Neuronal Networks: A Discrete Model - Mathematical Concepts and Methods in Modern Biology - page 201

of stochasticity both in the structure of contacts and in the actual transmission of the
disease that results from a given contact. We can build these effects into our model by
altering it as follows. Assume that during each time period of length T 0 any two given
individuals interact with probability q 0 , and these interactions are independent. If a
given interaction involves a susceptible and an infectious individual, a transmission
will result with probability q . Thus we can model the dynamics of a random discrete
dynamical system N 1 whose updating rules are identical to the model N above, except
that the digraph D t will not be fixed but will be randomly drawn anew at each time
step from the distribution

of Erdos-Rényi random digraphs.
Notice that N 1 treats the ratio E

D n (

q 0 q )

exactly as an integer and treats the
“small standard deviation” of T 1 that we get from the data as practically zero. This
may be permissible, but one needs to be careful about such simplifications. Project
8[ 1 ] invites the reader to explore whether or not these simplifications will lead
to false predictions. More precisely, the project proposes a suite N 1 ,

( T 1 )/ E ( T 0 )

N 4 of
progressively more elaborate and apparently more realistic models of the same natural
system. In particular, the constructions used for models N 2 through N 4 incorporate
the standard deviation of T 1 that is ignored by model N 1 , and allow us to deal with
situations where the empirically observed ratio E

N 2 , N 3 ,

(

T 1 )/

(

T 0 ) is only approximately,

but not exactly an integer, as will almost certainly be the case for real data sets. The
more elaborate models are more difficult to study though. We will explore which of
these models, if any, incorporates just the right level of detail.
Without giving away the correct answer for Project 8 [ 1 ], let us assume for the
sake of argument that our simulations for model N 3 confirm the predictions of well-
established models from the literature. Would this imply that model N 3 is “good
enough” to make accurate predictions about the course of an actual disease? This
kind of question cannot be answered in the affirmative with certainty; Nature always
may hold some surprises in the formof hidden features of the real system that influence
the dynamics but that a modeler may have overlooked. But it might be possible to
prove mathematical theorems to the effect that a given simpler model such as our
discrete model N 3 will give us the exact same answer to a given question than a
more detailed one, such as an ODE version of the corresponding SIR model. We
will not address this question for disease models, but in the next section we will
describe such a theorem for models of certain neuronal networks. Theorems of this
type would be extremely valuable, since discrete models are often easier to study,
at least by simulations. For example, in disease dynamics the underlying network of
contacts plays an important role, with some individuals having frequent contact with
many others, whereas others may be more socially isolated. This phenomenon can be
easily modeled in our framework by drawing D from a given distribution with unequal
probabilities for different potential arcs

E

i , j , but it is more difficult to incorporate

into a differential equations framework.
We want to point out though that such theorems could only address the suitability of
a given simplification for a specified set of questions about model dynamics. Notice
that there is at least one aspect of disease dynamics in which all our models are
blatantly wrong: Assume, for example, that E ( T 0 ) is one week, which corresponds to