2.2.1- Probabilistic ApproximationLet p(I,t) denote the probability distribution for there to be I infected individuals at time t. Many quantities of interest can be calculated from this distribution. The probability that the infection is extinct at time t is represented by p(0,t), and the expected number of infected individuals and its variance are easily computed by appropriate sums over the p(I,t). The time-evolution of this distribution is given by:
where
The rates
where
Substituting these approximations for the rates into Eq. 3, we obtain:
where
Figure 2 compares the expected number of infected individuals as a
function of time as obtained from Eq. 6 with the deterministic
result given by Eq. 2. The graph contains 100 nodes
with connectivity
Figure 2: Comparison of the number of infected nodes I as a function of time in the deterministic and stochastic models. The total population is 100 nodes. The average rate at which a node attempts to infect its neighbors is  ,
and the cure rate is
 . Thus the system is above the classical threshold
for an epidemic by a factor of 5.
Black curve: deterministic I(t). White curve: stochastic average
of I(t). Gray area: One
standard deviation about the stochastic average. The final equilibrium
values differ by only
0.3%. For t ;SPMgt; 15, the gray area can be interpreted as the magnitude
of fluctuations about the
equilibrium.
The same dynamics are presented from a different point of view in Figure 3, which shows snapshots of p(I,t) at successive stages in its evolution. The parameters are the same as in Fig. 2. Initially, at time t=0, the probability distribution is a delta function at I=1 (i.e., there is exactly one infected node). As time passes, the probability distribution splits into two components: a delta function at I=0 (corresponding to extinction of the virus) and a ``survival'' component which is initially distributed exponentially (t=1). At first, the mean of the survival component increases exponentially in time, and the standard deviation grows quite large, reaching a maximum of 20 at t=6.3. Soon, however, the population becomes saturated with infected individuals, and the survival component is nearly gaussian with a mean of 79.75 and a standard deviation of 4.51 at t=20. This ``metastable'' phase is extremely long-lived, but the extinction component grows extremely slowly at the expense of the survival component until finally it is all that remains. In general, any population of viruses will eventually die out, but the time scale on which this takes place is so long as to be unobservable unless the graph is quite small. Eventual extinction is inevitable because there is a very tiny probability that all infected individuals will detect and cure their infection at approximately the same time.
Figure 3: Evolution of the probability distribution for the number of infected individuals in the stochastic approximation. All parameters are the same as in Fig. 2. Starting from a state in which one individual is infected at t=0, the distribution splits into an ``extinction'' component (I=0) and a ``survival component'' which eventually assumes a gaussian form with the same average and standard deviation as in Fig. 2. The survival component lasts for an extremely long time, but decays with a time constant of  . Note: vertical scales are not all the same.
The fact that the metastable phase has a finite lifetime means that we cannot
define the probability that the virus becomes extinct without specifying the time period
of interest. However, in practice the choice of a ``time limit'' has little effect on the measured
extinction probability provided that it is somewhere within the wide timespan of the metastable
regime. For
those epidemics which have not died out by a certain time limit, we are interested in the form of
the survival component -- the distribution p(I,t) for I;SPMgt;0. Although p(I,t) itself approaches 0
as
The survival component is then
Given that an epidemic is still active
after a given amount of time, we can calculate from
|
,
, and
denotes the rate at
which transitions occur from state a to state b.
is the average total rate at which a
node attempts to infect its neighbors. The probability per unit time
that an infected node will be cured is simply the number of infected nodes times the rate at
which each is cured:
and
.
For a graph with N nodes, this is a set of N+1 coupled linear differential equations which
is relatively simple to solve because the matrix is tridiagonal.
, and the infection and cure rates are
at t=6.3 and then diminish to
in equilibrium. The large deviations during
the exponential rise can be attributed to the extreme sensitivity of the number of infected
individuals at a particular time to random jitter in when the exponential rise occurs. Thus, one
would expect a lot of variance in the number of infected individuals from one simulation run to
another. However, in equilibrium the size of the infected population is completely insensitive
to the moment at which the exponential rise occurred, and the variations from one run to another
are much smaller. In equilibrium, the ergodic hypothesis [
, the conditional probability for there to be I infections given
that there is at least one infection approaches a well-defined metastable distribution:
the expected
number of infected individuals and the fluctuations about that expectation, and these quantities
approach a well-defined asymptote.