28 



THE STRUGGLE FOR EXISTENCE 



frequency of the sickness among them. Ross has expressed in mathe- 

 matical terms this uninterrupted and simultaneous dependence of 

 the infection of the first component on the second, and that of the 

 second on the first, with the aid of what is called in mathematics 

 simultaneous differential equations. These equations are very simple 

 and we shall examine them at once. In a quite general form they 

 can be represented as follows : 



Rate of increase of 

 affected individuals 

 among the human 

 population 



Rate of increase of 

 infected individuals 

 among the mosquito 

 population 



> =1 



> =\ 



New infections 

 per unit of time. 

 Depends on in- 

 fected mosqui- 

 toes 



New infections 

 per unit of time. 

 Depends on the 

 infected humans 



[ Recoveries] * 

 • — \ per unit of 

 time 



)- 



>■ 



(1) 



Deaths of 

 infected 

 mosquitoes 

 per unit of 

 time 



t 



z l = 



(2) Translating these relations into mathematical language we 

 shall obtain the simultaneous differential equations of Ross. Let us 

 introduce the following notation (that of Lotka (51)) : 



total number of human individuals in a given locality. 



total number of mosquitoes in a given locality. 



total number of people infected with malaria. 



total number of mosquitoes containing malaria parasites. 



total number of infective malarians (number of persons 

 with gametocytes in the blood ; a certain fraction of the 

 total number of malarians). 



total number of infective mosquitoes (with matured para- 

 sites ; a certain fraction of the total number of mosquitoes 

 containing parasites). 



recovery rate, i.e., fraction of infected population that re- 

 verts to noninfected (healthy) state per unit of time. 



r = 



* The human death rate is not taken into consideration. To simplify the sit- 

 uation we assume that it is negligible and counterbalanced by the birth rate, 

 t We assume that the recovery of the infected mosquitoes does not occur. 



