May 1 8, 191 6] 



NATURE 



243 



Tied away from the winter quarters, and the relief 

 •expedition ought to be able to search independently 

 for the ice-bound Endurance and for the party or 

 parties left on shore. There would obviously be 

 a much better chance of success if two vessels 

 could be employed — one to search the coast- 

 lands, and the other to scour the sea along the 

 probable lines of drift of the Weddell Sea pack. 

 From the observations of the Scotia in the Wed- 

 dell Sea the prevalent wind direction there appears 

 to be from the east, so that some belt of "land 

 Tvater " may be fairly persistent off Coats Land 

 and the drift of the ice may be westward; but 

 "knowledge of meteorology in the Weddell Sea is 

 so scanty that forecasts as to the usual drift of the 

 ice would command but little confidence and may 

 be falsified by an unusual season. The com- 

 mander of the relief expedition should be at liberty 

 to select his own route. 



Sir Ernest Shackleton has met with very bad 

 iuck from the wxather. His proposed transcon- 

 tinental sledge journey was a daring and difficult 

 undertaking. He had, however, considered all its 

 p>ossibilitIes, and it promised a fair chance of suc- 

 cess ; but his plans may have been deranged at the 

 outset by the exceptionally unfavourable season. 

 The ice conditions in the Weddell Sea may have 

 prevented his starting forth on his great adventure. 

 No time must be lost in organising the expedition 

 to take him the help which he and his colleagues 

 may sorely need. In addition to the return of the 

 Aurora to ^facmurdo Sound, tw'o vessels, if pos- 

 sible, should be sent to the Weddell Sea, for the 

 area that will have to be searched is vast, the 

 clues are uncertain, and the season is short. 



THE APPLICATION OF MATHEMATICS TO 

 EPIDEMIOLOGY. 



IT may seem remarkable that serious attempts 

 to elucidate the mysteries of epidemic disease 

 with the help of mathematical methods should 

 only have been made within the last sixty years, 

 and, even when made, should have been confined 

 to the efforts of a very small number of students. 

 In the seventeenth and early eighteenth centuries, 

 the school of which Borelli was the most famous 

 exponent endeavoured to bring much less pro- 

 mising medical fields under mathematical culti- 

 vation, while Sydenham's exposition of the prin- 

 cipia of epidemiology would, one might have 

 thought, have suggested to the founders of our 

 modern calculus of probabilities that here was 

 indeed an opportunity for them. No doubt, how- 

 ever, the explanation is to be found in the absence 

 of statistical data, without which mathematical 

 mills are forced to stand idle. It is of interest 

 to recall the fact that the solution of a problem 

 which took its rise in the failure to publish cer- 

 tain detailed statistics reveals a method w-hich 

 might have been generalised. We allude to 

 Daniel Bernoulli's work on smallpox. ^ 



His solution was as follows : — 



If -v denote the age in years, ^ the number 

 who survive at that age out of a given number 



J See To<Jhii'iter"« " Hi-trrv o'' th" Tfi-ory of Probability,' p- 225- 



NO. 2429, VOL. 97] 



born, 5 the number of these survivors who have 

 not had smallpox, and if in a year smallpox 

 attacks i out of every n who have not had the 

 disease, while i out of every m attacked dies, 

 then the number attacked in element of time 

 dx is sdx/n and we have : — 



, sdx s/j^, sdx\ sd$-ids $dx dx 



-ds= -M^+ — ) or -H -=^ 



n (\ mnj s^ ns mn 



Substituting q for ^/j, we have da = ^~ dx, so that 



mn 



n log {tnq- i) = jc + constant, and ultimately, since 



when x — o, s = $, 



X 



(in - I j^" + I 



This investigation contains the germ of a 

 method which, as Sir Ronald Ross has brilliantly 

 demonstrated, might be applied to the study of 

 the succession of cases in an epidemic. Nobody, 

 however, took the hint, and the . real history of 

 mathematical epidemiology begins with Farr, 

 whose work on these lines has been made familiar 

 to the present generation by Dr. John Brownlee. 

 Modern researches fall into one of two classes. 

 On one hand, those directly or indirectly inspired 

 by the epoch-making discoveries of Prof. Karl 

 Pearson in the theory of mathematical statistics ; 

 on the other, the independent investigations of 

 Sir Ronald Ross. 



Prof, Pearson's development of a family of fre- 

 quency curves, including the Gauss-Laplace or 

 normal curve as a particular case and capable 

 of describing effectively distributions very far 

 indeed from normal, enabled statisticians to deal 

 with a wide range of frequency systems, and it 

 naturally occurred to some to use this method in 

 the study of epidemics. Frequency curves have 

 been fitted by Brownlee, ^ Greenwood, ^ and other 

 medical statisticians to different epidemics, the 

 most extensive work in this direction having been 

 that of Brownlee, Much of this work was de- 

 scriptive; that is to say, the object was in the 

 first place to graduate the statistics, and, if pos- 

 sible, to classify epidemics on the basis of the 

 type of curve found. So far as graduation is 

 concerned, the results have been fairly satis- 

 factory, but it proved to be impossible to effect 

 any useful classification, the only result that 

 emerged being that Pearson's Type IV curve was 

 more commonly encountered than any other. The 

 more fundamental problem of epidemiology, viz., 

 that of discovering the law of which the epidemic, 

 whether viewed in its temporal or spatial rela- 

 tions, is an expression, could scarcely be solved in 

 this way. Brownlee, however, was by no means 

 content with the mere graduation of statistics. 

 Following Farr, he surmised, for reasons ex- 

 plained in his papers, that the theoretical curve of 

 an epidemic in time or space should be normal, 

 and that any practical departure from normalit}' 

 should be susceptible of an explanation capable 

 of expression in terms of a function of the 



- Proc. Roy. 5m>c. Edin., 1906, xxvi., 484 ; ihid., 1911, xxxi., 262. 

 ^ J oum. Hygiene, 1911, xi.,96; Proc. lyth Inter. Congress Med., iqn. 

 Sect. 18. 



