  We extend a recent study of susceptible-infected-removed epidemic processes
with long range infection (referred to as I in the following) from
1-dimensional lattices to lattices in two dimensions. As in I we use hashing to
simulate very large lattices for which finite size effects can be neglected, in
spite of the assumed power law $p({\bf x})\sim |{\bf x}|^{-\sigma-2}$ for the
probability that a site can infect another site a distance vector ${\bf x}$
apart. As in I we present detailed results for the critical case, for the
supercritical case with $\sigma = 2$, and for the supercritical case with $0<
\sigma < 2$. For the latter we verify the stretched exponential growth of the
infected cluster with time predicted by M. Biskup. For $\sigma=2$ we find
generic power laws with $\sigma-$dependent exponents in the supercritical
phase, but no Kosterlitz-Thouless (KT) like critical point as in 1-d. Instead
of diverging exponentially with the distance from the critical point, the
correlation length increases with an inverse power, as in an ordinary critical
point. Finally we study the dependence of the critical exponents on $\sigma$ in
the regime $0<\sigma <2$, and compare with field theoretic predictions. In
particular we discuss in detail whether the critical behavior for $\sigma$
slightly less than 2 is in the short range universality class, as conjectured
recently by F. Linder {\it et al.}. As in I we also consider a modified version
of the model where only some of the contacts are long range, the others being
between nearest neighbors. If the number of the latter reaches the percolation
threshold, the critical behavior is changed but the supercritical behavior
stays qualitatively the same.
