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

IF 2.2 3区 物理与天体物理 Q2 MECHANICS Journal of Statistical Mechanics: Theory and Experiment Pub Date : 2024-08-30 DOI:10.1088/1742-5468/ad6975
M Ali Saif
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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":null,"pages":null},"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}
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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+II+I with probability λ, IR after infection time τI and RI 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 τR0, 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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二维网格上从动态渗流类到有向渗流类的交叉
我们研究了二维晶格上疾病传播模型 "易感-感染-难治-易感(SIRS)"中从动态渗流类(DyP)到有向渗流类(DP)的交叉现象。在该模型中,网格上 S、I 和 R 三种病原体的反应如下:S+I→I+I,概率为 λ;感染时间 τI 后,I→R;恢复时间 τR 后,R→I。根据参数 τR 的取值,SIRS 模型可以简化为以下两种众所周知的特殊情况。一方面,当 τR→0 时,SIRS 模型简化为 SIS 模型。另一方面,当 τR→∞ 时,该模型会简化为 SIR 模型。众所周知,SIS 模型属于 DP 普遍性类别,而 SIR 模型属于 DyP 普遍性类别。我们可以从模型动力学推导出,在任何有限的 τR 值下,SIRS 都将表现为 SIS 模型。只有当 τR=∞ 时,模型才会表现为 SIR。通过蒙特卡罗模拟,我们发现只要 τR 是有限的,SIRS 就属于 DP 大学类。我们还研究了相图,并分析了该模型沿临界线的缩放行为。通过数值模拟和分析论证,我们发现从 DyP 到 DP 的交叉可以用交叉指数 1/j=0.67(2) 来描述。
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来源期刊
CiteScore
4.50
自引率
12.50%
发文量
210
审稿时长
1.0 months
期刊介绍: JSTAT is targeted to a broad community interested in different aspects of statistical physics, which are roughly defined by the fields represented in the conferences called ''Statistical Physics''. Submissions from experimentalists working on all the topics which have some ''connection to statistical physics are also strongly encouraged. The journal covers different topics which correspond to the following keyword sections. 1. Quantum statistical physics, condensed matter, integrable systems Scientific Directors: Eduardo Fradkin and Giuseppe Mussardo 2. Classical statistical mechanics, equilibrium and non-equilibrium Scientific Directors: David Mukamel, Matteo Marsili and Giuseppe Mussardo 3. Disordered systems, classical and quantum Scientific Directors: Eduardo Fradkin and Riccardo Zecchina 4. Interdisciplinary statistical mechanics Scientific Directors: Matteo Marsili and Riccardo Zecchina 5. Biological modelling and information Scientific Directors: Matteo Marsili, William Bialek and Riccardo Zecchina
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