Stable populations based on a life table and assumed growth rate (cont.)

Constructing a stable population from a stationary (life table) population and an assumed growth rate

We will construct a stable population with growth rate 3% from a stationary population based on the following life table, based on a radix of 1000 births:

age group a to a+n Internal link nLa
0  –  5 700 Internal link
5 – 10 500
10 – 15 300
15 – 20 200
20 – 25 100
25 – 30 0
total person-years lived in stationary population 1800

If the population is growing at 3% per year, and there were 1,000 births in the base year,
then the following year the number of births will be 1,000 x exp(0.03) = 1,000 x 1.03 = 1,030,
and 5 years later, the number of births will be 1,000 x exp{0.03 x 5} = 1,000 x 1.162 = 1162
whereas the number of births in the year before the base year was 1,000 x exp{-0.03} = 970,
similarly the number of births 15 years earlier was 1,000 x exp{-0.03 x15} = 638.

The next table shows how the relative size of the birth cohorts born Y years before the base year is calculated, and then derives the number of survivors at each age group in the stable population.

age group a to a+n nLa survivors born on average Y years earlier relative size of birth cohort born Y years earlier exp{-0.03 x Y} stable population age group a to a+n nLa x exp{-0.03 x Y} proportion aged a to a+n in stable population
0  –  5 700 2.5 0.928 649 45.6%
5  – 10 500 7.5 0.799 399 28.0%
10 – 15 300 12.5 0.687 206 14.5%
15 – 20 200 17.5 0.592 118 8.3%
20 – 25 100 22.5 0.509 51 3.6%
25 – 30 0 27.5      
      stable population total 1,424 100.0%

For example those aged 10-15 will have been born on average 12.5 years ago, so that their birth cohort was exp{-0.03 x 12.5} = 0.687 times smaller than the cohort born in the base year.  Since 300 of 1000 births survive to ages 10-15, the stable population aged 10-15 is 300 x 0.687 = 206

Life table person-year function.
From a cohort of 1000 births, 700 person-years will be lived between ages 0 and 5.