The existence and occurrence, especially by a backward bifurcation, of
endemic equilibria is of utmost importance in determining the spread and
persistence of a disease. In many epidemiological models, the equation
for the endemic equilibria is quadratic, with the coefficients
determined by the parameters of the model. Despite its apparent
simplicity, such an equation can describe an amazing number of dynamical
behaviours. In this paper, we shall provide a comprehensive survey of
possible bifurcation patterns, deriving explicit conditions on the
equation's parameters for the occurrence of each of them, and discuss
illustrative examples.