Models & Qualitative Behavior · Superchallenge problem

In the SIR model $S' = -\beta SI$, $I' = \beta SI - \gamma I$ with $R_0 = \beta/\gamma = e$, an outbreak starts from $S(0) = 1$, $I(0) \approx 0$ (fractions of the population). What is the largest fraction of the population that is infected at any one time? Give an exact answer.