /F5 11.955 Tf 12.24 0 Td[(a2b1>0.Let)]TJ /F13 11.955 Tf 10.1 0 Td[(beaperiodicorbitwithperiodT>0thatsurrounds(x,y).Iftr(J(x,y))<0then)]TJ /F13 11.955 Tf 10.1 0 Td[(isorbitallyasymptoticallystable.Iftr(J(x,y))>0then)]TJ /F13 11.955 Tf 10.1 0 Td[(isunstable. Proof. Notethat(x,y)isuniquebytheconditiona1b2)]TJ /F5 11.955 Tf 12.31 0 Td[(a2b1>0.Lethxidenotetheaverageofx(t)along)]TJ /F1 11.955 Tf 6.78 0 Td[(,i.e.hxi=1 TRT0x(t)dt.Then,integratingalong)]TJ /F1 11.955 Tf 6.77 0 Td[(,8>>><>>>:0=ZT0_x xdt=a1hxi+b1hyi+c10=ZT0_y ydt=a2hxi+b2hyi+c2whichimpliesthathxi=xandhyi=ybyuniquenessoftheendemicequilibrium.Itfollowsthat htr(J(x,y))i=a1hxi+b2hyi=a1x+b2y=tr(J(x,y)).(1)Furthermore,ifhtr(J(x,y))i<0then)]TJ /F1 11.955 Tf 10.1 0 Td[(isorbitallyasymptoticallystableandifhtr(J(x,y))i>0then)]TJ /F1 11.955 Tf 10.1 0 Td[(isunstable[ 6 ,Proposition2.130,page220].Thus,byequation( 1 ),theproofiscomplete. Thefollowingtwotheoremsareusefulinprovingtheexistenceanduniquenessofsolutionsofdifferentialequations.Wesupposethat(X,d)isametricspace. 17

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Denition6. Supposethat:X!X,andisarealnumbersuchthat0<1.Thefunctioniscalledacontraction(withcontractionconstant)ifd((x),(y))d(x,y)forallx,y2X. Theorem1.1(ContractionMappingTheorem). Ifthefunctionisacontractiononthecompletemetricspace(X,d),thenhasauniquexedpointx2X. Theorem1.2(Gronwall'sInequality). Supposethata
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Alongthecharacteristic,thesolutionthensatises d dt(x(t),t)=@ @t+@ @xdx dt=Q(,x(t),t).(1)Givenaninitialcondition,onecanthensolve( 1 )byintegratingalongthecharacteristic.ThestatementsintheremainderofthissectioncanbefoundinthetextsbyEvansandLeung[ 14 25 ].Considerasystemoflinearrst-orderPDEoftheform @~u @t+nXj=1Bj@~u @xj=~f(1)inRn(0,1)where~u:Rn[0,1)!Rm,~f:Rn[0,1)!Rm,andBj:Rn[0,1)!Mmmforj=1,...,nwhereMmmisthesetofreal-valuedmmmatrices. Denition8. Thesystem( 1 )ishyperbolicifthemmmatrixB(x,t;y)=nXj=1yjBj(x,t)isdiagonalizableforeachx,y2Rnandt0.NowconsidertheoperatorLunXi,j=1aij(x,t)@2u @xi@xj+nXi=1bi(x,t)@u @xi+c(x,t)u)]TJ /F3 11.955 Tf 13.15 8.08 Td[(@u @tinT=(0,T],withT>0,andinRn,openandbounded.WeassumethatthecoefcientsinLareboundedfunctionsinT. Denition9. Lis(uniformly)parabolicinTifthereexists>0suchthatforevery(x,t)2Tandforanyrealvector6=0,nXi,j=1aij(x,t)ij>jj2. 19

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Theorem1.3(ParabolicComparisonPrinciple). SupposeLisaparabolicoperatorwithc0.Ifw,v2C2,1(T)\C( T)satisfy8>>>>>><>>>>>>:LwLvinT@v @@w @on@(0,T]v(x,0)w(x,0)forx2 thenv(x,t)w(x,t)on [0,T]. 1.3.3NumericalMethodsAnintroductiontonumericalmethodsisprovidedbyEpperson[ 13 ]aswellasKincaidandCheney[ 22 ].Wetakethefollowingdenitionsfromthesetexts.Therstgivesaquadratureformulaforapproximatingtheintegralofafunctionoveraninterval[a,b]. Denition10. ThetrapezoidruleisgivenbyZbaf(x)dxb)]TJ /F5 11.955 Tf 11.96 0 Td[(a 2[f(a)+f(b)].Nowsupposewewanttonumericallysolvethedifferentialequation dy dt=f(t,y),(1)onthetimeinterval[0,T].Lettbethetimestepandconsidertimenodestn=nt(n=0,...,M)whereMt=T.Letyn=y(tn). Denition11. Euler'sMethodforequation( 1 )isgivenbyyn+1=yn+tf(tn,yn). 20

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Euler'smethod,arisingfromthestandardtangentlineapproximation,isrst-orderaccurate[ 13 ,Theorem6.3,Page333].Anothermethodtosolveequation( 1 )isamulti-stepmethodbasedontheintegrationofinterpolatingpolynomials.Supposethevaluesy(tn)]TJ /F7 7.97 Tf 6.59 0 Td[(k)fork=0,1,...,pareknown. Denition12. TheLagrangefunctions(fortheAdams-Bashforthmethod)areLk(s)=pYi=0,i6=ks)]TJ /F5 11.955 Tf 11.95 0 Td[(tn)]TJ /F7 7.97 Tf 6.59 0 Td[(i tn)]TJ /F7 7.97 Tf 6.59 0 Td[(k)]TJ /F5 11.955 Tf 11.96 0 Td[(tn)]TJ /F7 7.97 Tf 6.58 0 Td[(ifork=0,1,...,p. Denition13. TheAdams-Bashforthmethodoforderp+1forequation( 1 )isgivenbyyn+1=yn+pXk=0kf(tn)]TJ /F7 7.97 Tf 6.58 0 Td[(k,yn)]TJ /F7 7.97 Tf 6.58 0 Td[(k)wherek=Ztn+1tnLk(s)ds.Asimilarfamilyofmethods,calledtheBackwardDifferentiationFormula(BDF)family,isderivedinthecasethataninterpolatingpolynomialfory(t),insteadoff(t,y(t)),isconstructedatthenodestn)]TJ /F7 7.97 Tf 6.59 0 Td[(p+1,tn)]TJ /F7 7.97 Tf 6.59 0 Td[(p+2,...,tn+1. Denition14. TheLagrangefunctions(fortheBDFmethods)areLk(s)=p)]TJ /F9 7.97 Tf 6.59 0 Td[(1Yi=)]TJ /F9 7.97 Tf 6.59 0 Td[(1,i6=ks)]TJ /F5 11.955 Tf 11.96 0 Td[(tn)]TJ /F7 7.97 Tf 6.59 0 Td[(i tn)]TJ /F7 7.97 Tf 6.58 0 Td[(k)]TJ /F5 11.955 Tf 11.96 0 Td[(tn)]TJ /F7 7.97 Tf 6.59 0 Td[(ifork=)]TJ /F4 11.955 Tf 9.29 0 Td[(1,0,...,p)]TJ /F4 11.955 Tf 11.95 0 Td[(1.Thentheinterpolatingpolynomialisdenedbyq(t)=p)]TJ /F9 7.97 Tf 6.59 0 Td[(1Xk=)]TJ /F9 7.97 Tf 6.59 0 Td[(1Lk(t)y(tn)]TJ /F7 7.97 Tf 6.59 0 Td[(k).Approximatingy0(tn+1)byq0(tn+1)impliesthefollowingmethod. 21

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Denition15. TheBackwardDifferentiationFormula(BDF)methodwithpstepsisgivenbyy(tn+1)=p)]TJ /F9 7.97 Tf 6.58 0 Td[(1Xk=0ky(tn)]TJ /F7 7.97 Tf 6.59 0 Td[(k)+f(tn+1,y(tn+1))wherek=)]TJ /F5 11.955 Tf 13.59 8.09 Td[(L0k(tn+1) L0)]TJ /F9 7.97 Tf 6.59 0 Td[(1(tn+1),=1 L0)]TJ /F9 7.97 Tf 6.59 0 Td[(1(tn+1).Fornumericalanalysisofpartialdifferentialequations,wemustadditionallydiscretizethespatialdomain.Letxbethemeshsizeandthespatialnodesbegivenbyxi=ix(i=0,...,J)forsomeJ>0.Finitedifferencesareusedtoapproximatespatialderivatives. Denition16. Spatialderivativesapproximatedbycentraldifferencesaregivenbyf0(xi)f(xi+1))]TJ /F5 11.955 Tf 11.95 0 Td[(f(xi)]TJ /F9 7.97 Tf 6.59 0 Td[(1) 2x,f00(xi)f(xi+1))]TJ /F4 11.955 Tf 11.95 0 Td[(2f(xi)+f(xi)]TJ /F9 7.97 Tf 6.59 0 Td[(1) x2. Denition17. TheLeapfrogschemeisanumericalmethodwhichusescentraldifferencesintimeandspace[ 35 ].Finally,animportantnecessary,butnotsufcient,conditionforthestabilityofnumericalmethodsforpartialdifferentialequationswasestablishedbyCourant,FriedrichsandLewyin1928[ 7 39 ].ConsideraPDEwithanalyticalsolutionu(x,t)anditscorrespondingapproximationv(x,t)determinedbysomenumericalmethod.Denotevni=v(xi,tn). Denition18. Themathematicaldomainofdependence,X(x,t),ofasolutionu(x,t)isthesetofallthepointsinspacewheretheinitialdataatt=0mayhaveaneffectonu(x,t). Denition19. Thenumericaldomainofdependenceforaxedvaluet,Xt(x,t),isthesetofallpointsxjwhoseinitialdatav0jenterintothecomputationofv(x,t).Dene 22

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takesintotheaccounttheprobabilityofsuccessofviralentry.Thederivationofthecorrespondingintegralexpressioninequations( 2 )and( 2 )willbediscussedinSection 2.1.3 .Wemakethesimplifyingassumptionthatthereisnochangeingeometryduringsyncytiaformation.Thatis,weconsideracellthatissyncytia-incorporatedtostillbesphericalofradiusrc.Anuninfectedcellcanfuseintoasyncytiaifitisincontactwitheitheranuninfectedcellorasyncytia-incorporatedcell.Weassumethissyncytiaincorporationoccursataratewithcoefcientandisproportionaltotheaveragedensityofneighboringuninfectedandsyncytia-incorporatedcells.WewillderiveinSection 2.1.3 theexactformulationofthecorrespondingintegralterminequations( 2 )and( 2 ).Asingleinfectedcellcanbeincorporatedintoasyncytiathroughsurfacecontactwithacellofanyothertype,againatarateproportionalto.Sinceweneglectnecrosisweassumethatimmediatelyupondeathacellisremoved.Forinfectedcellsthisprocessoccursatrateandforsyncytiaatrate.Weallowfreeviralparticlestobegeneratedthroughtwomechanisms,buddingandlysis.Aninfectedorsyncytia-incorporatedcellreleasesviralparticlesthroughbuddingatarate.Itishypothesizedthatsyncytiaareremovedviaanon-apoptoticmechanismthatdoesn'tallowviralrelease[ 4 ].Thereforeweassumethatonlyinfectedcellsundergolysisupondeath,releasingNviralparticles.Moredetailonthecorrespondingbuddingandlysistermsinequation( 2 )isdiscussedinSection 2.1.3 .Wefurtherassumethatfreeviralparticlesareremovedatrate.Therefore,for00,thedynamicsofthestatevariablesaredeterminedby@x(r,t) @t=x(r,t))]TJ /F3 11.955 Tf 14.81 8.09 Td[(x(r,t) jIrc(r,t)jZIrc(r,t)v(s,t)ds)]TJ /F3 11.955 Tf 17.64 8.09 Td[(x(r,t) jI2rc(r,t)jZI2rc(r,t)y(s,t)+z(s,t)ds)]TJ /F4 11.955 Tf 15.23 8.09 Td[(1 r2@ @r)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(r2u(r,t)x(r,t), (2)@y(r,t) @t=x(r,t) jIrc(r,t)jZIrc(r,t)v(s,t)ds)]TJ /F4 11.955 Tf 11.95 0 Td[((+)y(r,t))]TJ /F4 11.955 Tf 15.24 8.09 Td[(1 r2@ @r)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(r2u(r,t)y(r,t), (2) 25

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@z(r,t) @t=x(r,t) jI2rc(r,t)jZI2rc(r,t)y(s,t)+z(s,t)ds+y(r,t))]TJ /F3 11.955 Tf 11.96 0 Td[(z(r,t))]TJ /F4 11.955 Tf 15.23 8.09 Td[(1 r2@ @r)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(r2u(r,t)z(r,t), (2)@v(r,t) @t=N jJrc(r,t)jZIrc(r,t)[r2c)]TJ /F4 11.955 Tf 11.95 0 Td[((r)]TJ /F5 11.955 Tf 11.96 0 Td[(s)2]y(s,t)ds+ jIrc(r,t)jZIrc(r,t)y(s,t)+z(s,t)ds)]TJ /F3 11.955 Tf 11.96 0 Td[(v(r,t)+1 r2@ @rr2@v(r,t) @r (2)whereIrc(r,t)=(max[0,r)]TJ /F5 11.955 Tf 11.96 0 Td[(rc],min[R(t),r+rc])andjJrc(r,t)j=ZIrc(r,t)[r2c)]TJ /F4 11.955 Tf 10.11 0 Td[((r)]TJ /F5 11.955 Tf 10.11 0 Td[(s)2]ds.Thelasttermineachofequations( 2 ),( 2 ),and( 2 )followsfromtheassumptionoftheadvectiveforceactingonthetumorcells.Noticethattheviralparticles,beingofnegligiblevolume,donotundergoadvection.However,theyareassumedtodiffusewithdiffusioncoefcient.Treatingthetumorasanincompressibleuid,weassumethatthetotaltumorcelldensityhasaconstantvalue.Thatis, x(r,t)+y(r,t)+z(r,t)=(2)for00,@x(r,t) @t+u(r,t)@x(r,t) @r=x(r,t))]TJ /F3 11.955 Tf 14.81 8.09 Td[(x(r,t) jIrc(r,t)jZIrc(r,t)v(s,t)ds)]TJ /F3 11.955 Tf 17.64 8.09 Td[(x(r,t) jI2rc(r,t)jZI2rc(r,t))]TJ /F5 11.955 Tf 11.96 0 Td[(x(s,t)ds 26

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)]TJ /F5 11.955 Tf 13.15 8.08 Td[(x(r,t) [x(r,t))]TJ /F3 11.955 Tf 11.96 0 Td[(y(r,t))]TJ /F3 11.955 Tf 11.95 0 Td[(()]TJ /F5 11.955 Tf 11.96 0 Td[(x(r,t))]TJ /F5 11.955 Tf 11.95 0 Td[(y(r,t))], (2)@y(r,t) @t+u(r,t)@y(r,t) @r=x(r,t) jIrc(r,t)jZIrc(r,t)v(s,t)ds)]TJ /F4 11.955 Tf 11.96 0 Td[((+)y(r,t))]TJ /F5 11.955 Tf 13.15 8.08 Td[(y(r,t) [x(r,t))]TJ /F3 11.955 Tf 11.96 0 Td[(y(r,t))]TJ /F3 11.955 Tf 11.95 0 Td[(()]TJ /F5 11.955 Tf 11.96 0 Td[(x(r,t))]TJ /F5 11.955 Tf 11.96 0 Td[(y(r,t))], (2)@v(r,t) @t=N jJrc(r,t)jZIrc(r,t)[r2c)]TJ /F4 11.955 Tf 11.96 0 Td[((r)]TJ /F5 11.955 Tf 11.95 0 Td[(s)2]y(s,t)ds+ jIrc(r,t)jZIrc(r,t))]TJ /F5 11.955 Tf 11.96 0 Td[(x(s,t)ds)]TJ /F3 11.955 Tf 11.96 0 Td[(v(r,t)+1 r2@ @rr2@v(r,t) @r, (2)u(r,t)=1 r2Zr0s2[)]TJ /F3 11.955 Tf 9.3 0 Td[(+(+)x(s,t)+()]TJ /F3 11.955 Tf 11.95 0 Td[()y(s,t)]ds, (2)dR(t) dt=u(R(t),t). (2)WeassumeaStefan,ormoving,boundaryconditionwhichgivesequation( 2 );theradiusofthetumorR(t)ismovingatexactlytherateofthelocaladvectivevelocity. 2.1.2BoundaryandInitialConditionsTheboundaryconditionsatthetumorcenter, @x(r,t) @r=@y(r,t) @r=@v(r,t) @r=u(r,t)=0atr=0,(2)followfromradialsymmetry.WeadditionallyassumetheNeumannboundarycondition @v(r,t) @r=0atr=R(t).(2)InSection 2.2 ,wewillmakeatransformationtoxthemovingboundaryandnootherconditionswillbeneededatthetumorboundary. 27

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Theinitialconditionsaregivenby x(r,0)=x0(r),y(r,0)=y0(r),v(r,0)=v0(r),R(0)=R0(2)for0rR(t)wherex0(r)0,y0(r)0,x0(r)+y0(r),v0(r)0,R0>0andx0(r),y0(r)andv0(r)arecontinuouslydifferentiablefunctionson[0,R0]. 2.1.3DerivationofIntegralTerms A BFigure2-1. Localcoordinatesystemswiththeiroriginsatthecenterofatumorcelladistancerfromthetumorcenter.A)Spherical(~r,~,~)andCartesian(~x,~y,~z).B)Cylindrical(~,~,~z)andCartesian(~x,~y,~z). 28

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Wuetal.[ 46 ]describeindetailtheformulationofsimilarintegraltermsforinfectionandlysiswhere,instead,thelimitsofintegrationarer)]TJ /F5 11.955 Tf 12.51 0 Td[(rcandr+rc.However,hereweacknowledgethenecessityofIrc(r,t)andJrc(r,t),asdenedinSection 2.1.1 ,tomaintainthewell-posednessofthemodel.UsingIrc(r,t)ensuresthatthelimitsofintegrationarewithinthephysicalboundsofthetumor,i.e.weareonlyintegratingover0rR(t).Weadopttheirnotationandlikewiseconstructlocalspherical(~r,~,~),Cartesian(~x,~y,~z),andcylindrical(~,~,~z)coordinateswiththe~zaxisparalleltotheradialdirectionsofthetumorasshowninFigure 2-1 .Forthetermdescribingthesyncytia-incorporationofanuninfectedcellataradialdistancerfromthetumorcenter,weseektond w2rc(r,t),thespatially-weightedaverageofitsneighboringinfectedandsyncytia-incorporatedcells.Weobservethatcellsareneighboringifandonlyifthedistancebetweentheircentersisexactly2rc(Figure 2-2A ).LetW2rc(r,t)beitstotalnumberofneighboringinfectedandsyncytia-incorporatedcells.Itholdsthats=r+~z,ds=d~z,~z=2rccos~,andd~z=)]TJ /F4 11.955 Tf 9.3 0 Td[(2rcsin~d~.LetImin=max[0,r)]TJ /F4 11.955 Tf 12.53 0 Td[(2rc]andImax=min[R(t),r+2rc].Subsequently,wedenea=arccosImax)]TJ /F5 11.955 Tf 11.96 0 Td[(r 2rcandb=arccosImin)]TJ /F5 11.955 Tf 11.95 0 Td[(r 2rc.Since0~2,itfollowsthatab.ThereforewecalculatethatW2rc(r,t)=Z20Zba(y(r+~z,t)+z(r+~z,t))(2rc)2sin~d~d~=8r2cZba(y(r+~z,t)+z(r+~z,t))sin~d~=)]TJ /F4 11.955 Tf 9.3 0 Td[(4rcZImin)]TJ /F7 7.97 Tf 6.58 0 Td[(rImax)]TJ /F7 7.97 Tf 6.59 0 Td[(r(y(r+~z,t)+z(r+~z,t))d~z=4rcZImax)]TJ /F7 7.97 Tf 6.59 0 Td[(rImin)]TJ /F7 7.97 Tf 6.59 0 Td[(r(y(r+~z,t)+z(r+~z,t))d~z 29

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=4rcZImaxImin(y(s,t)+z(s,t))ds=4rcZI2rc(s,t)()]TJ /F5 11.955 Tf 11.95 0 Td[(x(s,t))ds. ASyncytia-incorporation BBudding CInfection DLysisFigure2-2. Schematicdrawingsofcellarrangementspertainingtoderivationofintegraltermsforsyncytia-incorporation(A),viralbudding(B),viralinfection(C),andlysis(D).Foracellofinterest(gray),thederivationincludestheappropriatecells(blue)thatareaprescribeddistanceaway(red).Celltypesindicatedareuninfectedcells(x),infectedcells(y),andsyncytia-incorporatedcells(z).vrepresentsafreeviralparticle. 30

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Likewise,thetotalsurfacearea,A2rc(r,t),thatweareintegratingoveriscalculatedbyA2rc(r,t)=Z20Zba(2rc)2sin~d~d~=4rcjI2rc(r,t)j.Thespatially-weightedaverage w2rc(r,t)isthengivenbyW2rc(r,t) A2rc(r,t)=1 jI2rc(r,t)jZI2rc(r,t)()]TJ /F5 11.955 Tf 11.96 0 Td[(x(s,t))ds.Henceweobtaintheintegraltermforsyncytiaformationinequation( 2 ).Notethat,becauseweassumeinfectedcellscanformsyncytiawithanyothertypeofcell,theanalogousrateofsyncytiaformationforinfectedcellsinequation( 2 )reducesto.Theterminequation( 2 )forthegenerationoffreeviralparticlesbybuddingisderivedsimilarly.Afreeviralparticlegeneratedatadistancerfromthetumorcentermustbereleasedfromaninfectedcellorsyncytia-incorporatedcellthatisadistancercaway(Figure 2-2B ).Therefore,analogoustoabove,thespatially-weightedaverage, wrc(r,t),ofsuchcellsisgivenbyWrc(r,t) Arc(r,t)=1 jIrc(r,t)jZIrc(r,t)()]TJ /F5 11.955 Tf 11.96 0 Td[(x(s,t))dsandwethusobtaintheintegraltermforbudding.Likewise,theinfectionofatumorcelloccursbythepresenceofafreeviralparticleonthesurfaceofthecell,i.e.atadistancercawayfromthecellcenter(Figure 2-2C ),sotheintegraltermforinfectioninequation( 2 )followsanalogously.Weassumethatlysisofaninfectedcellreleasesfreeviralparticlesuniformlythroughoutthevolumeofthecell.Soifafreeviralparticleisgeneratedbylysisatadistancerfromthetumorcenter,thenitmusthavebeenreleasedfromaninfectedcellwhosecenterislessthanorequaltorcawayfromit(Figure 2-2D ).Hence,toderivethelysisintegralterm,weconsiderthespatially-weightedvolumeaverage, yrc(r,t),ofinfectedcellswithinadistancercfromtheviralparticle.Letimin=max[0,r)]TJ /F5 11.955 Tf 12.63 0 Td[(rc]andimax=min[R(t),r+rc].Usingthecylindricalcoordinates,asinFigure 2-1B ,wecalculate 31

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Yrc(r,t),thetotalnumberofinfectedcellswithinadistancercfromtheviralparticle,tobeYrc(r,t)=Z20Zimax)]TJ /F7 7.97 Tf 6.58 0 Td[(rimin)]TJ /F7 7.97 Tf 6.58 0 Td[(rZp r2c)]TJ /F9 7.97 Tf 6.67 0 Td[(~z20y(r+~z,t)~d~d~zd~=Zimax)]TJ /F7 7.97 Tf 6.59 0 Td[(rimin)]TJ /F7 7.97 Tf 6.59 0 Td[(r(r2c)]TJ /F4 11.955 Tf 12.1 0 Td[(~z2)y(r+~z,t)d~z=ZIrc(r,t)[r2c)]TJ /F4 11.955 Tf 11.96 0 Td[((s)]TJ /F5 11.955 Tf 11.96 0 Td[(r)2]y(s,t)ds.ThetotalvolumeVrc(r,t)thatweareintegratingoverisVrc(r,t)=Z20Zimax)]TJ /F7 7.97 Tf 6.59 0 Td[(rimin)]TJ /F7 7.97 Tf 6.59 0 Td[(rZp r2c)]TJ /F9 7.97 Tf 6.66 0 Td[(~z20~d~d~zd~=ZIrc(r,t)[r2c)]TJ /F4 11.955 Tf 11.96 0 Td[((s)]TJ /F5 11.955 Tf 11.95 0 Td[(r)2]ds=jJrc(r,t)j.Therefore, yrc(r,t)=Yrc(r,t) Vrc(r,t)=1 jJrc(r,t)jZIrc(r,t)[r2c)]TJ /F4 11.955 Tf 11.95 0 Td[((s)]TJ /F5 11.955 Tf 11.95 0 Td[(r)2]y(s,t)dsfromwhichthelysisterminequation( 2 )follows. 2.1.4TumorGrowthintheAbsenceofInfectionTogetabettergrasponthetumorgrowthbehaviorasdeterminedbyequations( 2 )and( 2 ),letusconsiderthesimplecasewherethereisnoinfection.Letx(r,t)=for0rR(t)andt0.Thenbyequation( 2 )wehaveu(r,t)=1 r2Zr0s2ds=r 3.Then,byequation( 2 ),wecandeterminethatthetumorradiusattimetisR(t)=R0et=3.Thatis,intheabsenceofviralinfection,thetumorgrowsexponentiallywithgrowthrate 3. 2.2TransformationoftheSystemBeforeprovingthemainexistenceanduniquenessresultofthischapterwewillperformatransformationtothesystem( 2 )-( 2 )inordertoxthemovingboundary.ThenowroutinetransformationwasrstintroducedbyLandau[ 24 ]in1950.Wedene 32

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newspaceandtimevariables,(r,t):=r R(t)and(r,t):=t.Weobservethat@ @r=1 R(t)and@ @=@ @t@t @=)]TJ /F5 11.955 Tf 9.3 0 Td[(r (R(t))2dR(t) dt=)]TJ /F3 11.955 Tf 9.3 0 Td[( R()dR() d.Wedene~x(,):=x((r,t)R(t),t)andlet~y(,),~v(,),and~u(,)bedenedsimilarly.Thensincex(r,t)=~x(,)itfollowsthat@x(r,t) @t=@~x(,) @t=@~x(,) @@ @t+@~x(,) @@ @t.Therefore @~x(,) @=@x(r,t) @t)]TJ /F3 11.955 Tf 13.15 8.09 Td[(@~x(,) @@ @t=@x(r,t) @t+ R()dR() d@~x(,) @.(2)Similarly,equation( 2 )holdsfor~yand~v.Theviraldiffusiontermistransformedby r2@ @rr2@v(r,t) @r= 2R2(t)1 R(t)@ @2R2(t)1 R(t)@~v(,) @= R2()1 2@ @2@~v(,) @.TransformationoftheintegraltermsrequiresthedenitionofIrc(,)=max0,)]TJ /F5 11.955 Tf 21.11 8.09 Td[(rc R(),min1,+rc R()andjJrc(,)j=ZIrc(,)[r2c)]TJ /F5 11.955 Tf 11.95 0 Td[(R2()(!)]TJ /F3 11.955 Tf 11.95 0 Td[()2]d!. 33

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Thereforethetransformedmodelbecomes,for2(0,1]and>0,@~x(,) @+)]TJ /F3 11.955 Tf 19.54 8.09 Td[( R()dR() d+~u(,) R()@~x(,) @=~x(,))]TJ /F3 11.955 Tf 14.81 8.09 Td[(~x(,) jIrc(,)jZIrc(,)~v(!,)d!)]TJ /F3 11.955 Tf 17.64 8.09 Td[(~x(,) jI2rc(,)jZI2rc(,))]TJ /F4 11.955 Tf 12.14 0 Td[(~x(!,)d!)]TJ /F4 11.955 Tf 13.34 8.09 Td[(~x(,) [~x(,))]TJ /F3 11.955 Tf 11.96 0 Td[(~y(,))]TJ /F3 11.955 Tf 11.96 0 Td[(()]TJ /F4 11.955 Tf 12.14 0 Td[(~x(,))]TJ /F4 11.955 Tf 12.24 0 Td[(~y(,)], (2)@~y(,) @+)]TJ /F3 11.955 Tf 19.54 8.08 Td[( R()dR() d+~u(,) R()@~y(,) @=~x(,) jIrc(,)jZIrc(,)~v(!,)d!)]TJ /F4 11.955 Tf 11.95 0 Td[((+)~y(,))]TJ /F4 11.955 Tf 13.44 8.09 Td[(~y(,) [~x(,))]TJ /F3 11.955 Tf 11.96 0 Td[(~y(,))]TJ /F3 11.955 Tf 11.96 0 Td[(()]TJ /F4 11.955 Tf 12.14 0 Td[(~x(,))]TJ /F4 11.955 Tf 12.24 0 Td[(~y(,)], (2)@~v(,) @)]TJ /F3 11.955 Tf 22.2 8.09 Td[( R()dR() d@~v(,) @=N jJrc(,)jZIrc(,)[r2c)]TJ /F5 11.955 Tf 11.95 0 Td[(R2()()]TJ /F3 11.955 Tf 11.95 0 Td[(!)2]~y(!,)d!+ jIrc(,)jZIrc(,))]TJ /F4 11.955 Tf 12.14 0 Td[(~x(!,)d!)]TJ /F3 11.955 Tf 11.95 0 Td[(~v(,)+ R2()1 2@ @2@~v(,) @, (2)~u(,)=R() 2Z0!2[)]TJ /F3 11.955 Tf 9.3 0 Td[(+(+)~x(!,)+()]TJ /F3 11.955 Tf 11.96 0 Td[()~y(!,)]d!, (2)dR() d=~u(1,). (2)Theboundaryandinitialconditionsaregiven,for>0,by@~v @(0,)=@~v @(1,)=0, (2)~v(,0)=~v0()for01, (2)@~x @(0,)=@~y @(0,)=0 (2)~x(,0)=~x0(),~y(,0)=~y0()for01. (2) 34

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~u(0,)=0, (2)R(0)=R0. (2)Weadditionallywillassumethat~x0()0,~y0()0,~x0()+~y0(),~v0()0,R0>0,@~x0 @(0)=@~y0 @(0)=@~v0 @(0)=@~v0 @(1)=0,andthat~x0(),~y0(),and~v0()arecontinuouslydifferentiablefunctionson[0,1].Wenotethatwhen=1thecoefcientof@~x(,) @inequation( 2 )is1 R())]TJ /F5 11.955 Tf 10.5 8.09 Td[(dR() d+~u(1,)whichbyequation( 2 )iszero.Thuswedonotneedtoimposeaboundaryconditionfor@~x(1,) @.Thesamejusticationholdsfor@~y(1,) @. 2.3ExistenceandUniquenessofSolutionsThemainresultofthischapterestablishestheexistenceanduniquenessofsolutionstothesystem( 2 )-( 2 ).FriedmanandTaoprovedexistenceanduniquenessofsolutionstoasimplermodelforoncolyticvirotherapyin[ 16 ].ThegeneraloutlineoftheproofpresentedinthissectionfollowsthatbyFriedmanandTaobutthedetailscontainadditionalconsiderationsforthenonlocalintegraltermsandthepresenceofsyncytiaandbudding.Webeginbydeningtwocompletemetricspacesoffunctions.Lemma 1 thenestablishestheexistenceofasolutiontotheparabolicsystem( 2 ),( 2 ),( 2 )givenxed~u,~x,and~y.Lemma 2 ,ontheotherhand,providesasolutiontothehyperbolicsystem( 2 ),( 2 ),( 2 ),( 2 )forxed~uand~v. 2.3.1PreliminaryDenitionsWeusekkthroughouttodenotethesupnorm.Thatis,forafunctionf:X!R,weletkfk=supx2Xjf(x)j 35

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andforf:X!R2,givenbyx!(f1(x),f2(x)),weletkfk=max(kf1k,kf2k).WedenotebyC1,0thespaceoffunctionsthatarecontinuouslydifferentiablewithrespecttothespatialvariableandcontinuouswithrespecttotime.ForT>0weconsiderthecompletemetricspaceoffunctionsETC1,0([0,1][0,T])givenbyET=~u(,)2C1,0([0,1][0,T]):~u(0,)=0,~u(,0)=~u0(),k~ukL,whereL>R0max+ 3, 3,and~u0()=R0 2Z0!2[)]TJ /F3 11.955 Tf 9.3 0 Td[(+(+)~x0(!)+()]TJ /F3 11.955 Tf 11.95 0 Td[()~y0(!)]d!.WenotethatETisnonemptysince~u(,)~u0()2ET.Foragiven~u2ETwedeneR()by R()=R0+Z0~u(1,s)ds.(2)Weobservethatfor,any~u2ET, 0><>>:d d=)]TJ /F3 11.955 Tf 20.23 8.08 Td[( R()dR() d+~u(,) R()(s;,s)=.(2) 36

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Theexistenceanduniquenessofforevery(,s)2[0,1][0,T]followsfromastandardtheoremofordinarydifferentialequationssincetherighthandsideofequation( 2 )iscontinuouslydifferentiablewithrespecttoandcontinuouswithrespectto[ 6 ,Theorem1.261,Page138].Notethat0isacharacteristicsince~u(0,)=0and1isalsoacharacteristicbyequation( 2 ).Thusbyuniqueness, 0<(;,s)<1(2)forany(,s)2(0,1)[0,T].Wewillsuppresstwoargumentsandusethenotation()=(;,s).ForT>0wealsowillconsiderthecompletemetricspaceofpairsoffunctionsgivenbyST=f(x((),),y((),)):x((),),y((),)2C[0,T],0x((),),0y((),)g.WemustmakeonenalobservationwhichisthatjIrc((),)j,thelengthoftheintervalIrc((),),isuniformlyboundedaboveandbelowforany2[0,T]and2[0,1].Indeed,itfollowsfromthedenitionthatjIrc((),)j22rc R(),1,1)]TJ /F3 11.955 Tf 11.95 0 Td[(()+rc R(),()+rc R().Furthermore,recallinginequalities( 2 )and( 2 ),min2rc R(),1,1)]TJ /F3 11.955 Tf 11.95 0 Td[(()+rc R(),()+rc R()min1,rc R()min1,rc R0+TLandmax2rc R(),1,1)]TJ /F3 11.955 Tf 11.95 0 Td[(()+rc R(),()+rc R()max2rc R(),1+rc R()max2rc R0)]TJ /F5 11.955 Tf 11.95 0 Td[(TL,1+rc R0)]TJ /F5 11.955 Tf 11.95 0 Td[(TL. 37

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Therefore,forany2[0,T]and2[0,1] min1,rc R0+TLjIrc((),)jmax2rc R0)]TJ /F5 11.955 Tf 11.96 0 Td[(TL,1+rc R0)]TJ /F5 11.955 Tf 11.95 0 Td[(TL.(2) 2.3.2ExistenceandUniquenessfortheParabolicandHyperbolicSubsystems Lemma1. Consider~u2ETand(^x,^y)2ST.Thenthereexistsaunique,continu-oussolution~vtothesystem( 2 ),( 2 ),( 2 )with^xand^yreplacing~xand~y,respectively.Furthermore,forany(,)2[0,1][0,T] 0~v(,)max(N+) ,k~v0k.(2) Proof. LetR()bedenedbyequation( 2 ).WedenetheparabolicoperatorL~v= R2()@2~v @2+2 + R()dR() d@~v @)]TJ /F3 11.955 Tf 11.95 0 Td[(~v)]TJ /F3 11.955 Tf 13.15 8.09 Td[(@~v @andthefunctionf(,)=N jJrc(,)jZIrc(,)[r2c)]TJ /F5 11.955 Tf 11.96 0 Td[(R2()()]TJ /F3 11.955 Tf 11.95 0 Td[(!)2]^y(!,)d!+ jIrc(,)jZIrc(,))]TJ /F4 11.955 Tf 12.14 0 Td[(^x(!,)d!.Thenequation( 2 )isequivalenttoL~v+f(,)=0foranysolution~vtoequation( 2 )with^xand^yreplacing~xand~y,respectively.IfweconvertbacktoCartesiancoordinates,weseethatallthecoefcientsoftheparabolicoperatorarebounded,andthustheexistenceanduniquenessofaclassicalsolution~vfollowsfromstandardparabolictheory[ 15 ,Theorem2,Page144].Furthermore,wenotethatthecoefcientsoftheoperatorLareboundedoneveryclosedballcontainedin(0,1][0,T]andf(,)0since^y0and^x.Thereforetheparaboliccomparisonprinciple[ 25 ,Theorem1.2-4,Page18]impliesthat~v0. 38

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Now,letk=max(N+) ,k~v0k.ThenLk+f(,)0and~v(,0)kfor2[0,1].Thereforewederivetheupperboundgivenby( 2 )againbytheparaboliccomparisonprinciple. Lemma2. Consider~u2ETand~v0continuousandbounded.ThenifT>0issufcientlysmall,thereexistunique,continuoussolutions~xand~ytothesystem( 2 )-( 2 ),( 2 )-( 2 )on[0,1][0,T]. Proof. LetR()bedenedbyequation( 2 )and()beasdenedbysystem( 2 ).LetZ((),)=(~x((),),~y((),))T.LetM>max(k~x0k,k~y0k).WeconsiderthecompletemetricspaceoffunctionsBTC[0,T]denedasBT=fZ((),)2C[0,T]:kZkMandZ((0),0)=Z0((0))g.NotethatBTisnonemptysinceZ((),)Z0((0))2BT.Alongthecharacteristiccurve()wecanwritethesystem( 2 )-( 2 )as d dZ((),)=f(Z((),),(),)+h(Z((),),(),)ZI2rc((),)Z(!,)d!(2)wheref(Z,(),)=2664+)]TJ /F3 11.955 Tf 11.96 0 Td[()]TJ /F3 11.955 Tf 39.9 8.09 Td[( jIrc((),)jZIrc((),)~vd!0 jIrc((),)jZIrc((),)~vd!)]TJ /F3 11.955 Tf 11.96 0 Td[()]TJ /F3 11.955 Tf 11.96 0 Td[(3775Z+Z2664)]TJ /F3 11.955 Tf 10.5 8.09 Td[(+ )]TJ /F3 11.955 Tf 10.49 8.08 Td[()]TJ /F3 11.955 Tf 11.96 0 Td[( 3775TZandh(Z,(),)=264 jI2rc((),)j000375Z.Then,recallingtheboundsforjIrc((),)jgivenbyinequality( 2 )andthefactthat~visbounded,itfollowsthatfandhareLipschitzcontinuousinZandbounded 39

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onBT[0,1][0,T].SothereexistpositiveconstantsL1andL2suchthat,foranyZ1,Z22BT,2[0,T],()2[0,1],jf(Z1((),),(),))]TJ /F5 11.955 Tf 11.95 0 Td[(f(Z2((),),(),)jL1kZ1)]TJ /F5 11.955 Tf 11.95 0 Td[(Z2k,jh(Z1((),),(),))]TJ /F5 11.955 Tf 11.95 0 Td[(h(Z2((),),(),)jL2kZ1)]TJ /F5 11.955 Tf 11.95 0 Td[(Z2k.Wedenethemapping:BT!C[0,T]by(Z)((),)=Z0((0))+Z0 f(Z((s),s),(s),s)+h(Z((s),s),(s),s)ZI2rc((s),s)Z(!,s)d!!ds. (2)andobservethatasolutiontoequation( 2 )mustsatisfy(Z)=Z.Weclaimthat:BT!BT.Indeed,letZ((),)2BT.Clearly(Z)((),)2C[0,1].Itisalsoimmediatethat(Z)((0),0)=Z0((0)).Fromequation( 2 )weseethat,becauseMwaschosensothatkZ0k
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)]TJ /F5 11.955 Tf 9.3 0 Td[(h(Z2((s),s),(s),s)ZI2rc((s),s)Z2(!,s)d!!dsZ0jf(Z1((s),s),(s),s))]TJ /F5 11.955 Tf 11.96 0 Td[(f(Z2((s),s),(s),s)jds+Z0h(Z1((s),s),(s),s)ZI2rc((s),s)Z1(!,s)d!)]TJ /F5 11.955 Tf 9.29 0 Td[(h(Z1((s),s),(s),s)ZI2rc((s),s)Z2(!,s)d!ds+Z0h(Z1((s),s),(s),s)ZI2rc((s),s)Z2(!,s)d!)]TJ /F5 11.955 Tf 9.29 0 Td[(h(Z2((s),s),(s),s)ZI2rc((s),s)Z2(!,s)d!dsZ0jf(Z1((s),s),(s),s))]TJ /F5 11.955 Tf 11.96 0 Td[(f(Z2((s),s),(s),s)jds+Z0jh(Z1((s),s),(s),s)j ZI2rc((s),s)jZ1(!,s))]TJ /F5 11.955 Tf 11.96 0 Td[(Z2(!,s)jd!!ds+Z0jh(Z1((s),s),(s),s))]TJ /F5 11.955 Tf 11.95 0 Td[(h(Z2((s),s),(s),s)j ZI2rc((s),s)jZ2(!,s)jd!!dsZ0L1kZ1)]TJ /F5 11.955 Tf 11.96 0 Td[(Z2kds+Z0khkZ10kZ1)]TJ /F5 11.955 Tf 11.95 0 Td[(Z2kd!ds+Z0L2kZ1)]TJ /F5 11.955 Tf 11.95 0 Td[(Z2kZ10Md!ds(L1+khk+L2M)TkZ1)]TJ /F5 11.955 Tf 11.96 0 Td[(Z2k.Therefore,ifTissufcientlysmall,k(Z1))]TJ /F4 11.955 Tf 11.95 0 Td[((Z2)kKkZ1)]TJ /F5 11.955 Tf 11.96 0 Td[(Z2k 41

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forsomeK2[0,1).Thatis,isacontractiononBTforsufcientlysmallT.BytheContractionMappingTheoremthereexistsauniquexedpointofinBTwhichisasolutiontoequation( 2 ). 2.3.3ExistenceandUniquenessfortheHyperbolic-ParabolicSubsystemGivenaxed~u2ETwearenowabletoprovetheexistenceanduniquenessofasolutiontothecoupledhyperbolic-parabolicsystem( 2 )-( 2 ),( 2 )-( 2 )byconsideringacontractiononthespaceSTdenedinSection 2.3.1 Theorem2.1. IfT>0issufcientlysmall,thenforany~u2ETthereexistsaunique,continuoussolution(~x,~y,~v)ofthesystem( 2 )-( 2 ),( 2 )-( 2 )on[0,1][0,T].Furthermore,forany(,)2[0,1][0,T],0~x(,),0~y(,),0~v(,)max(N+) ,k~v0k. Proof. Let~u(,)2ET.DeneR()byequation( 2 ).Observethatforany(,)2[0,1][0,T],(~x((),),~y((),))(~x0(0),~y0(0))2STwhere(0;,)=0.ThereforeSTisnonempty.For^Z=(^x,^y)2ST,deneamapHbythefollowing.Given^Z=(^x,^y)2STrstapplyLemma 1 tosolvefor~v.Thengiven~v,solvefor~xand~ybyLemma 2 .LetH(^Z)=ZwhereZ=(~x,~y)T.NowweshowthatH:ST!ST.Let(^x,^y)2STand~vbethecorrespondingsolutionfromLemma 1 .Fromequation( 2 )itisclearthat,alongthecharacteristiccurve(),d d(~x((),))=A1(,(),~x,~y)~x((),)foranappropriatelychosenboundedfunctionA1.Therefore~x00impliesthat~x((),)0.Recallingthat~v0itfollowsthenfromequation( 2 )thatd d(~y((),))A2(~x,~y)~y((),) 42

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foranappropriatelychosenboundedfunctionA2.Therefore~y00impliesthat~y((),)0.Let~z=)]TJ /F4 11.955 Tf 12.14 0 Td[(~x)]TJ /F4 11.955 Tf 12.25 0 Td[(~y.Thend d(~z((),))=~x jI2rc((),)jZI2rc((),))]TJ /F4 11.955 Tf 12.14 0 Td[(~xd!+~y)]TJ /F3 11.955 Tf 11.96 0 Td[(~z)]TJ /F4 11.955 Tf 13.29 8.09 Td[(~z (~x)]TJ /F3 11.955 Tf 11.96 0 Td[(~y)]TJ /F3 11.955 Tf 11.96 0 Td[(~z).If~x+~y=thend d(~z((),))0implying~x+~yforall.Therefore(~x,~y)2ST.ItremainstoshowthatHisacontractionforsufcientlysmallT.Let^Z1,^Z22STandZi=H^Zifori=1,2.Let~vibethesolutionfromLemma 1 correspondingtoZi.WerstclaimthatthereexistsC0>0suchthat k~v1)]TJ /F4 11.955 Tf 12.24 0 Td[(~v2kC0k^Z1)]TJ /F4 11.955 Tf 15.12 2.66 Td[(^Z2k.(2)Indeed,forLdenedasinLemma 1 wehaveL(~v1)]TJ /F4 11.955 Tf 14.02 0 Td[(~v2)+~f(,)=0where~f(,)=N jJrc(,)jZIrc(,)[r2c)]TJ /F5 11.955 Tf 11.37 0 Td[(R2()()]TJ /F3 11.955 Tf 11.37 0 Td[(!)2](^y1)]TJ /F4 11.955 Tf 11.66 0 Td[(^y2)d!)]TJ /F3 11.955 Tf 31.79 8.09 Td[( jIrc(,)jZIrc(,)(^x1)]TJ /F4 11.955 Tf 11.55 0 Td[(^x2)d!.Letk=N k^y1)]TJ /F4 11.955 Tf 12.66 0 Td[(^y2k+ k^x1)]TJ /F4 11.955 Tf 12.56 0 Td[(^x2k.ThenLk+~f(,)0and(~v1)]TJ /F4 11.955 Tf 14.44 0 Td[(~v2)(,0)=0for2[0,1].Thereforetheparaboliccomparisonprinciple[ 25 ,Theorem1.2-4,Page18]impliesthatforall(,)2[0,1][0,T],j(~v1)]TJ /F4 11.955 Tf 12.25 0 Td[(~v2)(,)jkandthuswederiveinequality( 2 ). 43

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Byformallyintegratingequations( 2 )-( 2 )alongthecharacteristic,weseethatfori=1,2,ZimustsatisfyZi((),)=Z0((0))+Z0f(Zi((s),s))+g(Zi((s),s),(s),s)ZIrc((s),s)~vi(!,s)d!+h(Zi((s),s),(s),s)ZI2rc((s),s)Zi(!,s)d!#ds (2)withf(Z)=264+)]TJ /F3 11.955 Tf 11.96 0 Td[(00)]TJ /F3 11.955 Tf 11.95 0 Td[()]TJ /F3 11.955 Tf 11.95 0 Td[(375Z+Z2664)]TJ /F3 11.955 Tf 10.49 8.09 Td[(+ )]TJ /F3 11.955 Tf 10.49 8.09 Td[()]TJ /F3 11.955 Tf 11.96 0 Td[( 3775TZ,g(Z,(s),s)=2664)]TJ /F3 11.955 Tf 36.63 8.08 Td[( jIrc((s),s)j0 jIrc((s),s)j03775Zandh(Z,(s),s)=264 jI2rc((s),s)j000375Z.Itfollowsthatf,g,andhareLipschitzcontinuousinZandboundedonBT,BT[0,1][0,T]andBT[0,1][0,T],respectively.ThereforethereexistpositiveC1,C2,andC3suchthatforanyZ1,Z22BT,2[0,T],and()2[0,1],jf(Z1((),)))]TJ /F5 11.955 Tf 11.95 0 Td[(f(Z2((),))jC1kZ1)]TJ /F5 11.955 Tf 11.96 0 Td[(Z2k,jg(Z1((),),(),))]TJ /F5 11.955 Tf 11.95 0 Td[(g(Z2((),),(),)jC2kZ1)]TJ /F5 11.955 Tf 11.96 0 Td[(Z2k,jh(Z1((),),(),))]TJ /F5 11.955 Tf 11.95 0 Td[(h(Z2((),),(),)jC3kZ1)]TJ /F5 11.955 Tf 11.96 0 Td[(Z2k.Fromequation( 2 )wecanthendeterminethatforany2[0,T],jZ1((),))]TJ /F5 11.955 Tf 11.96 0 Td[(Z2((),)j=Z0f(Z1))]TJ /F5 11.955 Tf 11.96 0 Td[(f(Z2)+g(Z1,(s),s)ZIrc((s),s)~v1d!)]TJ /F5 11.955 Tf 11.95 0 Td[(g(Z2,(s),s)ZIrc((s),s)~v2d!+h(Z1,(s),s)ZI2rc((s),s)Z1d!)]TJ /F5 11.955 Tf 11.95 0 Td[(h(Z2,(s),s)ZI2rc((s),s)Z2d!!ds=Z0f(Z1))]TJ /F5 11.955 Tf 11.96 0 Td[(f(Z2)+g(Z1,(s),s)ZIrc((s),s)(~v1)]TJ /F4 11.955 Tf 12.24 0 Td[(~v2)d! 44

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+(g(Z1,(s),s))]TJ /F5 11.955 Tf 11.95 0 Td[(g(Z2,(s),s))ZIrc((s),s)~v2d!+h(Z1,(s),s)ZI2rc((s),s)(Z1)]TJ /F5 11.955 Tf 11.95 0 Td[(Z2)d!+(h(Z1,(s),s))]TJ /F5 11.955 Tf 11.95 0 Td[(h(Z2,(s),s))ZI2rc((s),s)Z2d!!dsZ0jf(Z1))]TJ /F5 11.955 Tf 11.95 0 Td[(f(Z2)j+jg(Z1,(s),s)jZ10j~v1)]TJ /F4 11.955 Tf 12.25 0 Td[(~v2jd!+jg(Z1,(s),s))]TJ /F5 11.955 Tf 11.96 0 Td[(g(Z2,(s),s)jZ10j~v2jd!+jh(Z1,(s),s)jZ10jZ1)]TJ /F5 11.955 Tf 11.95 0 Td[(Z2jd!+jh(Z1,(s),s))]TJ /F5 11.955 Tf 11.95 0 Td[(h(Z2,(s),s)jZ10jZ2jd!dsZ0(C1kZ1)]TJ /F5 11.955 Tf 11.96 0 Td[(Z2k+kgkk~v1)]TJ /F4 11.955 Tf 12.24 0 Td[(~v2k+C2k~v2kkZ1)]TJ /F5 11.955 Tf 11.96 0 Td[(Z2k+khkjjZ1)]TJ /F5 11.955 Tf 11.95 0 Td[(Z2k+C3kZ2kkZ1)]TJ /F5 11.955 Tf 11.96 0 Td[(Z2k)ds.Itfollowsfrominequality( 2 ),andthefactthathandZ2arebounded,thatkZ1)]TJ /F5 11.955 Tf 11.95 0 Td[(Z2kkgkk~v1)]TJ /F4 11.955 Tf 12.25 0 Td[(~v2kT+ZT0C4kZ1)]TJ /F5 11.955 Tf 11.96 0 Td[(Z2kdsforsomeC4>0.ThereforebyGronwall'sinequalityandinequality( 2 ),kZ1)]TJ /F5 11.955 Tf 11.96 0 Td[(Z2kkgkk~v1)]TJ /F4 11.955 Tf 12.25 0 Td[(~v2kTexpZT0C4dskgkk~v1)]TJ /F4 11.955 Tf 12.25 0 Td[(~v2kTexp(C4T)kgkexp(C4T)TC0k^Z1)]TJ /F4 11.955 Tf 13.36 2.66 Td[(^Z2k.ThereforeHisacontractionforTsufcientlysmallsincekgkexp(C4T)TC0!0asT!0.BytheContractionMappingTheoremthereexistsauniquexedpoint(~x,~y)ofHinSTandacorresponding~vsuchthat(~x,~y,~v)isasolutiontothesystem( 2 )-( 2 ),( 2 )-( 2 ). 45

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2.3.4ExistenceandUniquenessfortheFullSystemTheorem 2.1 providesamapfrom~u2ETtoasolution(~x,~y,~v)ofthehyperbolic-parabolicsubsystem.Thus,byconsideringacontractiononET,wearereadytoprovethemainresultofthischapter,existenceanduniquenessoflocalsolutionstothefullsystem( 2 )-( 2 ). Theorem2.2. IfT>0issufcientlysmall,thenthereexistsaunique,continuoussolution(~x,~y,~v,~u,R)tothesystem( 2 )-( 2 )on[0,1][0,T]. Proof. For~u2ET,wedeneamapF:~u(,)!~w(,)byrstsolvingfor(~x,~y,~v)byTheorem 2.1 .Thenwedene ~w(,)=R() 2Z0!2[)]TJ /F3 11.955 Tf 9.3 0 Td[(+(+)~x(!,)+()]TJ /F3 11.955 Tf 11.95 0 Td[()~y(!,)]d!,~w(0,)=0.(2)WeclaimthatF:ET!ET.Let~u2ETand~w=F~u.Thenclearly~w(,)2C1,0([0,1][0,T])since~x,~y,andR()arecontinuousand~w(,0)=~u0().Furthermore,)]TJ /F4 11.955 Tf 9.3 0 Td[((+))]TJ /F3 11.955 Tf 21.92 0 Td[(+(+)~x(!,)+()]TJ /F3 11.955 Tf 11.95 0 Td[()~y(!,)impliesthat)]TJ /F4 11.955 Tf 10.49 8.09 Td[((+) 3R()~w(,) 3R().Byinequality( 2 )weconcludethatk~wkmax+ 3, 3(R0+LT)
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fori=1,2.LetCithroughoutbeanappropriatelychosenpositiveconstant.Forany2(0,1]and2[0,T],itfollowsfromequation( 2 )thatj~w1(,))]TJ /F4 11.955 Tf 13.64 0 Td[(~w2(,)j=R1() 2Z0!2[)]TJ /F3 11.955 Tf 9.3 0 Td[(+(+)~x1(!,)+()]TJ /F3 11.955 Tf 11.96 0 Td[()~y1(!,)]d!)]TJ /F5 11.955 Tf 10.49 8.09 Td[(R2() 2Z0!2[)]TJ /F3 11.955 Tf 9.3 0 Td[(+(+)~x2(!,)+()]TJ /F3 11.955 Tf 11.96 0 Td[()~y2(!,)]d!(R1())]TJ /F5 11.955 Tf 11.96 0 Td[(R2()) 2Z0!2[)]TJ /F3 11.955 Tf 9.3 0 Td[(+(+)~x1(!,)+()]TJ /F3 11.955 Tf 11.96 0 Td[()~y1(!,)]d!+R2() 2Z0!2[(+)(~x1(!,))]TJ /F4 11.955 Tf 12.13 0 Td[(~x2(!,))+()]TJ /F3 11.955 Tf 11.95 0 Td[()(~y1(!,))]TJ /F4 11.955 Tf 12.24 0 Td[(~y2(!,))]d!:=I1+I2.Byequation( 2 )wehavejI1jjR1())]TJ /F5 11.955 Tf 11.96 0 Td[(R2()j 2Z0!2j)]TJ /F3 11.955 Tf 12.61 0 Td[(+(+)~x1(!,)+()]TJ /F3 11.955 Tf 11.96 0 Td[()~y1(!,)jd!=R0~u1(1,s))]TJ /F4 11.955 Tf 12.2 0 Td[(~u2(1,s)ds 2Z0!2j)]TJ /F3 11.955 Tf 12.62 0 Td[(+(+)~x1(!,)+()]TJ /F3 11.955 Tf 11.96 0 Td[()~y1(!,)jd!Tk~u1)]TJ /F4 11.955 Tf 12.2 0 Td[(~u2k 2Z0!2(+(+)k~x1k+j)]TJ /F3 11.955 Tf 11.95 0 Td[(jk~y1k)d!TC1 3k~u1)]TJ /F4 11.955 Tf 12.2 0 Td[(~u2kC2Tk~u1)]TJ /F4 11.955 Tf 12.2 0 Td[(~u2k.ToestimateI2wewillrstconsiderkZ1)]TJ /F5 11.955 Tf 12.82 0 Td[(Z2kwithZi=(~xi,~yi)Tandsatisfyingequation( 2 )asinLemma 2 .For(,)2[0,1][0,T]andfori=1,2,leti(s)=i(s;,)bethebackwardscharacteristiccurveforZithroughthepoint(,).Let=2rc R().ItfollowsthatZ1(,))]TJ /F5 11.955 Tf 11.96 0 Td[(Z2(,)=Z0(1(0)))]TJ /F5 11.955 Tf 11.96 0 Td[(Z0(2(0)) 47

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+Z0[f(Z1(1(s)),1(s),s))]TJ /F5 11.955 Tf 11.95 0 Td[(f(Z2(2(s)),2(s),s)]ds+Z0"h(Z1(1(s)),1(s),s)ZI2rc(1(s),s)Z1d!)]TJ /F5 11.955 Tf 9.3 0 Td[(h(Z2(2(s)),2,s)ZI2rc(2(s),s)Z2ds!#ds:=J1+J2+J3.ToestimateJiweneedtoestimatej1(s;,))]TJ /F3 11.955 Tf 11.33 0 Td[(2(s;,)j.Fromsystem( 2 )wehaved(1(s))]TJ /F3 11.955 Tf 11.96 0 Td[(2(s)) ds=)]TJ /F3 11.955 Tf 11.94 8.09 Td[(1(s) R1(s)dR1(s) ds+~u1(1(s),s) R1(s))]TJ /F8 11.955 Tf 11.96 16.86 Td[()]TJ /F3 11.955 Tf 11.95 8.09 Td[(2(s) R2(s)dR2(s) ds+~u2(2(s),s) R2(s)=)]TJ /F4 11.955 Tf 21.61 8.09 Td[(1 R1(s)dR1(s) ds(1(s))]TJ /F3 11.955 Tf 11.95 0 Td[(2(s)))]TJ /F3 11.955 Tf 11.96 0 Td[(2(s)1 R1(s)dR1(s) ds)]TJ /F4 11.955 Tf 24.27 8.09 Td[(1 R2(s)dR2(s) ds+~u1(1(s),s) R1(s)R2(s)(R2(s))]TJ /F5 11.955 Tf 11.96 0 Td[(R1(s))+1 R2(s)(~u1(1(s),s))]TJ /F4 11.955 Tf 12.2 0 Td[(~u1(2(s),s))+1 R2(s)(~u1(2(s),s))]TJ /F4 11.955 Tf 12.2 0 Td[(~u2(2(s),s)).Recallinginequalities( 2 )and( 2 )weobservethateachtermontherighthandsideisboundedbyaconstantmultipleofeitherk~u1)]TJ /F4 11.955 Tf 12.85 0 Td[(~u2kork1)]TJ /F3 11.955 Tf 12.6 0 Td[(2k.Therefore,byintegratingwehavefor0sj1(s;,))]TJ /F3 11.955 Tf 11.95 0 Td[(2(s;,)jZsC3k~u1)]TJ /F4 11.955 Tf 12.2 0 Td[(~u2k+C4k1)]TJ /F3 11.955 Tf 11.96 0 Td[(2kdtandthereforek1)]TJ /F3 11.955 Tf 11.96 0 Td[(2kC3Tk~u1)]TJ /F4 11.955 Tf 12.2 0 Td[(~u2k+Z0C4k1)]TJ /F3 11.955 Tf 11.96 0 Td[(2kdt.ThenGronwall'sinequalityimplies k1)]TJ /F3 11.955 Tf 11.96 0 Td[(2kC5Tk~u1)]TJ /F4 11.955 Tf 12.2 0 Td[(~u2k.(2) 48

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Z0()isLipschitzcontinuoussinceitiscontinuouslydifferentiableon[0,1].Therefore, jJ1jC6k1)]TJ /F3 11.955 Tf 11.95 0 Td[(2k.(2)Wenowdenethefunctionj(g,(s),s):=ZI2rc((s),s)gd!fors2[0,T],(s)2[0,1]andg2L1([0,1]).Notethatj(g,(s),s)isclearlyLipschitzcontinuousing.Additionally,weclaimthatjisLipschitzcontinuousin.Indeed,jj(g,1(s),s))]TJ /F5 11.955 Tf 11.96 0 Td[(j(g,2(s),s)j=ZI2rc(1(s),s)gd!)]TJ /F8 11.955 Tf 11.96 16.27 Td[(ZI2rc(2(s),s)gd!ZI2rc(1(s),s)gd!)]TJ /F8 11.955 Tf 11.96 16.27 Td[(ZI2rc(1(s),s)\I2rc(2(s),s)gd!+ZI2rc(1(s),s)\I2rc(2(s),s)gd!)]TJ /F8 11.955 Tf 11.96 16.27 Td[(ZI2rc(2(s),s)gd!=ZI2rc(1(s),s)nI2rc(2(s),s)gd!+ZI2rc(2(s),s)nI2rc(1(s),s)gd!IfI2rc(1(s),s)andI2rc(2(s),s)aredisjoint,thenjI2rc(i(s),s)j2rc R()j2(s))]TJ /F3 11.955 Tf 12.07 0 Td[(1(s)jfori=1,2.IfI2rc(1(s),s)\I2rc(2(s),s)6=;,thennecessarilyjI2rc(1(s),s)nI2rc(2(s),s)j
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LetL1,L2,...,L5bepositiveconstantssuchthatforanyZ,Z1,Z2,,1,2,s,jf(Z,1(s),s))]TJ /F5 11.955 Tf 11.96 0 Td[(f(Z,2(s),s)jL1j1(s))]TJ /F3 11.955 Tf 11.96 0 Td[(2(s)j,jf(Z1,(s),s))]TJ /F5 11.955 Tf 11.95 0 Td[(f(Z2,(s),s)jL2jZ1)]TJ /F5 11.955 Tf 11.95 0 Td[(Z2j,jh(Z1,1(s),s))]TJ /F5 11.955 Tf 11.95 0 Td[(h(Z2,2(s),s)jL3jZ1)]TJ /F5 11.955 Tf 11.95 0 Td[(Z2j,jj(Z,1(s),s))]TJ /F5 11.955 Tf 11.96 0 Td[(j(Z,2(s),s)jL4j1(s))]TJ /F3 11.955 Tf 11.95 0 Td[(2(s)j,jj(Z1,(s),s))]TJ /F5 11.955 Tf 11.96 0 Td[(j(Z2,(s),s)jL5jZ1)]TJ /F5 11.955 Tf 11.95 0 Td[(Z2j.TheexistenceofL3followsfrominequality( 2 )whichprovesthatjI2rc((s),s)jisboundedindependentof.Wealsorecallthathisbounded.ToestimatejJ2jandjJ3jwemustrstconsiderthatbyequation( 2 )wehave,forany1and2,jZ(1(),))]TJ /F5 11.955 Tf 11.95 0 Td[(Z(2(),)j=Z0(1(0)))]TJ /F5 11.955 Tf 11.95 0 Td[(Z0(2(0))+Z0[f(Z(1(s),s),1(s),s))]TJ /F5 11.955 Tf 11.96 0 Td[(f(Z(2(s),s),2(s),s)]ds+Z0[h(Z(1(s),s),1(s),s)j(Z,1(s),s))]TJ /F5 11.955 Tf 9.3 0 Td[(h(Z2(2(s),s),2(s),s)j(Z,2(s),s)]dsC6k1)]TJ /F3 11.955 Tf 11.96 0 Td[(2k+Z0jf(Z(1(s),s),1(s),s))]TJ /F5 11.955 Tf 11.96 0 Td[(f(Z(1(s),s),2(s),s)jds+Z0jf(Z(1(s),s),2(s),s))]TJ /F5 11.955 Tf 11.96 0 Td[(f(Z(2(s),s),2(s),s)jds+Z0jh(Z(1(s),s),1(s),s)jjj(Z,1(s),s))]TJ /F5 11.955 Tf 11.95 0 Td[(j(Z,2(s),s)jds+Z0jj(Z,2(s),s)jjh(Z(1(s),s),1(s),s))]TJ /F5 11.955 Tf 11.95 0 Td[(h(Z(2(s),s),2(s),s)jdsC6k1)]TJ /F3 11.955 Tf 11.96 0 Td[(2k+Z0L1j1(s))]TJ /F3 11.955 Tf 11.95 0 Td[(2(s)j+L2jZ(1(s),s))]TJ /F5 11.955 Tf 11.96 0 Td[(Z(2(s),s)jds+Z0khkL4j1(s))]TJ /F3 11.955 Tf 11.95 0 Td[(2(s)j+ML3jZ(1(s),s))]TJ /F5 11.955 Tf 11.96 0 Td[(Z(2(s),s)jds. 50

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ThereforejZ(1(),))]TJ /F5 11.955 Tf 11.95 0 Td[(Z(2(),)jC8k1)]TJ /F3 11.955 Tf 11.95 0 Td[(2k+Z0C9jZ(1(s),s))]TJ /F5 11.955 Tf 11.96 0 Td[(Z(2(s),s)jdsandGronwall'sinequalitygivespositiveL6suchthat,foranyZand2[0,1], jZ(1(),))]TJ /F5 11.955 Tf 11.95 0 Td[(Z(2(),)jL6k1)]TJ /F3 11.955 Tf 11.96 0 Td[(2k.(2)NowtoestimatejJ2jwecalculatethatjf(Z1(1(s),s),1(s),s))]TJ /F5 11.955 Tf 11.96 0 Td[(f(Z2(2(s),s),2(s),s)jjf(Z1(1(s),s),1(s),s))]TJ /F5 11.955 Tf 11.95 0 Td[(f(Z1(1(s),s),2(s),s)j+jf(Z1(1(s),s),2(s),s))]TJ /F5 11.955 Tf 11.95 0 Td[(f(Z1(2(s),s),2(s),s)j+jf(Z1(2(s),s),2(s),s))]TJ /F5 11.955 Tf 11.95 0 Td[(f(Z2(2(s),s),2(s),s)jL1j1(s))]TJ /F3 11.955 Tf 11.95 0 Td[(2(s)j+L2jZ1(1(s),s))]TJ /F5 11.955 Tf 11.95 0 Td[(Z1(2(s),s)j+L2jZ1(2(s),s))]TJ /F5 11.955 Tf 11.96 0 Td[(Z2(2(s),s)jL1j1(s))]TJ /F3 11.955 Tf 11.95 0 Td[(2(s)j+L2L6j1(s))]TJ /F3 11.955 Tf 11.96 0 Td[(2(s)j+L2kZ1)]TJ /F5 11.955 Tf 11.95 0 Td[(Z2kfromwhichitfollowsthat jJ2jC7Tk1)]TJ /F3 11.955 Tf 11.96 0 Td[(2k+Z0L2kZ1)]TJ /F5 11.955 Tf 11.96 0 Td[(Z2kds.(2)FinallywecanestimatejJ3j.Itholdsthath(Z1(1(s),s),1(s),s)ZI2rc(1(s),s)Z1d!)]TJ /F5 11.955 Tf 11.96 0 Td[(h(Z2(2(s),s),2(s),s)ZI2rc(2(s),s)Z2d!=jh(Z1(1(s),s),1(s),s)j(Z1,1(s),s))]TJ /F5 11.955 Tf 11.95 0 Td[(h(Z2(2(s),s),2(s),s)j(Z2,2(s),s)jjh(Z1(1(s),s),1(s),s)j(Z1,1(s),s))]TJ /F5 11.955 Tf 11.95 0 Td[(h(Z1(2(s),s),1(s),s)j(Z1,1(s),s)j+jh(Z1(2(s),s),1(s),s)j(Z1,1(s),s))]TJ /F5 11.955 Tf 11.95 0 Td[(h(Z2(2(s),s),2(s),s)j(Z1,1(s),s)j+jh(Z2(2(s),s),2(s),s)j(Z1,1(s),s))]TJ /F5 11.955 Tf 11.95 0 Td[(h(Z2(2(s),s),2(s),s)j(Z2,1(s),s)j 51

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+jh(Z2(2(s),s),2(s),s)j(Z2,1(s),s))]TJ /F5 11.955 Tf 11.95 0 Td[(h(Z2(2(s),s),2(s),s)j(Z2,2(s),s)jjj(Z1,1(s),s)jjh(Z1(1(s),s),1(s),s))]TJ /F5 11.955 Tf 11.96 0 Td[(h(Z1(2(s),s),1(s),s)j+jj(Z1,1(s),s)jjh(Z1(2(s),s),1(s),s))]TJ /F5 11.955 Tf 11.96 0 Td[(h(Z2(2(s),s),2(s),s)j+jh(Z2(2(s),s),2(s),s)jjj(Z1,1(s),s))]TJ /F5 11.955 Tf 11.96 0 Td[(j(Z2,1(s),s)j+jh(Z2(2(s),s),2(s),s)jjj(Z2,1(s),s))]TJ /F5 11.955 Tf 11.96 0 Td[(j(Z2,2(s),s)jML3jZ1(1(s),s))]TJ /F5 11.955 Tf 11.96 0 Td[(Z1(2(s),s)j+ML3jZ1(2(s),s))]TJ /F5 11.955 Tf 11.95 0 Td[(Z2(2(s),s)j+L5khkkZ1)]TJ /F5 11.955 Tf 11.96 0 Td[(Z2k+L4khkk1)]TJ /F3 11.955 Tf 11.96 0 Td[(2kC10k1)]TJ /F3 11.955 Tf 11.96 0 Td[(2k+C11kZ1)]TJ /F5 11.955 Tf 11.96 0 Td[(Z2kwherethelastinequalityholdsby( 2 ).Hence, jJ3jC10Tk1)]TJ /F3 11.955 Tf 11.96 0 Td[(2k+Z0C11kZ1)]TJ /F5 11.955 Tf 11.95 0 Td[(Z2kds.(2)Bysumminginequalities( 2 ),( 2 ),and( 2 ),andrecalling( 2 ),wecalculatethatforTsufcientlysmall,kZ1)]TJ /F5 11.955 Tf 11.95 0 Td[(Z2kC12Tk~u1)]TJ /F4 11.955 Tf 12.19 0 Td[(~u2k+Z0C13kZ1)]TJ /F5 11.955 Tf 11.96 0 Td[(Z2kdsfromwhichGronwall'sinequalityimpliesthatkZ1)]TJ /F5 11.955 Tf 11.95 0 Td[(Z2kC12Texp(C13T)k~u1)]TJ /F4 11.955 Tf 12.2 0 Td[(~u2k.Finallywehave,jI2jjR2()j 2Z0!2C14Tk~u1)]TJ /F4 11.955 Tf 12.2 0 Td[(~u2kd!C15Tk~u1)]TJ /F4 11.955 Tf 12.2 0 Td[(~u2k 52

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andthereforek~w1)]TJ /F4 11.955 Tf 15.42 0 Td[(~w2kC16Tk~u1)]TJ /F4 11.955 Tf 12.2 0 Td[(~u2k.ThusFisacontractionforTsufcientlysmall.BytheContractionMappingTheoremthereexistsauniquexedpoint~uofFinETandcorresponding~x,~y,~v,Rwhichsolvethesystem( 2 )-( 2 ),( 2 )-( 2 )on[0,1][0,T]. 53

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CHAPTER3SPATIALLYHOMOGENEOUSCASE 3.1SimplicationsandAssumptionsWebeginanalysisofourmodelinthecaseofaviralinjectionadministeredattime=0whichisspatiallyuniformthroughoutthetumor.Thatis,forsome00theinitialconditionsarex(r,0)=(1)]TJ /F5 11.955 Tf 11.96 0 Td[(q),y(r,0)=q,v(r,0)=v0for0
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or,equivalently,_x=x+)]TJ /F3 11.955 Tf 11.96 0 Td[()]TJ /F3 11.955 Tf 13.15 8.09 Td[( +)]TJ /F3 11.955 Tf 11.95 0 Td[()]TJ /F3 11.955 Tf 11.95 0 Td[(+ x +)]TJ /F3 11.955 Tf 11.96 0 Td[()]TJ /F3 11.955 Tf 13.15 8.09 Td[(N y (3)_y=y)]TJ /F3 11.955 Tf 11.96 0 Td[()]TJ /F3 11.955 Tf 11.96 0 Td[(+N )]TJ /F3 11.955 Tf 11.95 0 Td[()]TJ /F3 11.955 Tf 11.95 0 Td[(x +()]TJ /F3 11.955 Tf 11.95 0 Td[()y + x()]TJ /F5 11.955 Tf 11.95 0 Td[(x), (3)_R=R 3[(+)x+()]TJ /F3 11.955 Tf 11.95 0 Td[()y)]TJ /F3 11.955 Tf 11.95 0 Td[(]. (3)Let=f(x,y):x0,y0,x+yg.Thatisinvariantundertheowdeterminedbyequations( 3 )-( 3 )followsfromtheresultsofSection 2.3 .Wenotethatequations( 3 )and( 3 )aredecoupledfromequation( 3 ).Therefore,ifwecandeterminethebehaviorofsolutionsto( 3 )-( 3 ),wecandeterminefrom( 3 )thelong-termtumorbehavior.Ifeventually_R<0thenthetumorexponentiallydecaysinsizeandweconsiderthetreatmentsuccessful.Ifeventually_R>0thenthetumorexponentiallygrowsandthetreatmentfails. 3.2EquilibriaandStabilitySinceequations( 3 )and( 3 )aredecoupledfromequation( 3 ),tondtheequilibria(x,y)ofthesystemweneedonlyset( 3 )and( 3 )equaltozero.Weeasilyobtainthetrivialequilibrium(0,0)andtheinfection-freeequilibrium(,0).Inthecasethat<)]TJ /F3 11.955 Tf 12.48 0 Td[(therealsoexistsanequilibriumwhereonlyinfectedcellsarepresent,0,1)]TJ /F15 7.97 Tf 16.75 5.26 Td[( )]TJ /F15 7.97 Tf 6.59 0 Td[(.Solvingforanendemicequilibriumamountstosolvingasystemofonelinearandoneseconddegreeequationinthesimplexx+y.Weobservethattheequilibrium(,0)isonesuchsolutionandthustherecanexistatmostonepositiveendemicequilibrium(xs,ys)ofthesystem.They-interceptofthelinearequationresultingfromequation( 3 )isy1=+)]TJ /F3 11.955 Tf 11.96 0 Td[()]TJ /F15 7.97 Tf 13.16 5.26 Td[( N +)]TJ /F15 7.97 Tf 6.59 0 Td[( 55

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andfurthercalculationrevealsthatthex-coordinateofthepositiveendemicequilibriumisxs=y1 ()]TJ /F3 11.955 Tf 11.95 0 Td[())]TJ /F4 11.955 Tf 11.96 0 Td[(()]TJ /F3 11.955 Tf 11.96 0 Td[()]TJ /F3 11.955 Tf 11.95 0 Td[() N )]TJ /F15 7.97 Tf 13.15 5.26 Td[(+ +y1()]TJ /F15 7.97 Tf 6.58 0 Td[() 2+ y1.Itfollowsthat(xs,ys)existsifandonlyif0
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3.3TheCase=0(NoViralBudding)Forsomeviruses,buddingmaynotoccurortheratemaybenegligiblecomparedtolysisandsyncytiaformation.Therefore,inthissectionwewillneglectthebuddingmechanismofviralrelease.With=0theODEmodel( 3 )-( 3 )hastheformofageneralizedLotka-Volterrasystem.Thelocalasymptoticstabilityconditionfor(0,0)reducestothesingleinequality +)]TJ /F3 11.955 Tf 11.96 0 Td[(<0(3)and0,1)]TJ /F15 7.97 Tf 16.76 5.25 Td[( )]TJ /F15 7.97 Tf 6.58 0 Td[(isl.a.s.if+)]TJ /F3 11.955 Tf 13.15 8.09 Td[(N 1)]TJ /F3 11.955 Tf 20.89 8.09 Td[( )]TJ /F3 11.955 Tf 11.96 0 Td[(<0.Theinfectionfreeequilibrium(,0)isdeterminedtobel.a.s.if +)]TJ /F3 11.955 Tf 11.96 0 Td[(>0and++)]TJ /F3 11.955 Tf 13.15 8.08 Td[(N >0.(3)Inaddition,weareabletodetermineamoreusefulformulationoftheconditionfortheexistenceofapositiveendemicequilibrium.Thatis,apositiveendemicequilibrium(xs,yx)existsifandonlyif 0<+)]TJ /F3 11.955 Tf 11.96 0 Td[( N +)]TJ /F3 11.955 Tf 11.96 0 Td[(<1(3)and 1 )]TJ /F3 11.955 Tf 11.96 0 Td[(26641 N +)]TJ /F3 11.955 Tf 11.96 0 Td[(3775+)]TJ /F3 11.955 Tf 13.15 8.09 Td[(N 1)]TJ /F3 11.955 Tf 20.89 8.09 Td[( )]TJ /F3 11.955 Tf 11.96 0 Td[(++)]TJ /F3 11.955 Tf 13.15 8.09 Td[(N <0.(3)Wearenowabletodeterminethelong-termbehaviorofthesysteminalmostallpossiblecases. 3.3.1TheCase>)]TJ /F3 11.955 Tf 11.96 0 Td[(Inthiscase,theequilibrium0,1)]TJ /F15 7.97 Tf 16.75 5.26 Td[( )]TJ /F15 7.97 Tf 6.58 0 Td[(doesnotexistin.If+)]TJ /F3 11.955 Tf 12.26 0 Td[(<0thenitfollowsfrom( 3 )and( 3 )that(0,0)isl.a.s.and(,0)isunstable.Wewill 57

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useinequalities( 3 )and( 3 )toshowthat(xs,ys)doesnotexistinthiscase.LetC1,C2,C3andC4betherst,second,thirdandfourthterms,respectively,ontheleft-handsideofinequality( 3 ).Supposethat(xs,ys)doesexist.Since+)]TJ /F3 11.955 Tf 12.58 0 Td[(<0,itfollowsfrom( 3 )thatN +)]TJ /F3 11.955 Tf 12.39 0 Td[(<0andsincethersttermisstrictlypositiveitmustbethat)]TJ /F3 11.955 Tf 12.38 0 Td[(<0.ThereforeC1<0andC2<0.Now,multiplying( 3 )byN +)]TJ /F3 11.955 Tf 11.96 0 Td[(impliesthat+)]TJ /F3 11.955 Tf 11.96 0 Td[()]TJ /F3 11.955 Tf 13.15 8.08 Td[(N >0fromwhichitfollowsthatC4>0.Hence,C3<0necessarilysince( 3 )holds.Then,C3<0<+)]TJ /F3 11.955 Tf 11.96 0 Td[()]TJ /F3 11.955 Tf 13.15 8.09 Td[(N impliesN ()]TJ /F3 11.955 Tf 11.95 0 Td[()<)]TJ /F3 11.955 Tf 9.29 0 Td[(whichinturnimplies,since)]TJ /F3 11.955 Tf 11.96 0 Td[(<0,thatN +)]TJ /F3 11.955 Tf 11.95 0 Td[(>0.Thisisacontradiction,andthereforetheendemicequilibriumdoesnotexist.Itfollowsthat(0,0)isgloballyasymptoticallystableinasthephaseportraitinFigure 3-1A demonstrates.Furthermore,equation( 3 )impliesthateventually_R<0andthetumorsizedecaysexponentially.If+)]TJ /F3 11.955 Tf 12.93 0 Td[(>0then(0,0)isunstable.Thelong-termbehaviordependsonthesignofC4.IfC4>0then( 3 )guaranteesthat(,0)isl.a.s.Itcanagainbeshownthat(xs,ys)doesnotexist.Itfollowsthat(,0)isgloballyasymptoticallystableinnfx=0g(Figure 3-1B ).Then( 3 )impliesthateventually_R>0andthetumorgrowsexponentially. 58

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A B CFigure3-1. Phaseportraitswith=0and>)]TJ /F3 11.955 Tf 11.96 0 Td[(forsystem( 3 )-( 3 ).A)(0,0)isstablewhen=====N==1,=3.B)(,0)isstablewhen======1,=3.C)(xs,ys)isstablewhen=N=====1,=3,=2. If,ontheotherhand,C4<0then(,0)isunstableand(xs,ys)exists.Weclaimthattheendemicequilibriumisgloballyasymptoticallystablein+.Byconsideringthexandynullclinesandtheinvariantsetswherex=0ory=0wecandeterminethephasediagramgiveninFigure 3-2A .Thereweassume)]TJ /F3 11.955 Tf 12.61 0 Td[(<0whichimpliesthaty:=1)]TJ /F15 7.97 Tf 16.75 5.25 Td[( )]TJ /F15 7.97 Tf 6.59 0 Td[(>butthefollowingargumentholdsalsoif)]TJ /F3 11.955 Tf 12.16 0 Td[(>0.WedeterminefromthediagramthatthedeterminantoftheJacobianmatrixevaluatedat(xs,ys)ispositive.Supposethereexistsaperiodicorbit)]TJ /F1 11.955 Tf 10.09 0 Td[(surrounding(xs,ys).ThenbyProposition 1.1 ,)]TJ /F1 11.955 Tf 10.1 0 Td[(hasthesamestabilityas(xs,ys).Ifbotharestable,thennecessarilythere 59

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existsanunstableperiodicorbitbetween)]TJ /F1 11.955 Tf 10.1 0 Td[(and(xs,ys)whichviolatesProposition 1.1 .Suppose)]TJ /F1 11.955 Tf 10.1 0 Td[(isunstable.Consideranorbitwithaninitialconditionoutsideof)]TJ /F1 11.955 Tf 6.78 0 Td[(.Theomegalimitsetcannotcontain)]TJ /F1 11.955 Tf 6.77 0 Td[(,(0,0),or(,0)sincetheyareunstable.Theorbitalsocannotapproachasaddleconnectionsince(0,y)=2andisforwardinvariant.Therefore,theomegalimitsetofthispointisemptywhichisacontradiction.Therefore,theredoesnotexistanyperiodicorbitsfromwhichitfollowsthatanypositivesolutionconvergesto(xs,ys)asshowninFigure 3-1C .By( 3 ),thesignof(+)xs+()]TJ /F3 11.955 Tf -436.76 -23.91 Td[()ys)]TJ /F3 11.955 Tf 11.96 0 Td[(determinesthelong-termgrowthbehaviorofthetumor. y1 y y x A y1 y y x . BFigure3-2. Phasediagramswith=0forsystem( 3 )-( 3 ).Thex-nullcline(red)andy-nullcline(blue)areshown.A)Thecase>)]TJ /F3 11.955 Tf 11.95 0 Td[(,+)]TJ /F3 11.955 Tf 11.96 0 Td[(>0,and++)]TJ /F15 7.97 Tf 13.16 5.26 Td[(N <0.B)Thecase<)]TJ /F3 11.955 Tf 11.96 0 Td[(,++)]TJ /F15 7.97 Tf 13.15 5.26 Td[(N <0,and+)]TJ /F15 7.97 Tf 13.16 5.26 Td[(N 1)]TJ /F15 7.97 Tf 16.76 5.26 Td[( )]TJ /F15 7.97 Tf 6.58 0 Td[(>0. 3.3.2TheCase<)]TJ /F3 11.955 Tf 11.96 0 Td[(Theequilibrium0,1)]TJ /F15 7.97 Tf 16.75 5.26 Td[( )]TJ /F15 7.97 Tf 6.59 0 Td[(existsinthiscase.Furthermore,<)]TJ /F3 11.955 Tf 11.95 0 Td[(impliesthat+)]TJ /F3 11.955 Tf 11.97 0 Td[(>0so(0,0)isunstable.Necessarily,)]TJ /F3 11.955 Tf 11.97 0 Td[(>0andthereforeC1>0andC2>0.TheresultingbehaviorofsolutionsdependsonthesignsofC3andC4.IfC4>0then(,0)isl.a.s.IfitalsoholdsthatC3>0then0,1)]TJ /F15 7.97 Tf 16.76 5.26 Td[( )]TJ /F15 7.97 Tf 6.59 0 Td[(isunstableand(xs,ys)doesnotexist.Itfollowsthat(,0)isgloballyasymptoticallystable 60

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A B C DFigure3-3. Phaseportraitswith=0and<)]TJ /F3 11.955 Tf 11.96 0 Td[(forsystem( 3 )-( 3 ).A)(,0)isstablewhen===N==1,=3,=2,=0.5.B)BistabilityoccurswhenN==1,=10,=20.5,=2,==0.5,=0.2.C)(xs,ys)isstablewhen=N====1,=2,=4 13,=1 2.D)0,1)]TJ /F15 7.97 Tf 16.76 5.26 Td[( )]TJ /F15 7.97 Tf 6.59 0 Td[(isstablewhen====1,=3,=2,N=3,=0.5. innfx=0g(Figure 3-3A ).Then( 3 )impliesthateventually_R>0andthetumorgrowsexponentially.However,ifC3<0then0,1)]TJ /F15 7.97 Tf 16.75 5.26 Td[( )]TJ /F15 7.97 Tf 6.58 0 Td[(isalsol.a.s.andthesystemexhibitsbistability(Figure 3-3B ).Apositivesolutioncanconvergeto(,0)or0,1)]TJ /F15 7.97 Tf 16.76 5.26 Td[( )]TJ /F15 7.97 Tf 6.58 0 Td[(dependingontheinitialconditionasdemonstratedinFigure 3-3B .Thetumorwilltheneventuallyexponentiallygroworshrink,respectively.IfC4<0then(,0)isunstable.If,inaddition,C3>0then0,1)]TJ /F15 7.97 Tf 16.76 5.25 Td[( )]TJ /F15 7.97 Tf 6.58 0 Td[(isunstableand(xs,ys)exists.Wedeterminefromthephasediagram(Figure 3-2B )that 61

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thedeterminantoftheJacobianmatrixJ(xs,ys)ispositive.Furthermore,wecalculatetr(J(xs,ys))=)]TJ /F4 11.955 Tf 9.29 0 Td[((+)]TJ /F3 11.955 Tf 11.95 0 Td[()xs )]TJ /F4 11.955 Tf 11.95 0 Td[(()]TJ /F3 11.955 Tf 11.96 0 Td[()ys <0.Theendemicequilibriumisthereforel.a.s.andwecanruleoutthepossibilityofperiodicorbitsbyProposition 1.1 .Therefore,(xs,ys)isgloballyasymptoticallystablein+(Figure 3-3C ).Thelong-termgrowthbehaviorofthetumorisagaindeterminedbythesignof(+)xs+()]TJ /F3 11.955 Tf 11.96 0 Td[()ys)]TJ /F3 11.955 Tf 11.95 0 Td[(.IfC3<0inthecaseC4<0then0,1)]TJ /F15 7.97 Tf 16.75 5.25 Td[( )]TJ /F15 7.97 Tf 6.59 0 Td[(isstableandtheendemicequilibriumdoesnotexist.Itfollowsthatallsolutionsinny=0convergeto0,1)]TJ /F15 7.97 Tf 16.75 5.26 Td[( )]TJ /F15 7.97 Tf 6.59 0 Td[((Figure 3-3D ).Eventually,_R<0andthetumorshrinks. 62

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CHAPTER4NUMERICALMETHODSInthischapterwedevelopanitedifferencenumericalschemeforthefullmodel( 2 )-( 2 )basedonthemethodforasimilarvirotherapymodelformulatedbyWangandTianin[ 41 ].Weintroducethenewnotation, A=)]TJ /F3 11.955 Tf 11.45 8.09 Td[( RdR d+U R,A1=)]TJ /F8 11.955 Tf 11.29 16.86 Td[( RdR d+2 R2,A2=)]TJ /F3 11.955 Tf 14.09 8.09 Td[( R2(4)andF=1 [X)]TJ /F3 11.955 Tf 11.95 0 Td[(Y)]TJ /F3 11.955 Tf 11.96 0 Td[(()]TJ /F5 11.955 Tf 11.95 0 Td[(X)]TJ /F5 11.955 Tf 11.96 0 Td[(Y)] (4)F1=X)]TJ /F3 11.955 Tf 27.67 8.09 Td[(X jIrc(,)jZIrc(,)V(!,)d!)]TJ /F3 11.955 Tf 30.51 8.09 Td[(X jI2rc(,)jZI2rc(,))]TJ /F5 11.955 Tf 11.95 0 Td[(X(!,)d!)]TJ /F5 11.955 Tf 11.95 0 Td[(FX (4)F2=X jIrc(,)jZIrc(,)V(!,)d!)]TJ /F4 11.955 Tf 11.95 0 Td[((+)Y)]TJ /F5 11.955 Tf 11.96 0 Td[(FY (4)F3=N jJrc(,)jZIrc(,)[r2c)]TJ /F5 11.955 Tf 11.96 0 Td[(R2()()]TJ /F3 11.955 Tf 11.96 0 Td[(!)2]Y(!,)d!+ jIrc(,)jZIrc(,))]TJ /F5 11.955 Tf 11.96 0 Td[(X(!,)d!)]TJ /F3 11.955 Tf 11.96 0 Td[(V. (4)Themodel,for0<1,>0isthengivenby@X @+A@X @=F1 (4)@Y @+A@Y @=F2 (4)@V @+A1@V @+A2@2V @2=F3 (4)1 R2@ @(2U)=F (4)dR d=U(1,) (4) 63

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withboundaryconditions@V @(0,)=@V @(1,)=0, (4)@X @(0,)=@Y @(0,)=0, (4)U(0,)=0. (4) 4.1TheNumericalAlgorithmWeaimtonumericallysolvethesystem( 4 )-( 4 )for2[0,1]and2[0,T]withT>0.Letbethemeshsizeforthespatialdomain.Thatis,ourspatialnodesaregivenbyi=i(i=0,...,J)whereJ=1.Wechooseatimestepandconsidertimenodesn=n(n=0,...,M)whereM=T.LetXni=X(i,n)anddeneYni,Uni,andVnisimilarly.LetRn=R(n).Wewillnowpresentthenumericalalgorithmwherewecalculate(Rn,Xni,Yni,UnI,Vni)fori=0,...,Jateachtimestepn. 4.1.1Then=0StepWespecifyinitialconditionsforthevariableswhenn=0.Unlessotherwisenoted,weassumethesameinitialconditionsasin[ 41 ]whichwerechosentomatchtheexperimentalconditionsin[ 19 ].TheinitialvirusproleisgivenbyaGaussiandistributiontorepresentaninjectionofvirusintothecenterofthetumor.Thatis,R0=2mmX0i=0.84(i=0,...,J)Y0i=0.10(i=0,...,J)V0i=ce)]TJ /F16 5.978 Tf 5.76 0 Td[(42i c2(i=0,...,J) 64

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wherecZ102e)]TJ /F16 5.978 Tf 5.76 0 Td[(42 c2d=0.45.ThecalculationsforUandallotherquantitiesthenfollowfromtheformulasinSection 4.1.3 withn=0. 4.1.2Then=1StepForn=1weusetheforwardEulermethodwithcentraldifferencesinspacetoadvanceX,Y,VandR.Indeed,R1=R0+U0Jandfori=0,...,J,X1i=X0i+(F1)0i)]TJ /F5 11.955 Tf 11.96 0 Td[(A0i(X0i+1)]TJ /F5 11.955 Tf 11.95 0 Td[(X0i)]TJ /F9 7.97 Tf 6.59 0 Td[(1) 2Y1i=Y0i+(F2)0i)]TJ /F5 11.955 Tf 11.96 0 Td[(A0i(Y0i+1)]TJ /F5 11.955 Tf 11.95 0 Td[(Y0i)]TJ /F9 7.97 Tf 6.59 0 Td[(1) 2whereweobservethat,foralln,An0=0byboundarycondition( 4 )whileAnJ=0byequation( 4 ).Therefore,attheendpointstheequationsforXandYreducetoordinarydifferentialequations.ForV,werstcalculatetheinternalnodes,i=1,...,J)]TJ /F4 11.955 Tf 11.96 0 Td[(1,byV1i=V0i+(F3)0i)]TJ /F4 11.955 Tf 11.95 0 Td[((A1)0i(V0i+1)]TJ /F5 11.955 Tf 11.95 0 Td[(V0i)]TJ /F9 7.97 Tf 6.59 0 Td[(1) 2)]TJ /F4 11.955 Tf 11.96 0 Td[((A2)0i(V0i+1)]TJ /F4 11.955 Tf 11.96 0 Td[(2V0i+V0i)]TJ /F9 7.97 Tf 6.59 0 Td[(1) 22andthenenforcetheboundaryconditions( 4 )bysettingV10=V11andV1J=V1j)]TJ /F9 7.97 Tf 6.59 0 Td[(1.ThecalculationsforUandallotherquantitiesagainfollowfromtheformulasinSection 4.1.3 withn=1. 4.1.3ThenthStepSupposewehavedeterminedallvariablesandquantitiesfortimesteps0mn)]TJ /F4 11.955 Tf 11.26 0 Td[(1.WebeginatthenthtimestepbyusingthesecondorderAdams-BashforthmethodtocalculateRn,Rn=Rn)]TJ /F9 7.97 Tf 6.59 0 Td[(1+ 2(3Un)]TJ /F9 7.97 Tf 6.58 0 Td[(1J)]TJ /F5 11.955 Tf 11.96 0 Td[(Un)]TJ /F9 7.97 Tf 6.59 0 Td[(2J). 65

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TheLeapfrogscheme,whichassumescentraldifferencesintimeandspace,isusedtoadvanceXandY,Xni=Xn)]TJ /F9 7.97 Tf 6.58 0 Td[(2i+2"(F1)n)]TJ /F9 7.97 Tf 6.59 0 Td[(1i)]TJ /F5 11.955 Tf 11.96 0 Td[(An)]TJ /F9 7.97 Tf 6.58 0 Td[(1i)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(Xn)]TJ /F9 7.97 Tf 6.58 0 Td[(1i+1)]TJ /F5 11.955 Tf 11.96 0 Td[(Xn)]TJ /F9 7.97 Tf 6.59 0 Td[(1i)]TJ /F9 7.97 Tf 6.59 0 Td[(1 2#Yni=Yn)]TJ /F9 7.97 Tf 6.59 0 Td[(2i+2"(F2)n)]TJ /F9 7.97 Tf 6.59 0 Td[(1i)]TJ /F5 11.955 Tf 11.95 0 Td[(An)]TJ /F9 7.97 Tf 6.59 0 Td[(1i)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(Yn)]TJ /F9 7.97 Tf 6.58 0 Td[(1i+1)]TJ /F5 11.955 Tf 11.96 0 Td[(Yn)]TJ /F9 7.97 Tf 6.59 0 Td[(1i)]TJ /F9 7.97 Tf 6.59 0 Td[(1 2#fori=0,...,J,recallingAn0=AnJ=0.Notingthecommentin[ 41 ],everytentimestepsweimplementanaverageintimewithXn)]TJ /F9 7.97 Tf 6.59 0 Td[(1=1 2(Xn+Xn)]TJ /F9 7.97 Tf 6.59 0 Td[(2)Yn)]TJ /F9 7.97 Tf 6.59 0 Td[(1=1 2(Yn+Yn)]TJ /F9 7.97 Tf 6.59 0 Td[(2)inordertoavoidameshdriftinginstabilitycausedbytheLeapfrogscheme.NextwecalculateFni=1 [Xni)]TJ /F3 11.955 Tf 11.95 0 Td[(Yni)]TJ /F3 11.955 Tf 11.95 0 Td[(()]TJ /F5 11.955 Tf 11.96 0 Td[(Xni)]TJ /F5 11.955 Tf 11.95 0 Td[(Yni)]fori=0,...,Jbyequation( 4 ).Recallingtheboundarycondition( 4 )wesetUn0=0.Applyingthetrapezoidalruletoequation( 4 ),Unfori=1,...,JisgivenbyUni=1 2i2i)]TJ /F9 7.97 Tf 6.59 0 Td[(1Uni)]TJ /F9 7.97 Tf 6.58 0 Td[(1+Rn 2(2iFni+2i)]TJ /F9 7.97 Tf 6.58 0 Td[(1Fni)]TJ /F9 7.97 Tf 6.59 0 Td[(1).Nowwecandeterminefori=1,...,J,Ani=)]TJ /F3 11.955 Tf 12.61 8.09 Td[(i RnUnJ+Uni Rn(A1)ni=)]TJ /F8 11.955 Tf 11.29 16.86 Td[(i RnUnJ+2 (Rn)2i(A2)n=)]TJ /F3 11.955 Tf 21.5 8.09 Td[( (Rn)2bytheequationsin( 4 ). 66

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CalculationofVnwillbedoneimplicitlyand,therefore,wemustrstapproximatetheintegraltermsinF3.Asanexample,wewilldiscussthesecondintegraltermappearingin( 4 ).RecallthatXnis,atthistime,alreadyknown.First,wedeterminewhichnodesarecontainedinIrc(i,n),andthenweemploythetrapezoidalruletoapproximatethedeniteintegral.LetKnidenotetheapproximation,Kni jIrc(i,n)jZIrc(i,n))]TJ /F5 11.955 Tf 11.95 0 Td[(X(!,n)d!Recall,Irc(,)=[Imin,Imax]whereImin=maxh0,)]TJ /F7 7.97 Tf 18.65 4.7 Td[(rc R()iandImax=minh1,+rc R()i.LetiminandimaxbesuchthatIminimin0,weapproximatethevalueX(Imin,n)bylinearlyinterpolatingbetweenXniminandXnimin)]TJ /F9 7.97 Tf 6.58 0 Td[(1.Likewise,weinterpolateX(Imax,n)whenimax><>>:(imin)]TJ /F5 11.955 Tf 11.95 0 Td[(Imin) 2(2)]TJ /F5 11.955 Tf 11.95 0 Td[(X(Imin,n))]TJ /F5 11.955 Tf 11.95 0 Td[(Xnimin)imin>00imin=0andCimax=8>><>>:(Imax)]TJ /F3 11.955 Tf 11.95 0 Td[(imax) 2(2)]TJ /F5 11.955 Tf 11.95 0 Td[(X(Imax,n))]TJ /F5 11.955 Tf 11.96 0 Td[(Xnimax)imax
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NowwearereadytocalculateVfromtheparabolicequation( 4 ).Following[ 41 ],wewillemploythesecond-orderimplicitbackwarddifferentiationformula(BDF)methodintimeandcentraldifferencesinspace,whichgives3Vni)]TJ /F4 11.955 Tf 11.96 0 Td[(4Vn)]TJ /F9 7.97 Tf 6.59 0 Td[(1i+Vn)]TJ /F9 7.97 Tf 6.59 0 Td[(2i 2+(A1)niVni+1)]TJ /F5 11.955 Tf 11.95 0 Td[(Vni)]TJ /F9 7.97 Tf 6.58 0 Td[(1 2+(A2)niVni+1)]TJ /F4 11.955 Tf 11.95 0 Td[(2Vni+Vni)]TJ /F9 7.97 Tf 6.59 0 Td[(1 2=(F3)nifori=1,...,J)]TJ /F4 11.955 Tf 11.96 0 Td[(1.Thissetofequationsformsalinear,tridiagonalsystemgivenbybiVni)]TJ /F9 7.97 Tf 6.58 0 Td[(1+diVni+aiVni+1=Si,i=0,...,Jwherebi=2 2(A2)ni)]TJ /F4 11.955 Tf 13.49 8.09 Td[( (A1)nidi=3)]TJ /F4 11.955 Tf 13.15 8.09 Td[(4 2(A2)ni+2ai=2 2(A2)in+ (A1)niSi=2(Lni+Kni)+4Vn)]TJ /F9 7.97 Tf 6.59 0 Td[(1i)]TJ /F5 11.955 Tf 11.95 0 Td[(Vn)]TJ /F9 7.97 Tf 6.58 0 Td[(2ifori=1,...,J)]TJ /F4 11.955 Tf 11.96 0 Td[(1andd1=1,a1=)]TJ /F4 11.955 Tf 9.29 0 Td[(1,S1=0,bJ=1,dJ=)]TJ /F4 11.955 Tf 9.3 0 Td[(1,SJ=0,inordertoenforcetheboundaryconditions( 4 ).Aftercalculationofthesecoefcients,thesystemissolvedusingastandardtridiagonalmatrixalgorithmrstpresentedbyThomasin[ 37 ].Finally,wemustupdate(F1)ni,(F2)ni,(F3)nibyequations( 4 ),( 4 ),and( 4 )wherewerstapproximatetheintegraltermsthereinwithanalogousmethodstothatdescribedaboveforKni. 4.2AnalysisoftheNumericalMethod 4.2.1ParameterEstimationWeassumeandtohaveunitsofhour(h)andmillimeter(mm),respectively.Unlessotherwisenoted,wetaketheparametervaluesfor,,,,,,from[ 17 ], 68

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whichwereestimatedfromexperimentaldataonthetreatmentofgliomas[ 19 ].Thesevalues,alsoassumedbyWangandTianin[ 41 ],are=210)]TJ /F9 7.97 Tf 6.58 0 Td[(2,=3.510)]TJ /F9 7.97 Tf 6.59 0 Td[(2,=5.610)]TJ /F9 7.97 Tf 6.59 0 Td[(2,=2.510)]TJ /F9 7.97 Tf 6.58 0 Td[(2,=2.110)]TJ /F9 7.97 Tf 6.58 0 Td[(2withunitsofh)]TJ /F9 7.97 Tf 6.58 0 Td[(1and=1106cellspermm3.Atypicalcelldiameterisalsostatedtherefromwhichweestimaterc=0.510)]TJ /F9 7.97 Tf 6.58 0 Td[(2mm.RatesofsyncytiaformationareestimatedbyBajzeretal.tottheirmodeltovirotherapydataforatreatmentofmultiplemyelomawithafusogenicstrainofmeaslesvirus[ 2 ].Theyestimatetobebetween0.4810)]TJ /F9 7.97 Tf 6.59 0 Td[(2h)]TJ /F9 7.97 Tf 6.59 0 Td[(1and2.510)]TJ /F9 7.97 Tf 6.58 0 Td[(2h)]TJ /F9 7.97 Tf 6.58 0 Td[(1fordifferentcases.Theviralburstsize,N,ishighlyvirus-dependent.In[ 41 ],burstsizesareinvestigatedwithintherangeof50to400.Inthefollowingsimulationswewillchoosehypotheticalvaluesfor,,Ntodemonstratearangeofdynamicalbehaviors. 4.2.2IntegralTermsandSpatialHomogeneityOurnumericalmethodmaintainstheaveragingpropertyoftheintegraltermsdescribinginfection,syncytiaformation,buddingandlysis.Therefore,solutionswithaspatiallyhomogeneousinitialconditionshouldremainspatiallyhomogeneous;thisbehaviorwasconrmedbynumericalsimulations.Wenotethattheintegralsarecalculatedusingmorethanonenodeifimin
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A B C D E FFigure4-1. Densityplotscomparingnumericalalgorithmsusingmassaction(left-handside)versustheoriginalintegralcalculations(right-handside).A,B)X,Uninfectedtumorcelldensity.C,D)Y,Infectedtumorcelldensity.E,F)V,Freevirusdensity. 70

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Y0i=be)]TJ /F16 5.978 Tf 5.76 0 Td[(42i b2(i=0,...,J)wherea=0.84andb=0.50.Eveninthisnon-spatiallyhomogenouscase,densityplotsforX,Y,andZshownoperceptibledifferencebetweenthetwomethods(Figure 4-1 ).However,theoriginalmethodismorecomputationallyintensiveandtakessignicantlylongertorun.Therefore,toincreaseefciency,wewillusethenumericalalgorithmwhichassumesmassactionfortheremainingsimulations.AsinFigure 4-1 ,anysimulationsweattemptedresultedinconvergencetospatiallyhomogeneoussolutions.Thisobservationmotivatesthefollowingconjecture.Conjecture.ForanypiecewisesmoothinitialconditionswithX0,Y0,V0andX+Y,thesolution(X,Y,V)ofthesystem( 4 )-( 4 )convergestoaspatiallyhomogeneoussolution. 4.2.3SimulationsofTumorGrowthBehaviorRecallfromequation( 3 )that,inthespatiallyhomogeneouscase,thegrowthofthetumorradiusisgivenby_R=R 3[(+)X+()]TJ /F3 11.955 Tf 11.96 0 Td[()Y)]TJ /F3 11.955 Tf 11.95 0 Td[(].Notethatthisequationholdsevenifwedonotmakethequasi-steadystateassumptionaswedidinSection 3.1 .Supposeasolution(X,Y,V)ofsystem( 4 )-( 4 )convergestoaspatiallyhomogeneoussolution(~X,~Y,~V).Then,long-termtumorgrowthisexponentialwiththerateofgrowthordecaydeterminedby~F:=1 3[(+)~X+()]TJ /F3 11.955 Tf 11.96 0 Td[()~Y)]TJ /F3 11.955 Tf 11.95 0 Td[(].Ifasolutionconvergestotheinfection-freeequilibrium(IFE)(X,Y,V)=(,0,0),thenthetumorispredictedtoeventuallygrowexponentiallywithgrowthconstant~F= 3.Indeed,withparametervalues=0.510)]TJ /F9 7.97 Tf 6.58 0 Td[(2,=0.110)]TJ /F9 7.97 Tf 6.58 0 Td[(2andN=0.8thenumericalsolutionconvergestotheIFE(Figure 4-2 ).Furthermore,linearregressionofthelog 71

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A B CFigure4-2. Densityplotsdemonstratingconvergencetotheinfection-freeequilibriumwhen=0.510)]TJ /F9 7.97 Tf 6.59 0 Td[(2,=0.110)]TJ /F9 7.97 Tf 6.59 0 Td[(2andN=0.8.A)X,Uninfectedtumorcelldensity.B)Y,Infectedtumorcelldensity.C)V,Freevirusdensity. 72

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A BFigure4-3. Exponentialgrowthofthetumorradiuswhen=0.510)]TJ /F9 7.97 Tf 6.58 0 Td[(2,=0.110)]TJ /F9 7.97 Tf 6.58 0 Td[(2andN=0.8. plotofR(Figure 4-3 )for1500t2000revealsagrowthrateof0.0067.Recall=210)]TJ /F9 7.97 Tf 6.59 0 Td[(2,so~F=2 310)]TJ /F9 7.97 Tf 6.58 0 Td[(2,thusconrmingtheaccuracyofournumericalmethod.Simulationsalsodemonstratethecasewherethenumericalsolutionconvergestotheendemicequilibrium.InFigure 4-4 theparametersinclude=110)]TJ /F9 7.97 Tf 6.59 0 Td[(2,=0.510)]TJ /F9 7.97 Tf 6.59 0 Td[(2,andN=1.4;thesolutionconvergesto(0.4255,0.1844,0.6933)(Figure 4-4 ).Inthiscase,~F=)]TJ /F4 11.955 Tf 9.3 0 Td[(0.0033andthetumordecaysexponentially(Figure 4-5A ).Ifinstead,N=1withallotherparametersequal,convergenceisto(0.8322,0.0925,0.2410),andthetumorgrowsexponentiallysince~F=0.0033(Figure 4-5B ).WearealsoabletoisolateaspecialcasewithN=1.1015where~F0andthetumorradiusstabilizesatapproximately0.82mm(Figure 4-5C ). 4.2.4StabilityandAccuracyAswehavedemonstrated,themethodpresentedinSection 4.1 allowsustosuccessfullysimulatethesystem( 4 )-( 4 )inmanycases.TheCFLconditionisnoteasilycalculatedforoursystem,butwenotethat,foreachsetofinitialconditionsexplored,themethodwasstableaslongas waschosensufcientlysmallandT 73

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A B CFigure4-4. Densityplotsdemonstratingconvergencetotheendemicequilibriumwhen=110)]TJ /F9 7.97 Tf 6.59 0 Td[(2,=0.510)]TJ /F9 7.97 Tf 6.59 0 Td[(2,andN=1.4.A)X,Uninfectedtumorcelldensity.B)Y,Infectedtumorcelldensity.C)V,Freevirusdensity. 74

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wasnottoolarge.InsomecaseswithlargeT,themethoddemonstratedunexpectedbehavior(Figure 4-6 ).Thesolutionappearedtoconvergetothepredictedequilibriumbutthenoscillatorybehavioroccurredatlargetimes.TheerrorappearstohappenwhenRdecaysveryclosetozero,perhapsbeyondthelimitofprecisecalculationinMATLAB.Furtherinspectionofthisinstabilityisneeded. Figure4-6. Instabilityofthenumericalmethodatlargetimes. WangandTian[ 41 ]claimthattheirsimilarmethodissecond-orderaccurateinbothtimeandspace.TheLeapfrogscheme,BDFmethod,andtrapezoidalruleareallsecond-orderaccuratesowehypothesizethatourmethodisalsosecond-orderaccurate.However,aproofofsuchaccuracyisnotinvestigatedfurtherhere. 76

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tothevirusitselfcanseverelyhinderthetherapy,thereisalsorecentevidencethatthedeathofsyncytiabooststheimmuneresponsetothetumorcells[ 26 ]andcould,infact,contributetotumorcontrol.Amodelwhichincorporatestheseconictingeffectscouldprovideavaluableperspectiveonthepossibilitiesofsuccessfulclinicaltherapy. 78

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