diff_eq¶

View page source

The Dismod_at Differential Equation¶

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)¶

We normalize the functions \(S(a, t)\) and \(C(a, t)\) so that the initial population \(S(0, t) + C(0, t)\) is equal to one. The initial prevalence \(P(0, t)\) is the faction of the population that is born with the condition. It follows that \(C(0, t) = P(0, t)\) and \(S(0, t) = 1 - P(0, t)\)

The Deterministic Dismod_at ODE¶

We use \(c\) to denote the time of birth for a cohort. The ordinary differential equation for a fixed cohort is

\begin{split} \frac{d}{d a} S(a,c+a) = & - S(a,c+a) ~ \left[ \iota (a,c+a) + \omega (a,c+a) \right] \\ & + C(a,c+a) ~ \rho (a,c+a) \\ \frac{d}{d a} C(a,c+a) = & + S(a,c+a) ~ \iota (a,c+a) \\ & - C(a,c+a) ~ \left[ \rho (a,c+a) + \chi (a,c+a) + \omega (a,c+a) \right] \end{split}

The Deterministic Dismod_at PDE¶

Using the Chain rule, the left hand sides in the equations above are also given by:

\begin{split} \frac{d}{d a} S(a,c+a) = \frac{ \partial }{ \partial a } S(a, c+a) + \frac{ \partial }{ \partial t } S(a, c+a) \\ \frac{d}{d a} C(a,c+a) = \frac{ \partial }{ \partial a } C(a, c+a) + \frac{ \partial }{ \partial t } C(a, c+a) \end{split}

The Stochastic Dismod_at PDE¶

The stochastic version of this equation would include random drift for the rates \(\iota , \rho , \chi\) and \(\omega\) . There is a further complication here because the random drift is for versions of the rates that have the covariate effects removed.