Crossover from dynamical percolation class to directed percolation class on a two dimensional lattice

Abstract

We study the crossover phenomena from the dynamical percolation class (DyP) to the directed percolation class (DP) in the model of diseases spreading, Susceptible-Infected-Refractory-Susceptible (SIRS) on a two-dimensional lattice. In this model, agents of three species S, I, and R on a lattice react as follows: $S+I\rightarrow I+I$ with probability $λ$, $I\rightarrow R$ after infection time $τ_I$ and $R\rightarrow I$ after recovery time $τ_R$. Depending on the value of the parameter $τ_R$, the SIRS model can be reduced to the following two well-known special cases. On the one hand, when $τ_R \rightarrow 0$, the SIRS model reduces to the SIS model. On the other hand, when $τ_R \rightarrow \infty$ the model reduces to SIR model. It is known that, whereas the SIS model belongs to the DP universality class, the SIR model belongs to the DyP universality class. We can deduce from the model dynamics that, SIRS will behave as an SIS model for any finite values of $τ_R$. SIRS will behave as SIR only when $τ_R=\infty$. Using Monte Carlo simulations we show that as far as the $τ_R$ is finite the SIRS belongs to the DP university class. We also study the phase diagram and analyze the scaling behavior of this model along the critical line. By numerical simulation and analytical argument, we find that the crossover from DyP to DP is described by the crossover exponent $1/ϕ=0.67(2)$.