The crossover from a dynamical percolation class to a 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 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: <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mo>+</mml:mo> <mml:mi>I</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mi>I</mml:mi> <mml:mo>+</mml:mo> <mml:mi>I</mml:mi> </mml:mrow> </mml:math> with probability λ , <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>I</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> </mml:math> after infection time τ I and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>R</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mi>I</mml:mi> </mml:mrow> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>τ</mml:mi> <mml:mi>R</mml:mi> </mml:msub> <mml:mo stretchy="false">→</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , the SIRS model reduces to the SIS model. On the other hand, when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>τ</mml:mi> <mml:mi>R</mml:mi> </mml:msub> <mml:mo stretchy="false">→</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>τ</mml:mi> <mml:mi>R</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> </mml:math> . 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>ϕ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0.67</mml:mn> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> .