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 .
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 .
A
process
R
an SIR spatial
a spatial SIR
R
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
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
D
developed in previous work
A
\alpha
R
proposed
obtained
A
obtained
R
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
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>
R
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
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