  The relationship between a screening tests' positive predictive value,
$\rho$, and its target prevalence, $\phi$, is proportional - though not linear
in all but a special case. In consequence, there is a point of local extrema of
curvature defined only as a function of the sensitivity $a$ and specificity $b$
beyond which the rate of change of a test's $\rho$ drops precipitously relative
to $\phi$. Herein, we show the mathematical model exploring this phenomenon and
define the $prevalence$ $threshold$ ($\phi_e$) point where this change occurs
as:
  $\phi_e=\frac{\sqrt{a\left(-b+1\right)}+b-1}{(\varepsilon-1)}$ where
$\varepsilon$ = $a$+$b$.
  Using its radical conjugate, we obtain a simplified version of the equation:
  $\frac{\sqrt{1-b}}{\sqrt{a}+\sqrt{1-b}}$.
  From the prevalence threshold we deduce a more generalized relationship
between prevalence and positive predictive value as a function of
$\varepsilon$, which represents a fundamental theorem of screening, herein
defined as:
  $\displaystyle\lim_{\varepsilon \to 2}{\displaystyle
\int_{0}^{1}}{\rho(\phi)d\phi} = 1$
  Understanding the concepts described in this work can help contextualize the
validity of screening tests in real time, and help guide the interpretation of
different clinical scenarios in which screening is undertaken.
