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Statistical properties and dynamical disease propagation have been studied numerically using a percolation model in a one dimensional small world network. The parameters chosen correspond to a realistic network of school age children. It has been found that percolation threshold decreases as a power law as the shortcut fluctuations increase. It has also been found that the number of infected sites grows exponentially with time and its rate depends logarithmically on the density of susceptibles. This behavior provides an interesting way to estimate the serology for a given population from the measurement of the disease growing rate during an epidemic phase. The case in which the infection probability of nearest neighbors is different from that of short cuts has also been examined. A double diffusion behavior with a slower diffusion between the characteristic times has been found.

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Propagation properties in a two-dimensional network with local small-world effects corresponding to the influence zone of each active site are studied. Two different weights based on characteristic times are introduced. The propagation of the front (here a forest fire front) in such a network exhibits two thresholds: the first one is geometrical corresponding to the percolation threshold and the second one is dynamical and results from the weighting procedure. The geometrical threshold is found to be a second-order phase transition as for regular networks. Further results are provided on the fractal dimension of the area covered during the propagation below the percolation threshold.

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