MATHEMATICAL BIOSCIENCES http: www mbejournal org AND ENGINEERING Volume Number April pp

Fabien Crauste

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Niveau: Supérieur, Doctorat, Bac+8
MATHEMATICAL BIOSCIENCES AND ENGINEERING Volume 3, Number 2, April 2006 pp. 325–346 GLOBAL ASYMPTOTIC STABILITY AND HOPF BIFURCATION FOR A BLOOD CELL PRODUCTION MODEL Fabien Crauste Laboratoire de Mathematiques Appliquees UMR 5142, Universite de Pau et des Pays de l'Adour, Avenue de l'universite, 64000 Pau, France, ANUBIS project, INRIA Futurs (Communicated by Yasuhiro Takeuchi) Abstract. We analyze the asymptotic stability of a nonlinear system of two differential equations with delay, describing the dynamics of blood cell produc- tion. This process takes place in the bone marrow, where stem cells differen- tiate throughout division in blood cells. Taking into account an explicit role of the total population of hematopoietic stem cells in the introduction of cells in cycle, we are led to study a characteristic equation with delay-dependent coefficients. We determine a necessary and sufficient condition for the global stability of the first steady state of our model, which describes the popula- tion's dying out, and we obtain the existence of a Hopf bifurcation for the only nontrivial positive steady state, leading to the existence of periodic solutions. These latter are related to dynamical diseases affecting blood cells known for their cyclic nature. 1. Introduction. The blood cell production process is based on the differentiation of so-called hematopoietic stem cells, located in the bone marrow. These undiffer- entiated and unobservable cells have unique capacities of differentiation (the ability to produce cells committed to one of the three blood cell types: red blood cells, white cells or platelets) and self-renewal (

• blood cell


• has been


• asymptotic stability


• unique solution


• stem cells


• production dynamics


• blood cells known

Sujets

• Blood cell
• Has Been
• Lyapunov stability
• Célula madre

Informations

• Publié par: mijec
• Nombre de lectures: 20
• Langue: English

Extrait

MATHEMATICALBIOSCIENCES V A o N lu D m E e N 3, GI N N u E m E b R er IN 2, G April2006 http://www.mbejournal.org/ pp.325–346 GLOBALASYMPTOTICSTABILITYANDHOPFBIFURCATION FORABLOODCELLPRODUCTIONMODEL FabienCrauste LaboratoiredeMathe´matiquesApplique´esUMR5142, A U v n e i n v u er e si d t e e´ l d ’u e n P iv a e u rs e i t te´ d , e 6 s 40 P 0 a 0 ys P d a e u, l’ F A r d a o n u c r e , , ANUBISproject,INRIAFuturs (CommunicatedbyYasuhiroTakeuchi) Abstract.Weanalyzetheasymptoticstabilityofanonlinearsystemoftwo differentialequationswithdelay,describingthedynamicsofbloodcellproduc- tion.Thisprocesstakesplaceinthebonemarrow,wherestemcellsdifferen- tiatethroughoutdivisioninbloodcells.Takingintoaccountanexplicitrole ofthetotalpopulationofhematopoieticstemcellsintheintroductionofcells incycle,weareledtostudyacharacteristicequationwithdelay-dependent coefficients.Wedetermineanecessaryandsufficientconditionfortheglobal stabilityofthefirststeadystateofourmodel,whichdescribesthepopula- tion’sdyingout,andweobtaintheexistenceofaHopfbifurcationfortheonly nontrivialpositivesteadystate,leadingtotheexistenceofperiodicsolutions. Theselatterarerelatedtodynamicaldiseasesaffectingbloodcellsknownfor theircyclicnature. 1.Introduction.Thebloodcellproductionprocessisbasedonthedifferentiation ofso-calledhematopoieticstemcells,locatedinthebonemarrow.Theseundiffer- entiatedandunobservablecellshaveuniquecapacitiesofdifferentiation(theability toproducecellscommittedtooneofthethreebloodcelltypes:redbloodcells, whitecellsorplatelets)andself-renewal(theabilitytoproducecellswiththesame properties). Mathematicalmodellingofhematopoieticstemcelldynamicswasintroducedat theendoftheseventiesbyMackey[20].Heproposedasystemoftwodifferential equationswithdelaywherethetimedelaydescribesthecellcycleduration.Inthis model,hematopoieticstemcellsareseparatedinproliferatingandnonproliferating cells,theselatterbeingintroducedintheproliferatingphasewithanonlinearrate dependingonlyuponthenonproliferatingcellpopulation.Theresultingsystem ofdelaydifferentialequationsisthenuncoupled,withthenonproliferatingcells equationcontainingthewholeinformationaboutthedynamicsofthehematopoietic stemcellpopulation.Thestabilityanalysisofthemodelin[20]highlightedthe existenceofperiodicsolutions,throughaHopfbifurcation,describinginsomecases diseasesaffectingbloodcells,characterizedbyperiodicoscillations[18]. 2000MathematicsSubjectClassification.34K20,92C37,34D05,34C23,34K99. Keywordsandphrases.asymptoticstability,delaydifferentialequations,characteristicequa- tion,delay-dependentcoefficients,Hopfbifurcation,bloodcellmodel,stemcells. 523

326F.CRAUSTE Mackey’smodel[20]hasbeenstudiedbymanyauthors,mainlysincethebe- ginningofthenineties.MackeyandRey[22,23,24]numericallystudiedthebe- haviorofastructuredmodelbasedonthemodelin[20],stressingtheexistenceof strangebehaviorsofthecellpopulations(suchasoscillationsorchaos).Mackey andRudnicky[25,26]developedthedescriptionofbloodcelldynamicsthrough anage-maturitystructuredmodel,stressingtheinfluenceofhematopoieticstem cellsonbloodproduction.TheirmodelhasbeenfurtherdevelopedbyDysonetal. [12,13,14],AdimyandPujo-Menjouet[6],AdimyandCrauste[1,2]andAdimyet al.[3].Recently,Adimyetal.[4,5]studiedthemodelproposedin[20],takinginto accountthatcellsincycledivideaccordingtoadensityfunction(usuallygamma distributionsplayanimportantroleincellcycledurations),contrarytowhathas beenassumedintheworkscitedabove,wherethedivisionhasalwaysbeenassumed tooccuratthesametime. Morerecently,Pujo-MenjouetandMackey[29]andPujo-Menjouetetal.[28] gaveabetterinsightintotheMackey’smodel[20],highlightingtheroleofeach parameterofthemodelontheappearanceofoscillationsand,moreparticularly,of periodicsolutions,whenthemodelisappliedtothestudyofchronicmyelogenous leukemia[15]. Contrarytotheassumptionusedinalloftheworkscitedabove,westudy, inthispaper,themodelintroducedbyMackey[20],consideringthattherate ofintroductionintheproliferatingphase,whichcontainsthenonlinearityofthis model,dependsuponthetotalpopulationofhematopoieticstemcellsandnot onlyuponthenonproliferatingcellpopulation.Theintroductionincellcycleis partlyknowntobeaconsequenceofactivationofhematopoieticstemcellsdue tomoleculesfixingonthem.Hence,theentirepopulationisincontactwiththese molecules,anditisreasonabletothinkthatthetotalnumberofhematopoieticstem cellsplaysaroleintheintroductionofnonproliferatingcellsintheproliferating phase. Thefirstconsequenceisthatthemodelisnotuncoupled,andthenonprolif- eratingcellpopulationequationdoesnotcontainalltheinformationaboutthe dynamicsofbloodcellproduction,contrarytothemodelin[20,28,29].Therefore, weareledtothestudyofamodifiedsystemofdelaydifferentialequations(system (3)–(4)),wherethedelaydescribesthecellcycleduration,withanonlinearpart dependingononeofthetwopopulations. Second,whilestudyingthelocalasymptoticstabilityofthesteadystatesofour model,wehavetodeterminerootsofacharacteristicequationtakingtheformof afirst-degreeexponentialpolynomialwithdelay-dependentcoefficients.Forsuch equations,BerettaandKuang[8]developedaveryusefulandpowerfultechnique, whichwewillapplytoourmodel. Ouraimistoshow,throughthestudyofthesteadystates’stability,thatour model,describedin(3)–(4),exhibitspropertiessimilartothosein[20]andthatit canbeusedtomodelbloodcellproductiondynamicswithgoodresults,particularly whenoneisinterestedintheappearanceofperiodicsolutionsinbloodcelldynamics models.Wewanttopointoutthattheusuallyacceptedassumptionaboutthe introductionratemaybelimitativeandthatourmodelcandisplayinteresting dynamics,suchasstabilityswitches,thathaveneverbeennotedbefore. Thepresentworkisorganizedasfollows.Inthenextsectionwepresentour model,statedinequations(3)and(4).Wethendeterminethesteadystatesofthis model.Insection3,welinearizethesystem(3)–(4)aboutasteadystate,andwe