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3.1- Homogeneous MixingIn the homogeneous mixing scenario, each cell is a neighbor of every other cell. Such a world can be pictured as a fully-connected graph in which the nodes represent cells and the arcs represent connections (neighbor-relationships) between them. This topology and numerous variations on it have been favored by theorists for many decades -- mainly for reasons of analyzability rather than realism. In this topology, the question of which nodes are infected has no bearing on the overall population dynamics; all that matters is how many are infected. Thus, as was stated earlier, topology is virtually eliminated as a consideration. If the number of nodes is very large, stochastic effects can be ignored, and the problem can be treated deterministically. In this case, the fraction of infected nodes as a function of time, a(t), can be obtained by solving a simple nonlinear differential equation [1]:
the solution to which is
where
The solution given by Eq. 2
reveals that, when the death rate
Thus there is a sharp ``epidemic threshold'' such that the population survives
if the birth rate exceeds the death rate and is driven to extinction
otherwise
In systems of finite size, probabilistic analysis reveals that,
even above the threshold, the population has some chance of becoming extinct,
but only if the initial number of infected
nodes is small enough to be vulnerable to statistical fluctuations [1].
For example, starting from an initial
condition in which one cell is occupied by A, the probability that the population will
survive is
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and
.
exceeds the birth rate
(
),
the fractional population of A decays exponentially from its initial value.
Thus extinction is inevitable.
However, when the birth rate exceeds the death rate, the fraction of infected
nodes grows exponentially at first and then saturates at the value
.
An example of such behavior is illustrated in Fig. 
and
, starting from an initial
condition a(0)=0.0001.
(If the initial fractional population exceeds the equlibrium value,
a(t) decays to the equilibrium at an exponential rate.)
