McKendrick, Anderson Gray

(1876-1943), mathematician, epidemiologist, and army officer

by Warren M. Hirsch

© Oxford University Press 2004 All rights reserved

McKendrick, Anderson Gray (1876-1943), mathematician, epidemiologist, and army officer, was born on 8 September 1876 at 1 Chester Street, Edinburgh, youngest of the five children of John Gray McKendrick (1841-1926), professor of physiology in Glasgow, and his wife, Mary Souttar (1842-1896), eldest daughter of William Souttar, Great North of Scotland Railway accountant, and his wife, Mary Mearns. His second forename carried forward in the family the name of his father's benefactor when a student in Aberdeen.

McKendrick was educated at Kelvinside Academy, Glasgow (1882-91), studied at Jena University, Germany (1894-5), and completed medical studies at the University of Glasgow in 1900, graduating MB ChB with high commendation. With a first on the entrance examination, he entered the Indian Medical Service (IMS) and trained in military and tropical medicine at the Army Medical School, Netley, where he studied under the bacteriologist Almroth Wright.

In June 1901 McKendrick was dispatched to Sierre Leone on the fifth Liverpool malarial expedition to study anti-malarial techniques with Ronald Ross, whose example profoundly influenced the direction of McKendrick's life work. Ross impressed on the young physician lieutenant, not yet trained in mathematics, the tremendous power of mathematical methods in medical research.

McKendrick saw distinguished military service in the Somaliland campaign (1903-4), then, posted to India, was appointed in 1905 to the research department of the Government of India Pasteur Institute at Kasauli, the provincial bacteriological and pathological laboratory for the Punjab. There he concentrated on investigating the disease of rabies, treating patients, preparing vaccine, and conducting experimental work, with evenings reserved for study of J. W. Mellor's Higher Mathematics, for Students of Chemistry and Physics (1902).

On 3 February 1910 McKendrick married Mildred Macleod Wylie (1886-1952), daughter of Major-General Henry Wylie, former resident of the Gurkha state of Nepal. They had two sons and two daughters. His son Gordon was a naval officer, related by marriage to Admiral Andrew Browne Cunningham; his other son, John Gray, was an army officer (died of wounds in 1945).

During his forty-year research career McKendrick made many important contributions to biology, medicine, epidemiology, demography, mathematics, and statistics, and to the intricate embroidery that weaves these fields together. In an early statistical study in immunology (with W. F. Harvey, 1907), McKendrick showed that the quantity of virus and the length of time over which it is administered are the critical factors in anti-rabic immunization. In time he became the acknowledged British authority on rabies, publishing important critical reviews of rabies literature in the Tropical Disease Bulletin (1924-34) and influential annual reviews of rabies treatment statistics for the League of Nations (1927-37).

For some years McKendrick searched for a theoretical way of comparing different anti-malarial measures. He succeeded by mathematically modelling the interaction between infected humans and infected mosquitos ('The rise and fall of epidemics', 1912), showing that when the ratio of the size of the mosquito population to the human is large, quinine prophylaxis gives better results, whereas if it is small, anti-mosquito measures are more efficient.

In 1912, in a letter to his mentor Ronald Ross, McKendrick reported a mathematical breakthrough related to leukocyte effectiveness, later published in Science Progress ('The physical aspect of the opsonic experiment', 1914). McKendrick had derived a differential-difference equation satisfied by the distribution function of the number of bacteria neutralized by each leukocyte. Amazingly, in solving this equation McKendrick independently discovered the important Poisson distribution, and his equation may well be the earliest characterization of the Poisson process by a differential-difference equation. In a subsequent paper ('Studies on the theory of continuous probabilities, with special reference to its bearing on natural phenomena of a progressive nature', 1914), McKendrick derived and studied equations for a stochastic process now called the pure birth process and for a particular birth-death process, which appears to be the earliest treatment of these processes in probability theory. This paper foreshadowed McKendrick's later extraordinary work on mathematical epidemiology, having in it his first mathematical model of an epidemic.

On home leave in 1913, McKendrick finally undertook formal training in mathematics at Aberdeen, but was recalled to the Pasteur Institute in 1914, where he served for six years as its director. In 1920, for health reasons, McKendrick was forced to give up his post in India to return to Scotland, initially leaving his family in India. In the same year he was appointed superintendent of the Royal College of Physicians' laboratory, Edinburgh, where he actively pursued his research until retirement. While in Edinburgh he was an elder of the Pleasance church and lived in Succoth Gardens, Murrayfield.

McKendrick was awarded the DSc degree by the University of Aberdeen in 1927. His pathbreaking dissertation in 1926, 'Applications of mathematics to medical problems', contains the first formulation of a partial differential equation for the age distribution of a population and the first stochastic treatment of an epidemic. It is a tour de force, cited continually.

This work was followed by five major papers (with W. O. Kermack) on a deterministic theory of epidemics, in which several fundamental problems in epidemiology were addressed. For example: what starts an epidemic and what brings it to an end? One of the celebrated results is the classic Kermack-McKendrick threshold theorem: if a contagious disease confers permanent immunity to a survivor, introduction of infectious cases into a closed population of susceptibles (no births, immigration, or emigration) will give rise to an epidemic if and only if the density of susceptibles is above a certain critical value (threshold). The epidemic ends when the susceptible density is about as far below the threshold as it was above it initially. This result was extended, with appropriate modifications, when there is a continuous supply of new susceptibles, and immunity is partial.

In addition to his pioneering work in mathematical epidemiology, McKendrick also made contributions of lasting consequence to the development of mathematical analysis in demography, especially on generation mortality projection and applications, showing in particular that the important factor in lifetime health is the environment up to age fifteen, with obvious implications for health policy (W. Kermack, A. McKendrick, and P. McKinlay, 'Death-rates in Great Britain and Sweden: expressions of specific mortality rates as products of two factors and some consequences thereof', Journal of Hygiene, 34, 1934, 433-57).

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. He was fluent in German and French and a fine musician who played the cello, organ, piano, and ocarina. He was a fellow of the Royal Society of Edinburgh and a fellow of the Royal College of Physicians, Edinburgh. He was awarded the Cullen prize by the Royal College of Physicians in 1934.

In 1941, due to poor health, McKendrick retired to Carrmoor, Carrbridge, Inverness-shire. He was an elder of the Church of Scotland there, but also sometimes attended Free Church services. He died at Carrmoor of coronary heart disease on 30 May 1943 and was buried in Carrbridge cemetery.

WARREN M. HIRSCH

Sources  
W. F. H., Edinburgh Medical Journal, 3rd ser., 50 (1943), 500-06 [incl. list of publications]
Year Book of the Royal Society of Edinburgh (1942-3), 23-4
BMJ (19 June 1943), 771-2
Glasgow Medical Journal, 140 (1943), 21-2
The Lancet (10 July 1943), 59
J. Aitchison and G. S. Watson, 'A not-so-plain tale from the Raj: A. G. McKendrick, IMS', The influence of Scottish medicine, ed. D. A. Dow (1988), 113-28
J. Gani, 'The early use of stochastic methods: a historical note on McKendrick's pioneering papers', Statistics and probability: essays in honor of C. R. Rao, ed. G. Kallianpur, P. R. Krishnaiah, and J. K. Ghosh (1982), 263-8
J. O. Irwin, 'The contributions of G. U. Yule and A. G. McKendrick to stochastic process methods in biology and medicine', Stochastic models in medicine and biology, ed. J. Gurland (1964), 147-65
F. Yates, Memoirs FRS, 17 (1971), 416-20 [appx to J. N. Davidson, obit. of W. O. Kermack]
J. G. McKendrick, The story of my life (1919)
R. Ross, Memoirs (1923), 434-54
J. O. Irwin, 'Mathematics in medical and biological statistics', Journal of the Royal Statistical Society: series A, 126 (1963), 18-29
F. A. Haight, Handbook of the Poisson distribution (1967), 102-121
F. C. Hoppensteadt, 'Some influences of population biology on mathematics', Essays in the history of mathematics, ed. A. Schlissel (1984), 25-9
M. S. Bartlett, Stochastic population models (1960), 54-61
W. Feller, An introduction to probability theory and its applications, 3rd edn (1968), 450
M. S. Bartlett, An introduction to stochastic processes, with special reference to methods and applications, 3rd edn (1978), 58
private information (2004) [Joyce Matthew]

Archives  
Royal College of Physicians of Edinburgh, MSS |  London School of Hygiene and Tropical Medicine, Ross MSS
UCL, Karl Pearson MSS
Wellcome L., Ross MSS

Wealth at death  
£6116 8s. 3d.: confirmation, 24 Sept 1943, CCI


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