The spread of an epidemic is considered in the context of a 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,  the value of p turns out to have a decisive influence on the existence or not of a major outbreak. A deterministic approximation of the stochastic model , developed by the authors in previous work , is considered. The existence of a threshold value of p for exponential epidemic spread is checked in this deterministic context . An analytical expression, that defines a function of the quotient between the transmission and recovery rates, is obtained to approximate this threshold. Different analyses based on intensive stochastic simulations show 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.
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.
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