Approaches to representing mortality (cont.)
Mathematical models (cont.)
Gompertz’ Law (cont.)
The force of mortality
The force of mortality is one of the most important concepts in formal demography. It describes the behaviour of a mortality rate, nmx, over an infinitely small duration n. Mathematically, then,
| μx= | lim n→0 |
nmx |
An alternative, but equivalent, representation of the force of mortality is to note that if the number of survivors at each age x (that is, lx) is a continuous function of age x, then the force of mortality is defined as the ratio of the rate of decrease of lx (in other words the instantaneous effect of mortality) at that age to the value of lx. Algebraically,
| μx | = | - | d | (lx) |
| dx | ||||
| lx | ||||
In this representation, d/dx represents differentiation with respect to age, x, and indicates the rate of change of lx over a infinitesimally small increment of age. (Since lx is a decreasing function of age, the minus sign is introduced to make μx positive). The theory of differential calculus, however, reminds us that the derivative of the natural log of a function of x, ln(f(x)), is
| d | ( ln ( f(x) ) ) | = | 1 | d | ( f(x) ) |
| dx | f(x) | dx |
, implying that
| μx | = | - | d | ( ln(lx) ) |
| dx |
Integrating both sides over some range [0,n) results in
| n ∫ 0 |
μx+t dt | = | n ∫ 0 |
- | d | ( ln(lx+t) )dt | = | [ | - | ln(lx+t) | n ] 0 |
| dt |
| = | - | ln(lx+n) | - | ln(lx) | = | - | ln | ( | lx+n | ) |
| lx |
| => | n px | = | exp | ( | - | n ∫ 0 |
μx+t dt | ) |
One implication of this fundamental relationship is that the ratio of survivors at any age x+n to the survivors at age x is determined solely by the aggregate mortality between those two ages. This result, and several other identities connecting the force of mortality to other life table quantities, notably lx, dx and Lx and ex are of immense importance in mathematical demography, and in the theory of stable populations. These are outside the scope of this course.
Substituting Gompertz’ formula for mx into the equation above results in the identity,
| n px | = | exp | ( | - | n ∫ 0 |
BC x+t dt | ) | = | exp | ( |
|
.C x (C n - 1) | ) |
in other words, expressing survival between two ages x and x+n in terms of the parameters B and C. For application to most human populations, values of B typically lie in the range 10-6<B<10-3 while C tends to lie in the range 1.06<C<1.12.
The Gompertz curve still finds use in the determination of life table survivors for ages beyond which there is empirical data.