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

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x List of Figures 8.3 Phase space plot of M, Ma and Ba, showing a stable spiral together with the corresponding time-plot . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 8.4 Time- and phase space plots of an unstable spiral. . . . . . . . . . . . . . . 78 8.5 Plots of the real and imaginary parts of the eigenvalues with respect to f1 on the NODf1-LUB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 8.6 Simulation of the DuCa model setting f1 = 2.7 × 10 −5 and f2 = 1 × 10 −5 . 82 8.7 An illustrative figure of the flow in phase space after the Hopf bifurcation has occurred. The point named “optimal initial condition” would flow along the unstable manifold of the NODf1-LUB to the USB. . . . . . . . . . . . . 84 8.8 The difference between chronic inflammation or not comes down to one active macrophage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 8.9 The UUB added to figure 8.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 86 8.10 The flow of Ma represented on a circle. . . . . . . . . . . . . . . . . . . . . 87 8.11 Bifurcation diagram for the DuCa model using f1 as the bifurcation parameter and Balb/c parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 88 8.12 Bifurcation diagram for the DuCa model using f2 as the bifurcation parameter, and NOD parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 89 8.13 Overview of the eigenvalues with f2 as the bifurcation parameter with f1 as in NOD-mice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 8.14 The real and imaginary parts of the eigenvalues of the fixed point along the NODf2-LUB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 8.15 Bifurcation diagram for the DuCa model using f2 as the bifurcation parameter, and Balb/c parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 92 8.16 Simulation of the DuCa model setting f1 = 2 × 10 −5 and f2 = 2.54 × 10 −5 93 8.17 Simulations of the DuCa versus the reduced DuCa model . . . . . . . . . . 94 8.18 Bifurcation diagram for the three dimensional version of the full model using NOD parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 11.1 Behavior of the concentrations for Balb/c- and NOD-mice with the inclusion of a compartment of healthy β-cells; Model A. . . . . . . . . . . . . . . . . 108 11.2 Behavior of the concentrations for Balb/c-mice with the inclusion of a compartment of healthy β-cells; Model B. . . . . . . . . . . . . . . . . . . . . . 110 11.3 Behavior of the concentrations for NOD-mice with the inclusion of a compartment of healthy β-cells; Model B. . . . . . . . . . . . . . . . . . . . . . 111 11.4 Behavior of the concentrations for Balb/c-mice based on Model C. . . . . . 115 11.5 Behavior of the concentrations for NOD-mice based on Model C. . . . . . . 115 B.1 Plot of the eigenvalues for the Jacobian evaluated in fixed points in the upper inflamed region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 B.2 Plot of the eigenvalues for the Jacobian evaluated in fixed points in the lower inflamed region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 B.3 Plot of the eigenvalues for the Jacobian evaluated in the healthy rest state. 140 B.4 A classical diagram of the imaginary part versus real part of the eigenvalues for the Jacobian evaluated in fixed points in the upper inflamed region. . . 141 B.5 Plot of the eigenvalues for the Jacobian evaluated in fixed points in the upper inflamed region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
B.6 Plot of the eigenvalues for the Jacobian evaluated in fixed points in the lower inflamed region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 B.7 Plot of the eigenvalues for the Jacobian evaluated in the healthy rest state. 143 B.8 Plot of the eigenvalues for the Jacobian evaluated in the fixed points along the dotted (black) line seen in figure 8.11. . . . . . . . . . . . . . . . . . . . 144 B.9 Plot of the eigenvalues for the Jacobian evaluated in the fixed points along the dashed-dotted (magenta) line seen in figure 8.11. . . . . . . . . . . . . . 144 B.10 A classical diagram of the imaginary part versus real part of the eigenvalues for the Jacobian evaluated in fixed points in the upper inflamed region. . . 145 B.11 Plot of the eigenvalues for the Jacobian evaluated in fixed points in the upper inflamed region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 B.12 Plot of the eigenvalues for the Jacobian evaluated in fixed points in the lower inflamed region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 B.13 Plot of the eigenvalues for the Jacobian evaluated in the healthy rest state. 147 B.14 A classical diagram of the imaginary part versus real part of the eigenvalues for the Jacobian evaluated in fixed points in the upper inflamed region. . . 148 B.15 Plot of the eigenvalues for the Jacobian evaluated in fixed points in the upper inflamed region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 B.16 Plot of the eigenvalues for the Jacobian evaluated in fixed points in the lower inflamed region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 B.17 Plot of the eigenvalues for the Jacobian evaluated in the healthy rest state. 150 B.18 A classical diagram of the imaginary part versus real part of the eigenvalues for the Jacobian evaluated in fixed points in the upper inflamed region. . . 151 B.19 Plot of the eigenvalues for the Jacobian evaluated in fixed points in the upper inflamed region (setting f2 = 1 × 10 −5 ). . . . . . . . . . . . . . . . . . . . . 152 B.20 Plot of the eigenvalues for the Jacobian evaluated in fixed points in the lower inflamed region (setting f2 = 1 × 10 −5 ). . . . . . . . . . . . . . . . . . . . . 153 B.21 Plot of the eigenvalues for the Jacobian evaluated in the healthy rest state (setting f2 = 1 × 10 −5 ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 B.22 A classical diagram of the imaginary part versus real part of the eigenvalues for the Jacobian evaluated in fixed points in the upper inflamed region (setting f2 = 1 × 10 −5 ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 B.23 Stable spirals for f1 = 2.56 × 10 −5 , and f1 = 2.566 × 10 −5 . . . . . . . . . . 155 B.24 Stable spirals for f1 = 2.5661 × 10 −5 , and f1 = 2.5662 × 10 −5 . . . . . . . . 155 B.25 Stable spirals for f1 = 2.5663 × 10 −5 , and f1 = 2.566305 × 10 −5 . . . . . . . 156 xi
Page 1 and 2: - I, OM OG MED MATEMATIK OG FYSIK E
Page 3 and 4: Exploring Treatment Strategies for
Page 5: Preface iii questions that were out
Page 8 and 9: vi Contents Significance of the Apo
Page 10 and 11: List of Tables 5.1 summarizes the m
Page 14 and 15: xii
Page 16 and 17: 2 Introduction In the following dec
Page 18 and 19: 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
Page 49 and 50: 5.7 Discussion of Marée et al. (20
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
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48 The Intermediated Model expected
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50 The Intermediated Model Stabilit
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52 The Intermediated Model restrict
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54 The Intermediated Model We see t
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56 The Intermediated Model paramete
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58 The Intermediated Model ✻ Ma(t
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60 The Intermediated Model 6.2 The
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62 The Intermediated Model ✻ Ma(t
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7 A Brief Introduction to Bifurcati
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7.2 The Hopf bifurcation 67 the gen
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7.3 Biological Relevance of Bifurca
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7.5 Locating the Fixed Points 71 Ev
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74 Bifurcation Diagrams and Analysi
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98 Discussion of the Bifurcation An
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100 Discussion of the Bifurcation A
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102
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104
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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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122
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124
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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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