If you have N cases, growing at rate r i.e. each person affects r others in one day, then the number of new cases is:
\[
\Delta N = r \times N
\]
Soon the infected run out of uninfected people to infect. The probability of any given person being infected is just N/P, so (1-N/P) is the probability of encountering an uninfected person.
Then the number of new cases is:
\[
\Delta N = r \times N \times (1-N/P)
\]
where P is the total population.
