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

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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 (

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MATHEMATICALBIOSCIENCESVAoNluDmEeN3,GINNuEmEbRerIN2,GApril2006http://www.mbejournal.org/pp.325–346GLOBALASYMPTOTICSTABILITYANDHOPFBIFURCATIONFORABLOODCELLPRODUCTIONMODELFabienCrausteLaboratoiredeMathe´matiquesApplique´esUMR5142,AUvneinvueresidtee´lduenPivaeurseitte´d,e6s40P0a0ysPdaeu,lFArdaonucre,,ANUBISproject,INRIAFuturs(CommunicatedbyYasuhiroTakeuchi)Abstract.Weanalyzetheasymptoticstabilityofanonlinearsystemoftwodifferentialequationswithdelay,describingthedynamicsofbloodcellproduc-tion.Thisprocesstakesplaceinthebonemarrow,wherestemcellsdifferen-tiatethroughoutdivisioninbloodcells.Takingintoaccountanexplicitroleofthetotalpopulationofhematopoieticstemcellsintheintroductionofcellsincycle,weareledtostudyacharacteristicequationwithdelay-dependentcoefficients.Wedetermineanecessaryandsufficientconditionfortheglobalstabilityofthefirststeadystateofourmodel,whichdescribesthepopula-tion’sdyingout,andweobtaintheexistenceofaHopfbifurcationfortheonlynontrivialpositivesteadystate,leadingtotheexistenceofperiodicsolutions.Theselatterarerelatedtodynamicaldiseasesaffectingbloodcellsknownfortheircyclicnature.1.Introduction.Thebloodcellproductionprocessisbasedonthedifferentiationofso-calledhematopoieticstemcells,locatedinthebonemarrow.Theseundiffer-entiatedandunobservablecellshaveuniquecapacitiesofdifferentiation(theabilitytoproducecellscommittedtooneofthethreebloodcelltypes:redbloodcells,whitecellsorplatelets)andself-renewal(theabilitytoproducecellswiththesameproperties).MathematicalmodellingofhematopoieticstemcelldynamicswasintroducedattheendoftheseventiesbyMackey[20].Heproposedasystemoftwodifferentialequationswithdelaywherethetimedelaydescribesthecellcycleduration.Inthismodel,hematopoieticstemcellsareseparatedinproliferatingandnonproliferatingcells,theselatterbeingintroducedintheproliferatingphasewithanonlinearratedependingonlyuponthenonproliferatingcellpopulation.Theresultingsystemofdelaydifferentialequationsisthenuncoupled,withthenonproliferatingcellsequationcontainingthewholeinformationaboutthedynamicsofthehematopoieticstemcellpopulation.Thestabilityanalysisofthemodelin[20]highlightedtheexistenceofperiodicsolutions,throughaHopfbifurcation,describinginsomecasesdiseasesaffectingbloodcells,characterizedbyperiodicoscillations[18].2000MathematicsSubjectClassification.34K20,92C37,34D05,34C23,34K99.Keywordsandphrases.asymptoticstability,delaydifferentialequations,characteristicequa-tion,delay-dependentcoefficients,Hopfbifurcation,bloodcellmodel,stemcells.523
326F.CRAUSTEMackey’smodel[20]hasbeenstudiedbymanyauthors,mainlysincethebe-ginningofthenineties.MackeyandRey[22,23,24]numericallystudiedthebe-haviorofastructuredmodelbasedonthemodelin[20],stressingtheexistenceofstrangebehaviorsofthecellpopulations(suchasoscillationsorchaos).MackeyandRudnicky[25,26]developedthedescriptionofbloodcelldynamicsthroughanage-maturitystructuredmodel,stressingtheinfluenceofhematopoieticstemcellsonbloodproduction.TheirmodelhasbeenfurtherdevelopedbyDysonetal.[12,13,14],AdimyandPujo-Menjouet[6],AdimyandCrauste[1,2]andAdimyetal.[3].Recently,Adimyetal.[4,5]studiedthemodelproposedin[20],takingintoaccountthatcellsincycledivideaccordingtoadensityfunction(usuallygammadistributionsplayanimportantroleincellcycledurations),contrarytowhathasbeenassumedintheworkscitedabove,wherethedivisionhasalwaysbeenassumedtooccuratthesametime.Morerecently,Pujo-MenjouetandMackey[29]andPujo-Menjouetetal.[28]gaveabetterinsightintotheMackey’smodel[20],highlightingtheroleofeachparameterofthemodelontheappearanceofoscillationsand,moreparticularly,ofperiodicsolutions,whenthemodelisappliedtothestudyofchronicmyelogenousleukemia[15].Contrarytotheassumptionusedinalloftheworkscitedabove,westudy,inthispaper,themodelintroducedbyMackey[20],consideringthattherateofintroductionintheproliferatingphase,whichcontainsthenonlinearityofthismodel,dependsuponthetotalpopulationofhematopoieticstemcellsandnotonlyuponthenonproliferatingcellpopulation.Theintroductionincellcycleispartlyknowntobeaconsequenceofactivationofhematopoieticstemcellsduetomoleculesfixingonthem.Hence,theentirepopulationisincontactwiththesemolecules,anditisreasonabletothinkthatthetotalnumberofhematopoieticstemcellsplaysaroleintheintroductionofnonproliferatingcellsintheproliferatingphase.Thefirstconsequenceisthatthemodelisnotuncoupled,andthenonprolif-eratingcellpopulationequationdoesnotcontainalltheinformationaboutthedynamicsofbloodcellproduction,contrarytothemodelin[20,28,29].Therefore,weareledtothestudyofamodifiedsystemofdelaydifferentialequations(system(3)–(4)),wherethedelaydescribesthecellcycleduration,withanonlinearpartdependingononeofthetwopopulations.Second,whilestudyingthelocalasymptoticstabilityofthesteadystatesofourmodel,wehavetodeterminerootsofacharacteristicequationtakingtheformofafirst-degreeexponentialpolynomialwithdelay-dependentcoefficients.Forsuchequations,BerettaandKuang[8]developedaveryusefulandpowerfultechnique,whichwewillapplytoourmodel.Ouraimistoshow,throughthestudyofthesteadystates’stability,thatourmodel,describedin(3)–(4),exhibitspropertiessimilartothosein[20]andthatitcanbeusedtomodelbloodcellproductiondynamicswithgoodresults,particularlywhenoneisinterestedintheappearanceofperiodicsolutionsinbloodcelldynamicsmodels.Wewanttopointoutthattheusuallyacceptedassumptionabouttheintroductionratemaybelimitativeandthatourmodelcandisplayinterestingdynamics,suchasstabilityswitches,thathaveneverbeennotedbefore.Thepresentworkisorganizedasfollows.Inthenextsectionwepresentourmodel,statedinequations(3)and(4).Wethendeterminethesteadystatesofthismodel.Insection3,welinearizethesystem(3)–(4)aboutasteadystate,andwe