Within-Host Virus Models with Periodic Antiviral Therapy

22 Pages·2008·0.45 MB·English

BulletinofMathematicalBiology(2009)71:189–210
DOI10.1007/s11538-008-9359-5
ORIGINAL ARTICLE
Within-Host Virus Models with Periodic Antiviral Therapy
PatrickDeLeenheer
DepartmentofMathematics,UniversityofFlorida,Gainesville,USA
Received:28January2008/Accepted:11September2008/Publishedonline:13December2008
©SocietyforMathematicalBiology2008
Abstract Thispaperinvestigatestheeffectofdrugtreatmentonthestandardwithin-host
virusmodel,assumingthattherapyoccursperiodically.Itisshownthateradicationispos-
sibleundertheseperiodicregimens,andwequantitativelycharacterizesuccessfuldrugs
ordrugcombinations,boththeoreticallyandnumerically.Wealsoconsidercertainopti-
mizationproblems,motivatedforinstance,bythefactthateradicationshouldbeachieved
atacceptable toxicity levelstothepatient.Itturnsoutthattheseoptimization problems
canbesimplifiedconsiderably,andthismakescalculationsoftheoptimaafairlystraight-
forwardtask.AllourresultswillbeillustratedonanHIVmodelbymeansofnumerical
examplesbasedonup-to-dateknowledgeofparametervaluesinthemodel.
Keywords Within-hostvirusmodels·HIV·HBV·HCV·Influenza·Malaria·
Periodicdrugtreatment·Optimization
1. Introduction
Forthepasttwodecades,within-hostvirusmodelsdescribingviralinfectionshaveplayed
animportantroleintheunderstandingofviruses,andthewaysinwhichtheyescapenot
only the immune system, but also the various drugs that have been developed to sup-
pressviralreplication.Testingspecifichypothesesbasedonclinicaldataisoftendifficult
sincesamplescannotalwaysbetakentoofrequentlyfrompatients,orbecausedetection
techniquesofthevirusmaynotbeaccurate.Thisjustifiesthecentralroleplayedbymath-
ematicalmodelsinthisareaofresearch.
In this paper, we will revisit the standard model of within-host virus infections
(Perelson and Nelson, 1999; Nowak and May, 2000) which encompasses several im-
portantinfectionssuchasHIV(Richman, 2004),hepatitis B(GanemandPrince, 2004;
LocarniniandLai,2003)andC(SpecialissueonvirologyandclinicaladvancesofHCV
infection,2006),influenza(Earnetal.,2002),andeventhemalariaparasiteP.falciparum
(Molineaux and Dietz, 2000), and we will explore the consequences of periodic antivi-
ral treatment. We will characterize analytically and numerically the periodic treatment
schedulesthatleadtoviraleradication.However,ourresultsshouldbeinterpretedwith
E-mailaddress:[email protected]fl.edu.
SupportedinpartbyNSFgrantDMS-0614651.
190 DeLeenheer
a healthy dose of caution. Indeed, in the case of HIV, for example, we know that viral
eradicationwiththecurrentlyavailabledrugsisnotpossible,evenwhenpatientsadhere
tothestricttreatmentregimens.Thismeansthateitherthemathematicalmodeldoesnot
capturealltherelevantdynamicsoftheinterplayofthevirusanditshostcells,orthatthe
drugsarenotpotentenoughforviraleradication.Itseemsreasonabletospeculatethata
mixofbothofthesefactorsistoblame,andadditionalmodelingeffortsarenecessary.
In the case of HIV, several explanations for treatment failure have been proposed,
suchasthepre-existenceofdrug-resistantstrains(RibieroandBonhoeffer,2000),orthe
emergence of resistant strains after initiation of drug therapy (Bonhoeffer and Nowak,
1997;Larderetal.,1989;Richmanetal.,1994).Theabilityofthevirustomutatequickly
intoformswhichmaybelesssensitivetodrugshasbeen,andcontinuestobe,thefocusof
muchattention;seetherecentcontributions(Balletal.,2007;DeLeenheerandPilyugin,
2008) that study the behavior of multistrain viral models. According to Siliciano et al.
(2003), treatment failure is due to the presence of a latent reservoir of HIV in resting
memoryCD4+Tcells.
Otherresearchhasgravitatedaroundthefactthattheperiodicregimeninwhichdrugs
are taken daily (or more frequently), puts a very high strain on the patient, calling for
therapiesminimizingthetreatmentburdenonthepatient(Kirschneretal.,1997),andalso
leadingtoinvestigationsoftheuseofStructuredTreatmentInterruptions(STIs)(Bajaria
etal.,2004;KrakovskaandWahl,2007;Ortizetal.,2001).
Recallthatthestandardmodel(PerelsonandNelson,1999;NowakandMay,2000)is
athree-dimensionalnonlinearODE.InthecaseofHIV,itsstateconsistsoftheconcentra-
tionsofhealthyCD4+Tcells(thetargetsoftheHIV),infectedTcells,andviralparticles.
UponinfectionofahealthyTcell,oneofthefirstordersofbusinessistomakeacopyof
theviralRNA,usingtheenzymereversetranscriptase.Thisstep,whichiserror-proneand
leads to mutations, can beblocked by aclass of drugs called reverse transcriptase (RT)
inhibitors.Oncetheviralcopyhasbeenproduced,thedoublestrandedviralDNAinte-
gratesinthecell’snucleusasprovirus.Theusualgeneexpressionnowdoestherest,and
viralproteinsareproducedaccordingtothegeneticinformationencodedintheprovirus.
These proteins are assembled, mature, and ultimately new viruses bud off from the in-
fected cell’s surface which go on to infect other T cells. During the maturation stage,
theproteaseenzymeisusedtocleavelongproteinchains,andtheso-calledprotease(P)
inhibitorsaredrugsthattargetthisstep.Ifeffective,theygiverisetodefectivevirus.
Forotherviruses,despitedifferencesintheinfectionandreplicationmechanisms,and
in the drugs used against them, the standard model is still a popular model for describ-
ingthedynamicsoftheinfection.ForhepatitisBorC,thetargetcellsarelivercellsand
someofthedrugsusedarelamivudine,adefovirandentecavir(forHBV),ribavirin(for
HCV)andinterferon(forHBVandHCV).Theinfluenzavirusinfectsepithelialcellsand
istreatedwithneuraminidaseinhibitors(suchasoseltamivirandzanamivir)andM2in-
hibitors(suchasamantadineandrimantadine).Finally,themalariaparasiteP.falciparum
infectserythrocytes(redbloodcells)andmalariahascommonlybeentreatedbychloro-
quine.
The purpose of this paper is to assess theoretically and quantitatively what the im-
pactisofperiodicdrugtreatmentonthedynamicbehaviorofthestandardmodel,andin
particular to determine what it takes to get rid of the infection. Mathematically, we ob-
tainanonlinearperiodicODE,forwhichingeneralitisdifficulttoproveglobalstability
Within-HostVirusModelswithPeriodicAntiviralTherapy 191
and this explains why much research has traditionally resorted to simulations. Surpris-
ingly though, solutions to the standard model ultimately are bounded by solutions of a
monotonesystem,aspointedoutbyd’Onofrio(2005),andthisallowstoconcludeglobal
stabilityforthenonlinearperiodicmodel.
We will first consider a simple case of HIV treatment, where only RT inhibitors are
administered,andwhereitisassumedthatthedrugisofthebang-bangtype,i.e.,ateach
momentduringtheperiodofthetreatmentcycle,thedrugisinoneoftwostates:Either
itisactiveatafixedefficiencylevel,oritisinactive.Thedrugisthuscharacterized by
twoparameters:itsefficiencylevelwhenactive,andthedurationoftheactivity.Amajor
roleinouranalysisisplayedbythespectralradiusofanon-negativematrix(namelyof
thefundamentalmatrixsolution,evaluatedoveronetreatmentcycle,ofthelinearization
attheinfection-freeequilibrium),whichisshowntopossessexpectedmonotonicityprop-
ertiesintermsofthetwoparametersthatcharacterizethedrug.Specifically,thisspectral
radius—whichalsocontrolsthespeedofconvergencetotheinfection-freeequilibrium—
islowerwhenthedrugismorepotentorwhenitisactivelonger.Equivalently,conver-
gencetotheinfection-freeequilibriumisfasterwithamorepotentdrug,oradrugwhose
activitylastslonger.WewillseethattheseresultscanbegeneralizedtothecaseofPin-
hibitors,ortoamixofbothRTandPinhibitors.Thislatterscenarioreflectsmoreclosely
thestandardpracticeofadministeringcocktailsofdrugstoHIVinfectedpatients.
Inreality,theefficiencyofadrugisnotofthebang-bangtype.Infact,currentresearch
isinvestigatingtheeffectofincludingpharmacokineticsintothepicture,andhasrevealed
that the efficiency is a periodic signal with an initial steep rise right after drug intake,
followed by a slower decay over a period; see the work of Dixit and Perelson (2004),
Rongetal.(2007)fordetailedmodels.Therefore,weturntothismoregeneralcase,by
approximatingtheefficiencybyamoregeneralpiecewiseconstantperiodicsignal.Itturns
outthatthepreviousresultsremainvalid.
Finally,weturntooptimizationproblemsthatinvolveeithermaximizingthespeedof
convergencetotheinfection-freeequilibriumwhilemakingsurethatacceptabletoxicity
levelsarenotexceeded,orbyminimizingtoxicitylevels,whilemakingsurethespeedof
convergencedoesnotfallbelowacertainthreshold.
Allourresultswillbeillustratedbymeansofnumericalexamplesofwithin-hostHIV
models whose parameters are chosen in accordance with current prevailing knowledge
basedonclinicaldataandextensiveexperimentalevidence.Ourresultshavethepotential
to suggest which drug, or which combination of drugs, are optimal for a given patient.
Theycanalsobeusedtoexploretheconsequencesofchangingthetreatmentfrequency.
Theinvestigationoftheimpactofperiodictreatmentcyclesonmulti-strainmodels,orthe
effectofSTIsisthesubjectofongoingresearch.
Notation FormatricesAandB,0≤A,0<AmeansthatAisa(entry-wise)nonneg-
ative,positivematrix,respectively,andA≤B meansthat0≤B−A.Amatrixiscalled
quasi-positiveifallitsoff-diagonalentriesarenonnegative.Thespectralradiusofama-
trix A is defined as the largest modulus of all eigenvalues of A and will be denoted by
ρ(A).Wewillalsousethem(cid:2)atrixexponentialofasquarematrixA,whichisdefinedby
theconvergentmatrixseries ∞ 1Ai,andwillbedenotedbyEXP[A].
i=0i!
192 DeLeenheer
2. Within-hostvirusmodelwithtreatment
Webrieflyrecallthewell-knownstandardmodel(PerelsonandNelson,1999;Nowakand
May,2000).Let
T˙ =f(T)−kVT,
T˙∗=kVT −βT∗, (1)
V˙ =NβT∗−γV,
whereT,T∗,V denotetheconcentrationsofhealthyandinfectedcells,andvirusparti-
cles, respectively. The notation T,T∗,V is borrowed from the HIV literature (Perelson
andNelson,1999;NowakandMay,2000),wheretheyrepresentconcentrationsofCD4+
T-cells,infectedT-cells,andHIVvirus.Butwestressthatothervirusandparasiteinfec-
tionsaredescribedwiththesamemodelasexplainedintheIntroduction.
Allparametersareassumedtobepositive.Theparametersβ andγ arethedeathrates
ofinfectedcellsandvirusparticles,respectively.Theinfectionisrepresentedbyamass
actiontermkVT,andN istheaveragenumberofvirusparticlesbuddingoffaninfected
cellduringitslifetime.The(net)growthrateoftheuninfectedcellpopulationisgivenby
thesmoothfunctionf(T):R+→R,whichisassumedtosatisfythefollowing:
∃T >0:f(T)(T −T )<0 forT (cid:7)=T , and f(cid:8)(T )<0. (2)
0 0 0 0
We have chosen to make the class of allowable f(T)’s as large as possible, since the
growthrateishardtodetermine.Inaddition,mostmathematicalresultsapparentlyremain
validforthislargeclass.Finally,wenoticethatthetwomostpopularchoicesforf(T),
namelya−bT forsomepositivea andb;seeNowakandMay(2000)ands+rT(1−
T/T ) for some positive s,r and T , see Perelson and Nelson (1999) (here, s is a
max max
sourcetermmodelingcellproduction—whichinthecaseofHIVoccursinthethymus—
andr andT arethemaximalpercapitagrowthrateandcarryingcapacityrespectively
max
describinglogisticgrowthofcells),satisfytheprecedingconditions.
Sincecontinuityoff impliesthatf(T )=0,itfollowsthat
0
E =(T ,0,0),
0 0
isanequilibriumof(1),andwewillrefertoitastheinfection-freeequilibrium.
Asecond,positiveequilibrium(correspondingtoaninfection)mayexistifthefollow-
ingquantitiesarepositive:
¯ ¯
T¯ = γ , T¯∗= f(T), V¯ = f(T). (3)
¯
kN β kT
Notethatthisisthecaseifff( γ )>0,orequivalentlyby(2)thatT¯ = γ <T .Interms
kN kN 0
ofthebasicreproductionnumber
kN
R := T ,
0 γ 0
Within-HostVirusModelswithPeriodicAntiviralTherapy 193
existenceofapositiveequilibriumisthereforeequivalentwith
1<R , (4)
0
whichwillbeastandingassumptionthroughouttherestofthispaper.Indeed,ifwewould
assume that R <1, then it would follow from De Leenheer and Smith (2003) that the
0
infection-freeequilibriumE isgloballyasymptoticallystable(GAS),andhenceinthis
0
casetheinfectionwouldalwaysbeclearedwithouttreatment.
WedenotethepositiveequilibriumthatcorrespondstoaninfectionbyE=(T¯,T¯∗,V¯).
LinearizationatE showsthatitisunstable,andconditionsonf(T)areknownthatguar-
0
anteethatEisGAS(excludingofcourseinitialconditionscorrespondingtoahealthy,un-
infectedindividual;thesecoincidewiththeT-axis,whichisthestablemanifoldofE ).
0
However,itisalsopossiblethatthemodelexhibitssustainedoscillatorysolutionswhich
canbeasymptoticallystable.Regardlessofthedynamicalcomplexityofthesolutionsof
the model, in general, if left untreated, the infection will persist in a patient. All these
resultsfollowfromDeLeenheerandSmith(2003).
Obviously, the purpose of treatment is to clear the infection, hopefully by making
E GASbysuitablemodificationsofmodel(1)whichreflecttheeffectofdrugs.Letus
0
specializetoHIVbyexaminingtheeffectofRTinhibitors.TheeffectofusingPinhibitors
willbeconsideredlater.UsingmonotherapybasedonRTinhibitors,model(1)ismodified
to:
(cid:3) (cid:4)
T˙ =f(T)−k 1−(cid:5)(t) VT,
(cid:3) (cid:4)
T˙∗=k 1−(cid:5)(t) VT −βT∗, (5)
V˙ =NβT∗−γV,
where(cid:5)(t)∈[0,1]isthe(time-varying)drugefficiencyoftheRTinhibitors.Thedrugis
noteffectivewhen(cid:5)(t)=0and100%effectivewhen(cid:5)(t)=1.NoticethatE isstillan
0
equilibriumofthemodifiedmodel(5),regardlessofthedrugefficiency.
Assuming that the efficiency is constant over time, we set (cid:5)(t)=e∈(0,1]. Then to
cleartheinfection,itsufficestochooseesuchthatthemodifiedbasicreproductionnum-
berR ((cid:5))islessthan1,where
0
k(1−e)N
R ((cid:5)):= T .
0 γ 0
Indeed,theresultsmentionedpreviouslyareapplicabletothismodifiedmodel,andthey
implythatifR ((cid:5))<1,thenE isGASfor(5).Equivalently,iftheefficiencyesatisfies
0 0
1
e>e :=1− , (6)
crit R
0
thentreatmentwillbesuccessfulinthiscase.Ifthedrugwouldbeeffective100%sothat
e=1,thentreatmentwouldalwaysbesuccessful.CurrentRTinhibitorsclearlydonotfit
this profile. Moreover, in practice, the drug efficiency is not constant through time, and
themainpurposeofthispaperistoinvestigatethequantitativeconsequencesofthisfact.
194 DeLeenheer
3. Periodicdrugefficiency
Wenowmaketheassumptionthat(cid:5)(t)isperiodic:
(cid:5)(t)=(cid:5)(t+τ), forallt,
for some period τ >0. This is closer to reality where patients ideally adhere to a strict
periodic treatment schedule, taking medication daily (τ =1 day) or twice a day (τ =
0.5day)forinstance.
The shape of (cid:5)(t) over one treatment cycle is determined by the pharmacokinetics
which will not be modeled here. Roughly speaking, pharmacokinetics describes what
happenstoadrugafterthemomentofintake,butbeforeitstartsbeingactiveatthein-
fectionsite.Thestandardmodelcanbecoupledwithadetailedpharmacokineticsmodel;
see,forinstance,theworkofDixitandPerelson(2004),Rongetal.(2007),whereitwas
shownthatatleastqualitatively,thegraphoftheperiodicfunction(cid:5)(t)isroughlylikethe
onedepictedinFig.1.Itischaracterizedbyaquickriseoftheefficiencytoapeakvalue
rightafterdrugintake,followedbyaslowerdecay.Thisissignificantlydifferentfromthe
casewheretheefficiencyisconstant,thesituationwedescribedintheprevioussection.
Inpharmacokinetics,theefficiency(cid:5)(t)istraditionallydefinedas
y(t)
(cid:5)(t)= ,
K+y(t)
forsomepositiveconstantK,wherey(t)isastatecomponentofacompartmentallinear
system. For our purposes, we do not need to know the details of the pharmacokinetics
process,exceptforthefactthatthesignaly(t)(andhencealso(cid:5)(t))convergestoaperi-
odicsignalwiththesameperiodasthetreatmentschedule.
Thus,wewillassumethattheefficiency (cid:5)(t) isperiodic,andwestartbylinearizing
system(5)attheequilibriumE :
0
x˙=B(t)x, (7)
where
⎛ ⎞
f(cid:8)(T ) 0 −k(1−(cid:5)(t))T
0 0
⎜ ⎟
B(t)=⎜⎝ 0 −β k(1−(cid:5)(t))T0 ⎟⎠.
0 Nβ −γ
Fig.1 Periodicdrugefficiency(cid:5)(t).
Within-HostVirusModelswithPeriodicAntiviralTherapy 195
Itiswellknownthatthestabilitypropertiesoftheoriginof(7)(andgenericallythelocal
stabilitypropertiesoftheequilibrium E forsystem(5))aredeterminedbytheFloquet
0
multipliersof(7).Theblock-triangularstructureofB(t)impliesthattheseare
ef(cid:8)(T0)τ and λ , λ ,
2 3
whereλ andλ aretheFloquetmultipliersoftheplanarτ-periodicsystem:
2 3
(cid:11) (cid:12) (cid:11) (cid:12)(cid:11) (cid:12)
x˙2 = −β k(1−(cid:5)(t))T0 x2 . (8)
x˙3 Nβ −γ x3
In particular, since f(cid:8)(T )<0 by (2), it follows that the three Floquet multipliers of
0
system (7) are contained in the interior of the unit disk of the complex plane, which in
turnimpliesthat E islocallyasymptoticallystableforsystem(5)—if |λ |,|λ |<1.In
0 2 3
fact,byabeautifulargumentduetod’Onofrio(2005),itturnsoutthatthesameconditions
implythemuchstrongerresultofglobalasymptoticstabilityofE forsystem(5).
0
Proposition1(d’Onofrio,2005). LettheFloquetmultipliersofsystem(7)becontained
intheinterioroftheopenunitdiskofthecomplexplane.ThenE isGASforsystem(5),
0
hencetheinfectioniscleared.
ThisresultshowshowrelevantandimportantitistodeterminetheFloquetmultipliersof
system(7).Unfortunately,forgeneral functions (cid:5)(t),thisisanotoriouslydifficulttask.
Therefore, we will consider the simpler case where (cid:5)(t) is piecewise constant, bearing
in mind that piecewise constant functions are often good approximations to continuous
functions.Wewillstartwithanevensimplercasewhere(cid:5)(t)isofthebang-bangtype.
3.1. Periodicdrugefficiencyofthebang-bangtype
We make the following simplifying assumption regarding the shape of the graph of the
τ-periodicfunction(cid:5)(t),whichisillustratedinFig.2:
(cid:13)
e, t∈[0,p],
(cid:5)(t)= (9)
0, t∈(p,τ),
Fig.2 Periodicdrugefficiency(cid:5)(t)ofthebang-bangtype.
196 DeLeenheer
where p∈(0,τ) is the time duration during which the drug is supposed to be active
with efficiency e∈[0,1].During the remaining part of the treatment period the drug is
assumedtobetotallyinefficient.Clearly,thisisaverycrudewayofapproximatingthe
morerealisticshapeof(cid:5)(t)depictedinFig.1,butsomekeypropertiesaretobelearned
fromthiscase,andtheycarryovertomoregeneralcasesthatdescriberealitybetter,as
wewilldiscoverlater.
Therearetwopossibleparameterswhichcanbevariedin(9),namelyeandp,andthe
purposeoftherestofthissubsectionistoinvestigatetheireffectontheFloquetmultipliers
of system (8) with (9). These Floquet multipliers are the eigenvalues of the following
matrix
(cid:14) (cid:15) (cid:14) (cid:15)
Φ(e,p)=EXP (τ−p)A(0) EXP pA(e) , (10)
where
(cid:16) (cid:17)
−β k(1−e)T
A(e):= 0 . (11)
Nβ −γ
SincebothA(e)andA(0)arequasi-positivematricestheirmatrixexponentialsarenon-
negativematrices.1 Thus, Φ(e,p) isanonnegativematrixandbythePerron–Frobenius
theorem(BermanandPlemmons,1994);itsspectralradiusρ(Φ(e,p))isaneigenvalueof
Φ(e,p).Thus,theFloquetmultipliersofsystem(8)with(9)arecontainedintheinterior
oftheunitdiskofthecomplexplaneifandonlyifρ(Φ(e,p))<1.Thisguaranteesthat
theinfectioniscleared(globally)byProposition1.
The following proposition—whose proof is deferred to the Appendix—reveals that
ρ(Φ(e,p))hastheexpectedmonotonicityproperties:itdecreaseswithe(moreefficient
treatment)andwithp(drugiseffectivelonger).
Proposition2. Lete,e(cid:8)∈[0,1]andp,p(cid:8)∈[0,τ].Thenthemap(e,p)→ρ(Φ(e,p))is
continuous,
(cid:3) (cid:4) (cid:3) (cid:4)
e<e(cid:8), p(cid:7)=0⇒ρ Φ(e(cid:8),p) <ρ Φ(e,p) , (12)
and
(cid:3) (cid:4) (cid:3) (cid:4)
p<p(cid:8), e(cid:7)=0⇒ρ Φ(e,p(cid:8)) <ρ Φ(e,p) . (13)
Moreover,
(cid:3) (cid:4) (cid:3) (cid:4) (cid:3) (cid:14) (cid:15)(cid:4)
ρ Φ(e,0) =ρ Φ(0,τ) =ρ EXP τA(0) >1
foralle∈[0,1]andallp∈[0,τ](notreatment) (14)
and
1Proof: Let A be quasi-positive. Then B=A+αI is a nonnegative matrix for all sufficiently large
values of α, implying that EXP[tB] is a nonnegative matrix for all t ≥0. But since EXP[tA]=
EXP[−αt]EXP[tB],thesameconclusionholdsforEXP[tA].
Within-HostVirusModelswithPeriodicAntiviralTherapy 197
Fig.3 Spectralradiusofρ(Φ(e,p))asafunctionofefficiencyeandtreatmentdurationp.Thehorizontal
surfacecorrespondstoρ=1.
(cid:3) (cid:4) (cid:3) (cid:14) (cid:15)(cid:4) (cid:18) (cid:19)
ρ Φ(1,τ) =ρ EXP τA(1) =max EXP[−βτ],EXP[−γτ] <1
(constant,100%effectivetreatment). (15)
Sinceρ(Φ(e,p))(provideditislessthan1)isameasureofhowfastsolutionsof(5)
with(9)approachE (atleastlocallynearE ),thisresultmaybeinterpretedasfollows:
0 0
Letthetreatmentbeperiodic,ofthebang-bangtype,andcapableofclearingtheinfection.
Ifitismoreefficient,orlastslonger,thentheinfectionisclearedmorequickly.Weillus-
trateProposition2inFigs.3and4.TheparametersusedaretakenfromRongetal.(2007),
and they are as follows: f(T)=a−bT with a=104ml−1day−1 and b=0.01day−1
(whichimpliesthat T =106ml−1), k=2.4×10−8mlday−1, β=1day−1, N =3000,
0
γ =23day−1.Theperiodofthetreatmentτ is1day.
InRongetal.(2007),forarelatedmodel,itisarguedthattheaveragedrugefficiency,
(cid:20)
1 τ ep
(cid:5)(t)dt= ,
τ τ
0
isagoodindicatortoassessthesuccessorfailureofperiodictreatmentregimens,inthe
sensethattheoutcomeofconstant(nonperiodic)treatmentwithdrugefficiencyatalevel
equaltotheaveragedrugefficiencyoftheperiodicregimen,willpredicttheoutcomeof
theperiodicregimen.
WeshowinFigs.5and6thatthisclaimshouldbeinterpretedwithsomecaution,asit
isonlyapproximatelytrue.
Usingthesamemodelparameters asbefore,weseeforinstancethat thelevelcurve
ep =0.67 for the average drug efficiency (recall that τ =1 day) intersects the level
198 DeLeenheer
Fig.4 Contourplotofthespectralradiusρ(Φ(e,p))asafunctionofefficiencyeandtreatmentduration
p:ρ(Φ(e,p))>1inred(dark)regionand<1ingreen(light)region.
curve ρ(Φ(e,p))=1. Thus, all points on this level curve and above (below) the curve
ρ(Φ(e,p))=1correspondtotreatmentsuccess(failure).Oneparticularpointforwhich
treatment fails isthepoint (e,p)=(0.67,1) which correspondsto aconstant (nonperi-
odic)treatmentregimen.Recalling(6),wenoticethattreatmentfailureforthisparticular
pointcouldalsohavebeenconcludedfromthefactthat
1
0.67<e =1− ≈0.681.
crit R
0
Wecaninfactgeneralizethisconclusion,anddeterminenumericallyalllevelcurves
fortheaveragedrugefficiencywhichcontainpointscorrespondingtobothtreatmentfail-
ureandsuccess;seeFig.6.Thesearethelevelcurvesep=c,wherec∈(0.659,0.681),
sincetheyintersectthecurveρ(Φ(e,p))=1.Pointsontheselevelcurvesthatarebelow
(above) the latter curve correspond to treatment failure (success). The efficiency levels
e=0.659ande=e ≈0.681constituteanintervalof
crit
e −0.659
crit ≈3.3%
e
crit
neare forwhichcautionshouldbeexertedinclaimingthattheaveragedrugefficiency
crit
isagoodmeasurefortreatmentfailureorsuccess.Fortunately,thisisonlyasmallinterval
forthisexample,andthustheaveragedrugefficiencyisapproximatelyagoodmeasure
fortreatmentsuccessorfailure.
Forcompleteness,wealsopresentinFigs.7and8theresultsofasimulationrunthat
illustratewhathappenswhenbothaperiodicdrugtreatmentregimen,anditscorrespond-
ingconstant(nonperiodic)treatmentregimenwiththesameaveragedrugefficiencyfail.