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Prevalence Does Not Depend On Other Cause Mortality
###################################################

Lemma
*****
Suppose :math:`\iota (t) \geq 0`, :math:`\omega (t) \geq 0` and
:math:`\chi(t) \geq 0` are known functions.
Define :math:`S(t)` by
:math:`S(0) = s_0 > 0` and

.. math::

   S'(t) =  - [ \iota (t) + \omega (t) ] S(t)

Define :math:`C(t)` by
:math:`C(0) = c_0 > 0` and

.. math::

   C'(t) =  + \iota (t) S(t) - [ \chi (t) + \omega (t) ] C(t)

Define :math:`P(t)` by :math:`P(t) = C(t) / [ S(t) + C(t) ]`
It follows that :math:`P(t)` does not depend on the value of
:math:`\omega (t)`.

Proof
*****
It follows that :math:`S(t) > 0`, :math:`C(t) > 0` for all :math:`t` and

.. math::

   P(t) = 1 / [ 1 + S(t) / C(t) ]

Define :math:`Q(t) = C(t) / S(t)`.
It suffices to show that :math:`Q(t)` does not depend on :math:`\omega(t)`.
Taking the derivative of :math:`Q(t)` we have

.. math::

   Q'(t) = [ C'(t) S(t) - S'(t) C(t) ] / S(t)^2

Dropping the dependence on :math:`t` we have

.. math::
   :nowrap:

   \begin{eqnarray}
   Q'
   & = &
   [ + \iota S S - ( \chi + \omega ) C S + (  \iota  + \omega ) S C ] / S^2
   \\
   & = &
   [ + \iota S - ( \chi + \omega ) C + (  \iota  + \omega ) C ] / S
   \\
   & = &
   \iota  + ( \iota - \chi ) Q
   \end{eqnarray}

So :math:`Q(t)` satisfies the equation
:math:`Q(0) = c_0 / s_0` and

.. math::

   Q'(t) = \iota(t) + [ \iota(t) - \chi (t) ] Q(t)

If follows that :math:`Q(t)` does not depend on :math:`\omega (t)`
which completes the proof.

{xrst_end prev_dep}