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This is dismod_at-20221105 documentation: Here is a link to its
current documentation
.
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{array}{rcl}
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{array}
@]@
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.
Input File: omh/model/prev_depend.omh
