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

8 Bifurcation Diagrams and Analysis
Bifurcation diagrams provide a nice overview of how the stable and unstable solutions
change as a bifurcation parameter changes. We have used matlab to produce our
bifurcation diagrams for the DuCa model. 1 Please note that we have not included
the apoptotic wave in the simulations behind these bifurcation diagrams since we are
interested in investigating other regions besides those the apoptotic wave pushes the
system into. We are aware that the apoptotic wave is the initiator of inflammation so
in lieu of the wave as a catalyst we apply appropriate initial conditions to stimulate the
system.
The bifurcation diagrams are plotted with the fixed points of the activated macrophages
M ∗ a as a function of the phagocytosis rate constants; f1 or f2 (e.g figure 8.1). From a
mathematical point of view we could have chosen any of the variables M, Ma, Ba, Bn, C
to be the dependent variable in the bifurcation diagram. However from a medical point
of view the presence of activated macrophages – which implies an activated immune
system – is a more direct measure of inflammation in the DuCa model, and thus the
development of T1D.
8.1 Codimension One Bifurcations
In the next two sections we deal with codimension one bifurcations based on f1 first
and f2 second. One thing this analysis can tell us is: given that f2 (or f1), is kept at
a given value. Can we make the flow in NOD-mice approach the healthy stable rest
state? And if so: what should the value of f1 (or f2) be before this happens? Such an
event will imply some kind of bifurcation, so in other words, this analysis can provide
us with (an approximation of) such a bifurcation value of f1, respectively f2.
The bifurcation diagrams based on NOD values will be given the most extensive scrutiny.
Dealing with the Balb/c bifurcation diagram to the same extent as these would not yield
any fundamentally new knowledge – had there been some interesting behavior associated
with the Balb/c diagram, then we would naturally have presented this, but the
behavior is qualitatively similar to the bifurcation diagrams based on NOD parameters.
Furthermore, from a physiological point of view, it is most interesting to find out if we
can induce a healthy state in NOD-mice by simply tweaking the phagocytosis rates, or
the amount of active macrophages. 2
The bifurcation analysis is primarily done on the full DuCa model, but in section 8.4
we also look at a bifurcation diagram (figure 8.18) that is based on a reduced three
1 A template of the matlab code for the bifurcation diagrams can be found in appendix C.1.
2 As mathematicians doing mathematical modelling of biological systems we are allowed to dream up
solutions to problems, unaffected by the applicability of said solutions to the real world – though we
are not inimical to solutions that are readily applicable.
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