2.1- Deterministic Approximation
It is often the case that a deterministic analysis can provide a reasonably accurate
picture of many aspects of the dynamics of an epidemic [27], [32].
For the sake of simplicity, we shall assume that the infection rate along each edge is
Now consider a particular infected node. Since the expected number of edges in the graph is
pN(N-1), the expected number of edges emanating from this node (which we shall refer to as the
connectivity
The solution to Eq. 1 is:
where
If
|
and the cure rate for each node is
. If the population N is sufficiently
large, we can convert I(t), the number of infected nodes in the population
at time t, to
, a continuous quantity representing the fraction of
infected nodes. Then,
if we assume that the details of the graph's connectivity are fairly unimportant,
the dynamics
of infection depend only upon how many nodes are infected (rather than which
particular nodes are infected). Later, in section 2.3, when we use simulations to test
the validity of this assumption, we shall find that it works well when there are many edges in
the graph but fails miserably when there are only a few edges per node.
. The fraction of neighbors that are susceptible to infection is
1-i, so the expected number of uninfected nodes which can be infected by this node is
. Therefore, on average we expect the total system-wide rate at which infected
nodes infect uninfected nodes to be
, where
is
the average total rate at which a node attempts to infect its neighbors. The system-wide rate at
which infected nodes are cured is simply
. By ignoring stochastic variation in the number
of branches emanating from each node and in the average infection and cure rates, we obtain a
deterministic differential equation describing the time evolution of i(t):
is the
average ratio of the rate at which an infected node is cured to that at which it
infects other nodes, and
is the initial fraction of
infected nodes.
, the fraction of infected individuals decays exponentially from
the initial value
to 0, i.e., there is no epidemic.
If
, the
fraction of infected individuals grows from the initial value
) and eventually saturates at the value
. This result has a simple intuitive interpretation: if the average number of neighbors
that an individual can infect during the time that it is infected exceeds one, there will be an
epidemic; if this number is less than one, the infection will die out. The existence of this
threshold was first established for homogeneous interactions by Kermack and McKendrick
[
at which an infected node transmits infection to other nodes -- not the details of how
it distributes that infection.