@(@\newcommand{\B}[1]{ {\bf #1} }
\newcommand{\R}[1]{ {\rm #1} }
\newcommand{\W}[1]{ \; #1 \; }@)@This is dismod_at-20221105 documentation: Here is a link to its
current documentation
.
Susceptible, S(a, t)
We use @(@
S(a, t)
@)@ to denote the population that is susceptible
to a condition as a function of age and time.
With Condition, C(a, t)
We use @(@
C(a, t)
@)@ to denote the population that has the
condition.
Prevalence, P(a, t)
The prevalence @(@
P(a, t)
@)@ is the fraction of the population that
has the condition; i.e.,
@[@
P(a, t) = \frac{ C(a, t) }{ S(a, t) + C(a, t) }
@]@
Incidence, iota(a, t)
We use @(@
\iota (a, t)
@)@ to denote the probability (per unit time)
that a susceptible individual will get the condition.
Note that age and time have the same units.
Remission, rho(a, t)
We use @(@
\rho (a, t)
@)@ to denote the probability (per unit time)
that an individual will be cured of the condition.
Excess Mortality, chi(a, t)
We use @(@
\chi (a, t)
@)@ to denote the probability (per unit time)
that an individual with die due to the condition.
Other Cause Mortality, omega(a, t)
We use @(@
\omega (a, t)
@)@ to denote the probability (per unit time)
that an individual with die from a cause other than the specific
condition we are modeling.
Initial Prevalence, pini(t)
The initial prevalence @(@
P(0, t)
@)@
is the faction of the population that is born with the condition.
We normalize the function @(@
S(a, t)
@)@ and @(@
C(a, t)
@)@ so
that the initial population @(@
S(0, t) + C(0, t)
@)@ is equal to one.
It follows that @(@
C(0, t) = P(0, t)
@)@ and
@(@
S(0, t) = 1 - P(0, t)
@)@
The Dismod_at ODE
Fix @(@
c
@)@ the time of birth for one cohort.
Given a function @(@
f(a, t)
@)@ we use the notation
@(@
f_c (a) = f(a, c + a)
@)@.
The ordinary differential equation for this cohort is
@[@
\begin{array}{rcl}
S_c '(a)
& = &
- [ \iota_c (a) + \omega_c (a) ] S_c (a) + \rho_c (a) C_c (a)
\\
C_c '(a)
& = &
+ \iota_c (a) S_c (a) - [ \rho_c (a) + \chi_c (a) + \omega_c (a) ] C_c (a)
\end{array}
@]@
Input File: omh/math/ode.omh
