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The spread of an epidemic process is considered in the context of a spatialSIR stochastic model that includes a parameter $0\le p\le 1$ that assignsweights $p$ and $1- p$ to global and local infective contacts respectively. Themodel was previously studied by other authors in different contexts. In thiswork we characterized the behavior of the system around the threshold forepidemic spreading. We first used a deterministic approximation of thestochastic model and checked the existence of a threshold value of $p$ forexponential epidemic spread. An analytical expression, which defines a functionof the quotient $\alpha$ between the transmission and recovery rates, isobtained to approximate this threshold. We then performed different analysesbased on intensive stochastic simulations and found that this expression isalso a good estimate for a similar threshold value of $p$ obtained in thestochastic model. The dynamics of the average number of infected individualsand the average size of outbreaks show a behavior across the threshold that iswell described by the deterministic approximation. The distributions of theoutbreak sizes at the threshold present common features for all the casesconsidered corresponding to different values of $\alpha>1$. These features areotherwise already known to hold for the standard stochastic SIR model at itsthreshold, $\alpha=1$: (i) the probability of having an outbreak of size $n$goes asymptotically as $n^{-3/2}$ for an infinite system, (ii) the maximal sizeof an outbreak scales as $N^{2/3}$ for a finite system of size $N$.

Exploring the threshold of epidemic spreading for a stochastic SIR model with local and global contacts