  We investigate a non-Markovian analogue of the Harris contact process in a
finite connected graph G=(V,E): an individual is attached to each site x in V,
and it can be infected or healthy; the infection propagates to healthy
neighbors just as in the usual contact process, according to independent
exponential times with a fixed rate lambda>0; however, the recovery times for
an individual are given by the points of a renewal process attached to its
timeline, whose waiting times have distribution mu such that mu(t,infty) =
t^{-alpha}L(t), where 1/2 < alpha < 1 and L is a slowly varying function; the
renewal processes are assumed to be independent for different sites. We show
that, starting with a single infected individual, if |V| < 2 + (2 alpha
-1)/[(1-alpha)(2-alpha)], then the infection does not survive for any lambda;
and if |V| > 1/(1-alpha), then, for every lambda, the infection has positive
probability to survive
