  The spread of an epidemic process is considered in the context of a spatial
SIR stochastic model that includes a parameter $0\le p\le 1$ that assigns
weights $p$ and $1- p$ to global and local infective contacts respectively. The
model was previously studied by other authors in different contexts. In this
work we characterized the behavior of the system around the threshold for
epidemic spreading. We first used a deterministic approximation of the
stochastic model and checked the existence of a threshold value of $p$ for
exponential epidemic spread. An analytical expression, which defines a function
of the quotient $\alpha$ between the transmission and recovery rates, is
obtained to approximate this threshold. We then performed different analyses
based on intensive stochastic simulations and found that this expression is
also a good estimate for a similar threshold value of $p$ obtained in the
stochastic model. The dynamics of the average number of infected individuals
and the average size of outbreaks show a behavior across the threshold that is
well described by the deterministic approximation. The distributions of the
outbreak sizes at the threshold present common features for all the cases
considered corresponding to different values of $\alpha>1$. These features are
otherwise already known to hold for the standard stochastic SIR model at its
threshold, $\alpha=1$: (i) the probability of having an outbreak of size $n$
goes asymptotically as $n^{-3/2}$ for an infinite system, (ii) the maximal size
of an outbreak scales as $N^{2/3}$ for a finite system of size $N$.
