J. Alexander Mitchell went to school at Foyle College, Londonderry, and then qualified as Bachelor of Medicine (MB) and Bachelor of Surgery (ChB) at the University of Glasgow in 1898. Thereafter he spent a year and a half in the Army Medical Service, Edinburgh district. At some time he obtained the Diploma in Tropical Medicine (DTM) at the University of Edinburgh. He came to the Cape Colony in September 1899 to take up an appointment as government bacteriologist in charge of leprosy research on Robben Island, and was licensed to practice at the Cape in June 1900. His "Report on leprosy investigation" was issued in 1901. When bubonic plague broke out at the Cape in 1900 he was appointed medical officer in charge of the plague quarantine stations at Saldanha Bay and Izeli (Eastern Cape). During the severe epidemic of 1901 he was medical superintendant of the plague hospital and quarantine station at Maitland, Cape Town, and in July that year was appointed assistant medical officer of health of the Cape Colony. He remained on special duty in connection with bubonic plague until 1903 and thereafter was engaged on public health work throughout the colony. In 1906 he was awarded the Diploma in Public Health (DPH) by the University of Cambridge. Around this time he was chairman of the Robben Island and Emjanyana Leprosy Commissions.
Mitchell was a member of the South African Association for the Advancement of Science from 1903. When the association met jointly with its British counterpart in South Africa in 1905 he presented a paper on "Bubonic Plague in the Cape Colony". It dealt with the introduction of plague into the colony around 1900, its epidemiology and the measures taken to combat it, and was included in the Addresses and papers... (Vol. 3, pp. 447-483) published after the meeting. In 1908 he read a paper before the association on "The growth of the native races of Cape Colony and some factors affecting it" (Report, 1908, pp. 253-262). This paper dealt mainly with the rapid population growth of the Bantu speaking peoples and how it was affected by disease.
After the formation of the Union of South Africa in 1910 Mitchell was appointed assistant medical officer of health for the Union, stationed in Cape Town. During World War I (1914-1918) he served in the South African Medical Corps, reaching the rank of lieutenant-colonel. From 1911 he was a member of the Colonial Medical Council (to 1918), then of the Transvaal Medical Council (to 1928) and finally of the Union Medical Council (to 1932). In 1917 he became a member of the board of management of the South African Institute for Medical Research. When the Department of Health was established in 1919 he was appointed as the first secretary for public health and chief health officer of the Union, a post he held until he retired in April 1932. During this period he was largely responsible for drafting the Public Health Act of 1919 and the Medical, Dental and Pharmacy Act of 1928. He was also chairman from its inception of the Union Leprosy Commission and served on the Commission on Tuberculosis of the League of Nations at Geneva.
Meanwhile his interest in bubonic plague continued, as shown by his paper on "Plague in South Africa: Perpetuation and spread of infection by wild rodents" (Journal of Hygiene, Cambridge, 1922). This was followed a few years later by Plague in South Africa: Historical summary (South African Institute for Medical Research, Publication No. 20, 1927) in which he showed that the disease was first recorded in South Africa early in 1899. Other publications by him dealt with, among others, "The problem of venereal disease" (South African Medical Journal,1921), "Smallpox and 'Amaas' in South Africa" (The Lancet, 1922), "The public ill-health" (Medical Journal of South Africa, 1922), Snake-bite and its treatment (Pretoria, 1926), and "Epizootic among veld rodents in De Aar and neighbouring districts" (Epidemiology and Infection, 1930). In January 1931 he represented South Africa at a conference of leprologists at Manila, in the Philippines. He was married to Ellen Mary Thompson, with whom he had six children.