Now we illustrate another fundamental concept of
epidemiology. Using the definition of 
, rewrite
equation (5a),
Suppose we immunize 90% of the population; then when we
introduce a few infecteds, Z is 90% and X is 10%
of the population, so X/N=.1. In that case, so long as
, 
and the epidemic cannot take off.
If you know , you can use equation (5b)
to calculate the minimum fraction of the population that
must be immunized to prevent an epidemic. Again, the
general principle holds for more complicated
transmission systems although it may be difficult to
calculate the required immunization exactly for more
complicated systems. The larger the larger the fraction of the
population that must be immunized to prevent an
epidemic.