We investigate the effects of risk perception in a simple model of epidemic spreading. We assume that the perception of the risk of being infected depends on the fraction of neighbors that are ill. The effect of this factor is to decrease the infectivity, that therefore becomes a dynamical component of the model. We study the problem in the mean-field approximation and by numerical simulations for regular, random, and scale-free networks. We show that for homogeneous and random networks, there is always a value of perception that stops the epidemics. In the “worst-case” scenario of a scale-free network with diverging input connectivity, a linear perception cannot stop the epidemics; however, we show that a nonlinear increase of the perception risk may lead to the extinction of the disease. This transition is discontinuous, and is not predicted by the mean-field analysis.

In this paper we study the interplay between epidemic spreading and risk perception on multiplex networks. The basic idea is that the effective infection probability is affected by the perception of the risk of being infected, which we assume to be related to the fraction of infected neighbors, as introduced by Bagnoli et al. [ Phys. Rev. E 76, 061904 (2007)]. We rederive previous results using a self-organized method that automatically gives the percolation threshold in just one simulation. We then extend the model to multiplex networks considering that people get infected by physical contacts in real life but often gather information from an information network, which may be quite different from the physical ones. The similarity between the physical and the information networks determines the possibility of stopping the infection for a sufficiently high precaution level: if the networks are too different, there is no means of avoiding the epidemics.

We study the synchronization of totalistic one-dimensional cellular automata (CA). The CA with a nonzero synchronization threshold exhibit complex nonperiodic space time patterns and vice versa. This synchronization transition is related to directed percolation. We also study the maximum Lyapunov exponent for CA, defined analogous to continuous dynamical systems as the exponential rate of expansion of the linear map induced by the evolution rule of CA, constructed with the aid of the Boolean derivatives. The synchronization threshold is strongly correlated to the maximum Lyapunov exponent and we propose approximate relations between these quantities. The value of this threshold can be used to parametrize the space time complexity of CA.

We present a probabilistic cellular automaton with two absorbing states, which can be considered a natural extension of the Domany-Kinzel model. Despite its simplicity, it shows a very rich phase diagram, with two second-order and one first-order transition lines that meet at a bicritical point. We study the phase transitions and the critical behavior of the model using mean field approximations, direct numerical simulations and field theory. The second-order critical curves and the kink critical dynamics are found to be in the directed percolation and parity conservation universality classes, respectively. The first–order phase transition is put in evidence by examining the hysteresis cycle. We also study the “chaotic” phase, in which two replicas evolving with the same noise diverge, using mean field and numerical techniques. Finally, we show how the shape of the potential of the field-theoretic formulation of the problem can be obtained by direct numerical simulations.