Biology Reference
In-Depth Information
The analysis of a simple diffusion process immediately
highlights the importance of network topology. In a random
network with arbitrary degree distribution the stationary
state reached by a swarm of particles diffusing with the
same diffusive rate yields N
k
N
we assume a diffusion rate d for each individual, and that
the single population reproductive number of the SIR
model is R
0
>
1, we can easily identify two different limits.
If d
0 any epidemic occurring in a given subpopulation
will remain confined: no individual could travel to
a different subpopulation and spread the infection across
the system. In the limit d
¼
k and the probability of
finding a single diffusing walker in a node of degree k is
1 we have individuals constantly
wandering from one subpopulation to the other and the
system is in practice equivalent to a well-mixed single
population. In this case, since R
0
>
¼
1
V
where V is the total number of nodes in the network.
This expression tell us that the larger the degree of the
nodes, the larger the probability of being visited by the
walker. Evidently, the topology of the system has a large
impact in the way contagion processes will spread in large
meta-population networks.
Consider, for instance, a simple epidemic process such
as the SIR model in a meta-population context
[56,61
k
<
k
>
p
k
¼
1, the epidemic will
spread across the entire system. A transition point between
these two regimes is therefore occurring at a threshold
value d
c
of the diffusion rate, identifying a global invasion
threshold. Interestingly, this threshold cannot be uncovered
by continuous models as it is related to the stochastic
diffusion rate of single individuals. Furthermore, the global
invasion threshold is affected by the topological fluctua-
tions of the meta-population network. In particular, the
larger the network heterogeneity, the smaller the value of
the diffusion rate above which the epidemic may globally
invade the meta-population system. This result assumes
68]
.
In this case each node of the network is a subpopulation
(ideally an urban area) connected by a transportation
system (the edges of the network) that allows individuals to
move from one subpopulation to another (
Figure 27.6
). If
e
FIGURE 27.6
Illustration of the global threshold in reaction
e
diffusion processes. (a) Schematic of the simplified modeling framework based on
the particle-network scheme. (b) Within each subpopulation individuals can mix homogeneously or according to a subnetwork, and can diffuse with rate
d from one subpopulation to another, following the edges of the network. (c) A critical value d
c
of the diffusion strength for individuals or particles
identifies a phase transition between a regime in which the contagion affects a large fraction of the system and one in which only a small fraction is
affected.
