  We present a numerical study of a reaction-diffusion model on a small-world
network. We characterize the model's average activity $F_T$ after $T$ time
steps and the transition from a collective (global) extinct state to an active
state in parameter space. We provide an explicit relation between the
parameters of our model at the frontier between these states. A collective
active state can be associated to a global epidemic spread, or to a persistent
neuronal activity. We found that $F_T$ does not depends on disorder in the
network if the transmission rate $r$ or the average coordination number $K$ are
large enough. The collective extinct-active transition can be induced by
changing two parameters associated to the network: $K$ and the disorder
parameter $p$ (which controls the variance of $K$). We can also induce the
transition by changing $r$, which controls the threshold size in the dynamics.
In order to operate at the transition the parameters of the model must satisfy
the relation $rK=a_p$, where $a_p$ as a function of $p/(1-p)$ is a stretched
exponential function. Our results are relevant for systems that operate {\it
at} the transition in order to increase its dynamic range and/or to operate
under optimal information-processing conditions. We discuss how glassy
behaviour appears within our model.
