常微分方程病毒模型

BasicModelsforViralDynamics-ODE,唐思铭,BasicModelsforViralDynamics-ODE,A.Perelsonetal,Science,271(1996),1582,M.NowakandC.R.M.Bangham,Science,272(1996),74,V.HerzandM.Nowaketal,Proc.Acad.Sci.USA.,93(1996),7247,Virusreplication,Basicmodelofviraldynamicswithvirusreplicationcontainsthreevariables:,x(t):thepopulationsofuninfectedcellsattimety(t):thepopulationsofinfectedcellsattimet,whichproducevirus,v(t):freevirusparticlesattimet,ky(t),x(t)v(t)+,dx(t),uv(t),ay(t),Uninfectedcells,Virus,Infectedcellsproducingvirus,x(t),v(t),y(t),4,:thedeathrateatwhichvirusparticlesareclearedfromsystem:therateconstantcharacterizinginfectionofcellsofvirusorimmunesystemd:thedeathrateofuninfectedtargetcells:thedeathrateofinfectedcelleitherduetoaction,ky(t),x(t)v(t)+,dx(t),x(t)=,uv(t),dx(t),ay(t),x(t)v(t),y(t)=x(t)v(t)ay(t)(1),v(t)=ky(t)uv(t),Thebasicreproductivenumber:,R,0,=k/adu,NowakandBangham,Science,272(1996),74,A.Korobeinikov,Bull.Math.Biol.,66(2004),879,If,R,0,1,inthebeginningofinfection,eachvirus-infectedcell,produceonaveragelessthanonenewlyinfectedcell.Theinfectioncannotspread,andthatthesystemreturnstotheuninfectedstate:,E,=(/d,0,0).,0,IfR1,theinitiallyeachvirus-infectedcellproduceonaverage,0,morethanonenewlyinfectedcell.Theinfectedcellpopulationwillincrease,andwhereastheuninfectedcellpopulationwilldeclineand,thereforeprovidelessopportunityforthevirustoinfectnewcells.,*,Thesystemwillconvergetotheinfectedstate:E(x,y,*,v).,+=,LASofEand0,GASofEand,0,E+:NowakandBangham(1996),E+:Korobeinikov(2004),xxa,V(x,y,v)=,x,0,ln,xx,+y+v,k,V(x,y,z)=,*,x,0,x,*,ln,0,x,*,+y,*,y,*,ln,y,*,a,+,*,v,v,*,ln,v,*,xx,yy,kvv,BasicModelsforViralDynamics-ODE,A.Perelsonetal,Science,271(1996),1582,M.NowakandC.R.M.Bangham,Science,272(1996),74,V.HerzandM.Nowaketal,Proc.Acad.Sci.USA.,93(1996),7247,Y.Iwasa,F.Michor,M.A.Nowak,J.Theor.Biol.,232(2005),17,Immuneresponses,Basicmodelofviraldynamicswithimmuneresponsescontainsfourvariables:,x(t):thepopulationsofuninfectedcellsattimety(t):thepopulationsofinfectedcellsattimet,whichproducevirus,v(t):freevirusparticlesattimet,z(t):thepopulationsofCTLresponseattimet,ky(t),x(t)v(t)+,dx(t)uv(t),UninfectedcellsVirus,ay(t),Infectedcellsproducingvirus,Proliferationandkilling,+,+,Infected,cells,CTLComplex,b,Proliferation,nokillingKilling,noproliferation,Abortiveinteraction,+,W.Liu,Theor.Popul.Biol.,52(1997),224,M.NowakandC.R.M.Bangham,Science,272(1996),74,T.Kajiwaraetal,DiscreteConti.Dyn.Sys.,4(2004),615,cyz:therateofCTLproliferationinresponsetoantigen,c:,theCTLresponsiveness,pyz:therateatwhichinfectedcellsarekilledbyCTLsbz:therateatwhichCTLsdecayintheabsenceofstimulation,Ifcyb,thesystemshowsdampedoscillationstothe,equilibrium,(,x,y,v,z,),of(2),cy,b,*,x,y,v,v,CTLresponsecanreducevirusloadandincreasetheequilibriumabundanceofuninfectedcells,PanH.,WangW.andWangK.,J.SouthwestChinaNormalUni.,30(2005),796,Thebasicreproductivenumber:,R,0,=k/adu,Theimmuneresponsereproductivenumber:R,=c(kadu)/abk,0,If,R,1,thentheuninfectedstate,E,:GAS.,0,0,If,If,R,0,R,1,1,R1,thentheimmunefreestate,1,1,thentheendemicstateE+:GAS.,E,1,:GAS.,V=xx,a,x+y+,v+,p,Panetal,z,0,0lnx,2005,x,k,k,0,V=xx,1,x,1,x,ln,+,yy,1,y,1,ya,ln+,vvvln,11,v,+,p,z,V=xxx,ln,x,1,x,+yyy,y,1,ln,k,y,v,1,k,+,apz,+,x,vvv,ln,v,y,p,+,zzz,ln,z,kk,v,kz,14,BasicModelsforViralDynamics-ODE,A.Korobeinikov,Bull.Math.Biol.,66(2004),879,Virusreplicationandexposedstate,Basicmodelofviraldynamicswithvirusreplicationandexposedstatecontainsfourvariables:,x(t):thepopulationsofuninfectedcellsattimetz(t):thepopulationsofexposedcellsattimety(t):thepopulationsofinfectedcellsattimet,whichproducevirus,v(t):freevirusparticlesattimet,kyt),x(t)v(t),cz(t),+,dx(t),uv(t),bz(t),ay(t),UninfectedcellsVirus,Exposedcells,Infectedcells,producingvirus,x(t),v(t),z(t),y(t),A.Korobeinikov,Bull.Math.Biol.,66(2004),879,)=dx(t)x(t)v(t)x(,z(t)=xt)v(t,)bzt)czt),(4),y(t)=cz(t)ayt),v(t)=ky,(t)uv(t),xv,bz,cz,:therateatwhichuninfectedcellsareinfected:therateatwhichexposedcellsdecay,:therateatwhichexposedcellsbecomeuninfectedcells,i.e.,1/cistheaveragetimeofthelatent,state,Thebasicrepro