{"title":"The crossover from a dynamical percolation class to a directed percolation class on a two dimensional lattice","authors":"M Ali Saif","doi":"10.1088/1742-5468/ad6975","DOIUrl":null,"url":null,"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 <italic toggle=\"yes\">S</italic>, <italic toggle=\"yes\">I</italic>, and <italic toggle=\"yes\">R</italic> on a lattice react as follows: <inline-formula>\n<tex-math><?CDATA $S+I\\rightarrow I+I$?></tex-math><mml:math 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><inline-graphic xlink:href=\"jstatad6975ieqn1.gif\"></inline-graphic></inline-formula> with probability <italic toggle=\"yes\">λ</italic>, <inline-formula>\n<tex-math><?CDATA $I\\rightarrow R$?></tex-math><mml:math overflow=\"scroll\"><mml:mrow><mml:mi>I</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mi>R</mml:mi></mml:mrow></mml:math><inline-graphic xlink:href=\"jstatad6975ieqn2.gif\"></inline-graphic></inline-formula> after infection time <italic toggle=\"yes\">τ</italic><sub><italic toggle=\"yes\">I</italic></sub> and <inline-formula>\n<tex-math><?CDATA $R\\rightarrow I$?></tex-math><mml:math overflow=\"scroll\"><mml:mrow><mml:mi>R</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mi>I</mml:mi></mml:mrow></mml:math><inline-graphic xlink:href=\"jstatad6975ieqn3.gif\"></inline-graphic></inline-formula> after recovery time <italic toggle=\"yes\">τ</italic><sub><italic toggle=\"yes\">R</italic></sub>. Depending on the value of the parameter <italic toggle=\"yes\">τ</italic><sub><italic toggle=\"yes\">R</italic></sub>, the SIRS model can be reduced to the following two well-known special cases. On the one hand, when <inline-formula>\n<tex-math><?CDATA $\\tau_R \\rightarrow 0$?></tex-math><mml:math 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><inline-graphic xlink:href=\"jstatad6975ieqn4.gif\"></inline-graphic></inline-formula>, the SIRS model reduces to the SIS model. On the other hand, when <inline-formula>\n<tex-math><?CDATA $\\tau_R \\rightarrow \\infty$?></tex-math><mml:math 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><inline-graphic xlink:href=\"jstatad6975ieqn5.gif\"></inline-graphic></inline-formula> 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 <italic toggle=\"yes\">τ</italic><sub><italic toggle=\"yes\">R</italic></sub>. The model will behave as SIR only when <inline-formula>\n<tex-math><?CDATA $\\tau_R = \\infty$?></tex-math><mml:math 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><inline-graphic xlink:href=\"jstatad6975ieqn6.gif\"></inline-graphic></inline-formula>. Using Monte Carlo simulations, we show that as long as the <italic toggle=\"yes\">τ</italic><sub><italic toggle=\"yes\">R</italic></sub> 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 <inline-formula>\n<tex-math><?CDATA $1/\\phi = 0.67(2)$?></tex-math><mml:math 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><inline-graphic xlink:href=\"jstatad6975ieqn7.gif\"></inline-graphic></inline-formula>.","PeriodicalId":17207,"journal":{"name":"Journal of Statistical Mechanics: Theory and Experiment","volume":"45 1","pages":""},"PeriodicalIF":2.2000,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Statistical Mechanics: Theory and Experiment","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1088/1742-5468/ad6975","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MECHANICS","Score":null,"Total":0}
引用次数: 0
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: S+I→I+I with probability λ, I→R after infection time τI and R→I after recovery time τR. Depending on the value of the parameter τR, 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 τR. The model will behave as SIR only when τR=∞. Using Monte Carlo simulations, we show that as long as the τR 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).
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