Friday, September 15, 2017

The Gompertz Mortality Function

In Section 2.4 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss exponential decay with a variable rate. If the rate is constant, the fraction of a population remaining after a time t decays exponentially. This is not a good way to estimate the lifespan of humans, because as we age the likelihood of death increases. A simple model is to assume that the mortality rate increases exponentially, leading to the Gompertz mortality function. IPMB explores this behavior in a homework problem.
Problem 15. When we are dealing with death or component failure, we often write Eq. 2.17 in the form y(t) = y0 exp[-∫0t m(t') dt'] and call m(t) the mortality function. Various forms for the mortality function can represent failure of computer components, batteries in pacemakers, or the death of organisms. (This is not the most general possible mortality model. For example, it ignores any interaction between organisms, so it cannot account for effects such as overcrowding or a limited supply of nutrients.)
(a) For human populations, the mortality function is often written as m(t) = m1e b1t + m2 + m3e +b3t . What sort of processes does each of these terms represent?
(b) Assume that m1 and m2 are zero. Then m(t) is called the Gompertz mortality function. Obtain an expression for y(t) with the Gompertz mortality function. Time tmax is sometimes defined to be the time when y(t) = 1. It depends on y0. Obtain an expression for tmax.
I won’t solve this problem for you (after all, it's your homework problem). Instead, I will examine this behavior in a different way. First, let’s recast the governing differential equation in terms of dimensionless variables. Let p(t) = y(t)/y0 be the fraction surviving after time t, where y0 is the initial number at t = 0. Also, define a dimensionless time scale as T = m3t, and a dimensionless ratio of rates as X = b3/m3. The differential equation governing p(T) is then

dp/dT = - exp(XT) p

where p = 1 at T = 0. This form of the equation shows that, aside from scale factors, the behavior depends only on X.

The homework problem asks you to find an analytical expression for p(T). This is a valuable exercise, but you can also learn about the behavior by solving for p(T) numerically. The figure below shows p(T) for several values of X, calculated using Euler's method. If the increase in mortality is slow compared to the decay of p (that is, X is much less than 1), the decay is approximately exponential (the red X=0 curve). However, if X is large the decay starts exponentially (for T less than about 0.1 the curves in the figure are all nearly equal) but then accelerates as the rate grows.

An exponential decay of mortality was first analyzed by Benjamin Gompertz (1779-1865), an English mathematician and actuary. His 1825 article “On the Nature of the Function Expressive of the Law of Human Mortality” helped establish two fields of study: actuarial science and the biology of aging. Thomas Kirkwood’s 2015 paper describes Gompertz’s life and work. The title and abstract are below.
Deciphering death: a commentary on Gompertz (1825) ‘On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies’
In 1825, the actuary Benjamin Gompertz read a paper, ‘On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies’, to the Royal Society in which he showed that over much of the adult human lifespan, age-specific mortality rates increased in an exponential manner. Gompertz's work played an important role in shaping the emerging statistical science that underpins the pricing of life insurance and annuities. Latterly, as the subject of ageing itself became the focus of scientific study, the Gompertz model provided a powerful stimulus to examine the patterns of death across the life course not only in humans but also in a wide range of other organisms. The idea that the Gompertz model might constitute a fundamental ‘law of mortality’ has given way to the recognition that other patterns exist, not only across the species range but also in advanced old age. Nevertheless, Gompertz's way of representing the function expressive of the pattern of much of adult mortality retains considerable relevance for studying the factors that influence the intrinsic biology of ageing.

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