4.2- Simulations on Two-Dimensional Lattices
We have simulated epidemics on a two-dimensional square array wrapped around in both
dimensions to form a torus (so as to avoid edge effects). The neighborhood about
each node is an
Figure 11 illustrates the state of a typical simulation run during the
spreading phase of the epidemic. Initially, the central node in the 100 by 100 array was
infected. The size of the neighborhood was chosen to be
Figure 11: State of an epidemic in diffusion phase (t=50) as obtained from a simulation on a 100-by-100 array. The infection and cure rates are  and
 , as
usual. Each node can infect the 8 neighbors lying within a 3-by-3 square centered on
itself. Black and white squares represent infected and uninfected individuals, respectively.
As in the theoretical curves of Fig. 10, the boundary of
the expanding circle of infected nodes is roughly circular and fairly sharp (despite the fact
that the infection neighborhood is square).
Simulations verify the quadratic growth with time of the total number of infected
individuals. Figure 12 compares the relative rates at which the equilibrium is
attained in the two-dimensional lattice with that of the random and hierarchical graphs. In order to provide a fair comparison, the infection and cure rates were given their usual values of
Figure 12: Comparison of fraction i of infected individuals vs. time for random graph, hierarchical graph, and two-dimensional graph SIS models. The infection and cure rates have their usual values of and
. The number of nodes in the
simulations are 10000 for the random and spatial models and
 for the
hierarchical model. In order to ensure that the equilibrium was close to the homogeneous
limit, the connectivity of the random graph was chosen to be
 , and the
localization parameter for the hierarchical graph was r=0.225. The parameters for the
spatial graph are as given in Fig. 11. As predicted, the
growth in the infected population is quadratic in the spatial model, which is much slower
than the exponential growth exhibited by the random graph model. The functional form of the
growth of i in the hierarchical graph is not yet known.
It must be emphasized that the curves in Fig. 12 are not to be taken as an absolute comparison of the growth rates for these three models. The exact rates depend upon the diffusion coefficient in the spatial model and are extremely sensitive to the localization parameter in the hierarchical model. (For example, when r is increased by only 10%, the growth rate is increased by about 35%.) It is the functional form of the growth that is of the greatest importance. As expected, the random graph attains the equilibrium at an exponential rate, while the two-dimensional spatial model reaches it at a much slower, quadratic rate. The functional form of the growth rate for the hierarchical graph is not yet known, but in this and other simulations it appears to be quite slow. Thus both of the local models that we have studied exhibit very slow growth compared to the random graph model (e.g., polynomial rather than exponential in time). We expect that this will prove to be the case for other more complicated and realistic models that take locality into account. In local models, the majority of the infected nodes find themselves in a region that has already reached local equilibrium. It is only the relatively small number of nodes on the expanding front of the infected region that can spread the infection to the uninfected nodes lying outside the region. Since they can only infect nodes lying close by, the vast majority of uninfected nodes lying outside the region are unavailable. The spread of infection on a random graph is much more efficient because the boundary of the infected region expands so rapidly that it quickly encompasses the entire graph. It is likely that the real situation lies somewhere in between the extremes represented by the random graph model and the two local models that we have studied in this section and the previous one.
|
-by-
for each node within that
neighborhood and zero for all nodes outside of it. The dynamical details of the simulation are
identical to those which we have presented for the random graph; the only difference lies in
the choice of infection rates between pairs of nodes. Thus, if we were to expand the
neighborhood to include all of the nodes, our resultant system would be equivalent to a
fully-connected graph and its dynamics would be described by the homogeneous limit.
, i.e., it was composed
of the 8 nearest neighbors. It is interesting to note that, despite the fact that
is square in shape, the pattern of infection is roughly circular. In fact,
the greater generality of radial spreading is expected from details of the derivation of Eq.