152 JOURNAL OF THE WASHINGTON ACADEMY OF SCIENCES VOL. 13, No. 8 
BIOMETRICS.—Contribution to quantitative parasitology.: ALFRED 
J. Lorxa. 
An analytic study of the quantitative relations between a population 
of host organisms and a parasite species preying upon such a popu- 
lation has been made by W. R. Thompson.? The time unit employed 
by this author is virtually a “generation’’; thus, for example, he 
enquires how many generations must pass, after introduction of the 
parasite, to practically exterminate the host species. There is some- 
thing unsatisfactory in this, inasmuch as the delimitations of a genera- 
tion are, in the nature of things, ill-defined. So, for example, if we 
consider the progeny of a thousand human males born on January 1, 
1900, the births of their sons will stretch over a continuous period 
from about 1920 to 1950, the births of their grandsons from 1940 to 
2000, their great-grandsons from 1960 to 2050, andso on. Ina mixed 
population the limits of a generation are still more indefinite. 
For this and other reasons it seems desirable to approach from 
another angle the problem broached by W. R. Thompson. This 
may be done as follows: 
Let N, denote the number of “healthy” hosts, i.e., individuals not 
attacked by the parasite, and let N, be the number of parasites in 
the adult (free-living) state. The number N, of healthy hosts will be 
augmented, per unit of time, by b,N, births; it will be diminished, 
per unit of time, by two separate terms, namely first d,N, deaths 
from other causes, and, second, by a,N,N, attacks by parasites. 
We shall here consider the case in which attack is always fatal, so 
that no recoveries occur; then we have 
ae = b.N, — d,iNi — a.N,N, (1)8 
where the coefficients b,, d:, a; are, in general, functions of N, and Nb». 
The parasite considered by W. R. Thompson is one that deposits 
a single egg in each host attacked. We may at once proceed to the 
more general case, that k eggs are deposited in each larva attacked, 
out of which h are actually hatched. In that case the number of 
1 Papers from the Department of Biometry and Vital Statistics, School of Hygiene 
and Public Health, Johns Hopkins University, No.83. (Received Jan. 30, 1923). 
2 Comptes Rendus vol. 174,(1922) pp. 1201, 1433; vol. 175, p. 65. 
§ (1) might of course be written in the perfectly general form 
dN, 
= N,, N: 
dt e ( 1) 2) 
But this would fail to bring out the fact that the first two terms must vanish with N; 
and the third with N; and also with Ng. 
