Chapter 8.5, Problem 49E

Epidemic Model 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

${\displaystyle \int \frac{1}{(x+1)(n-x)}}dx={\displaystyle \int kdt}$

Solve for x as a function of t.