Spreading with immunization in high dimensions

We investigate a model of epidemic spreading with partial immunization which is controlled by two probabilities, namely, for first infections, $p_0$, and reinfections, $p$. When the two probabilities are equal, the model reduces to directed percolation, while for perfect immunization one obtains the general epidemic process belonging to the universality class of dynamical percolation. We focus on the critical behavior in the vicinity of the directed percolation point, especially in high dimensions $d>2$. It is argued that the clusters of immune sites are compact for $d\leq 4$. This observation implies that a recently introduced scaling argument, suggesting a stretched exponential decay of the survival probability for $p=p_c$, $p_0\ll p_c$ in one spatial dimension, where $p_c$ denotes the critical threshold for directed percolation, should apply in any dimension $d \leq 3$ and maybe for $d=4$ as well. Moreover, we show that the phase transition line, connecting the critical points of directed percolation and of dynamical percolation, terminates in the critical point of directed percolation with vanishing slope for $d<4$ and with finite slope for $d\geq 4$. Furthermore, an exponent is identified for the temporal correlation length for the case of $p=p_c$ and $p_0=p_c-\epsilon$, $\epsilon\ll 1$, which is different from the exponent $\nu_\parallel$ of directed percolation. We also improve numerical estimates of several critical parameters and exponents, especially for dynamical percolation in $d=4,5$.