  We revisit the classic Susceptible-Infected-Recovered (SIR) epidemic model
and one of its nonlocal variations recently developed in \cite{Guan}. We
introduce several new approaches to derive exact analytical solutions in the
classical situation and analyze the corresponding effective approximations in
the nonlocal setting. An interesting new feature of the nonlocal models,
compared with the classic SIR model, is the appearance of multiple peak
solutions for the infected population. We provide several rigorous results on
the existence and non-existence of peak solutions with sharp asymptotics.
