APPENDIX 



NOTE ON SOME MATHEMATICAL MORTALITY 



MODELS 



R. E. Beard 



Pearl Assurance Co. Ltd., London 



1. A satisfying basis for a law of mortality would be a formula 

 that, starting from some fundamental concepts about the biological 

 ageing process, led to a distribution of deaths by age which was 

 comparable with observational data. Such comparison would not be 

 simple and straightforward because environmental and secular 

 factors would introduce distortions as compared with the theoretical 

 underlying distribution. 



2. In the course of numerical work, extending over a number of 

 years, on the expression of human mortality functions by mathe- 

 matical formulae, various attempts have been made by the writer 

 to develop an approach on this basis. The results obtained have not 

 led to any satisfying formulae, but they are suggestive of different 

 lines of approach and have been summarized below in the hope they 

 may be of value to others interested in the subject. The note follows 

 the sequence in which the ideas have developed in the mind of the 

 writer and leads from considerations based on the force of mortality, 

 /x^, to those based on the curve of deaths, ^xjl^. 



3. The first mathematical expression which provided a reasonable 

 representation of the observed force of mortality in human data was 

 that first proposed by Gompertz (1825) and later modified by Make- 

 ham (1867). Basically the "law" was derived by postulating a 

 relationship between the rate of change of the force of mortality at 

 any age and its value at that age. The next significant modification 

 to the Makeham law was the system of curves devised by Perks in 

 1932 and of which the important formula was the logistic. Many 

 human life-tables have been graduated by this basic cur^'e, modified 

 in some instances to allow for special features of the data, particu- 

 larly at the younger and early middle ages, and the clear fact emerges 

 that adult human mortality can be very well represented by a 

 logistic curve of the form 



ix^- A = Be^^'Kl+De^') (1) 



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