Hazard RatesΒΆ

The hazard rate is defined first for an individual:

A hazard rate is the probability, per unit time, that an event will happen given that it has not yet happened.

For a population, the hazard rate is the sum of the hazard rates for all individuals in that population. For instance, the remission rate, as a function of age, averages over all the different times someone may have entered the with-condition state.

The Dismod-AT compartmental model has four Dismod-AT primary rates, all of which are hazard rates,

  • Susceptible Incidence rate, \(\iota\)

  • Remission rate, \(\rho\)

  • Excess mortality rate, \(\chi\)

  • Other-cause mortality rate, \(\omega\)

and an initial condition, birth prevalence, \(p_{ini}\). We call the primary rates hazard rates because they are the probability per unit time that an individual, age \(x\), moves from one compartment to another, given that they have not yet left their current compartment. Note that birth prevalence for a cohort is, when we look at it across years, a birth rate. That is why you will see birth prevalence called one of the Dismod-AT primary rates.

These primary rates are exactly the parameters in the Dismod-AT differential equation,

\[ \begin{align}\begin{aligned}\frac{dS(x)}{dx} = -\iota(x) S(x) +\rho(x) C(x) - \omega(x) S(x)\\\frac{C(x)}{dx} = \iota(x) S(x) - \rho(x) C(x) - \left(\omega(x) + \chi(x)\right) C(x)\end{aligned}\end{align} \]

where \(S(x)\) are susceptibles as a function of cohort age and \(C(x)\) are with-condition as a function of cohort age.