@(@\newcommand{\B}[1]{ {\bf #1} }
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This is dismod_at-20221105 documentation: Here is a link to its
current documentation
.
Theorem
If @(@
S
@)@ and @(@
C
@)@ satisfy the dismod_at
ordinary differential equation
then prevalence @(@
P = C / (S + C)
@)@ satisfies
@[@
P' = + \iota - ( \iota + \rho + \chi ) P + \chi P^2
@]@
Proof
Suppose that @(@
S(a)
@)@ and @(@
C(a)
@)@ satisfy the dismod_at
ordinary differential equation
@[@
\begin{array}{rrr}
S' =& - ( \iota + \omega ) S & + \rho C \\
C' =& + \iota S & - ( \rho + \chi + \omega ) C
\end{array}
@]@
It follows that
@[@
(S + C)' = - \omega S - ( \omega + \chi ) C
@]@
Using @(@
C = (S + C) P
@)@, we also have
@[@
\begin{array}{rcl}
C' & = & (S + C)' P + (S + C) P'
\\
(S + C) P' & = & C' - (S + C)' P
\\
(S + C) P' & = &
+ \iota S - ( \rho + \chi + \omega ) C
+ \omega S P + ( \omega + \chi ) C P
\\
P' & = &
+ \iota (1 - P) - ( \rho + \chi + \omega ) P
+ \omega (1 - P) P + ( \omega + \chi ) P^2
\\
P' & = & + \iota - ( \iota + \rho + \chi ) P + \chi P^2
\end{array}
@]@
Advantage
One advantage of this approach,
over the original ODE in @(@
(S, C)
@)@,
is that the solution is stable as @(@
S + C \rightarrow 0
@)@.
The @(@
(S, C)
@)@ approach computes @(@
P = C / (S + C)
@)@.
Integrands
All of the current integrands, except for
susceptible
and
withC
can be computed from @(@
P
@)@ (given that the rates are inputs to the ODE).
S and C
If we know all cause mortality @(@
\alpha = \omega + \chi P
@)@,
once we have solved for @(@
P
@)@,
we can compute @(@
\omega = \alpha - \chi P
@)@.
Furthermore
@[@
(S + C)' = - \alpha (S + C)
@]@
We can also compute @(@
S + C
@)@, and
@(@
C = P (S + C)
@)@, @(@
S = (1 - P)(S + C)
@)@.
Input File: omh/math/prevalence_ode.omh
