74 lotka: contribution to epidemiology 



pendent of the "duration" (age dated from inception) of the case, 

 and that Ff^ is constant, namely 



F:,s = Ft.o (2) 



over the range defined by 



qi < s < q-2 (3) 



while 



Ft,s = o (4) 



outside of this range. 



A solution can still be obtained, as indicated below, if we 

 abandon these restrictions,- retaining only the following as- 

 sumptions : 



1. Immigration and emigration are negligible. 



2. The affected population is always a small fraction of the 

 total and, the total population being given as constant, the 

 unaffected population also may therefore be considered as 

 practically constant.^ 



3. The coefficient of infectivity c, though a function of the 

 duration s, is independent of t. This is equivalent to saying that 

 the infectivity of the disease varies in each affected person 

 according to the "age" (duration) of the case, but is the same 

 function of this age from case to case, at all times. 



Equation (i) is then of the form 



c (s) Ft,s ds. (5) 



^ We shall, however, retain the condition expressed in Equation (4) . This is 

 virtually no restriction at all since qi and ^2 in (3) may be given any values from 

 to 00 , and in practice the duration of the disease is always limited to finite time. 

 At most the condition (4) might be construed as excluding infection by nonaffected 

 carriers. 



2 In the notation of Sir Ronald Ross's paper we have, in fact, in this case 



A P-Z 



— = = I since Z, the affected population, is small as compared with P, the total 



population. The coefficient K which occurs in (5) and subsequent equations is 

 thus practically unity. 



The case is here analogous to that of certain chemical reactions in which one of the 

 reacting substances is present in large excess, as for instance water, when used as a 

 solvent for the reacting substances in dilute solution, and when itself consumed or 

 formed in the process. The concentration of the water in such cases may be re- 

 garded as practically constant. 



