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lines 5-89 of file: xrst/math/prevalence_ode.xrst
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{xrst_begin prevalence_ode}
{xrst_spell
  rrr
}

The Prevalence Only ODE
#######################

Theorem
*******
If :math:`S` and :math:`C` satisfy the dismod_at
:ref:`avg_integrand@Ordinary Differential Equation` then prevalence :math:`P = C / (S + C)` satisfies

.. math::

    P' = + \iota  - ( \iota + \rho + \chi ) P  +  \chi  P^2

Proof
*****
Suppose that :math:`S(a)` and :math:`C(a)` satisfy the dismod_at
:ref:`avg_integrand@Ordinary Differential Equation`

.. math::

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

It follows that

.. math::

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

Using :math:`C = (S + C) P`, we also have

.. math::
    :nowrap:

    \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 :math:`(S, C)`,
is that the solution is stable as :math:`S + C \rightarrow 0`.
The :math:`(S, C)`  approach computes :math:`P = C / (S + C)`.

Integrands
**********
All of the current integrands, except for
:ref:`avg_integrand@Integrand, I_i(a,t)@susceptible` and
:ref:`avg_integrand@Integrand, I_i(a,t)@withC`
can be computed from :math:`P` (given that the rates are inputs to the ODE).

S and C
*******
If we know all cause mortality :math:`\alpha = \omega + \chi P`,
once we have solved for :math:`P`,
we can compute :math:`\omega = \alpha - \chi P`.
Furthermore

.. math::

    (S + C)' = - \alpha (S + C)

We can also compute :math:`S + C`, and
:math:`C = P (S + C)`, :math:`S = (1 - P)(S + C)`.

{xrst_end prevalence_ode}