PAPP103 - S09: Models of populations with variable growth rates

Application 2: Net Reproduction in the presence of migration

Starting with the general equation for the age structure of a population:

$$P_x=B_0{e}^{-{\displaystyle \sum }_{a=0}^{x-1}{r}_a-0.5r_x}\cdot L_x\cdot L_x^{*}$$

where l₀ = 1, by multiplying each side of the equation by the population's age-specific fertility rates f_x and dividing by B₀, we get:

$$v_x=\frac{P_x\cdot f_x}{B_0}=e^{-{\displaystyle \sum }_{a=0}^{x-1}{r}_a-0.5r_x}\cdot L_x\cdot L_x^{*}\cdot f_x$$

or

$$L_x\cdot L_x^{*}\cdot f_x=v_x{e}^{{\displaystyle \sum }_{a=0}^{x-1}{r}_a+0.5r_x}$$

where v_x is the proportion of births occurring at age x.

Denoting the net reproduction rate in the presence of migration as NRR*:

$${\text{NRR}}^{*}={\displaystyle \sum _{x=12}^{49}{L}_x\cdot L_x^{*}\cdot f_x}={\displaystyle \sum _{x=12}^{49}{v}_x{e}^{{\displaystyle \sum }_{a=0}^{x-1}{r}_a+0.5r_x}}$$

Thus, we can calculate NRR* from v_x, the proportional distribution of births by the age group of the mother and the age-specific growth rates of the population. The calculations are illustrated using data on England and Wales in the table on the following page.