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HAL Id: hal-00858208

https://hal.inria.fr/hal-00858208

Submitted on 4 Sep 2013

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Xavier Dupuis

To cite this version:

Xavier Dupuis. Optimal control of leukemic cell population dynamics. Mathematical Modelling of Natural Phenomena, EDP Sciences, 2014, Issue dedicated to Michael Mackey, 9 (1), pp.4-26.

�10.1051/mmnp/20149102�. �hal-00858208�

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0249-6399ISRNINRIA/RR--8356--FR+ENG

RESEARCH REPORT N° 8356

August 2013

Optimal control of

leukemic cell population dynamics

Xavier Dupuis

(3)
(4)

RESEARCH CENTRE SACLAY – ÎLE-DE-FRANCE 1 rue Honoré d’Estienne d’Orves Bâtiment Alan Turing

XavierDupuis

Projet-TeamsCOMMANDS

ResearhReport 8356August201325pages

Abstrat: We areinterestedin optimizingthe o-administrationoftwodrugsfor someaute

myeloidleukemias(AML),andwearelookingforinvitroprotoolsasarststep. Thisissuean

beformulatedasanoptimalontrolproblem. Thedynamisofleukemiellpopulationsinulture

isgivenbyage-struturedpartialdierentialequations,whihanbereduedtoasystemofdelay

dierential equations, and where the ontrols represent the ation of the drugs. The objetive

funtion relies on eigenelementsof the unontrolled model and on generalrelativeentropy, with

theideatomaximizetheeienyoftheprotools. Theonstraintstakeintoaountthetoxiity

ofthedrugs. Wepresentinthis paperthemodelingaspets,aswellastheoretialandnumerial

resultsontheoptimalontrolproblemthatweget.

Key-words: Aute myeloidleukemia,optimal ontrol,delay dierential equations, population

dynamis,generalrelativeentropy,invitrotherapeuti optimization

Thisworkispartof theDIMLSC projet ALMAon theAnalysisofAuteMyeloidLeukemia. Itbrings

together liniians andbiologistsof the Insermteam18ofUMRS872 (Jean-PierreMarie,Pierre Hirsh,Ruo-

PingTang, FannyFava,AnnabelleBallesta, FatenMerhi)andmathematiiansof theInria teamsBang(Jean

Clairambault,AnnabelleBallesta),Diso(CatherineBonnet,JoséLuisAvila)andCommands(J.FrédériBon-

nans,XavierDupuis).TheauthoralsothankFabienCrausteandThomasLepoutrefortheirusefulinsightsand

advie.

CMAP,EolePolytehnique&InriaSalay,91128Palaiseau,Frane.

E-mail:xavier.dupuismap.polytehnique.fr

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Résumé: Noussommesintéressésparoptimiserlao-administrationdedeuxmédiaments

pourertainesleuémiesaigüesmyéloïdes(LAM)etnousherhonsdesprotoolesd'administra-

tion in vitro dans un premier temps. Cela peut se formuler omme un problème de ontrle

optimal. Ladynamiquedepopulationsdeellulesleuémiquesenultureestdonnéepardeséqua-

tionsauxdérivéespartiellesstruturéesenâge,quipeuventserameneràunsystèmed'équations

diérentielles àretards, et lesontrlesreprésentent l'ationdes médiaments. La fontion

objetifestdénieàpartird'élémentspropresdumodèlenonontrléetd'unpriniped'entropie

relativegénéralisée,avel'idéedemaximiserl'eaitédesprotooles. Lesontraintesprennent

en ompte la toxiité desmédiaments. Nous présentonsdans e rapport lesaspets de mod-

élisation, ainsi que desrésultats théoriques et numériques sur le problèmede ontrle optimal

obtenu.

Mots-lés : Leuémies aigües myéloïdes, ontrle optimal, équations diérentielles à re-

tards,dynamique de populations, entropie relative généralisée, optimisation thérapeutique in

vitro

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1 Introdution

Autemyeloidleukemias(AML)areanersofthemyeloidlineageofwhitebloodells. Thepro-

essofbloodprodution,alledhematopoiesis,takesplaeinthebonemarrow,withhematopoi-

eti stemells(HSC)at itsroot. HSC havetheabiltytoself-renew, i.e. todivide withoutdif-

ferentiating,andtodierentiatetowardsanylineageofblood ellsbydividingintoprogenitors.

These progenitorsare ommitted stem ellswhih follow apath of dienrentiation, produing

ells whih are moreand more engagedinto one lineageand lose progressivelytheir ability to

self-renew. One they are fully mature and funtional, ells of eah lineage are released into

the bloodstream. Thehematopoiesis onsists in theregulation of theself-renewaland thedif-

ferentiationofellpopulations[22℄. InAML, thedierentiationisblokedat someearlystage,

leadingto theaumulationofimmaturewhitebloodells,alledblasts,ofthemyeloidlineage.

Thisblokadebeingassoiatedwithaproliferationadvantage,theblastsquiklyrowdthebone

marrowandareeventuallyreleasedintothebloodstream.

Oneoftherstmathematialmodelonhematopoiesiswasproposedin 1978byMakeyand

foused on the HSC population dynamis [18℄. Makey onsidered two phases in his model,

a resting phase and a proliferating phase, and desribed the dynamis of the two HSC sub-

populationsbyasystemofdelaydierentialequations;theseequations anbejustiedbyage-

strutured partial dierential equations. To represent the blokade of the dierentiation in

AML, Adimy etal. onsidered thedynamis ofellpopulationsofseveral maturitystages and

developpedamulti-ompartmentalmodel,whereeahompartmentrepresentsamaturitystage

and is again divided in two phases [2℄. Özbay et al. proeeded with the stabiliy analysis of

this delay dierential system in [21℄, and Avila et al. rened the model in [3℄ by onsidering

more than two phases per ompartement and modeling the fast proliferation in AML. Stiehl

and Mariniakalso proposed a multi-ompartmental model onleukemias[25℄; theyonsidered

healthy and leukemi ellpopulations,but did notdistinguishresting and proliferating phases

andthusdidnotgetdelays.

Thetreatment for mostof thetypes of AML is ahallenge [24℄. Cliniians of the depart-

ment of hematology at Saint-Antoine hospital in Paris would be interested for some ases in

o-administratingtwodrugs: aytotoxi(Araytin),whihenhaneselldeath,andaytostati

(AC200), whih slowsdownproliferation. A rststepis to determinehowsuh aombination

should be sheduled in in vitro experiments. Tothat purpose, biologists of the same hospital

havesampledbloodfrompatientswithAML,sortedanerblasts,andarriedoutleukemiell

ultures. Thenumberof ells,theirstate in theellyle, and theirmaturitystage havethen

beendailymeasuredduring5days,withoutandwitheahofthetwodrugsatdierentonstant

onentrationsin theulture[4℄.

Inthispaper,weidealizetheseexperimentsandonsiderleukemiellultureswithvarying

onentrationof bothdrugs. Weare lookingfor in vitroprotoolsof drugsadministration,i.e.

shedules of the onentration of both drugs during the experiment, whih are as eient as

possiblewithoutbeingtootoxi. Toformulatethisissueasanoptimalontrolproblem,astate

equation,anobjetivefuntion,andonstraintshavetobeenset.

Thestate equation models theell population dynamis under the ationof the drugs; we

onsideranage-struturedmodelwithonematurityompartement,dividedinonerestingphase

and one proliferating phase. Adimy and Crauste used suh a model in [1℄ to represent the

dependeneofelldeathandproliferationongrowthfators. Here,theationoftheytotoxion

elldeathisage-dependent,andthedrugonentrationsarenotsolutionsofevolutionequations

but are ontrol variableswhih denean in vitro protool. Gabriel et al. identied theation

of a drug induingquiesene (erlotinib) with afration of quiesentells in [11℄. Theation

of the ytostati in our model is also represented by a fration of resting ells, whih is here

(7)

time-dependent,andnotbyavaryingveloityin theproliferatingphaseasHinowetal. in[15℄.

Seealso[6℄aboutthemodelingoftheationofthedrugs.

The objetive funtion aims at minimizing the leukemi ell population at the end of the

experiment, in order to maximize the eieny of the orresponding protool. Its denition

atuallyrequiresalongtimeasymptotianalysisto avoidanhorizoneet. Thisanalysisrelies

onthespeializationofthegeneralrelativeentropyprinipleintroduedbyMiheletal. [20℄to

ourmodel. Variouskindsofobjetivefuntionsexistinthelitterature: nalormaximalnumber

oftumor ells[5℄, nal tumorvolume [16℄,performane index[17℄, oreigenvalue[7℄;the useof

anage-dependentweightgivenbyeigenelementsinthispaperseemstobenew.

The onstraintsome from biologialbounds ontheation ofthe drugsand from maximal

umulative doses that weimpose to limit the toxiity of the protools, asin [16℄; there is no

healthy populationin our model on whih weould set a toxiity threshold asin [5, 7℄. The

optimizationproblemthat wegetisequivalent,bythemethodof harateristis,toanoptimal

ontrol problem of delay dierential equations. Forsuh a problem, optimality onditions are

available in the form of Pontryagin's minimum priniple [14℄; it an also be redued to an

undelayedoptimalontrolproblem[12,13℄, andthensolvednumeriallybystandardsolvers.

Thepaperisorganizedasfollows. InSetion2,wemodelthepopulationdynamisunderthe

ationof the drugs. Setion 3ontains theanalysis of this model, inluding ageneral relative

entropy priniple and a longtime asymptoti analysis. The optimal ontrol problem is set in

Setion 4, and theoretial resultsand numerial optimal protools are presented in Setion 5.

Thepreise statement ofPontryagin'sminimumpriniplefor ourproblemhas beenpostponed

totheappendix, togetherwiththeparametersusedforthenumerialresolutions.

2 Modeling

Wepresenthere the dynamis of leukemi ell populations in ulture and under the ation of

thetwodrugs.

2.1 Cell populations

Weonsideraleukemiellpopulation,invitro,andwedistinguishtwosub-populations[1,18℄:

therestingells,whihareinative(

G

0 phase),andtheproliferatingells,whihareengagedin theiryle(

G

1

SG

2

M

phase).

Restingellsareintroduedintotheproliferatingphaseatarate

β

,independentlyofthetime spentintherestingphase. ConsideringthattheproliferationisunontrolledinaseofAML,we

donotrepresentanyfeedbakfrom aellpopulation[18,19℄oragrowthfator[1℄,and thus

β

isonstant in ourmodel.

Proliferating ells die by apoptosis at arate

γ

, and ifit doesnot die, aell divide during

mitosis,afteratime

spentinthephase,intwodaughterellswhihentertherestingphase.

Weonsider thatthe duration of the proliferating phase

is the same for all ells; this isnot

truebiologially[2℄but oneanthinkof

asanaverageduration[19℄.

We struture the proliferating population by an age variable

a

whih represents the time

spentin theproliferating phasebyaell. Wedenote by

R(t)

therestingpopulationat time

t

,

andby

p(t, a)

theproliferatingpopulationdensitywithage

a

attime

t

.

2.2 Ation of the drugs

Thetwodrugsareaytotoxi(Araytin)andaytostati(AC220).

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TheytotoxidamagestheDNAoftheellsduring the

S

sub-phaseoftheiryle;itresults

in anextradeathrate. Tosimplifyfurtheralulusandnumerialissues,weonsiderthat

u(t)

,

thedeathratedue totheytotoxiat time

t

, aetsthe seond-halfoftheproliferatingphase, i.e. proliferatingellswithage

a ∈ [τ, 2τ]

.

Theytostatiinhibitsareeptor tyrosinekynase (Flt3)ofthe ellsin theresting phase;it

resultsinafration

k(t)

ofinhibitedellsamongtheresting ells,whihannomoreenterthe

proliferating phase. The global introdution rate to the proliferating phase at time

t

is then

(1 − k(t))β

.

τ τ

(1 − k(t))β

γ γ + u(t)

R(t) p(t, a)

proliferating phase resting phase

×2

Figure1: Themodel.

Wedenote by

v(t)

theinhibition ratedue to the ytostatiat time

t

, and by

α

the rateof

naturaldis-inhibition. Weonsider thatthedynamisof

k

isgivenby

dk

dt (t) = v(t)(1 − k(t)) − αk(t).

(1)

Theationratesdue to thedrugsare inreasingfuntions oftheir onentration inthe ell

ulture, thelatterbeinghosenduring in vitroexperiments. Thusweonsider thatweontrol

diretly theationrates

u

and

v

.

2.3 The age-strutured model

Thedynamisoftheellpopulationsisgivenbythefollowingpartiallyage-struturedsystem:

dR

dt (t) = −(1 − k(t))βR(t) + 2p(t, 2τ)

(2)

∂p

∂t (t, a) + ∂p

∂a (t, a) = −(γ+χ

(τ,2τ)

(a)u(t))p(t, a) 0 < a < 2τ

(3)

p(t, 0) = (1 − k(t))βR(t)

(4)

The equation (2) isa balane equationfor the resting phasebetween theoutwardand inward

ow;thetransportequation(3) desribestheevolutionof theageohortsofproliferating ells,

sine they are aging with veloity

1

; the boundary ondition(4) givesthe inward ow to the

proliferatingphase.

(9)

2.4 A ontrolled version of Makey's model

Wedenoteby

P

and

P

2thetotalproliferatingpopulationandsub-populationintheseond-half ofthephase,respetively:

P (t) :=

Z

2τ 0

p(t, a)da, P

2

(t) :=

Z

2τ τ

p(t, a)da.

Formally,and this ouldbejustied withthe resultsofSetion 3, ifwe dierentiate

P, P

2 and

usethemethodof harateristisasin[1℄, wederivefrom (1) -(4)thefollowingsystemof delay

dierentialequations:

dR

dt (t) = −(1 − k(t))βR(t) + 2(1 − k(t − 2τ))βR(t − 2τ)e

(

γ2τ+Rttτu(s)ds

)

(5)

dP

dt (t) = −(γP (t) + u(t)P

2

(t)) + (1 − k(t))βR(t)

(6)

− (1 − k(t − 2τ))βR(t − 2τ)e

(

γ2τ+Rttτu(s)ds

) dP

2

dt (t) = −(γ + u(t))P

2

(t) + (1 − k(t − τ))βR(t − τ)e

γτ (7)

− (1 − k(t − 2τ))βR(t − 2τ)e

(

γ2τ+Rttτu(s)ds

) dk

dt (t) = v(t)(1 − k(t)) − αk(t)

(8)

Unsurprisingly,wegetaontrolledversionofMakey's1978model[18℄. Theoriginalmodelisa

systemoftwodierentialequationswithonedisretedelay,andanonlinearityin

β

;itisoneof

therstmathematial model ofthedynamis of hematopoietistemells(HSC), whih are at

therootof thehematopoiesis,theproessof bloodprodution. We havein (5)-(8) twoontrol

variables,

u

and

v

,andtwoextrastatevariables,

P

2 and

k

,beauseoftheontrols.

As we willexplain in Setion 4.1, the age-struturein theproliferating populationatually

matters,andthus itisofinterestto analysetheage-struturedmodel(2)-(4).

3 Analysis of the age-strutured model

3.1 Existene of solutions

Given

(β, γ) ∈ L

loc

(0, ∞) × L

loc

((0, ∞) × (0, 2τ))

and

(R

0

, p

0

) ∈ R × L

(0, 2τ)

,weonsiderthe

system

dR

dt (t) = −β(t)R(t) + 2p(t, 2τ) 0 < t

(9)

∂p

∂t (t, a) + ∂p

∂a (t, a) = −γ(t, a)p(t, a) 0 < t, 0 < a < 2τ

(10)

p(t, 0) = β(t)R(t) 0 < t

(11)

withtheinitialondition

R(0) = R

0

, p(0, ·) = p

0

.

(12)

Wefollow[10℄forthenotionofsolution.

(10)

Denition3.1. Wesaythat(10)holdsalongtheharateristisa.e. ifandonlyifthereholds,

fora.a.

(t, a) ∈ (0, ∞) × (0, 2τ)

,

p(s, a + s) = p(0, a) −

Z

s 0

(γp)(θ, a + θ)dθ

fora.a.

s ∈ (0, 2τ − a),

(13)

p(t + s, s) = p(t, 0) − Z

s

0

(γp)(t + θ, θ)dθ

fora.a.

s ∈ (0, 2τ).

(14)

Lemma3.2. If

p ∈ L

loc

((0, ∞) × (0, 2τ))

issuhthat (10)holdsalong the harateristisa.e., then

p

is Lipshitz along the harateristis

{t − a = c}

for a.a.

c

,

t 7→ R

0

p(t, a)da

is loally

Lipshitzandthere holdsa.e.

d dt

Z

2τ 0

p(t, a)da = p(t, 0) − p(t, 2τ) − Z

0

(γp)(t, a)da.

Proof. The rst assertion follows from (13) -(14). For the last two assertions, it is enough to

ompute

R

0

p(t, a)da

usingthesamerelations.

Denition3.3. Asolutionof (9)-(12)isany

(R, p) ∈ W

loc1,

(0, ∞) × L

loc

((0, ∞) × (0, 2τ))

suhthat(9)holdsa.e.,(10)holdsalongtheharateristisa.e.,(11)holdsa.e.,and(12)holds.

Lemma3.4. Given any

(β, γ)

and

(R

0

, p

0

)

,thereexistsauniquesolution

(R, p)

of (9)-(12). If

(β, γ)

and

(R

0

, p

0

)

arenon-negative, then

(R, p)

isnon-negative. Moreover, dening

Γ : (t, a) 7→

(R

t

0

γ(s, a − t + s)ds

if

0 < t < a < 2τ, R

a

0

γ(t − a + s, s)ds

if

0 < a < 2τ, a < t,

if

β

,

Γ

,

p

0 areloally Lipshitzand

p

0

(0) = β(0)R

0,then

p

isloally Lipshitzand

R ∈ W

loc2,.

Proof. Fora.a.

c ∈ (−2τ, 0)

,

p

isdeterminedon

{t − a = c}

by

p(t, a) = p

0

(a − t)e

R0tγ(s,at+s)ds

.

Then(9)beomesalinearODEon

(0, 2τ)

, fromwhih weget

R

,and then

p

on

{t − a = c}

for

a.a.

c ∈ (0, 2τ)

,andsoon. Thesignof

(R, p)

follows.

Observethata.e. on

{t − a > 0}

,

p(t, a) = β(t − a)R(t − a)e

R0aγ(t−a+s,s)ds

.

Theontinuityof

p

on

{t − a = 0}

isequivalentto

p

0

(0) = β(0)R

0.

3.2 General relative entropy

Weintroduethedualsystemassoiatedwith(9)-(11)

dt (t) = β(t)Ψ(t) − β(t)φ(t, 0) 0 < t

(15)

∂φ

∂t (t, a) + ∂φ

∂a (t, a) = γ(t, a)φ(t, a) 0 < t, 0 < a < 2τ

(16)

φ(t, 2τ) = 2Ψ(t) 0 < t

(17)

(11)

Solutionsof (15) -(17)aredenedasfortheprimal system,seeDenition 3.3.

Adapting[20℄ toourmodel,wegetageneralrelativeentropypriniple. Let

(β, γ)

bexed.

Given a solution

(R, p)

and apositive solution

( ˆ R, p) ˆ

of (9)-(11) , a positivesolution

(Ψ, φ)

of

(15)-(17) ,and

H ∈ L

loc

( R )

,wedene

H

by

H(t) := Ψ(t) ˆ R(t)H R(t)

R(t) ˆ

! +

Z

2τ 0

φ(t, a)ˆ p(t, a)H

p(t, a) ˆ p(t, a)

da.

Theorem3.5 (GeneralRelativeEntropy). Let

H

beloally Lipshitzand dierentiable every- where. Then

H

isloally Lipshitzandthereholdsa.e.

dH

dt (t) = φ(t, 2τ )ˆ p(t, 2τ )

H

p(t, 0) ˆ p(t, 0)

− H

p(t, 2τ) ˆ p(t, 2τ)

+H

p(t, 0) ˆ p(t, 0)

p(t, 2τ ) ˆ

p(t, 2τ ) − p(t, 0) ˆ p(t, 0)

.

(18)

Corollary3.6. Let

H

beonvex,possibly non dierentiable. Then

H

isnon-inreasing.

Proof ofCorollary 3.6. Let

H

be onvex. Then

H

is loally Lipshitz and has left and right

derivativeseverywhere. Then, as in the proof of Theorem3.5,

H

is loally Lipshitzand (18)

holdsa.e. ifwereplaethederivativeof

H

byitsrightderivativeandthederivativeof

H

byits

leftorrightderivative,dependingonthesignoftherightderivativeof

p(t,0) ˆ

p(t,0). Observenowthat

thisright-handsideof (18)isnon-positiveif

H

isonvex.

Proof ofTheorem3.5 . Observethat,alongtheharateristisa.e.,thereholds

∂t + ∂

∂a

φ(t, a)ˆ p(t, a) = 0, ∂

∂t + ∂

∂a

p(t, a) ˆ

p(t, a) = 0,

andthen

∂t + ∂

∂a

φ(t, a)ˆ p(t, a)H

p(t, a) ˆ p(t, a)

= 0.

ByLemma3.2, theseond termof

H

isloallyLipshitz,with derivativea.e.

φ(t, 0)ˆ p(t, 0)H

p(t, 0) ˆ p(t, 0)

− φ(t, 2τ)ˆ p(t, 2τ)H

p(t, 2τ) ˆ p(t, 2τ)

.

Thersttermof

H

isobviouslyloallyLipshitz,anda.e.

d

dt Ψ(t) ˆ R(t)

H R(t) R(t) ˆ

!

= − β(t)φ(t, 0) ˆ R(t) + Ψ(t)2ˆ p(t, 2τ)

H R(t) R(t) ˆ

!

= − φ(t, 0)ˆ p(t, 0) + φ(t, 2τ)ˆ p(t, 2τ) H

p(t, 0) ˆ p(t, 0)

,

and

Ψ(t) ˆ R(t) d

dt H R(t) R(t) ˆ

!

= Ψ(t) ˆ R(t)H

R(t) R(t) ˆ

! 1 R(t) ˆ

dR

dt (t) − R(t) R(t) ˆ

d ˆ R dt (t)

!

= Ψ(t)H

p(t, 0) ˆ

p(t, 0) 2p(t, 2τ) − p(t, 0) ˆ

p(t, 0) 2ˆ p(t, 2τ)

= φ(t, 2τ)ˆ p(t, 2τ)H

p(t, 0) ˆ p(t, 0)

p(t, τ) ˆ

p(t, τ) − p(t, 0) ˆ p(t, 0)

.

(12)

3.3 Eigenelements

Let

β > 0

and

γ ≥ 0

beonstant. Looking forpartiularsolutionsof (9)-(11)and (15)-(17)of

theform

R : t 7→ Re ¯

λt

Ψ : t 7→ Ψe ¯

λt

p : (t, a) 7→ p(a)e ¯

λt

φ : (t, a) 7→ φ(a)e ¯

λt

with

λ ∈ R

and

p ¯

,

φ ¯

dierentiable,wegetthefollowingeigenvalueproblem:

(λ + β) ¯ R = 2¯ p(2τ) (λ + β) ¯ Ψ = β φ(0) ¯

(19)

d¯ p

da (a) = −(λ + γ)¯ p(a) d ¯ φ

da (a) = (λ + γ) ¯ φ(a)

(20)

¯

p(0) = β R ¯ φ(2τ) = 2 ¯ ¯ Ψ

(21)

Equations(20)-(21)give

¯

p(a) = β Re ¯

−(λ+γ)a

, φ(a) = 2 ¯ ¯ Ψe

(λ+γ)(a−2τ)

.

(22)

If

R, ¯ Ψ ¯ 6= 0

,(19)isthenequivalentto

λ + β = 2βe

(λ+γ)2τ

.

(23)

Theorem 3.7 (First eigenelements).There exists a unique solution

(λ, R, ¯ p, ¯ Ψ, ¯ φ) ¯

of (19)-(21)

suhthat

R > ¯ 0, p > ¯ 0, R ¯ + Z

0

¯

p(a)da = 1,

Ψ ¯ > 0, φ > ¯ 0, Ψ ¯ ¯ R + Z

0

φ(a)¯ ¯ p(a)da = 1.

Proof. Itisenoughtoobservethat (23)hasauniquerealsolution.

3.4 Long time asymptoti

3.4.1 without theation ofthe drugs

We onsiderhere thatthere is noation ofthe drugsfor

t > 0

, i.e. that

β > 0

and

γ ≥ 0

are

onstant. Thersteigenelements

(λ, R, ¯ p, ¯ Ψ, ¯ φ) ¯

aregivenbyTheorem3.7.

Theorem3.8. Let

(R

0

, p

0

)

bean initial ondition,

C > 0

besuhthat

|R

0

| ≤ C R, ¯ |p

0

(·)| ≤ C p(·), ¯

and

(R, p)

bethe solutionof (9)-(12). Then,for all

t > 0

,

|R(t)| ≤ C Re ¯

λt

, |p(t, ·)| ≤ C p(·)e ¯

λt

,

(24)

ΨR(t) + ¯ Z

0

φ(a)p(t, a)da ¯

e

−λt

= ¯ ΨR

0

+ Z

0

φ(a)p ¯

0

(a)da := ρ,

(25)

Ψ|R(t)| ¯ + Z

0

φ(a)|p(t, a)|da ¯

e

λt

≤ Ψ|R ¯

0

| + Z

0

φ(a)|p ¯

0

(a)|da,

(26)

t

lim

→∞

Ψ|R(t)e ¯

λt

− ρ R| ¯ + Z

0

φ(a)|p(t, a)e ¯

λt

− ρ¯ p(a)|da

= 0.

(27)

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