The crossover from a dynamical percolation class to a directed percolation class on a two dimensional lattice

We study the crossover phenomena from the dynamical percolation class (DyP) to the directed percolation class (DP) in the model of disease 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→I+I with probability λ, I→R after infection time τ_I and R→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 τR→0, the SIRS model reduces to the SIS model. On the other hand, when τR→∞ the model reduces to the 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 the SIS model for any finite values of τ_R. The model will behave as SIR only when τR=∞. Using Monte Carlo simulations, we show that as long as the τ_R is finite the SIRS belong 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 simulations and analytical arguments, we find that the crossover from DyP to DP is described by the crossover exponent 1/ϕ=0.67(2).