nr. 477 - 2011 - Institut for Natur, Systemer og Modeller (NSM)
nr. 477 - 2011 - Institut for Natur, Systemer og Modeller (NSM)
nr. 477 - 2011 - Institut for Natur, Systemer og Modeller (NSM)
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List of Figures<br />
2.1 Relation between the number of autoantibodies in the blood and how likely<br />
the person is to develop diabetes . . . . . . . . . . . . . . . . . . . . . . . . 9<br />
3.1 Dependence of treatment strategy on β-cell mass . . . . . . . . . . . . . . . 13<br />
5.1 shows an example of a generic compartment model . . . . . . . . . . . . . . 22<br />
5.2 shows the compartment system as given by Marée et al. (2006) . . . . . . . 24<br />
5.3 dynamics of the Balb/c-mouse and NOD-mouse. . . . . . . . . . . . . . . . 28<br />
5.4 the non-necrotic part of our modified compartment model . . . . . . . . . . 30<br />
5.5 the necrotic part of our modified compartment model . . . . . . . . . . . . 31<br />
5.6 shows the compartment system corresponding to equations 5.17 to 5.20 . . 31<br />
5.7 Simulations of the DuCa model with a Hill function instead of the Michaelis-<br />
Menten function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34<br />
5.8 A plot based on the findings of Stoffels et al. (2004) that apoptotic as well<br />
as necrotic β-cells implies cytokine secretion . . . . . . . . . . . . . . . . . . 36<br />
5.9 simulations of the DuCa model with activation due to necrotic cells . . . . 37<br />
5.10 simulations of reversible and irreversible model with Balb/c phagocytosis<br />
parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39<br />
5.11 simulations of the reversible and irreversible model with NOD phagocytosis<br />
parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40<br />
5.12 shows simulations of the irreducible model with k = 0 . . . . . . . . . . . . 42<br />
5.13 shows a simulation made with matlab of the reversible model where a<br />
term of −cMa has been added to equation 5.7 to model turnover of active<br />
macrophages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42<br />
5.14 In the figures simulations of the DuCa model with turnover of the active<br />
macrophages are presented . . . . . . . . . . . . . . . . . . . . . . . . . . . 43<br />
6.1 A compartment representation of the CPH model. . . . . . . . . . . . . . . 45<br />
6.2 Shows the compartment system <strong>for</strong> the Intermediate Model . . . . . . . . . 47<br />
6.3 Sketch of the nullclines <strong>for</strong> the IM with dM/dt = 0. . . . . . . . . . . . . . . 57<br />
6.4 Sketch of the nullclines <strong>for</strong> the IM with dM/dt = 0. . . . . . . . . . . . . . . 58<br />
6.5 Sketch of the nullclines <strong>for</strong> the IM including crowding terms with QSS on M. 62<br />
7.1 A supercritical (I), a subcritical (II) and degenerate Hopf bifurcation (III). 68<br />
8.1 Bifurcation diagram <strong>for</strong> the DuCa model using f1 as the bifurcation parameter<br />
and NOD parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 74<br />
8.2 Plot of the real and imaginary parts of the eigenvalues on the NODf1-USB 76<br />
ix