nr. 477 - 2011 - Institut for Natur, Systemer og Modeller (NSM)

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56 The Intermediated Model parameters so let us assume the initial slope of the Ma-nullcline is less than the initial slope of the Ba nullcline. That is af1 ck < dc + af1 lc This can be rearranged to the familiar expression a d < ck f1(l − k) (6.57) (6.58) The constraints for a stable healthy rest state reappears, which means that in order for the model to show a physiological realistic behavior (see chapter 6) the relation between the initial slopes must in fact be as stated in equation 6.57. Another feature of the nullclines is that they both exhibit asymptotic behavior. This occurs when the denominator in the expression of the nullclines tend towards zero which happens at Ma nullcline asymptote: Ba nullcline asymptote: B a(Asymp1) = ck B a(Asymp2) = −(f1(b + k − l) + f2c) + (f1(b + k − l) + f2c) 2 + 4f1f2cl 2f1f2 bf1 (6.59) (6.60) Where the term on the right hand side is the positive root f1f2B 2 a + (f1(b + k − l) + f2c)Ba − cl Assuming that B a(Asymp1) ogether with the initial sloperelation (slope of Ma-nullcline must be less that slope of Ba-nullcline) leads to an intersection of the nullclines and hence a nontrivial fixed point; 7 cf. figure 6.3. This configuration implies the healthy rest state to be stable and the direction of the slope field is easily determined and indicated by a few arrows. Once the direction of the slope field is in place it is evident that the nontrivial fixed point at the intersection of the nullclines is a saddle point, which corresponds to what we found numerically to be the case for the IM without setting dM/dt = 0; cf. the subsection titled “Stability of the fixed points.” However for the IM with dM/dt = 0 we can claim it as a general statement. That is, as long as the parameters fits the constraints given in equation 6.37 the nontrivial fixed point must be a saddle. It is interesting to note that the stable manifold of the saddle point comprises a separatrix that when exceeded results in a escalating inflammation or, in the case of points below, a damped inflammation. So the saddle point separates the healthy regime from the inflamed regime. It should also be noted that the extensiveness of the inflammation is independent of the size of the apoptotic wave that is implemented in the IM to trigger the inflammation as long as it pushes the system above the given separatrix! In that case the densities of the macrophages and apoptotic β-cells will tend towards infinity. Now let us consider the phase plane where parameters that satisfy Balb/c dynamics 7 Under the assumption that the nullclines are otherwise well behaved; i.e. continuous between 0 and the aymptotes.
6.1 Fixed Points of the IM Model 57 ✻ Ma(t) ❅❘ ˙ Ma < 0 ˙ Ba > 0 Ma ˙ < 0 Ba ˙ < 0 ❄✛ ✠ ˙ Ma = 0 ✲ ✒ ˙ Ma > 0 ˙ Ba > 0 ✻ ❅■ Ma ˙ > 0 ˙ Ba < 0 ˙ Ba = 0 ✲ Asymp1 Asymp2 Ba(t) Figure 6.3 Sketch of the nullclines and some representative arrows that indicate the flow for the IM with dM/dt = 0. For the healthy rest state, (Ba, Ma) = (0, 0), to be stable the initial slope of the Ma-nullcline must be less than the initial slope of the Ba-nullcline. Then by assuming that the nullcline of Ma reaches its asymptotic point before the nullcline of Ba introduces a nontrivial fixed point at the intersection of the nullclines characteristic for the NOD dynamics. By the knowledge of the stability of the healthy rest state it is a simple matter to determine the direction of the slope field. This reveals that the nontrivial fixed point must be a saddle. are used. We have already concluded that the initial slope of the Ma nullcline must be less than the initial slope of the Ba nullcline for the healthy rest state to be stable. Thus this must still be the case when using Balb/c parameters. However unlike the NOD configuration there should be no nontrivial fixed point which implies that B a(Asymp1) > B a(Asymp2) so no intersection of the nullclines occur. This leads to the nullclines and slope field as seen on the sketch in figure 6.4. Here it is evident that any stimulated inflammation will be nonpermanent. Before we continue with the next subsection we would like to make a brief remark. Usually at the point where one is conducting a phase plane analysis the model under scrutiny has been recast in a dimensionless form, since this eases the job of deciding if some terms in the model can be disregarded, if some of the parameters are (relatively) significantly smaller than others or some parameters hold a greater impact on the behavior of the dynamic. This is often helpful if some or none of the parameters are
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- I, OM OG MED MATEMATIK OG FYSIK E
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Exploring Treatment Strategies for
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Preface iii questions that were out
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vi Contents Significance of the Apo
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List of Tables 5.1 summarizes the m
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x List of Figures 8.3 Phase space p
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xii
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2 Introduction In the following dec
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4 Introduction There are several sc
Page 20 and 21: 6 Introduction because I had to mak
Page 22 and 23: 8 Type 1 Diabetes and Its Etiology
Page 24 and 25: 10 Type 1 Diabetes and Its Etiology
Page 27 and 28: 3 Prospects for Therapy - A Mini Re
Page 29 and 30: 3.3 Gastrin 15 eration in situ Urus
Page 31 and 32: 4 Mathematical Modelling The langua
Page 33 and 34: 5 The DuCa Model In the following w
Page 35 and 36: 5.3 The DuCa Model - An Appetiser 2
Page 37 and 38: 5.4 The DuCa Model - Compartment Mo
Page 43 and 44: 5.6 Our Compartment Model 29 migrat
Page 45 and 46: 5.6 Our Compartment Model 31 a ✓
Page 47 and 48: 5.7 Discussion of Marée et al. (20
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Page 51 and 52: 5.8 Discussion of Parameters 37 Fig
Page 53 and 54: 5.8 Discussion of Parameters 39 Par
Page 55 and 56: 5.8 Discussion of Parameters 41 tha
Page 57: 5.8 Discussion of Parameters 43 Fig
Page 60 and 61: 46 The Intermediated Model The diff
Page 62 and 63: 48 The Intermediated Model expected
Page 64 and 65: 50 The Intermediated Model Stabilit
Page 66 and 67: 52 The Intermediated Model restrict
Page 68 and 69: 54 The Intermediated Model We see t
Page 72 and 73: 58 The Intermediated Model ✻ Ma(t
Page 74 and 75: 60 The Intermediated Model 6.2 The
Page 76 and 77: 62 The Intermediated Model ✻ Ma(t
Page 79 and 80: 7 A Brief Introduction to Bifurcati
Page 81 and 82: 7.2 The Hopf bifurcation 67 the gen
Page 83 and 84: 7.3 Biological Relevance of Bifurca
Page 85: 7.5 Locating the Fixed Points 71 Ev
Page 88 and 89: 74 Bifurcation Diagrams and Analysi
Page 112 and 113: 98 Discussion of the Bifurcation An
Page 114 and 115: 100 Discussion of the Bifurcation A
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106 Expanding and Modifying the DuC
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118 Discussion of the Expanded Mode
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120 Discussion of the Expanded Mode
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126 Bibliography Christen, U. and M
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128 Bibliography Notkins, A. L. and
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130 Bibliography Yu, J. and G. S. E
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132 Mathematical Appendices the com
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134 Mathematical Appendices in that
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136 Mathematical Appendices A.6 Dim
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138
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140 Additional Figures Figure B.2 P
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142 Additional Figures B.2 Eigenval
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144 Additional Figures Figure B.8 P
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148 Additional Figures Figure B.14
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158 matlab Code w(r+1)=w(r)+dw; g=w
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160 matlab Code end V_11(r)=E(1); V
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162 matlab Code n=675; w_5=0; dw_5=
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164 matlab Code M=M’; % end s_7(r
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166 matlab Code w_7,V_34,’.b’,w
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168 matlab Code M=M’; % M must be
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170 matlab Code %options = odeset(
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172
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174 Index value, 73 Bifurcation Ana
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176 Index of diabetes, 2 Michaelis
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nr. 477 - 2011 - Institut for Natur, Systemer og Modeller (NSM)

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