<Text-field style="Heading 1" layout="Heading 1">1. Implementation of the algorithm <Font size="14" style="Text"> <Font bold="true" family="Tlwg Typist">RelativeIdentifiabilityTree</Font> presented in </Font></Text-field><Text-field style="Text" size="14" layout="Normal"><Font size="14" encoding="UTF-8" italic="true">N. Verdi\303\250re and S. Orange. </Font><Font size="14" underline="true" italic="true">A systematic approach for doing an a priori identifiability study of dynamical nonlinear models.</Font><Font size="14" encoding="UTF-8" italic="true"> Mathematical Biosciences, 308:105\342\200\223113, 2019.
</Font><Font size="14">
</Font><Font size="11">The main procedure of the set of procedures hereafter is named <Font italic="true">RelativeIdentifiabilityTree</Font>. </Font><Font size="12">From the exhaustive summary of a parametrized dynamical system, </Font><Font size="11">it returns the identifiability tree <Font italic="true">T</Font> of this system, that is the set of all the possible lists of all the parameters such that, for any list <Font italic="true">l</Font> of <Font italic="true">T</Font> and any prefix<Font italic="true"> b</Font> of <Font italic="true">l</Font>, <Font italic="true">b</Font> is followed in <Font italic="true">l</Font> by an identifable parameter if there exists one. </Font><Font size="12">
From <Font italic="true">T</Font>, it is possible to determine the identifiability of any parameter when some others are supposed known.</Font><Font size="11" bold="true">
Procedure </Font><Font size="12" bold="true" italic="true">RelativeIdentifiabilityTree</Font><Font size="12" italic="true">
</Font><Font size="12" bold="true">Calling sequence
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* ES: an exhaustive summary of a system ;
*<Font italic="true"> Cond: </Font>a list possibly empty of parameters contrains and/or initial conditions;
* <Font italic="true">Prefix: </Font>an optional argument which is a list of parameters supposed to be known.
<Font bold="true">Description</Font>
The output is the identifiability tree corresponding to the system.
The following convention is used: for any list <Font italic="true">l</Font>, </Font><Font size="11"> the parameter <Font italic="true">p</Font> is denoted <Font italic="true">-p</Font> if it is not relatively identifiable with respect to the parameters appearing before it in <Font italic="true">l</Font>.</Font><Font size="12"> For example, if <Font italic="true">[p1, -p2, p3]</Font> is a list returned by this procedure, it means that
* <Font italic="true">p1</Font> is identifiable whereas neither <Font italic="true">p2</Font> and <Font italic="true">p3</Font> are identifiable;
* if <Font italic="true">p2</Font> is determined then <Font italic="true">p3</Font> is identifiable.</Font></Text-field>%!G-I%mrowG6#/I+modulenameG6"I,TypesettingGI(_syslibGF'6[\p-I#miGF$6(Q9IsRelativelyIdentifiableF'/%'familyGQ,Courier~NewF'/%'italicGQ&falseF'/%+foregroundGQ+[120,0,14]F'/%0font_style_nameGQ,Maple~InputF'/%,mathvariantGQ'normalF'-I#moGF$60Q"~F'F/F5F8F;/%&fenceGF4/%*separatorGF4/%)stretchyGF4/%*symmetricGF4/%(largeopGF4/%.movablelimitsGF4/%'accentGF4/%'lspaceGQ&0.0emF'/%'rspaceGFR-F?60Q*&coloneq;F'F/F5F8F;FBFDFFFHFJFLFN/FQQ,0.2777778emF'/FTFYF>-F?62Q%procF'F//%%boldGQ%trueF'F5F8/F<Q%boldF'/%+fontweightGF\oFBFDFFFHFJFLFNFPFS-I(mfencedGF$6)-F#6.-F,6(Q&thetaF'F/F2F5F8F;-F?60Q",F'F/F5F8F;FB/FEFjnFFFHFJFLFNFP/FTQ,0.3333333emF'F>-F,6(Q"lF'F/F2F5F8F;FgoF>-F,6(Q#ESF'F/F2F5F8F;FgoF>-F,6(Q%CondF'F/F2F5F8F;/%+executableGF4F;F/FhnF5F8F[oF]o-F,6#Q!F'-I'mspaceGF$6&/%'heightGQ&0.0exF'/%&widthGFR/%&depthGF`q/%*linebreakGQ(newlineF'-F\q6&F^qFaqFcq/FfqQ%autoF'-F?60Q"#F'F/F5F8F;FBFDFFFHFJFLFNFPFS-F,6(Qaq~Inputs:~~*~a~parameter,~theta,~which~the~relative~identifiabiability~with~respect~to~l~will~be~testedF'F/F2F5F8F;F[qFhqF\r-F,6(Qgn~~~~~~~~~~*~a~list~of~parameters,~l,~the~prefix~to~complete,F'F/F2F5F8F;F[qFhqF\r-F,6(Q]o~~~~~~~~~~*~the~list,~ES,~of~polynomials~of~the~exhaustif~summary,F'F/F2F5F8F;F[qFhqF\r-F,6(Q\o~~~~~~~~~~*~the~set~of~conditions~Cond~satified~by~the~parametersF'F/F2F5F8F;F[qFhqF\r-F,6(Qfp~Outputs:~a~boolean~equal~to~true~if~theta~is~identifiable~relatively~to~l,~false~otherwiseF'F/F2F5F8F;F[qFhq-F?62Q&localF'F/FhnF5F8F[oF]oFBFDFFFHFJFLFNFPFSF>-F,6(Q$varF'F/F2F5F8F;FgoF>-F,6(Q'newvarF'F/F2F5F8F;FgoF>-F,6(Q)ES_tildeF'F/F2F5F8F;FgoF>-F,6(Q(equasubF'F/F2F5F8F;FgoF>-F,6(Q"kF'F/F2F5F8F;FgoF>-F,6(Q&paramF'F/F2F5F8F;FgoF>-F,6(Q+Cond_tildeF'F/F2F5F8F;FgoF>-F,6(Q*eqns_CondF'F/F2F5F8F;FgoF>-F,6(Q(l_tildeF'F/F2F5F8F;FgoF>-F,6(Q'eqns_lF'F/F2F5F8F;FgoF>-F,6(Q+eqns_thetaF'F/F2F5F8F;FgoF>-F,6(Q$sasF'F/F2F5F8F;FgoF>-F,6(Q"RF'F/F2F5F8F;FgoF>-F,6(Q%videF'F/F2F5F8F;-F?60Q";F'F/F5F8F;FBFjoFFFHFJFLFNFPFZF[qFhqF>F>F>F>-F,6(Q%prinF'F/F2F5F8F;-F,6(Q#tfF'F/F2F5F8F;-F`o6'-F#6,Fhp-I#msGF$6#QRTesting~identifiability~of~%a~relatively~to~%a~\nF'FgoF>FdoFgoF>F]pFfpF;F/F5F8F;F[vF[qFhqF>F>F>F>FasF>FUF>-F`o6+-F#6*-F,6(Q#opF'F/F2F5F8F;-F`o6)-F#6&-F,6(Q'indetsF'F/F2F5F8F;-F`o6)-F#6%F`pFfpF;F/FhnF5F8F[oF]oFfpF;F/FhnF5F8F[oF]oFgoF>F`w-F`o6)-F#6&Fgw-F`o6)-F#6%FcpFfpF;F/FhnF5F8F[oF]oFfpF;F/FhnF5F8F[oF]oFfpF;F/FhnF5F8F[oF]o/%%openGQ"[F'/%&closeGQ"]F'F>F[vF>F>F>F>F\r-F,6(Qds~creation~of~the~new~variables,~and~of~ES_tilde~(resp.~Cond_tilde~and~l_tidle)~the~ES~(resp.~Cond~and~l)~evaluated~in~these~new~variablesF'F/F2F5F8F;F[qFhqF>F>F>F>FdsF>FUF>-F`o6+-F#6&-F,6(Q$seqF'F/F2F5F8F;-F`o6)-F#6--F,6(Q$catF'F/F2F5F8F;-F`o6)-F#6(-F,6(Q#t_F'F/F2F5F8F;FgoF>F`tFfpF;F/FhnF5F8F[oF]oFgoF>F`tF>-F?62Q#inF'F/FhnF5F8F[oF]oFBFDFFFHFJFLFNFPFSF>FasFfpF;F/FhnF5F8F[oF]oFfpF;F/FhnF5F8F[oF]oFfxFixF[vF[qFhqF>F>F>F>FjsFUF>-F,6(Q'EquateF'F/F2F5F8F;-F`o6)-F#6(FasFgoF>FdsFfpF;F/FhnF5F8F[oF]oF[vF[qFhqF>F>F>F>FgsF>FUF>-F,6(Q%evalF'F/F2F5F8F;-F`o6)-F#6'F`pFgoFjsFfpF;F/FhnF5F8F[oF]oF[vF[qFhqF>F>F>F>FctF>FUF>F^[l-F`o6)-F#6'FcpFgoFjsFfpF;F/FhnF5F8F[oF]oF[vF[qFhqF>F>F>F>FitF>FUF>F^[l-F`o6)-F#6'F]pFgoFjsFfpF;F/FhnF5F8F[oF]oF[vF[qFhqF>F>F>F>F\r-F,6(Q[o~Creation~of~the~semialgebraic~set,~sas,~used~for~emptiness~testF'F/F2F5F8F;F[qFhqF>F>F>F>FbuF>FUF>-F`o6+-F#65F`w-F`o6)-F#6&Fgz-F`o6)-F#6'F`pFgoFgsFfpF;F/FhnF5F8F[oF]oFfpF;F/FhnF5F8F[oF]oFgoF>F`wFbxFgoF>F`w-F`o6)-F#6&Fgz-F`o6)-F#6'F]pFgoFitFfpF;F/FhnF5F8F[oF]oFfpF;F/FhnF5F8F[oF]oFgoFdoF>-F?60Q+&NotEqual;F'F/F5F8F;FBFDFFFHFJFLFNFXFZF>F^[l-F`o6)-F#6)FdoFgoF>FgzFjzFfpF;F/FhnF5F8F[oF]oFfpF;F/FhnF5F8F[oF]oFfxFixF[vF[qFhqF>F>F>F>FeuF>FUF>-F,6(Q.RegularChainsF'F/F2F5F8F;-F?60Q#:-F'F/F5F8F;FBFDFFFHFJFLFNFPFS-F,6(Q/PolynomialRingF'F/F2F5F8F;-F`o6)-F#6%-F`o6+-F#6&F`w-F`o6)-F#6&Fgw-F`o6)-F#6%FbuFfpF;F/FhnF5F8F[oF]oFfpF;F/FhnF5F8F[oF]oFfpF;F/FhnF5F8F[oF]oFfxFixFfpF;F/FhnF5F8F[oF]oF[vF[qFhqF>F>F>F>F\r-F,6(Qen~returning~the~result~of~the~relative~identifiability~testF'F/F2F5F8F;F[qFhqF>F>F>F>-F?62Q'returnF'F/FhnF5F8F[oF]oFBFDFFFHFJFLFNFPFSF>F[^lF^^l-F,6(Q6SemiAlgebraicSetToolsF'F/F2F5F8F;F^^l-F,6(Q(IsEmptyF'F/F2F5F8F;-F`o6)-F#6(FbuFgoF>FeuFfpF;F/FhnF5F8F[oF]oF>F[vF[q-F?62Q$endF'F/FhnF5F8F[oF]oFBFDFFFHFJFLFNFPFSF>FenF>-F?60Q":F'F/F5F8F;FBFDFFFHFJFLFNFXFZF[q-F?6-FAF;FBFDFFFHFJFLFNFPFSF[qFhqFj`lF[q-F,6(Q<CompleteWithComputedSubTreeF'F/F2F5F8F;F>FUF>Fen-F`o6)-F#6+-F,6(Q%prefF'F/F2F5F8F;FgoF>-F,6(Q'TildeTF'F/F2F5F8F;FgoF>-F,6(Q(nbparamF'F/F2F5F8F;FfpF;F/FhnF5F8F[oF]oFhpF[qFhqF\r-F,6(Q`s~Inputs~~:~The~prefix~pref~to~complete~if~possible~with~the~set~TildeT~of~lists~(of~the~relative~idenfiability~tree)~already~computedF'F/F2F5F8F;F[qFhqF\r-F,6(Q[r~Outputs~:~The~set~of~lists~TildeT~completed~with~lists~of~the~relative~idenfiability~tree~admitting~l~as~prefixF'F/F2F5F8F;F[qFhqF^sF>F>-F,6(Q*TildeTauxF'F/F2F5F8F;FgoF>-F,6(Q#bbF'F/F2F5F8F;FgoF>F>-F,6(Q&bbauxF'F/F2F5F8F;F[vF[qFhqF>F>F>F>FbblF>FUFfalF[vF[qFhqF>F>F>F>-F?62Q#ifF'F/FhnF5F8F[oF]oFBFDFFFHFJFLFNFPFSF>-F,6(Q&evalbF'F/F2F5F8F;-F`o6)-F#6(-F,6(Q%nopsF'F/F2F5F8F;-F`o6)-F#6%FcalFfpF;F/FhnF5F8F[oF]o-F?60Q"=F'F/F5F8F;FBFDFFFHFJFLFNFXFZFialFfpF;F/FhnF5F8F[oF]oF>-F?62Q%thenF'F/FhnF5F8F[oF]oFBFDFFFHFJFLFNFPFSF[qFhqF>F>F>F>F\r-F,6(Q\q~all~the~parameters~appear~in~the~list~pref~which~is~therefore~a~list~of~the~identifiability~treeF'F/F2F5F8F;F[qFhqF>F>F>F>F>F>F>F>FbblF>FUF>-F`o6+-F#6)F`w-F`o6)-F#6%FbblFfpF;F/FhnF5F8F[oF]oFgoF>FcalFfpF;F/FhnF5F8F[oF]o/FgxQ"|frF'/FjxQ"|hrF'FhpF[qFhqF>F>F>F>-F?62Q%elseF'F/FhnF5F8F[oF]oFBFDFFFHFJFLFNFPFSF[qFhqF>F>F>F>F>F>F>F>-F?62Q$forF'F/FhnF5F8F[oF]oFBFDFFFHFJFLFNFPFSF>FeblF>FdzF>FfalF>-F?62Q#doF'F/FhnF5F8F[oF]oFBFDFFFHFJFLFNFPFSF[qFhqF>F>F>F>F>F>F>F>F\r-F,6(Qbo~searching~for~list~of~TildeT~admitting~a~permutation~of~pref~as~prefixF'F/F2F5F8F;F[qFhqF>F>F>F>F>F>F>F>F>F>F>F>F[clF>F^cl-F`o6)-F#6+Fgw-F`o6)-F#6&Febl-F`o6+-F#6*-I#mnGF$6'Q"1F'F/F5F8F;F>-F?60Q#..F'F/F5F8F;FBFDFFFHFJFLFN/FQQ,0.2222222emF'FSF>FeclFhclFfpF;F/FhnF5F8F[oF]oFfxFixFfpF;F/FhnF5F8F[oF]oF>F\dlF>FgwFhclFfpF;F/FhnF5F8F[oF]oF>F_dlF[qFhqF>F>F>F>F>F>F>F>F>F>F>F>F\r-F,6(Qhp~if~such~a~list,~bb,~is~found~completing~pref~into~a~list~of~the~identifiability~tree~with~bbF'F/F2F5F8F;F[qFhqF>F>F>F>F>F>F>F>F>F>F>F>FhblF>FUF>-F`o6+-F#6*F`wFhclFgoF>F`w-F`o6)-F#6&Febl-F`o6+-F#6/FeclFhclF>-F?60Q"+F'F/F5F8F;FBFDFFFHFJFLFNF`gl/FTFaglF>FiflF>F]glF>Fecl-F`o6)-F#6%FeblFfpF;F/FhnF5F8F[oF]oFfpF;F/FhnF5F8F[oF]oFfxFixFfpF;F/FhnF5F8F[oF]oFfpF;F/FhnF5F8F[oF]oFfxFixF[vF[qFhqF>F>F>F>F>F>F>F>F>F>F>F>FbblF>FUF>-F`o6+-F#6)F`wFidlFgoF>FhblFfpF;F/FhnF5F8F[oF]oF]elF_elFhpF[qFhqF>F>F>F>F>F>F>F>F>F>F>F>Fd`lF>F[clF[qFhqF>F>F>F>F>F>Fd`lF>FgelF[vF[qFhqF>F>F>F>Fd`lF>F[clF>F[vF[qFhqF>F>F>F>F\r-F,6(Qbq~returning~true~if~a~completion~of~pref~had~been~realized,~no~otherwise~and~TildeT~eventually~completedF'F/F2F5F8F;F[qFhqF>F>F>F>Fg_lF>-F`o6+-F#6)F^cl-F`o6)-F#6)FfalF>Fd]lF>FbblFfpF;F/FhnF5F8F[oF]oFgoF>FbblFfpF;F/FhnF5F8F[oF]oFfxFixFhpF[qFd`lF>FenF>Fg`lF[qFj`lF[qFhqFj`lF[qFhq-F,6(Q/CompletePrefixF'F/F2F5F8F;F>FUF>Fen-F`o6)-F#6.F]pFgoF>F`pFgoF>FcpFgoF>FfalFfpF;F/FhnF5F8F[oF]oFhpF[qF\r-F,6(Qbu~Inputs~~:~The~prefix~l~to~complete,~the~exhaustive~summary~ES,~the~set~of~conditions,~cond,~and~of~lists~TildeT~used~to~store~lists~of~the~relative~idenfiability~treeF'F//F3FjnF5F8/F<Q'italicF'F[qFhqF\rF_blF[qFhqF^sF>-F,6(Q$SPPF'F/F2F5F8F;FgoF>-F,6(Q(TildeTpF'F/F2F5F8F;FgoF>-F,6(Q#lpF'F/F2F5F8F;FgoF>FdoFgoF>-F,6(Q%cwepF'F/F2F5F8F;FgoF>-F,6(Q$lp2F'F/F2F5F8F;FgoF>FialF[vF[qFhqF>F>F>F>FejlF>FUF>-F`o6+F^wF/FhnF5F8F[oF]oF]elF_elF>-F?62Q&minusF'F/FhnF5F8F[oF]oFBFDFFFHFJFLFNFPFSF>Fgw-F`o6)-F#6%F]pFfpF;F/FhnF5F8F[oF]oF[vF[qFhqF>F>F>F>F\r-F,6(Qjn~SPP~is~the~set~of~parameters~which~can~be~used~to~complete~lp,F'F/F2F5F8F;F[qFhqF>F>F>F>FialF>FUF>F>Fecl-F`o6)-F#6%Fd[mFfpF;F/FhnF5F8F[oF]oF>F[vF[qFhqF>F>F>F>FhjlF>FUF>FfalF[vF[qFhqF>F>F>F>F[[mF>FUF>F]pF[vF[qFhqF>F>F>F>FdelF>FdoF>FdzF>FejlF>FgelF[qFhqF>F>F>F>F>F>F>F>F\r-F,6(QT~Searching~identifiable~parameters~relatively~to~lpF'F/F2F5F8F;F[qFhqF>F>F>F>F>F>F>F>F[clF>F+-F`o6)-F#6.FdoFgoF>F[[mFgoF>F`pFgoF>FcpFfpF;F/FhnF5F8F[oF]oF>F_dlF[qFhqF>F>F>F>F>F>F>F>F>F>F>F>FejlF>FUF>FejlF>Ff[mF>-F`o6+-F#6%FdoFfpF;F/FhnF5F8F[oF]oF]elF_elF[vF[qFhqF>F>F>F>F>F>F>F>F>F>F>F>F[[mF>FUF>-F`o6+-F#6)F`w-F`o6)-F#6%F[[mFfpF;F/FhnF5F8F[oF]oFgoF>FdoFfpF;F/FhnF5F8F[oF]oFfxFixF[vF[qFhqF>F>F>F>F>F>F>F>F>F>F>F>F\r-F,6(Q]o~if~such~a~parameter~is~found,~lp~is~completed~with~this~parameterF'F/F2F5F8F;F[qFhqF>F>F>F>F>F>F>F>F>F>F>F>F^[mF>FUF>F\al-F`o6)-F#6*F[[mFgoF>FhjlFgoFialFfpF;F/FhnF5F8F[oF]oF[vF[qFhqF>F>F>F>F>F>F>F>F>F>F>F>F[clF>F^cl-F`o6)-F#6&F^[m-F`o6+-F#6%FiflFfpF;F/FhnF5F8F[oF]oFfxFixFfpF;F/FhnF5F8F[oF]oF>F_dlF[qFhqF>F>F>F>F>F>F>F>F>F>F>F>F>F>F>F>F\r-F,6(Q[q~Testing~whether~the~computed~subset~TildeT~of~the~idenfiability~tree~can~be~used~to~complete~lpF'F/F2F5F8F;F[qFhqF>F>F>F>F>F>F>F>F>F>F>F>F>F>F>F>FejlF>FUF>-F`o6+-F#6%FhpFfpF;F/FhnF5F8F[oF]oF]elF_elF>F[vF[qFhqF>F>F>F>F>F>F>F>F>F>F>F>F>F>F>F>F\r-F,6(Q^p~This~affectation~stop~further~computations~since~lp~has~been~completed~with~TildeTF'F/F2F5F8F;F[qFhqF>F>F>F>F>F>F>F>F>F>F>F>F>F>F>F>FhjlF>FUF>F^[m-F`o6+-F#6%-Fjfl6'Q"2F'F/F5F8F;FfpF;F/FhnF5F8F[oF]oFfxFixF[vF[qFhqF>F>F>F>F>F>F>F>F>F>F>F>F>F>F>F>F\r-F,6(QK~TildeT~is~completed~with~completion~of~lpF'F/F2F5F8F;F[qFhqF>F>F>F>F>F>F>F>F>F>F>F>Fd`lF>F[clF[qFhqF>F>F>F>F>F>F>F>Fd`lF>F[clF[qF>F>F>F>Fd`lF>FgelF[vF[qF>F>F>F>F\r-F,6(Q[o~SPP~contains~only~non~identifiable~parameters~relatively~to~lp,F'F/F2F5F8F;FhqF[qF>F>F>F>FdelF>FdoF>FdzF>FejlF>FgelF[qF>F>F>F>F>F>F>F>F>F>-F,6(Fc[mF/FbjlF5F8FcjlF>FUF>-F`o6+-F#6*F`wFc]mFgoF>-F?60Q*&uminus0;F'F/F5F8F;FBFDFFFHFJFLFNF`glFdhlFdoFfpF;F/FhnF5F8F[oF]oFfxFixF[vF[qF>F>F>F>F>F>F>F>F>F>F\r-F,6(Qfq~Completion~of~prefix~lp~with~-theta~;~the~minus~sign~means~that~theta~is~not~identifiable~relatively~to~lpF'F/F2F5F8F;F[qF>F>F>F>F>F>F>F>F>F>F^[mF>FUF>F\al-F`o6)-F#6*Fa[mFgoF>FhjlFgoFialFfpF;F/FhnF5F8F[oF]oF[vF[qF>F>F>F>F>F>F>F>F>F>F[clF>F^[mFb^mF>F_dlF[qF>F>F>F>F>F>F>F>F>F>F>F>F>F>F>F>F\r-F,6(Qaq~Checking~now~whether~the~computed~subset~TildeT~of~the~idenfiability~tree~can~be~used~to~complete~lp2F'F/F2F5F8F;F[qFhqF>F>F>F>F>F>F>F>F>F>F>F>F>F>F>F>FejlF>FUF>Fi^mF>F>F>F>F>F[vF[qFhqF>F>F>F>F>F>F>F>F>F>F>F>F>F>F>F>F\rF]_mF[qFhqF>F>F>F>F>F>F>F>F>F>F>F>F>F>F>F>FhjlF>FUF>F^[mF`_mF[vF[qFhqF>F>F>F>F>F>F>F>F>F>F>F>F>F>F>F>F\rFg_mF[qF>F>F>F>F>F>F>F>F>F>FaelF[qF>F>F>F>F>F>F>F>F>F>F>F>F>F>F>F>FhjlF>FUF>FhjlF>-F?62Q&unionF'F/FhnF5F8F[oF]oFBFDFFFHFJFLFNFPFSF>Fhil-F`o6)-F#6.F_`mFgoF>F`pFgoF>FcpFgoF>FhjlFfpF;F/FhnF5F8F[oF]oF>F[vF[qF>F>F>F>F>F>F>F>F>F>F>F>F>F>F>F>F\r-F,6(Q`p~if~lp2~can~not~be~completed~with~TildeT,~a~recursive~call~is~realized~to~complete~itF'F/F2F5F8F;F[qF>F>F>F>F>F>F>F>F>F>Fd`lF>F[clF[qF>F>F>F>F>F>Fd`lF>FgelF[vF[qF>F>F>F>F>F>Fg_lF>FhjlF[qFd`lF>FenF>Fg`lF[qFhqFhqF[qFj`l-F?6-F^rF;FBFDFFFHFJFLFNFPFS-F,6%Q0~Main~function~F'F2F;F[qFhqF[qFhq-F,6(Q<RelativeIdentifiabilityTreeF'F/F2F5F8F;F>FUF>Fen-F`o6)-F#6-F`pFgoF>FcpFgoF>-F,6(Q#llF'F/F2F5F8F;FU-F`o6)F[_mF/F5F8F;FfxFixFfpF;F/FhnF5F8F[oF]oFhpF[qFhqF\r-F,6(Q\o~Inputs:~*~the~list,~ES,~of~polynomials~of~the~exhaustive~summaryF'F/F2F5F8F;F[qFhqF\r-F,6(Qjp~~~~~~~~~*~the~eventual~empty~list,~Cond,~of~contrains~on~the~parameters~and~initial~conditionsF'F/F2F5F8F;F[qFhqF\r-F,6(QY~Output:~the~corresponding~relative~identifiability~treeF'F/F2F5F8F;F[qF>F^sF>FfalF[vF[qFhqF>F>F>F>-F,6(Q+kerneloptsF'F/F2F5F8F;-F`o6)-F#6)-F,6(Q+printbytesF'F/F2F5F8F;F>F\dlF>-F,6(F4F/F2F5F8F;FfpF;F/FhnF5F8F[oF]oF[vF[qFhqF>F>F>F>F\r-F,6(QP~ll~is~the~empty~prefix~which~will~be~completedF'F/F2F5F8F;F[qFhqF>F>F>F>FfalF>FUF>Fi^mF[vF[qFhqF>F>F>F>F\r-F,6(Qjn~Computed~list~of~the~identifiability~tree~are~stored~in~TildeTF'F/F2F5F8F;F[qFhqF>F>F>F>Fg_lF>Fhil-F`o6)-F#6.FfbmFgoF>F`pFgoF>FcpFgoF>FfalFfpF;F/FhnF5F8F[oF]oF[vF[qFhqF>F>F>F>Fdcm-F`o6)-F#6)F[dmF>F\dlF>-F,6(FjnF/F2F5F8F;FfpF;F/FhnF5F8F[oF]oF>F[vF[qFd`lF>FenF>Fg`lFhpFfpF;<Text-field style="Heading 1" layout="Heading 1">2. Application to a model of microbial growth in a batch reactor </Text-field>This part of the worksheet is devoted to the application of the RelativeIdentifiabilityTree procedure to the parametric dynamical system presented in M.J. Chappell and K.R. Godfrey. Structural identifiability of the parameters of a nonlinear batch reactor model. Mathematical Biosciences, 108:245\342\200\223251, 1992.Brief description of the model :X(t) is the concentration of microorganisms and S(t) is the concentration of growth-limiting substrate; the parameters mu is the maximum velocity of the reaction; K_s is the Michaelis-Menten constant; a is the yield coefficient; and K_d the decay rate coefficient.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<Text-field style="Heading 2" layout="Heading 2">2.1 Relative identifiability tree without taking into account parameter contraints</Text-field>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
Which are the identifiable and non identifiable parameters ?
In order to turn the model into an identifiable one, which parameters should be estimated ?
<Text-field style="Heading 2" layout="Heading 2">2.2 Relative identifiability tree taking into account parameter contraints</Text-field>JSFHCond:=[JSFHmu > 0, K_s > 0, m >0, a > 0];
DecisionTree:=RelativeIdentifiabilityTree(ExhaustiveSummary, Cond);LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=JSFH
Compare this output with the last one.
<Text-field style="Heading 2" layout="Heading 2">2.3 Relative identifiability tree taking into account parameter contraints and generic initial conditions
<Font size="12" bold="false">In the next computation, initial conditions are supposed to be known and we suppose that s(0) >X(0). These assumptions are added to the set of conditions <Font italic="true">Cond.
</Font>Since only the branches of the identifiability tree admitting [</Font><Equation executable="true" style="2D Math" input-equation="" display="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">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</Equation><Equation executable="false" style="2D Math" input-equation="" 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