76 lyOTKA: CONTRIBUTION TO EPIDEMIOLOGY 



From the nature of the case q^, p (s), and c (s) are never nega- 

 tive. It follows that (id) has one and only one real root U, 

 which is I o according as 



K J'^ p(s)c(s)ds^i . (ii) 



It can easily be shown that any other root must have its 

 real part less than U. 



It follows that for large values of i the term with the real root 

 U outweighs all other terms in (9), and F {t) approaches the value 



F (0 = F (o) e""' . (12) 



Furthermore, 



Fit- s) = Fio)e ""''-'' (13) 



and 



Ft,s = F (t ~ s)p{s) = F{o)e "" ^'"^^ p (s) (14) 

 The function F,^ is thus determined. 



For Z (t), the total number of cases existing at time t, we have 

 by Equation 86 of Sir Ronald Ross 



Z (t) = JJ Fit- s)p is) ds (15) 



= F it) r e -""' p is) ds (16) 



= Ci^(0° iC = const.) (17) 



since we are assuming that p (5) is independent of t; and hence 

 by (12) 



Z it) = C F io) e^' (18) 



= Zio)e ^' (19) 



that is to say, the affected population increases in geometric 

 progression with the time, at the same proportional rate as do 

 the new cases per unit of time. 



In practice, if this state of affairs persists, a time must be 

 reached when, with a constant total population, the affected 

 population can no longer be regarded as a small fraction of the 

 total, and when, therefore, the solution here given no longer 

 applies.^ 



One point deserves special notice. It will be observed that 

 since U is determined by Equation, (10), it is wholly independent 



* See footnote 4. 



