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math_ode#
View page sourceThe Dismod_at Ordinary 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.,
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