2.3- Simulations
Both the deterministic and stochastic analyses of the model
required a number of assumptions which can only
be tested by simulation. We have simulated the model using a
straightforward event-driven implementation. A graph is generated
randomly according to the prescription given at
the beginning of this section, and a single initially-infected
node is selected randomly.
Then, the simulation proceeds one event (i.e., an
attempted infection or cure of a node) at a
time, using time steps generated randomly according to an
exponential distribution Figure 4 compares a typical simulation run on a 100-node graph to the corresponding deterministic solution, using the parameters of Figs. 2 and 3. The simulation run follows the deterministic solution reasonably well, except that the equilibrium appears to be lower.
Figure 4: Comparison between average number of infected individuals vs. time as given by deterministic theory and a typical simulation run on a randomly-generated graph with 100 nodes. The average number of edges emanating from each node is  , and all other parameters are as
given in Fig. 2. The magnitude of the fluctuations agrees reasonably well
with that predicted by the stochastic theory (compare with Fig. 2), but the
average number of infected individuals is slightly lower. This discrepancy can be attributed to
the low connectivity of the graph (
).
To investigate this discrepancy, we performed 2500 simulation runs using the same parameters but
different seeds for the random number generator. In
Why is the extinction probability a bit higher and the average number of infected individuals a bit
lower than predicted? The fault must lie in one or more of the approximations that were used to
derive the deterministic and stochastic theories. The most likely suspect is the neglect of the
particular details of how nodes are connected to one another. This assumption came into play in at
least two guises. First, it allowed us to assume that the dynamics could be expressed
solely in terms of how many nodes were infected, without having to delve into the details of
which were infected (a problem that would be completely intractable). Second, we neglected
variation in the number of nodes that a given node could infect, assuming that every node tried to
infect exactly
Imagine for a moment the extreme limit of tenuousness (below what is referred to by random graph theorists as the percolation threshold [35]), in which most nodes are isolated and a few are joined in small clusters. It is readily apparent that infection cannot spread beyond the small cluster in which the initially infected individual is located. Thus the equilibrium level should be depressed substantially below the homogeneous limit. If the infection is confined to very small clusters, it becomes much more likely that all infections in the cluster will be detected and cured at approximately the same time. In such a case, the lifetime of the metastable phase could become less than our chosen time limit, in which case the measured extinction probability would increase. Thus, infections should die out more easily in tenuous graphs.
This conjecture is borne out by Figure 5, in which we have varied the
connectivity of the graph
|
of the runs, the population
became extinct by t=1200 (a time limit that we chose to be comfortably within the metastable
regime). This is noticeably larger than the extinction probability of 0.20 predicted by
Eq. 
, which as we suspected from
Fig. 
and the stochastic prediction of 79.75. The average magnitude of the fluctuations within a run
was
, somewhat larger than the stochastic prediction of 4.508. The variation in the
equilibrium obtained across different simulation runs was only
, indicating that the
entire class of random graphs is in fact well-characterized by these measurements -- one of the
main assumptions made in the deterministic and stochastic analyses.
other nodes. Intuitively, we expect both of these assumptions to become
increasingly valid as the connectivity
.
For
, it is nearly certain that the
infection will die out quickly. The
extinction probability drops precipitously for
and slowly approaches the homogeneous
limit of 0.20 as
, in which case
the transition region is shifted upwards to approximately
.
. Each point
represents an average over 2500 simulation runs. For
, it was extremely rare for an
epidemic to survive beyond the time limit of 1200, despite the fact that the infection rate is 5
times the classical epidemic threshold. For higher connectivities, the extinction
probability and the average number of infected individuals approach the values predicted by the
classical homogeneous interaction theory. The dependence of these quantities upon the
connectivity changes very little when the number of nodes in the graph is increased from 100 to
1000; the transition region becomes slightly sharper. (Note: the measured equilibria for
1000-node graphs have been divided by 10 to scale them properly to the 100-node results.)