Global Asymptotic Stability and Hopf Bifurcation for a Blood Cell Production Model

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Niveau: Supérieur, Doctorat, Bac+8
Global Asymptotic Stability and Hopf Bifurcation for a Blood Cell Production Model? Fabien Crauste Year 2005 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 E-mail: Abstract We analyze the asymptotic stability of a nonlinear system of two differential equa- tions with delay describing the dynamics of blood cell production. This process takes place in the bone marrow where stem cells differentiate throughout divisions in blood cells. Taking into account an explicit role of the total population of hematopoietic stem cells on the introduction of cells in cycle, we are lead 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 population's dying out, and we obtain the existence of a Hopf bifurcation for the only nontrivial pos- itive steady state, leading to the existence of periodic solutions. These latter are related to dynamical diseases affecting blood cells known for their cyclic nature. Keywords: asymptotic stability, delay differential equations, characteristic equation, delay- dependent coefficients, Hopf bifurcation, blood cell model, stem cells. 1 Introduction Blood cell production process is based upon the differentiation of so-called hematopoietic stem cells, located in the bone marrow.

  • blood cell

  • production model

  • has been

  • steady state

  • unique solution

  • stem cells

  • been introduced

  • cycle duration

  • pos- itive steady

  • crauste stability


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GlobalAsymptoticStabilityandHopfBifurcationforaBloodCellProductionModelFabienCrausteYear2005LaboratoiredeMathe´matiquesApplique´es,UMR5142,Universite´dePauetdesPaysdel’Adour,Avenuedel’universite´,64000Pau,France.ANUBISproject,INRIA–FutursE-mail:fabien.crauste@univ-pau.frAbstractWeanalyzetheasymptoticstabilityofanonlinearsystemoftwodifferentialequa-tionswithdelaydescribingthedynamicsofbloodcellproduction.Thisprocesstakesplaceinthebonemarrowwherestemcellsdifferentiatethroughoutdivisionsinbloodcells.Takingintoaccountanexplicitroleofthetotalpopulationofhematopoieticstemcellsontheintroductionofcellsincycle,weareleadtostudyacharacteristicequationwithdelay-dependentcoefficients.Wedetermineanecessaryandsufficientconditionfortheglobalstabilityofthefirststeadystateofourmodel,whichdescribesthepopulation’sdyingout,andweobtaintheexistenceofaHopfbifurcationfortheonlynontrivialpos-itivesteadystate,leadingtotheexistenceofperiodicsolutions.Theselatterarerelatedtodynamicaldiseasesaffectingbloodcellsknownfortheircyclicnature.Keywords:asymptoticstability,delaydifferentialequations,characteristicequation,delay-dependentcoefficients,Hopfbifurcation,bloodcellmodel,stemcells.1IntroductionBloodcellproductionprocessisbaseduponthedifferentiationofso-calledhematopoieticstemcells,locatedinthebonemarrow.Theseundifferentiatedandunobservablecellshaveuniquecapacitiesofdifferentiation(theabilitytoproducecellscommittedtooneofthethreebloodcelltypes:redbloodcells,whitecellsorplatelets)andself-renewal(theabilitytoproducecellswiththesameproperties).MathematicalmodellingofhematopoieticstemcellsdynamicshasbeenintroducedattheendoftheseventiesbyMackey[21].Heproposedasystemoftwodifferentialequationswithdelaywherethetimedelaydescribesthecellcycleduration.Inthismodel,hematopoi-eticstemcellsareseparatedinproliferatingandnonproliferatingcells,theselatterbeingToappearinMathematicalBiosciencesandEngineering1
F.CrausteStabilityofabloodcellproductionmodelintroducedintheproliferatingphasewithanonlinearratedependingonlyuponthenon-proliferatingcellpopulation.Theresultingsystemofdelaydifferentialequationsisthenuncoupled,withthenonproliferatingcellsequationcontainingthewholeinformationaboutthedynamicsofthehematopoieticstemcellpopulation.Thestabilityanalysisofthemodelin[21]highlightedtheexistenceofperiodicsolutions,throughaHopfbifurcation,describinginsomecasesdiseasesaffectingbloodcells,characterizedbyperiodicoscillations[19].ThemodelofMackey[21]hasbeenstudiedbymanyauthors,mainlysincethebeginningofthenineties.MackeyandRey[23,24,25]numericallystudiedthebehaviorofastruc-turedmodelbasedonthemodelin[21],stressingtheexistenceofstrangebehaviorsofthecellpopulations(likeoscillations,orchaos).MackeyandRudnicky[26,27]developedthedescriptionofbloodcelldynamicsthroughanage-maturitystructuredmodel,stressingtheinfluenceofhematopoieticstemcellsonbloodproduction.TheirmodelhasbeenfurtherdevelopedbyDysonetal.[13,14,15],AdimyandPujo-Menjouet[7],AdimyandCrauste[2,3]andAdimyetal.[4].Recently,Adimyetal.[5,6]studiedthemodelproposedin[21]takingintoaccountthatcellsincycledivideaccordingtoadensityfunction(usuallygammadistributionsplayanimportantroleincellcyclesdurations),contrarytowhathasbeenassumedintheabove-citedworks,wherethedivisionhasalwaysbeenassumedtooccuratthesametime.Morerecently,Pujo-MenjouetandMackey[30]andPujo-Menjouetetal.[29]gaveabetterinsightintothemodelofMackey[21],highlightingtheroleofeachparameterofthemodelontheappearanceofoscillationsand,moreparticularly,ofperiodicsolutions,whenthemodelisappliedtothestudyofchronicmyelogenousleukemia[16].Contrarytotheassumptionusedinalloftheabove-citedworks,westudy,inthispaper,themodelintroducedbyMackey[21]consideringthattherateofintroductionintheprolifer-atingphase,whichcontainsthenonlinearityofthismodel,dependsuponthetotalpopulationofhematopoieticstemcells,andnotonlyuponthenonproliferatingcellpopulation.Thein-troductionincellcycleispartlyknowntobeaconsequenceofanactivationofhematopoieticstemcellsduetomoleculesfixingonthem.Hence,theentirepopulationisincontactwiththesemoleculesanditisreasonabletothinkthatthetotalnumberofhematopoieticstemcellsplaysaroleintheintroductionofnonproliferatingcellsintheproliferatingphase.Thefirstconsequenceisthatthemodelisnotuncoupled,andthenonproliferatingcellpopulationequationdoesnotcontainthewholeinformationaboutthedynamicsofbloodcellproduction,contrarytothemodelin[21,29,30].Therefore,weareleadtothestudyofamodifiedsystemofdelaydifferentialequations(system(3)–(4)),wherethedelaydescribesthecellcycleduration,withanonlinearpartdependingononeofthetwopopulations.Secondly,whilestudyingthelocalasymptoticstabilityofthesteadystatesofourmodel,wehavetodeterminerootsofacharacteristicequationtakingtheformofafirstdegreeexponentialpolynomialwithdelay-dependentcoefficients.Forsuchequations,BerettaandKuang[9]developedaveryusefulandpowerfultechnic,thatwewillapplytoourmodel.Ouraimistoshow,throughthestudyofthesteadystates’stability,thatourmodel,describedin(3)–(4),exhibitssimilarpropertiesthanthemodelin[21]andthatitcanbeusedtomodelbloodcellsproductiondynamicswithgoodresults,inparticularlywhenoneisinterestedintheappearanceofperiodicsolutionsinbloodcelldynamicsmodels.Wewanttopointoutthattheusuallyacceptedassumptionabouttheintroductionratemaybelimitativeandthatourmodelcandisplayinterestingdynamics,suchasstabilityswitches,thathaveneverbeennotedbefore.Thepresentworkisorganizedasfollows.Inthenextsectionwepresentourmodel,statedinequations(3)and(4).Wethendeterminethesteadystatesofthismodel.Insection3,welinearizethesystem(3)–(4)aboutasteadystateandwededucetheassociatedcharacteristicequation.Insection4,weestablishnecessaryandsufficientconditionsforthe2