  In this work, some new exact and approximate analytical solutions are
obtained for the SIR epidemic model, which is formulated in terms of
dimensionless variables and parameters. The susceptibles population (S) is in
this way explicitly related to the infectives population (I) using the Lambert
W function (both the principal and the secondary branches). A simple and
accurate relation for the fraction of the population that does not catch the
disease is also obtained. The explicit time dependences of the susceptibles,
infectives and removed populations, as well as that of the epidemic curve are
also modelled with good accuracy for any value of R0 (basic multiplication
number) using simple functions that are modified solutions of the R0 ->
infinity limiting case (logistic curve). It is also shown that for I0 << S0 the
effect of a change in the ratio I0/S0 on the population evolution curves
amounts to a time shift, their shape and relative position being unaffected.
