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

More documents

Recommendations

Info

62 The Intermediated Model ✻ Ma(t) ˙ Ma < 0 ˙ Ba > 0 ❅❘ ✠ ❄❪ ✓ Ma ˙ < 0 ✒Ba ˙ < 0 ✛ ✒ Ma ˙ > 0 Ba ˙ ✲ > 0 ✏ ✑ ❅■ ✻ ˙ Ma > 0 ˙ Ba < 0 Ba ˙ = 0 Ma ˙ ✠ = 0 ❄✛ ❪ ✓ Ma ˙ < 0 ✒Ba ˙ < 0 ✲ Ba(t) ✏ ✑ Figure 6.5 Sketch of the nullclines and arrows that indicate the flow for the IM including crowding terms with dM/dt = 0. It is assumed that the inclusion of crowding terms only have a minuscule effect on the position of the saddle point compared to figure 6.3, in the limit case close to the healthy rest state. Beyond the saddle point the crowding terms start to effect the behavior of the nullclines. Through analysis we have found that the crowding terms gradually reduce the slope of the nullclines, albeit more effectively for the nullcline of Ma. Thereby the asymptotic behavior fails to happen and instead the continuity and the relation between the slope of the nullclines causes a third intersection to take place. This additional nontrivial fixed point is stable and works as an upper bound for the inflammation.
6.3 Recapitulation of The Analysis of the IM 63 6.3 Recapitulation of The Analysis of the IM On the basis of the analysis of the IM we have reached an understanding of what kind of dynamics that the people of Marée et al. (2006) were looking for when modifying the CPH model and at the same time gained insight into the problems they were facing in this process. To summarize: the CPH model was extended to the IM for the purpose of testing the hypothesis that impaired phagocytosis makes the difference between health and disease. For the hypothesis to be confirmed the model should be able to exhibit two kinds of dynamics by changing the values of the parameters f1 and f2. The two types of dynamics were dubbed Balb/c and NOD dynamics respectively and refers to the classification of the mice that are prone to develop T1D (NOD-mice) and those who are not (Balb/c-mice). In the case where the estimated Balb/c phagocytosis rates were applied, the model was expected to show Balb/c dynamics. This implies the existence of a healthy rest state, where there is no activated macrophages and no apoptotic βcells, as the only fixed point in the physiological relevant region of the phase space. This fixed point should furthermore be stable such that any stimulated inflammation will be nonpermanent. When using the estimated phagocytosis rates for NOD-mice the model was expected to exhibit NOD dynamics which is more eventful than the former. Here the possibility for a permanent and nonpermanent inflammation should be open, since not all NOD-mice develop T1D. In addition the chronic inflammation should be bounded such that the concentration of neither the activated macrophages nor the apoptotic β-cells tends towards infinity. These expectations are reached when, besides the stable healthy rest state, there is a saddle point that together with its stable manifold comprises a separatrix. Below the separatrix the inflammation is damped, i.e. nonpermanent, whereas, if the separatrix is traversed, the inflammation will escalate. The escalating inflammation needs to be bounded, so a third fixed point must exist that has to be stable. These are the two types of dynamics that the model should be able to exhibit when using the estimated Balb/c and NOD parameters. However as discussed in section 6.1 this is not the case, since the IM also exhibits Balb/c dynamics when applying the estimated NOD parameters. By this realization the people of Marée et al. (2006) suggest to implement the role of necrosis in the model. We will proceed by embarking on an analysis of this model, i.e. the DuCa model, in the following sections.
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 12 and 13:
x List of Figures 8.3 Phase space p
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
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 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 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
Page 116 and 117: 102
Page 118 and 119: 104
Page 120 and 121: 106 Expanding and Modifying the DuC
Page 126 and 127:
112 Expanding and Modifying the DuC
Page 128 and 129:
Page 130 and 131:
Page 132 and 133:
118 Discussion of the Expanded Mode
Page 134 and 135:
120 Discussion of the Expanded Mode
Page 136 and 137:
122
Page 138 and 139:
124
Page 140 and 141:
126 Bibliography Christen, U. and M
Page 142 and 143:
128 Bibliography Notkins, A. L. and
Page 144 and 145:
130 Bibliography Yu, J. and G. S. E
Page 146 and 147:
132 Mathematical Appendices the com
Page 148 and 149:
134 Mathematical Appendices in that
Page 150 and 151:
136 Mathematical Appendices A.6 Dim
Page 152 and 153:
138
Page 154 and 155:
140 Additional Figures Figure B.2 P
Page 156 and 157:
142 Additional Figures B.2 Eigenval
Page 158 and 159:
144 Additional Figures Figure B.8 P
Page 160 and 161:
Page 162 and 163:
148 Additional Figures Figure B.14
Page 164 and 165:
Page 166 and 167:
Page 168 and 169:
Page 170 and 171:
Page 172 and 173:
158 matlab Code w(r+1)=w(r)+dw; g=w
Page 174 and 175:
160 matlab Code end V_11(r)=E(1); V
Page 176 and 177:
162 matlab Code n=675; w_5=0; dw_5=
Page 178 and 179:
164 matlab Code M=M’; % end s_7(r
Page 180 and 181:
166 matlab Code w_7,V_34,’.b’,w
Page 182 and 183:
168 matlab Code M=M’; % M must be
Page 184 and 185:
170 matlab Code %options = odeset(
Page 186 and 187:
172
Page 188 and 189:
174 Index value, 73 Bifurcation Ana
Page 190 and 191:
176 Index of diabetes, 2 Michaelis
show all

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

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?