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

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
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