  We present an analysis of six deterministic models for epidemic spreading.
The evolution of the number of individuals of each class is given by ordinary
differential equations of the first order in time, which are set up by using
the laws of mass action providing the rates of the several processes that
define each model. The epidemic spreading is characterized by the frequency of
new cases, which is the number of individuals that are becoming infected per
unit time. It is also characterized by the basic reproduction number, which we
show to be related to the largest eigenvalue of the stability matrix associated
with the disease-free solution of the evolution equations. We also emphasize
the analogy between the outbreak of an epidemic with a critical phase
transition. When the density of the population reaches a critical value the
spreading sets in, a result that was advanced by Kermack and McKendrick in
their study of a model in which the recovered individuals acquire permanent
immunization, which is one of the models analyzed here.
