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Prevalence Does Not Depend On Other Cause Mortality¶

Lemma¶

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

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

Define \(C(t)\) by \(C(0) = c_0 > 0\) and

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

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

Proof¶

It follows that \(S(t) > 0\), \(C(t) > 0\) for all \(t\) and

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

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

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

Dropping the dependence on \(t\) we have

\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 \(Q(t)\) satisfies the equation \(Q(0) = c_0 / s_0\) and

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

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