We investigate a model of epidemic spreading with partial immunization which
is controlled by two probabilities, namely, for first infections, p0, 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≤4. This observation implies that a
recently introduced scaling argument, suggesting a stretched exponential decay
of the survival probability for p=pc, p0≪pc in one spatial dimension,
where pc denotes the critical threshold for directed percolation, should
apply in any dimension d≤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≥4. Furthermore, an exponent is identified for the temporal correlation
length for the case of p=pc and p0=pc−ϵ, ϵ≪1, which
is different from the exponent ν∥ of directed percolation. We also
improve numerical estimates of several critical parameters and exponents,
especially for dynamical percolation in d=4,5.Comment: LaTeX, IOP-style, 18 pages, 9 eps figures, minor changes, additional
reference