Population Heterogeneity, Late-Life Mortality Deceleration,

Mortality Levelling-off, Mortality Plateaus

Here are some key references to pioneer scientific publications, which first introduced the idea of population heterogeneity (gamma-distributed individual risk) to explain a logistic function for mortality rate, late-life mortality deceleration, mortality levelling-off, and mortality plateaus:

• Beard, R.E. 1959. Note on some mathematical mortality models. In: G.E.W.Wolstenholme and M.O'Connor (eds.). The Lifespan of Animals. Little, Brown, Boston, 302-311

• Beard, R.E. 1964 Some observations on stochastic process with particular reference to mortality studies. International Congress of Actuaries, vol.3, p.463-477

This logistic model with gamma distributed individual risk and late-life mortality deceleration became widely known to biologists and demographers in 1971, when Dr. Beard published a detailed review of his studies in the book "Biological Aspects of Demography":

• Beard RE. 1971.  Some aspects of theories of mortality, cause of death analysis, forecasting and stochastic processes. In. Biological aspects of demography (Brass W., Ed.), Taylor & Francis, London, 57-68
  

The heterogeneity mortality model with gamma distributed individual risk (producing logistic function for mortality rate) was developed by famous British statistician and actuary Robert Eric Beard in 1959.   Dr. Beard also became famous for his book "Risk Theory", which had three editions:

• Risk Theory by Beard, Pentilkainen and Pesonen, Chapman and Hall.