Chapter III 



THE STRUGGLE FOR EXISTENCE FROM THE POINT OF 

 VIEW OF THE MATHEMATICIANS 



(1) In this chapter we shall make the acquaintance of the astonish- 

 ing theories of the struggle for existence developed by mathematicians 

 at a time when biologists were still far from any investigation of these 

 phenomena and had but just begun to make observations in the field. 



The first attempt at a quantitative study of the struggle for exist- 

 ence was made by Sir Ronald Ross ('08, '11). He undertook a theo- 

 retical investigation of the propagation of malaria, and came to con- 

 clusions which are of great interest for quantitative epidemiology 

 and at the same time constitute an important advance in the under- 

 standing of the struggle for existence in general. Let us examine 

 the fundamental idea of Ross. Our object will be to give an analysis 

 of the propagation of malaria in a certain locality under somewhat 

 simplified conditions. We assume that both emigration and immi- 

 gration are negligible, and that in the time interval we are studying 

 there is no increase of population or in other words the birth rate is 

 compensated by the death rate. In such a locality a healthy person 

 can be infected with malaria, according to Ross, if all the following 

 conditions are realized: (1) That a person whose blood contains a 

 sufficient number of gametocytes (sexual forms) is living in or near 

 the locality. (2) That an Anopheline capable of carrying the para- 

 sites sucks enough of that person's blood. (3) That this Anopheline 

 lives for a week or more afterwards under suitable conditions — long 

 enough to allow the parasites to mature within it, and (4) that it next 

 succeeds in biting another person who is not immune to the disease 

 or is not protected by quinine. The propagation of malaria in such 

 a locality is determined in its general features by two continuous and 

 simultaneous processes: on the one hand the number of new infec- 

 tions among people depends on the number and infectivity of the 

 mosquitoes, and at the same time the infectivity of the mosquitoes is 

 connected with the number of people in the given locality and the 



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