Models & Qualitative Behavior · Superchallenge problem

In the SIR epidemic model $S' = -\beta SI$, $I' = \beta SI - \gamma I$ (with $S, I$ fractions of the population), an outbreak starts with essentially everyone susceptible: $S(0) = 1$, $I(0) \approx 0$. For what value of the basic reproduction number $R_0 = \beta/\gamma$ does exactly half of the population escape infection, $S(\infty) = \tfrac12$? Give an exact answer.