# Anderson Gray McKendrick

Lt Col **Anderson Gray McKendrick** DSc FRSE (8 September 1876 – 30 May 1943) was a Scottish military 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."

## Life[edit]

McKendrick was born at 2 Chester Street^{[1]} in Edinburgh the fifth and last child of John Gray McKendrick FRS, a distinguished physiologist, and his wife, Mary Souttar. His older brother was John Souttar McKendrick FRSE (1874-1946).

He was educated at Kelvinside Academy then trained as a doctor at the University of Glasgow qualifying MB ChB in 1900.^{[2]} He then was commissioned in the British Army and joined the Indian Medical Service. At the rank of Lt Colonel he led an expedition into Somaliland in 1903/4 as part of what was then known as the Dervish Wars.^{[3]}

He later worked with Ronald Ross and eventually would continue his work on mathematical epidemiology. His primary interest was in research and he was director of the Pasteur Institute at Kasauli in the Punjab 1914–1920.^{[2]} 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.^{[4]} In 1912 he was elected a Fellow of the Royal Society of Edinburgh. His proposers were James Oliver, Diarmid Noel Paton, Ralph Stockman and Cargill Gilston Knott. He served as the Society's Vice President 1933-36.^{[5]}

In 1914 he published a paper in which he gave equations for the pure birth process and a particular birth-death process. In 1924 he was elected a Fellow of the Royal College of Physicians of Edinburgh. 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

Some of this paper's 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 and 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 William Ogilvy 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[edit]

- 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 “A Contribution to the Mathematical Theory of Epidemics,” Proceedings of the Royal Society of London. Series A, Vol. 115, (1927), pp. 700–721.
- 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.
- W. O. Kermack; A. G. McKendrick “Contributions to the Mathematical Theory of Epidemics. III. Further Studies of the Problem of Endemicity,” Proceedings of the Royal Society of London. Series A, Vol. 141, (1933), pp. 94–122.

## Commentary[edit]

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[edit]

- 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.

## References[edit]

**^**Edinburgh and Leith Post Office Directory 1876-77- ^
^{a}^{b}"Lieut. Colonel A. G. M'Kendrick".*The Glasgow Herald*. 31 May 1943. p. 4. Retrieved 1 May 2017. **^***Biographical Index of Former Fellows of the Royal Society of Edinburgh 1783–2002*(PDF). The Royal Society of Edinburgh. July 2006. ISBN 0 902 198 84 X.**^**McKendrick, A. G.; Kesava Pai, M. (January 1912). "XLV.—The Rate of Multiplication of Micro-organisms: A Mathematical Study".*Proceedings of the Royal Society of Edinburgh*.**31**: 649–653. doi:10.1017/S0370164600025426.**^***Biographical Index of Former Fellows of the Royal Society of Edinburgh 1783–2002*(PDF). The Royal Society of Edinburgh. July 2006. ISBN 0 902 198 84 X.

## External links[edit]

- O'Connor, John J.; Robertson, Edmund F., "Anderson Gray McKendrick",
*MacTutor History of Mathematics archive*, University of St Andrews.

There is a photograph at

- A. G. McKendrick on the Portraits of Statisticians page.

There is a modern presentation of one of the Kermack–McKendrick models in

McKendrick's father was elected to the Royal Society, as was Kermack his co-worker