Topological effects on the absorbing phase transition of the contact process in fractal media

In a recent paper [S.B. Lee, Physica A 387, 1567 (2008)] the epidemic spread of the contact process (CP) in deterministic fractals, already studied by I. Jensen [J. Phys. A 24, L1111 (1991)], has been investigated by means of computer simulations. In these previous studies, epidemics are started from randomly selected sites of the fractal, and the obtained results are averaged all together. Motivated by these early works, here we also studied the epidemic behavior of the CP in the same fractals, namely, a Sierpinski carpet and the checkerboard fractals but averaging epidemics started from the same site. These fractal media have spatial discrete scale invariance symmetry, and consequently the dynamic evolution of some physical observables may become coupled to the topology, leading to the logarithmic-oscillatory modulation of the corresponding power laws. In fact, by means of extensive simulations we shown that the topology of the substrata causes the oscillatory behavior of the epidemic observables. However, in order to observe these oscillations, which have not been reported in earlier works, the interference effect arising during the averaging of epidemics started from nonequivalent sites should be eliminated. Finally, by analyzing our data and those available on the literature for the dependence of the exponents eta and delta on the dimensionality of substrata, we conjectured that for integer dimensions (2 <= d <= d(c) = 4) the following exact relationship may hold: delta + eta =d+2/6.