262 Proceedings of the Royal Society of Edinburgh. [Sess. 
XIV. — The Mathematical Theory of Random Migration and 
Epidemic Distribution. By John Brownlee, M.D., D.Sc. 
(MS. received May 16, 1910. Read same date.) 
The general theory of epidemic disease I have already considered in a 
communication to this Society.* In that communication I showed that 
the course of epidemics of all forms of infectious disease obeyed certain 
very definite laws. In the same paper it was also shown that the distri- 
bution of epidemic disease in a uniformly populated area obeyed a law 
essentially similar. Certain reasons were given why the normal curve of 
_ X 2 
error y = y 0 e 2a ' 2 might be expected to give an approximate solution in 
both the cases considered, but why the distribution actually found (type iv.) 
should be the common form was not at all clear. I think, however, I have 
now arrived at the solution. 
The distribution of an epidemic in space is evidently a problem in 
chance. If there is an infective group in the middle of a uniformly dis- 
posed population, then the distance from which friends come to visit a sick 
person or the distance a sick person travels while developing the disease 
determines the subsequent distribution of cases — a distribution, therefore, 
obeying some law on the average. This problem has since been attacked 
and solved by Professor Pearson under the title of “The Problem of 
Random Migration.” The case which he considers refers specially to the 
prevention of malaria, which is now known to be spread through the 
agency of mosquitoes. The mathematical theory, which is very complex, 
leads to the determination that the normal surface of error gives a very 
close representation of this distribution. For epidemiological purposes 
the result is quite sufficiently close. To make the matter perfectly clear, 
the conditions of the problem solved are given in Professor Pearson’s 
own words : — 
“(1) Breeding grounds and food supply are supposed to have an 
average uniform distribution over the district under considera- 
tion. There is to be no special following of river beds or forest 
tracks. 
“(2) The species scattering from a centre is supposed to distribute 
itself uniformly in all directions. The average distance through 
which an individual of the species moves from habitat to habitat 
* Proc. Roy. Soc. Edin., June 1906. 
