2.2- Probabilistic Analysis
As we have just seen, the deterministic approximation is useful for estimating some important
characteristics of epidemics -- the conditions under which they occur, the rate at which they
grow, and the expected number of infections once they have reached equilibrium. However, since it
ignores the stochastic nature of an epidemic, it provides no information about other important
features of the dynamics, including the size of fluctuations in the number of infected individuals
and the possibility that fluctuations will result in extinction of the infection. Consider for a
moment the issue of the survival of the virus in a population. The deterministic analysis concludes
that there will be an epidemic if
As in the deterministic analysis of the previous section, we shall assume that the number of infected nodes sufficiently characterizes the state of a system, i.e., the details of which nodes are infected are relatively unimportant. Although it is very easy to construct particular graphs for which such details are important (e.g., small graphs with a large variation in the in-degree and out-degree of nodes), we assume that the properties of most members of the class of random graphs with N nodes and edge probability p will not be sensitive to them. Again, we must resort to simulation (in section 2.3) to test the validity of these assumptions. First, we shall describe the probabilistic approximation. Then, we shall use it to calculate various quantities of interest.
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and there will not be one if
. However, it
is intuitively clear that, even if