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View page sourcePrevalence 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
Define \(C(t)\) by \(C(0) = c_0 > 0\) and
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
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
Dropping the dependence on \(t\) we have
So \(Q(t)\) satisfies the equation \(Q(0) = c_0 / s_0\) and
If follows that \(Q(t)\) does not depend on \(\omega (t)\) which completes the proof.