Initially, the central node in the 100 by 100 array was infected. The size of the neighborhood was chosen to be , i.e. , it was composed of the 8 nearest neighbors. It is interesting to note that, despite the fact that is square in shape, the pattern of infection is roughly circular. The infection and cure rates are and , as usual. Black and white squares represent infected and uninfected individuals, respectively.
Simulations verify the quadratic growth with time of the total number of infected individuals.
Figure 12 compares the relative rates at which the equilibrium is attained in the two-dimensional lattice with that of the random and hierarchical graphs. In order to provide a fair comparison, the infection and cure rates were given their usual values of and , and the number of nodes in the simulation of each of the three models was chosen to be as close as possible to 10000. The number of nodes in the simulations are 10000 for the random and spatial models and for the hierarchical model. The parameters for the spatial graph are as given in Fig. As predicted, the growth in the infected population is quadratic in the spatial model, which is much slower than the exponential growth exhibited by the random graph model.