The spread of an epidemic is considered in the context of an SIR spatial
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. For
diseases with low values of the basic reproductive ratio, $R_0$, we found that
the value of $p$ has a decisive influence on the existence or not of a major
outbreak. We first used a deterministic approximation of the stochastic model
developed in previous work and checked the existence of a threshold value of
$p$ for exponential epidemic spread. An analytical expression, which defines a
function of the quotient between the transmission and recovery rates, is
proposed 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 $p$ in the stochastic
model. In this way, for $p$ values lower than the proposed one, the probability
of a major outbreak becomes negligible even when $R_0$ remains above 1. The
obtained results turn out to be relevant for infectious diseases with low $R_0$
but high mortality rates such as Ebola or H1N1 influenza. This study highlights
the importance of control measures that minimize the possibility of global
contacts, warning that a small reduction of them could produce a drastic
reduction in the probability of huge outbreaks.