An epidemic spreads through a population at a rate proportional to the product of the number of people already infected and the number of people susceptible, but not yet infected. Therefore, if $\displaystyle {S}$ denotes the total population of susceptible people and $\displaystyle {I}={I}{\left({t}\right)}$ denotes the number of infected people at time $\displaystyle {t}$, then

$\displaystyle {I}'={r}{I}{\left({S}-{I}\right)}$,

where $\displaystyle {r}$ is a positive constant.

A) Assuming that $\displaystyle {I}{\left({0}\right)}={I}_{{0}}$, find $\displaystyle {I}{\left({t}\right)}$ for $\displaystyle {t}>{0}$


$\displaystyle {I}{\left({t}\right)}=$ Preview Question 6 Part 1 of 2  

B) As the value of $\displaystyle {t}$ increases, can you see what $\displaystyle {I}{\left({t}\right)}$ become? In other words, calculate the following limit
$\displaystyle \lim_{{{t}\to\infty}}{I}{\left({t}\right)}=$ Preview Question 6 Part 2 of 2