lotka: contribution to epidemiology 75 



It will be convenient to adopt a somewhat different notation 

 to designate F,^ the "survivors" at time t, "of duration 5," 

 out of the original batch F< ^ of new cases. 



If we denote by F (t) the new cases per unit of time at time 

 t, and consequently by F (^ — s) the new cases per unit of time 

 at time (i — s), then let F {t — s) p (s) denote the survivors 

 at time t of the new cases per unit of time which originated at 

 time {t — s). Evidently the relation between this notation 

 and that in the original paper referred to above is 



Ft,s = F(t-s) p(s). (6) 



In this notation (5) becomes 



F (t) = K P F {t- s)p is) c is) ds (7) 



which, in view of (4), may indifferently be written 



F (0 = K J'^ F{t- s) p (5) c (s) ds. (8) 



The integral equation (8) is of the type dealt with by Hertz.'* 

 To solve it we must know the value of F {t) from / =0 to / =(72, 

 or what is the same thing, the number of cases at every "age" 

 (duration) between o and q-2 at time / = o. We may leave 

 out of account the number of cases of "age" (duration) above 

 g2 at time q-2, since they have ceased to be infective. 



We have then by Hertz 



trf Kj ^sp{s) c (5) e -%^ ds 



Where Ui, u^ . ■ ■ are the roots of the equation for m 



1= K J'^ pis) c (5) e -"' ds. (10) 



* Math. Ann. 65: 86. At the same time, it will also no longer be warranted to 

 make the assumption (implied in a constant population) that the death rate is not 

 appreciably affected by the progress of the disease (unless indeed the disease is 

 never fatal) . Moreover, if a large proportion of the population has become affected, 

 the unaffected portion of the population will in practice be inherently difTerent in 

 character from the total population. It will contain a larger proportion of persons 

 of low "susceptibility." So long as the number of persons affected forms but a 

 small fraction of the total population, we may neglect this "selective" effect, as has 

 been done above. But with a large "affected" proportion of the total population 

 this neglect is no longer justified. 



