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

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54 The Intermediated Model We see that Balb/c-parameters satisfy both conditions. To see if this is also the case when using NOD parameters these are substituted into equation 6.37 1 × 10 −5 0.5 ? < 0.1 × 0.4 1 × 10 −5 (0.41 − 0.4) 1 × 10 5 < 4 × 10 5 ≮ 9 × 10 3 ? < 0.09 5 × 10−5 (6.48) (6.49) We see that NOD parameters satisfy the conditions for a stable healthy rest state, but not the constraints for a nontrivial fixed point. That is, for both the estimated NOD and Balb/c parameters the IM will exhibit Balb/c dynamics – in both cases the inflammation will be nonpermanent. Based on this Marée et al. (2006) conclude that the CPH model is quantitatively incorrect (Marée et al., 2006, p.1280), and as a solution suggest adding necrotic β-cells and cytokines. Phase plane analysis Before we turn to the analysis of the DuCa model we will resort to a phase plane analysis of the IM. We have found this to be, a very useful tool to achieve a conceptual understanding of – what we believe to be – the kind of dynamics that Marée et al. (2006) aimed for, when modifying the CPH model. There is but one problem. In the end of last section we saw that the estimated NOD parameters do not satisfy the constraints for a nontrivial fixed point. This means that the use of the estimated NOD parameters do not bring the system to exhibit NOD dynamics. So for the purpose of providing qualitative illustrations we choose to use some arbitrary positive parameter values. Thus in the following assume that the parameters are chosen so that we have a set of NOD parameters that satisfy the constraints for NOD dynamics, and a set of Balb/c parameters that satisfy the constraints for Balb/c dynamics. A phase plane analysis implies looking at nullclines, and how the flow behaves around these. To specify what nullclines are, assume that we have a system ˙x = f(x, y), ˙y = g(x, y), where ˙x means dx/dt. Then the nullclines of this system are the curves given by ˙x = 0 and ˙y = 0. I.e. the nullclines define where the flow is horizontal or vertical (Lynch, 2004, p.186), and an intersection between nullclines defines a steady state also called a fixed point. The IM consists of three equations, and a phase plane analysis is as the name hints a two-dimensional undertaking. However because we are only interested in the nullclines, when we do the analysis, this does not pose a problem. 5 We simply pick one of the equations 6.5, 6.6 or 6.7 and equate it to zero – let us choose 6.5, so we set dM/dt = 0. Then we isolate M in this equation to obtain M = a + (k + b)Ma c + f1Ba (6.50) 5 We do not have to restrict ourselves to the plane! We could do a phase space analysis, but since we are interested in the nullclines we can do a phase plane analysis.
6.1 Fixed Points of the IM Model 55 and substitute the expression for M into the two other equations. 6 So we get dMa dt dBa dt a + (k + b)Ma = f1 Ba − kMa c + f1Ba a + (k + b)Ma = lMa − f1 Ba − f2MaBa − dBa c + f1Ba (6.51) (6.52) Linearizing and evaluating the Jacobian Matrix of this reduced system at Ma = Ba = 0 yields −k J = l f1a c − f1a c − d Which is similar to the 2 × 2 matrix shown in equation 6.24, thus the stability behavior and criteria for the healthy rest state is the same for the two dimensional system as the three dimensional system. The nullclines are found analytically by setting dMa/dt = 0 and dBa/dt = 0 in equation 6.51 and 6.52 and then solving with respect to Ma (or Ba) Ma nullcline : Ma = Baf1a ck − Babf1 Ba nullcline : dc + af1 + f1dBa Ma = − f2c + f1(b + k) − lf1 + f1f2Ba − lc Ba (6.53) (6.54) As it turns out, the nullcline for Ma is fairly easy to analyze whereas the nullcline for Ba is a little more tricky. They both approach the point (Ba, Ma) = (0, 0). To convince oneself of this in the case of the Ba-nullcline, let Ba → 0. Then the term lc/Ba → ∞. Thus the entire fraction will approach zero as Ba → 0. Now, by differentiating the two expressions for the nullclines (equations 6.53 and 6.54) we get the slope of the Ma nullcline: and the slope of the Ba nullcline: dMa dBa dMa af1 abf = + dBa ck − bf1Ba 2 1 Ba (ck − bf1Ba) 2 = (cdBa + af1Ba + df1B 2 a)(f1(b + k) + cf2 − f1l + 2f1f2Ba) (f1Ba(b + k) − cl + cf2Ba − f1f2B 2 a) 2 − (cd + af1 + 2df1Ba)(f1Ba(b + k) − cl + cf2Ba − f1f2B 2 a) (f1Ba(b + k) − cl + cf2Ba − f1f2B 2 a) 2 (6.55) (6.56) By setting Ba = 0, the initial slope is found to be af1/ck for the Ma-nullcline and (dc+af1)/lc for the Ba-nullcline. Remember that we are working with arbitrary (positive) 6 Here we are lucky enough that we are able to separate Ba from the other variables, which is not always the case. The implicit function theorem, see e.g. (Abraham et al., 1988, p.121), always tells us if such a separation is possible (in a given interval), and can be applied, if it is not apparent if it is possible to isolate one variable from the others.
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- I, OM OG MED MATEMATIK OG FYSIK E
Page 3 and 4:
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
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
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 70 and 71: 56 The Intermediated Model paramete
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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