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"Sir Ronald Ross's notes demand that my arguments on the first count (i.e., that relating to the symmetrical
form of the curve) should be reconsidered." It now transpires, in fact, that Dr. Brownlee
himself can obtain symmetrical curves on a hypothesis that the epidemic dies out from want of susceptible
persons, just as Sir Ronald has done, provided certain conditions are complied with. Thus,
if c be the number of susceptible persons "an infected person is likely to come into contact with at the
beginning of the epidemic," and d " the co-efficient of the power of infecting taken constant throughout
the epidemic," then, when the value of cd is "approximately equal to 1.5, very nearly symmetrical
curves arise." Such curves fit the actual statistics, but if (d be as great as 2 " these relationships
no longer hold."
Thus apparently so long as cd is in the neighbourhood of the value 1.5 Dr. Brownlee's mathematical
results admit of interpretation on an hypothesis of exhaustion of susceptibles. I made some
years ago (the Milroy Lectures, 1906) an attempt to define—without, however, any precise mathematical
analysis—the type to which the London measles wave conforms, and obtained a result which seems,
so far as it goes, to bear out the conclusion now reached by Dr. Brownlee. My attempted explanation
was based on an exhaustion of susceptibles hypothesis, and it seems to accord with some such range of
values for Dr. Brownlee's cd as he now postulates. On this view of the problem it was clear that
the susceptibles would at no point of time be completely exhausted (and this, of course, is in actual fact
the case) in a town like London. I concluded by saying: " This examination shows the absurdity of
assuming that an epidemic (in London) comes to an end because all the susceptibles have been attacked ;
and, again, in measles, at any rate, of explaining the decline by loss of virulence of the organism or of its
infecting power" (loc. cit., p. 52). Dr. Brownlee's latest views are summarised in his " Discussion of
the Results " (Proc. Roy. Soc Med, Vol XII, No 9, pp. 96 and 97). He again dismisses the exhaustion
of susceptibles hypothesis, but now not on the ground of the supposed necessarily symmetrical
form of the measles curve, but for the novel reason that he has been led to surmise that "epidemics of
different periods can be shown to exist in the same town at the same time" ; and he says : " The whole
evidence, therefore, seems to point to some condition in the organism which produces the disease, as the
potent cause of the difference, rather than to the number of susceptible children " He adds : " Compare
the paramacium which in natural conditions divides asexually for several hundred times and then dies
out unless conjugation takes place. The resting stage following conjugation persists for some time."
In the discussion on Dr. Brownlee's paper the strange inability of measles " to cross the bridges "
was commented upon ; for as already noted, Dr. Brownlee says the 97 weeks' epidemic is found not
to spread, while " with regard to the 87 weeks' epidemic, however, the case is different. This seems
to start in St. Saviour's parish and to spread thence to Camberwell, Lambeth, etc." The general question
was also raised as to the validity, from the mathematical and epidemiological standpoints, of the theory
of the paramount importance of loss of virulence on the part of the organism ; but it was felt that
the immediate object of interest emerging as the result of Dr. Brownlee's analysis was whether in fact
the alleged periodicities had anything more than an evanescent interest, and to this particular point
it is now proposed to restrict consideration.
It is clearly important to determine whether it is the case that here a periodicity of precisely
97 weeks, and there one of precisely 87 weeks, assume and maintain prominence, and so on. With a
view to throwing further light upon this subject an examination of the London mortality figures has been
made by Mr. B. E. Spear, with the following results.
Mr. Spear first notes that a source of difficulty arises where, in any series of prevalences analysed,
succeeding groups of high amplitudes or " peaks " are separated by an interval of time which stands in
near relation to the approximate " periods " characterising either of these groups ; for in such a case
the periodogram may practically superimpose the " peaks " of one group upon the "peaks " of the other,
and, piling Ossa on Pelion obtain a high amplitude for a " period " which, so far as the greater part of
the series of years examined is concerned, has no actual existence. He then investigates this phenomenon
by study of Dr. Brownlee's Table of Amplitudes, in groups of years for different " periods," in the
epidemics of measles in London, 1840-1912.
Number of weeks in " period."
Group of years. |
87 89½ 97 109½ 114
1840—1856 110 231 272 331 270
1856—1874 152 35 552 70 208
1874—1893 303 300 416 229 110
1893—1912 310 180 464 325 397
1840—1912 161 167 417 212 205
It will be seen that in 1840—56 the periodogram shows a high amplitude for 109½ weeks.
These years are, however, entirely dominated by the high amplitude or fluctuation in mortality
which occurred in the four years 1845-1848, as will be apparent from the first contour in the diagram
on page 89. The interval between the two "peaks" in these years measures about 108 weeks; and
the periodogram, owing to the absence of any marked " peaks " in the remaining years, assigns high
amplitudes to a period somewhat exceeding two years for the whole 17 years.
In the next series, the 19 years 1856-1874, the periodogram analysis shows a high amplitude
for a period of 97 weeks. The second and third contours of the diagram on page 89 reveal two epochs
of epidemic incidence—namely, from 1862 to 1866 inclusive, and from 1871 to 1874. The average