30 THE STRUGGLE FOR EXISTENCE 



each human being has been bitten. We have therefore bp = b l p l , 

 and finally 



b = tL (4) 



V 



Inserting the expression (4) into the formula (3) we obtain : 



b 1 p l fz , . A b x fz , , ., /rN 



—^f- (p 1 - z l ) = -i- (p 1 - z l ) (5) 



pp 1 V 



The expression (5) fills the second place in the lower line in Ross's 



differential equations of malaria (1). It gives the number of new 



infections of mosquitoes per unit of time. We can now put down the 



rate of increase of infected individuals among the human population 



dz 

 as -j- f and the rate of increase in the number of infected mosquitoes 



Lit 



dz 1 

 as — . The number of recoveries per unit of time among human in- 



(J/L 



dividuals will be rz, as z represents the number of people infected and 

 r the rate of recovery, i.e., the fraction of the infected population 

 recovering per unit of time. The number of infected mosquitoes 

 dying per unit of time can be put down as ik/V, since z 1 denotes the 

 number of mosquitoes infected, and M l the death rate in mosquitoes 

 per head per unit of time. We can now express the equation of Ross 

 in mathematical symbols instead of words: 



dz ,, n .p — z 



— = b l j l z l rz 



dt p 



dz 1 , , , p 1 — z l , , 



— = b l fz M 



dt J p 



\ 



1 2 1 



(6) 



These simultaneous differential equations of the struggle for exist- 

 ence express in a very simple and clear form the continuous depend- 

 ence of the infection of people on the infectivity of the mosquitoes and 

 vice versa. The increase in the number of sick persons is connected 

 with the number of bites made by infective mosquitoes on healthy 

 persons per unit of time, and at the same moment the increase in the 

 number of infected mosquitoes depends upon the bites made by 

 healthy mosquitoes on sick people. The equations (6) enable us to 



