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

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96 Bifurcation Diagrams and Analysis like the full DuCa model a natural question comes to mind: why did we devote so much of our time to analyzing the full DuCa model rather than the reduced 3-dimensional version? The answer is that in order to compare the reduced DuCa model to the full DuCa model we needed to produce bifurcation diagrams for the full DuCa model. These were immensely time-consuming to produce, especially due to the trial-and-error approach we used to make initial guesses when using the Newton-Raphson method. Another question one might ask is: why did we not use the reduced model to perform an analytical analysis based on f.x. the Routh-Hurwitz criteria? We actually tried to do this, but the Michaelis-Menten term in the reduced DuCa model makes it extremely hard to obtain anything that will provide new information.
9 Discussion of the Bifurcation Analysis In chapter 8.1 we presented bifurcation diagrams and some diagrams showing how the eigenvalues behaved with increasing f1 and f2 in the NOD-bifurcation diagrams. In this chapter we will give a summary of our analysis, discuss our findings, relate them to the underlying biological system and comment on question I-III of the thesis statement. Finally we will touch upon subjects for future work. By conducting a bifurcation analysis we obtained information about how much f1 and f2 needed to be tweaked to induce bifurcations. Both the NOD- and Balb/c-diagrams revealed the existence of (what inspection of the eigenvalues revealed to be) a Hopfbifurcation, which indirect results, e.g. hysteresis, suggest is sub-critical. In addition all figures also contained a bifurcation that consisted of two branches of saddle-points amalgamating, as well as the possibility of bistability – the Hopf-bifurcation point serves as the demarcation point between bi- and mono stability. When using NOD parameters the chosen bifurcation parameter, f1 or f2, lies within the parameter span where the model shows bistability: a stable healthy rest state (the HRS) and a stable upper nontrivial fixed point (the USB) that is associated with the chronic inflammation that is observed in NOD-mice; cf. figure 8.1 and 8.12. Between these is an unstable nontrivial fixed point (the LUB). This configuration accounts for the NOD behavior, i.e. they develop a chronic inflammation because the unstable fixed point is exceeded during the apoptotic wave, thus making the flow tend to the USB. The phagocytosis rates Marée et al. (2006) have used in the DuCa model come from several experiments. We discussed this in section 5.8 where we also provided different values of f1 and f2 as given in the different articles of Marée et al.; cf. table 5.2. The different values along with their standard errors are important in determining if any qualitative change occurs for physiologically realistic parameter-values. The standard error for f1 was < 0.005 × 10 −5 (not included in table 5.2), and the most extreme f2-value we could come across was 1.02×10 −5 in NOD-mice. This value was associated with a standard error of 0.01 × 10 −5 . When we compare these values to the f1- and f2-values at which the Hopf-bifurcations, ∼ 2.57 × 10 −5 and ∼ 8.5 × 10 −5 respectively, take place we find that no NOD-mouse that have phagocytosis rates in the intervals given in table 5.2 will escape chronic inflammation following the apoptotic wave. We know that not all NOD-mice become diabetic. But how can that be if the standard errors do not permit any NOD-mouse to supersede the Hopf-bifurcation value? For one thing the macrophages used in the various experiments of Marée et al. were from female mice – we learned in chapter 2 that 80 % of the female mice develop T1D against only 20 % of the male mice. In addition O’Brien et al. (2002) find that macrophages from male NOD-mice have a higher phagocytic capacity during the first 2 weeks of life (O’Brien et al., 2002, p.2483). At week 3 the phagocytic ability of the female NOD-mice is leading, but at this point the apoptotic wave has already set things in motion, and 97
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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
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6 Introduction because I had to mak
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8 Type 1 Diabetes and Its Etiology
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10 Type 1 Diabetes and Its Etiology
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3 Prospects for Therapy - A Mini Re
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3.3 Gastrin 15 eration in situ Urus
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4 Mathematical Modelling The langua
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5 The DuCa Model In the following w
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5.3 The DuCa Model - An Appetiser 2
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5.4 The DuCa Model - Compartment Mo
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5.6 Our Compartment Model 29 migrat
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5.6 Our Compartment Model 31 a ✓
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5.7 Discussion of Marée et al. (20
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5.7 Discussion of Marée et al. (20
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5.8 Discussion of Parameters 37 Fig
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5.8 Discussion of Parameters 39 Par
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5.8 Discussion of Parameters 41 tha
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5.8 Discussion of Parameters 43 Fig
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Page 62 and 63: 48 The Intermediated Model expected
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Page 79 and 80: 7 A Brief Introduction to Bifurcati
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Page 120 and 121: 106 Expanding and Modifying the DuC
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Page 140 and 141: 126 Bibliography Christen, U. and M
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146 Additional Figures B.3 Eigenval
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148 Additional Figures Figure B.14
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152 Additional Figures B.5 Eigenval
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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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