  This paper is dedicated to the memory of Dietrich Stauffer, who was a pioneer
in percolation theory and applications of it to problems of society, such as
epidemiology. An epidemic is a percolation process gone out of control, that
is, going beyond the critical transition threshold $p_c$. Here we discuss how
the threshold is related to the basic infectivity of neighbors $R_0$, for trees
(Bethe lattice), trees with triangular cliques, and in non-planar lattice
percolation with extended-range connectivity. It is shown how having a smaller
range of contacts increases the critical value of $R_0$ above the value
$R_{0,c}=1$ appropriate for a tree, an infinite-range system or a large
completely connected graph.
