The model for the spread of drug use in Exercise 5.8.26 of the textbook can be understood as follows. First consider a group of P people who have just started using drugs. If there is a constant per capita rate λ at which individuals are lost from this group as a result of death or the discontinuation of drug use, then from Section 3.6 of the textbook P will decay exponentially as described by the equation
Thus, from Section 3.6 of the textbook, the size of the group at time t will be
where P(0) is the size of the group at time t = 0. If we divide both sides of this equation by P(0), and define f(t)=P(t)/P(0) to be fraction of the group still remaining at time t, then we have
We can also interpret f(t) is the probability that any given individual that started in the group is still a drug user at time t. We will do so below.
Now suppose that each drug user occasionally encourages new users. We define S(t) to be the number of new users recruited by an individual, as a function of the time t since that individual started using drugs. As an individual spends more time as a drug user they often become increasingly enthusiastic and so S(t) might increase with time t. In fact, the researchers of this study used a biochemical model to suggest that the following function should apply:
where N is the population size, and c and k are positive constants related to the physiological response to the drug.
With the above considerations, the number of new users that an individual can expect to recruit over the entire time that they use drugs is just the number of new users recruited at time t (which is given by S(t)) multiplied by the probability that an individual is still a user at time t (which is given by f(t)) and integrated from their start of drug use at t = 0 over the entire duration of time. Calling this quantity γ we have
This is the integral given in Exercise 5.8.26.