prevalence_ode#

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The Prevalence Only ODE#

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

\[\begin{split}\begin{array}{rrr} S' =& - ( \iota + \omega ) S & + \rho C \\ C' =& + \iota S & - ( \rho + \chi + \omega ) C \end{array}\end{split}\]

It follows that

\[(S + C)' = - \omega S - ( \omega + \chi ) C\]

Using \(C = (S + C) P\), we also have

\begin{eqnarray} 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{eqnarray}

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 \(C = P (S + C)\), \(S = (1 - P)(S + C)\).