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

48 The Intermediated Model
expected. So on that basis it is possible, before digging deeper into a mathematical
analysis, to give an outline of what kind of dynamics that is expected from the IM
model.
First of all it must allow for the existence of a stable healthy rest state, for both NOD
as well as Balb/c-mice. That is, a stable steady state where no apoptotic β-cells are
present, and hence no activated macrophages and therefore just a constant density of
resting macrophages in the tissue, given by the fraction between influx and efflux, ( a/c),
of macrophages. This is a necessity since no unprovoked inflammation occurs, and when
inflammation does occur it is given that Balb/c-mice, and (in the real world) even some
NOD-mice 2 , return to the healthy state, so it has to be stable. This is a consequence
of the observation that not all NOD-mice develop T1D, as we stated in chapter 2.
Furthermore the model should hold the possibility for NOD-mice to acquire a chronic
inflammation, initiating the removal of pancreatic β-cells. This implies the existence
of, at least one, nontrivial fixed point, or a (at least in theory) stable limit-cycle in the
positive region, R 3 +. Information about the stability of this fixed point requires further
mathematical treatment.
6.1 Fixed Points of the IM Model
We now turn to the analysis of the IM which, once again, should be pointed out to
be an unfolding of what is already presented in Marée et al. (2006). From a modelling
point of view we are only concerned about what happens in the nonnegative region
since this is the only physiologically valid domain. Therefore we are only going to look
for fixed points in the nonnegative region. An equilibrium, or a fixed point of a system,
say ˙x = f(x), where the dot denotes the derivative with respect to time, satisfies ˙x = 0,
so for the IM we have
0 = dM
dt = a + (k + b)Ma − cM − f1MBa (6.8)
0 = dMa
dt
0 = dBa
dt
= f1MBa − kMa (6.9)
= lMa − f1MBa − f2MaBa − dBa (6.10)
Note that as in the full model all the parameters are positive. Looking at the healthy
rest state by setting (Ma, Ba) = (0, 0) and solving equation 6.8 with respect to M yields
M = a/c. So the healthy rest state is at
(M, Ma, Ba) = (a/c, 0, 0) (6.11)
This is furthermore a good approximation for the healthy rest state of the system where
crowding terms are included provided that e1 ≪ c2
/a; cf. appendix A.7). Now, turning
2 When we run simulations of the DuCa model using NOD parameters, naturally, we will never
witness that the concentrations return to the healthy rest state; cf. chapter 2. We model one mouse at
a time so to speak, thus if we use NOD parameters the concentrations will stabilize at the pathological
concentrations.