Spreading with immunization in high dimensions

Abstract

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 ≤ 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 ≪ p c in one spatial dimension, where p c 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 = p c and p 0 = p c − ǫ, ǫ ≪ 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.

Probing into the effectiveness of self-isolation policies in epidemic control

“…Despite the simplicity of these models, they have been successfully applied to a variety of cases [14,15,16,17,18,19,20,21] in the assessment of the propagation of an epidemic disease. In the form of differential equations, their dynamics is well-known in the literature of the theme where they are usually called general solutions.…”

An FPGA design for the stochastic Greenberg-Hastings cellular automata

“…The two models are thus expected to display different critical behavior. For instance the value of the critical exponent δ for DyP is much lower than its value in DP (0.092 versus 0.451) [9]. Hence, when M increases, the determination of p TC is complicated by the fact that, at some point, the universality class is expected to switch from DP to DyP.…”

“…Thus in effect, a cell can only be activated once in a finite time simulation, and once activated it remains insensible to its neighborhood. In this case, the GHCA model reduces to the so-called SIR model with perfect immunization (also called general epidemic process) [9], for which the critical transmission threshold is exactly 0.5 on a 2D square lattice [10]. Using this argument, one would expect an asymptotic value of 0.5 for p TC in the GHCA with 4-connected topology and not 0.51.…”

“…For instance, when the number of states (see below) is not too large, the critical behavior is known to be in the universality class of directed percolation [7]. At the limit when this number tends to infinity, the GHCA reduces to a SIR epidemic model with perfect immunization [9], which is in a different universality class (namely dynamic percolation), e.g. presents a different critical behavior [10].…”

A scaling analysis in the SIRI epidemiological model

“…For a recent investigation of reinfection models in the biological context, see for example [9]. In the physics literature, models with partial immunization have also found wide interest [1,5] due to their critical behaviour connecting directed percolation and dynamic percolation.…”