Supporting Information
The earliest concept of the effective reproduction number
Supplement to: The ideal reporting interval for an epidemic to objectively
interpret the epidemiological time course
Clare Oswald Stallybrass and his dispersibility ratio
In the main text, we discussed a historical study conducted by Stallybrass who is
known to have written one of the earliest epidemiologic textbooks, Principles of
Epidemiology in 1931 (Stallybrass, 1931), the contributions of which were discussed
elsewhere (Bracken, 2003; Lilienfeld, 2003). Here we give a brief biography of
Stallybrass in addition to those given by Lilienfeld (2003). Stallybrass was born in
Wallasey, Merseyside, 1881. After graduating in Medicine with honors at the
University of Liverpool in 1905, he was trained at Liverpool Royal Infirmary and the
Maternity Hospital, where he became interested in preventive medicine (Anonymous,
1951a). Following successful qualification of doctor of public health (D.P.H.) in
Liverpool while working as clinician at Parkhill Fever Hospital, he was appointed as
an Assistant Medical Officer in the Liverpool Port Medical Service in 1909
(Anonymous, 1951b). Whereas his career at Liverpool was interrupted during World
War I from 1914-18 to serve abroad as sanitary officer in Macedonia and later as a
specialist with Serbian Army, he thereafter returned to Liverpool and worked as
Deputy Medical Officer of Health for the City and Port of Liverpool from 1933-48.
Even after retirement, Stallybrass actively revised his books on epidemiology and
public health until his death on Jan 28, 1951. His first publication in 1931 is deemed
to be a departure of Epidemiology from tradition, and the medical societies admired
the attempt made by the medical doctor engaged in day-to-day work in a health
department to bring within one compass the sum of contemporary epidemiological
knowledge (Anonymous, 1951b). Whereas theoretical arguments of our interest are
also reviewed in the Chapter 14 of his textbook (Stallybrass, 1931, pp. 497-539) , an
original study suggesting “dispersibility” as one of the epidemiological markers
preceded that chapter in which the time course of various epidemics during 19th and
20th centuries was quantified (Stallybrass, 1928).
Although the textbook of Stallybrass (1931) was widely disseminated among
early epidemiologists during the mid- and late-20th century, a revision of the
dispersibility ratio has not been given for about 80 years, except for a short note
elsewhere (Fine, 1979). This may be in part owing to the lack of a rigorous
mathematical formulation of his theory and to the fact that the proposed correcting
factor was theoretically incorrect. Immediately after his first publication in 1928
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(Stallybrass, 1928), the crude measure (i.e. equation (1) in the main text) was
criticized for not allowing the comparison of ratios obtained for different diseases
(Stocks and Karn, 1928). Although he intended to make an adjustment of the
dispersibility ratio by using a correction ratio of the reporting interval to incubation
period, the denominator of which may have been assumed to reflect the
epidemiological characteristics of the disease, no reason was given as to why the
incubation period was used as the denominator. Many epidemiologists traditionally
assumed (incorrectly) that the incubation period, at least roughly, reflected the interval
between successive generations of cases (Sartwell, 1966; Fine, 2003). Moreover,
whereas he suggested that it is possible to roughly assess the time course of an
epidemic from the relative change of dispersibility ratios with respect to time, at least,
in the successive reporting intervals, his arguments were missing an epidemiological
interpretation of the absolute value of the ratio.
The average infectee number of Annet Nold
Nold (1979) defined the same dispersibility concept using more explicit notation,
which was most likely independent of the idea of Stallybrass. By taking the ratio of
infected individuals during the time interval (t1, t2), Nold referred to the quantity as
the “average infectee number” of an infectious disease, (t1, t2). In contrast to
Stallybrass, she clearly defined the generation time of a disease, which was originally
referred to as the “average transmission time” (Nold, 1979). Whereas the main
research interest of Nold was focused on a critical level (i.e. threshold condition) of
transmission in an endemic state where the infectee number becomes close to 1, she
also documented Rt using the mean generation time, , as a special case of the
average infectee number:
R (t) 
C(t,t   )
C(t  ,t)
(1)
where C(t1, t2) denotes the number of cases observed during the time interval between
t1 and t2. R(t) was originally referred to as “incidence ratio”, and she found that the
incidence trend is downward (respectively, upward) on an interval (t-, t+) when
R(t) < 1 (respectively, R(t) > 1). It should be noted that equation (1) implicitly
assumes that the generation-time distribution g() of length  is given by a delta
function (i.e. g() equals 1 if  =  and 0 otherwise).
References
Anonymous. 1951a. Clare Oswald Stallybrass, M.D. (Lond. and Liv.), D.P.H. Public
Health. 64, 104.
Anonymous. 1951b. Clare Oswald Stallybrass. Lancet. 1(6651), 415-416.
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Bracken MB. 2003. The first epidemiologic text. Am J Epidemiol. 157, 855-856.
Fine PEM. 1979. John Brownlee and the measurement of infectiousness: An historical
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Fine PE. 2003. The interval between successive cases of an infectious disease. Am J
Epidemiol. 158, 1039-1047.
Lilienfeld DE. 2003. The first epidemiology textbook, revisited. Am J Epidemiol. 157,
856-867.
Nold A. 1979. The infectee number at equilibrium for a communicable disease. Math
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Sartwell PE. 1966. The incubation period and the dynamics of infectious disease. Am
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Stallybrass CO. 1928. Season and disease. Proc R Soc Med (Section of Epidemiology
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Stocks P, Karn MN. 1928. A study of the epidemiology of measles. Ann Eugenics. 3,
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