VIRUSES 48 



Tlrua particles which are spread upon the surface of a leaf, but rather the nun- 

 ber of particles which come into contact with cells on the leaf surface injured 

 sufficiently but not too much. Perhaps as few as one in a million of the virus 

 particles spread upon the surface of the leaf actually is so favored. One might 

 be tempted to say that it takes a million particles on the surface of a leai" to 

 cause infection. On the contrary, only one of that million is involved in the 

 infection process. At ajiy rate it is possible to develop a theory of virus in- 

 fection based upon the sole assumption that infection depends upon the chance 

 occurrence of a single virus particle in a favorable region within the host. Siich 

 a theory does give a quantitative relationship between level of response and dose 

 which agrees in a general way, at least, with common experience. 



It can be shown by an application of the laws of chance that if the number 

 of virus particles per cc in a liquid is nx, the probability that one will not 

 find any virus particle in a small element of a volume of v cc is equal to 

 e-vnx. If the probability of not finding any particles in that volume is given 

 by this expression, then the probability of finding at least one but maybe more 

 than one is l _ e-vnx. If this probability is the same as the probability of 

 getting an infection, one can write the equation 



1=1- 9-^^ (1) 



where y is the niimber of infections observed and H is the maximum number possi- 

 ble. This equation was derived entirely from the laws of chance. Perhaps it 

 can best be illustrated by considering its application to a game of chance. In 

 the game familiarly known as craps, it is an expensive matter for the player to 

 roll a pair of ones with the dice. Gamblers usually call this combination "snake 

 eyes". Since a dice has six faces, there is one chance in six that a one will 

 turn up on one dice and one chance in 36 that it will turn up in both dice. A 

 gambler might have to roll the dice many times in sequence to try to make his 

 point. Suppose he has to roll the dice five times in sequence. V/hat is the pro- 

 bability that he will not obtain "snake-eyes" at least once? Well, if he rolls 

 the dice one time, the probability of not getting "snake-eyes" will be 35* If 

 he rolls them five times, the probability will be 35 raised to the ^5 fifth 

 power. This latter can easily be shown to be very 3^ nearly the same as g-5 x 1 . 

 Thus the probability of not obtaining "snake-eyes" in five rolls is 3^^ 



Q-5 X 1 . The probability of obtaining at least one pair of ones is 



3^ 



3^ 



This expression, potentially useful to crap shooters, is exactly analogous to 

 equation (1) derived for virus solutions. 



Another point of view can also lead to a similar end relationship between 

 virus concentration and host response. This is the second alternative theory. 

 It is argued by some that the vairiation in response to different levels of dos- 

 age is due primarily to variation of the susceptibility of the host to the virus. 

 This view, when investigated mathematically, leads to a prediction of the re- 

 sponse-dosage relationship shown by the broken curve in Figure 37» It is evident 

 that it will require reasonably precise data to differentiate between the two 

 theoretical possibilities. The fundamental reasoning underlying this second 

 approach is somewhat as follows. It is postulated that the susceptibility of 

 the host to the virus varies from individual to individual in such a way that 

 the logarithm of susceptibility is distributed according to the normal distri- 

 bution law, illustrated by the normal curve shown in Figure 38. 



