An epidemic occurs if dI/dt is greater than 0. Solving for the steady state solution reveals that there is a critical population of susceptible people equal to GAMMA/BETA. If the starting population is less than this value, no epidemic occurs (no one gets infected). On the other hand, if the starting population is greater than this critical value, then an epidemic occurs (people get infected).
The initial S value of 999 is greater than the critical value in all cases except for 3 and 4. Therefore, we should expect that the population experiences an epidemic in cases 1, 2, 5, 6, and 7, while no one gets infected in cases 3 and 4. The results are consistent with these expectations.