PAPP103 - S09: Models of populations with variable growth rates

Generalization to any population

This set of relationships between the age structure, vital rates and growth of any population at any point in time can be further generalized to encompass the impact of any other form of attrition that affects a population. The impact of a loss from the additional decrement will be the same as that of a death or emigrant of the same age. For example, the population of single women experiences attrition through first marriage as well as mortality and emigration. Using the age-specific first marriage rates, one can quantify the impact of those marriages on the population in exactly the same way that one can model the impact of mortality or migration.

To state this in the most general way possible, in any population structured by duration since entry and subject to n forces of accession and attrition with duration-specific rates at duration a of

μ

1 a

,μ

2 a

,...

,μ

n a

:

$$P_x=\frac{B_0}{l_0}{e}^{-{\displaystyle \sum }_{a=0}^{x-1}\left(r_a+{\mu }_a^1+{\mu }_a^2\mathrm{....}{\mu }_a^n\right)-0.5\left(r_x+{\mu }_x^1+{\mu }_x^2\mathrm{....}{\mu }_x^n\right)}$$

For conventional human populations, duration since birth is equivalent to age but the population in question need not be that of a country. It could be:

• the HIV positive population of the UK
• the residents of a long-stay nursing home
• the population of microcomputers owned by a government department.

For example, in a cohort study of a set of patients being followed up after a positive diagnosis of tuberculosis, the series of decrements operating would be recovery, mortality from TB, mortality from other causes, and loss to follow up.