A single infected individual enters a community of n susceptible individuals. Let x be the number of newly infected individuals at time t. The common epidemic model assumes that the disease spreads at a rate proportional to the product of the total number infected and the number not yet infected. So, dx/dt = k(x + 1)(n - x) and you obtain ∫ 1 / (x+1)(n-x) dx = ∫ k dt. Solve for x as a function of t.