# Anderson Gray McKendrick

Anderson Gray McKendrick (8 September 1876 – 30 May 1943) was a Scottish physician and epidemiologist pioneered the use of mathematical methods in epidemiology. Irwin (see below) commented on the quality of his work, "Although an amateur, he was a brilliant mathematician, with a far greater insight than many professionals."

McKendrick was born in Edinburgh the fifth and last child of John Gray McKendrick FRS, a distinguished physiologist. The son trained as a doctor at the University of Glasgow and joined the Indian Medical Service. He worked with Ronald Ross and eventually would continue his work on mathematical epidemiology. His primary interest was in research and he became director of the Pasteur Institute at Kasauli in the Punjab. He was invalided home to Britain in 1920 and settled in Edinburgh where he became Superintendent of the Laboratory of the Royal College of Physicians of Edinburgh. He held this post for the rest of his life.

McKendrick's career as a mathematical epidemiologist began in India. In 1911, McKendrick rediscovered the logistic equation and fit it to bacterial growth data.[1] In 1914 he published a paper in which he gave equations for the pure birth process and a particular birth-death process. After his return to Scotland he published more. His 1926 paper, 'Applications of mathematics to medical problems' was particularly impressive, including the widely used McKendrick–Von Foerster partial differential equation

$\frac{\partial n}{\partial t} + \frac{\partial n}{\partial a} = - \mu(t,a) n.$

Some of this papers other results for stochastic models of epidemics and population growth were rediscovered by William Feller in 1939. Feller remarks in his Introduction to the Theory of Probability & its Applications (3rd edition p. 450), "It is unfortunate that this remarkable paper passed practically unnoticed." The same paper is also the earliest reference in Dempster et al.'s 1977 paper that defined and popularized the EM algorithm (expectation-maximization algorithm) In 1927 McKendrick began a collaboration with W. O. Kermack (1898–1970) which produced a notable series of papers on the Kermack–McKendrick theory, a general theory of infectious disease transmission.

W. M. Hirsch gives this picture of the man: "McKendrick was a truly Christian gentleman, a tall and handsome man, brilliant in mind, kind and modest in person, a skilful counsellor and administrator who gave of himself and knew how to enable others."

## Selected works

• A. G. McKendrick Applications of mathematics to medical problems Kapil Proceedings of the Edinburgh Mathematical Society, vol 44, (1925–6), pp. 1–34. Reprinted with commentary in S. Kotz & N. L. Johnson (Editors) (1997) Breakthroughs in Statistics: Volume III New York Springer.
• W. O. Kermack; A. G. McKendrick “Contributions to the Mathematical Theory of Epidemics. II. The Problem of Endemicity,” Proceedings of the Royal Society of London. Series A, Vol. 138, (1932) pp. 55–83.

## Commentary

There is an account of McKendrick's Applications paper in

J. O. Irwin The Place of Mathematics in Medical and Biological Statistics, Journal of the Royal Statistical Society. Series A (General), Vol. 126, No. 1. (1963), pp. 1–45.

## Biography

• Warren M. Hirsch (2004) McKendrick, Anderson Gray (1876–1943), Oxford Dictionary of National Biography, Oxford University Press.
• Gani, J. (2001) Anderson Gray McKendrick, Statisticians of the Centuries (ed. C. C. Heyde and E. Seneta) pp. 323–327. New York: Springer.