Epidemic outbreaks on two-dimensional quasiperiodic lattices

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Epidemic outbreaks on two-dimensional quasiperiodic lattices G.B.M. Santos a , T.F.A. Alves a,∗ , G.A. Alves b , A. Macedo-Filho c , R.S. Ferreira d a

Departamento de Física, Universidade Federal do Piauí, 57072-970, Teresina - PI, Brazil Departamento de Física, Universidade Estadual do Piauí, 64002-150, Teresina - PI, Brazil c Campus Prof. Antonio Geovanne Alves de Sousa, Universidade Estadual do Piauí, 64260-000, Piripiri - PI, Brazil d Departamento de Ciências Exatas e Aplicadas, Universidade Federal de Ouro Preto, 35931-008, João Monlevade - MG, Brazil b

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Article history: Received 16 July 2019 Received in revised form 8 October 2019 Accepted 14 October 2019 Available online xxxx Communicated by M. Perc Keywords: Asynchronous SIR model Epidemic models on lattices Voronoi-Delaunay triangulation Markovian Monte Carlo process Finite size scaling

We present a novel kinetic Monte Carlo technique to study the susceptible-infected-removed model in order to simulate epidemic outbreaks on two quasiperiodic lattices, namely, Penrose and AmmannBeenker. Our analysis around criticality is performed by investigating the order parameter, which is defined as the probability of growing a spanning cluster formed by removed sites, evolving from an initial system configuration with a single random chosen infective site. This system is studied by means of the cluster size distribution, obtained by the Newman-Ziff algorithm. Additionally, we obtained the mean cluster size, and a cumulant ratio to estimate the epidemic threshold. In spite of the quasiperiodic order moves the transition point, compared to periodic lattices, this is not able to alter the universality class of the model, leading to the same critical exponents. In addition, our technique can be generalized to study epidemic outbreaks in networks and diffusing populations. © 2019 Published by Elsevier B.V.

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1. Introduction

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Nowadays, epidemic outbreaks have attracted the attention of the scientific community which has devoted efforts to studying these epidemics processes [1–5]. In this respect, Asynchronous SIR (susceptible-infected-removed) model [6–16] is a stochastic model useful to mimic an epidemic outbreak. The essential ingredients to simulate an epidemic outbreak are all intrinsic to the model. First, we have the spatial structure by means of a lattice where bonds can represent the contacts between individuals, that can be further generalized to random networks [8,17,9]. Moreover, the dynamics is composed of stoichiometric equations describing temporal evolution of a disease [14], taking into account recover and reproductive rates that can be inferred from the observation of a real epidemic outbreak. And last, fluctuations introduced by stochasticity which are an important distinction from deterministic models to simulate disease spreading. In general, epidemic models are an important tool to gain information on epidemic spreading and even, test some protocols to avoid or mitigate its destructive consequences by taking into account the aspects of collective dynamics and behavioral aspects of complex systems [18,19]. Thus, SIR model is one of the simplest models of epidemic spreading by contact, where infected

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*

Corresponding author. E-mail address: [email protected] (T.F.A. Alves).

https://doi.org/10.1016/j.physleta.2019.126063 0375-9601/© 2019 Published by Elsevier B.V.

subjects can be removed from dynamics by permanent immunity (or death). Besides that, another simple model of disease spreading is the SIS (susceptible-infected-susceptible) model (and its variant, the Contact Process model) [16,20–27,10,28–33]. The main difference between SIR and SIS models is the permanent immunity (or removal) present in SIR model, in a way that this main distinction let the two models falling into two distinct universality classes: dynamical percolation [7,16,11,12] and directed percolation [34,35], respectively. In addition, properties like permanent removal by immunity or death related to the SIR or SIS models are appropriate to simulate the epidemic outbreaks of diseases like Ebola, influenzas, SARS, AIDS, Chickenpox, etc., where death/removal or permanent immunity are one of the possible states of the individuals. Additionally, SIR model displays a continuous phase transition between an endemic state to an epidemic state on a square lattice [11,12], where epidemic and endemic phases are defined in a geometric fashion: the existence of an “infinite component”, identified by a percolating cluster formed by the activated sites (eventually removed ones at the end of dynamics) defines the epidemic phase, and for the converse we have the endemic phase. Ordinarily, reproductive and recover rates control the critical threshold between the epidemic (percolating) and endemic (non-percolating) phases. Consequently, every path starting from one seed, called the “patient zero” (one randomly chosen individual to be the first infected one of the epidemic outbreak) is driven by both reproductive and recover rates to an absorbing state in one of the two possible phases. In particular, absorbing states deny the detailed

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balance condition which indicates that the SIR model belongs to the category of non-equilibrium systems. In this paper, we considered SIR model on Penrose and Ammann-Beenker quasiperiodic lattices in such a way that our objectives are two-fold: first and foremost, we propose a new way to simulate an epidemic outbreak on non-periodic lattices by using Newman-Ziff algorithm [36] to track clusters generated by the dynamics and to obtain percolation observables like the mean cluster size. And second, we investigate if quasiperiodic order can affect the critical behavior of the model. There are some criteria trying to predict if quenched disorder, or quasiperiodic order, can affect the critical exponents of a given model, like Harris [37], or HarrisLuck [38] criteria, respectively. Both Harris and Harris-Luck criteria are superseded by Harris-Barghathi-Vojta criterion [39,40], however, even Harris-Barghathi-Vojta criterion is known to fail [33]. In order to accomplish our objectives, we redefined the asynchronous SIR model of refs. [11,12] to allow the application of its kinetic rules on non-homogeneous lattices by changing the kinetic Monte Carlo rules of the dynamics. First, we begin the simulation by a single random chosen infective node, i.e., a random seed, in a different way from refs. [11,12] where the seed is always the central node in a square lattice with fixed boundary conditions. Note that choosing a random seed is the right way to consider the non-homogeneity of the lattice nodes in full extent. In addition, we changed the way of detecting a percolating cluster by using Newman-Ziff algorithm with different boundary conditions of refs. [11,12]. We believe our redefined kinetic Monte Carlo rules can be applied to generalizations including non-sedentary populations and random networks. This paper is organized as follows: in section 2 we describe the SIR model and its dynamics, lattice building algorithm and details of calculated parameters by Newman-Ziff algorithm, in section 3 we discuss our numerical results and in section 4 we present our conclusions.

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2. Model and implementation

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We considered the Asynchronous SIR model coupled to the two-dimensional (2D) Penrose and Ammann-Beenker quasiperiodic lattices. SIR model is a compartmental model, where a sedentary population, composed of n vertexes in a lattice, is sorted in three compartments, namely: susceptible individuals, that are not infected, are not infecting and are not immune/dead; infected individuals, that can infect any susceptible neighbor, and are not immune/dead, and removed individuals, that are removed from dynamics because of permanent immunity/death. SIR model can be expressed as a reaction-diffusion process, where individuals can change its state and go from one compartment to another [41]. Transitions of the individuals between compartments are represented by the following stoichiometric reactions μi

S + I −→ 2I , μc

I− −→ R ,

(1) (2)

where μc and μi are the recover and reproductive rates, respectively. The stoichiometric equations (1) and (2) can be mapped in a lattice Markovian dynamics with the following rules:

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(1) We start with only one infected seed, the “patient zero”, randomly chosen from the lattice with a population of N individuals, attached to its respective lattice vertexes. Along the entire dynamics, two lists are updated: a list of infected individuals and a list of removed ones. The infected list starts with the “patient zero” and the list of removed sites begins empty.

(2) Next, we update the system state by randomly choosing an infected vertex i from the infected list and proceed as follows: (a) A random uniform number x in the interval [0, 1) is generated. If x ≤ λ, the infected site is removed from the infected list and placed in the removed list; (b) If x > λ, we pick a neighboring vertex randomly to add it on the infected list; (3) The step (2) is repeated several times until the system has no infected sites, i.e., reach any absorbing state. For each pass of the dynamics we can increment the dynamics clock by a time interval 1/ N i where N i is the number of infected individuals. Note that the update is made in an asynchronous way.

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From the above kinetic Monte Carlo rules, we can see that the recover and reproductive rates are given by μc = λ, and μi = 1 − λ, respectively. Some observables can be obtained at the absorbing state, for example, the number of removed vertexes N r as function of λ, expressed as

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Nr =

n 

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|ψi | .

(3)

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However, only N r is not sufficient to fully determine the critical behavior because it is not the order parameter. Following classical percolation theory [42,43], we need the observables related to the clusters formed by the removed sites and to accomplish that, we have to determine the cluster distribution ncluster (s), i.e., the number of clusters with s removed vertexes. The cluster distribution can be measured by using Newman-Ziff algorithm [36] with the feature of identifying if there is a percolating cluster as a result of the dynamics. In accordance with boundary conditions, the percolating cluster is the wrapping one, in the case of periodic boundaries. For non-periodic boundaries, the percolating cluster is the spanning cluster [36]. From the cluster distribution, we have the fraction of removed vertexes in the finite (non percolating) clusters with size s

Ps =

sncluster (s) Nr

(4)

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1  Nr

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Therefore, the following identity can be written

P∞ +

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sncluster (s) = 1,

(5)

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where P ∞ is the fraction of removed nodes in the percolating cluster and the summation excludes the percolating cluster. The epidemic phase is then defined in a geometrical way, where we should have a percolating cluster emulating an epidemic spreading reaching the most remote points of a lattice. At the epidemic threshold, size distribution of clusters should be a power law [11], i.e.

ncluster (s, λc ) ∝ s

−τ

(6)

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and, in general, we have the following scaling ansatz in the vicinity of the epidemic threshold −τ



 sσ ,

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(7)

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where F is a fast decaying function [42,43]. The epidemic outbreak should result in a percolating cluster at the epidemic phase, i.e. P ∞ = 0. Therefore, the order parameter is the percolating cluster density

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F (λ − λc )

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P =  P ∞

(8)

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Fig. 1. (Color Online.) Clusters resulting from SIR model dynamics in (a) Penrose lattice and (b) Ammann-Beenker lattice. In light gray (blue), darker gray (red) and black, we have susceptible, infected and removed vertexes, respectively. Note that the activity is restricted to cluster boundaries in a way that only one cluster is grown as a result of the dynamics.

where the average is done over an ensemble of dynamics realizations. Another relevant parameter is the mean cluster size  S = s s P s , expressed as

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S=

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Nr

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Nr

M =

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1 Nr

s3perc +



 s3 ncluster (s) .

(12)

s



P ≈ L −β/ν f P L 1/ν |λ − λc | ,

(13)

χ

(14)



 ≈ L γ /ν f χ L 1/ν |λ − λc | ,





S ≈ L γ /ν f  S L 1/ν |λ − λc | ,



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f M L

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|λ − λc | ,

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M

(16)



 S 2



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|λ − λc | .

(18)

Binder cumulant curve crossings for different lattice sizes give an estimate for the epidemic threshold λc , and scaling relations written in Eqs. (13), (14), and (18) can be used to estimate the critical exponent ratios 1/ν , β/ν , and γ /ν by standard data collapses, respectively, to completely determine the universality class. Concerning the particular case of SIR model, only one cluster of size s = N r is grown for each realization of the dynamics. Typical clusters are shown in Fig. 1. In this way, we have for SIR model at the endemic phase ncluster (s) = δs, N r and sperc = 0, where δ is a Kronecker index, and for the epidemic (percolating) phase we should have ncluster (s) = 0 and sperc = N r . Therefore S = N r and M = N r2 . On the other hand, the order parameter reduces to P ∞ = 0 if there is not a percolating cluster and to P ∞ = 1 for the converse. One should note that the average of the order parameter reduces to the probability of the dynamics to grow a percolating cluster, i.e., a single infective seed starts an epidemics. This particular simpler cluster structure can be simulated without Newman-Ziff algorithm [12], but at doing this, we can lose the power to generalize the model for non-periodic and random lattices. Indeed, Newman-Ziff algorithm can be applied with modifications to identify a spanning cluster in the case of non-periodic boundaries or a wrapping cluster, in the case of periodic boundaries [44]. Regarding the lattice structure, Penrose and Ammann-Beenker lattices were generated by projection algorithm [45,46,32], where we can define a generation parameter g for these lattices. The number of lattice vertexes is a function of g. We need to map all boundary nodes. They are sorted in four classes that should be searched in the following order:

(15)

according to classical percolation theory [42,43]. Combining Eqs. (13), (15), and (16), we find that the quantity

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(11)

s



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s2 ncluster (s) ,

In the infinite size lattice limit, sperc → ∞ and in the endemic phase, P ∞ = 0 and thus, s perc = 0, therefore observables S and M only make sense for finite lattice sizes. From the scaling ansatz written in Eq. (7), the relevant observables written in Eqs. (8), (10), (11), and (12) should scale as

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s2perc +



and the second is the mean quadratic cluster size

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S =

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(10)

We can define another two observables: first is the mean cluster size S where the summation now includes the percolating cluster, with size sperc

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s

χ = S .

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(9)

The mean cluster size S plays the role of the susceptibility in classical percolation theory, which is defined by the following average

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for ferromagnetic spin models [12]. Therefore, U obeys the following scaling

(17)

should be universal at the epidemic threshold where the scaling dependencies cancel out, being analogous to the Binder cumulant

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1: Vertexes with only two neighbors; 2: The two neighbors of class 1 vertexes; 3: In the case of two neighboring boundary rhombi sharing a vertex, the shared vertex is a boundary one; 4: In the case of two neighboring boundary rhombi sharing an edge, one of two nodes shared is a boundary one. This node has exactly three neighbors, all contained on boundary rhombi. If the two shared edge vertexes have three neighbors, the vertex located farther from the central lattice node is the boundary node.

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Table 1 Summary of the estimated critical properties.

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Lattice

Epidemic threshold

1/ν

β/ν

γ /ν

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Penrose Ammann-Beenker

λc = 0.1713(2) λc = 0.1732(5)

0.7883 ± 0.0267 0.7788 ± 0.0260

0.1012 ± 0.0067 0.1034 ± 0.0078

1.8094 ± 0.0079 1.7845 ± 0.0264

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Exact values

—

3/4

5/48

43/24

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Boundary rhombi can be identified by looking for extremal grid points in grid lines when calculate the multigrid crossings [32]. With boundary nodes, one can obtain the spanning cluster and identify the epidemic phase. For Penrose lattice, a spanning cluster is a cluster who contains at least one pair of vertexes symmetrical over a lattice rotation of ±4π /5 about the central node. However, for Ammann-Beenker lattice, a spanning cluster should be defined in a slightly different way because of rotational symmetry: a spanning cluster should contain at least one pair of vertexes symmetrical over a lattice rotation of π about the central node. Note that we used fixed boundaries and our results should be prone to finite size boundary effects, but they can be avoided by just using bigger lattice sizes as expected for amenable graphs. Generations from g = 4 to g = 10, used in our numerical results, give good data collapses for both lattices, indicating that these sizes are enough to avoid boundary effects. We repeated SIR model dynamics 106 times for every control parameter λ, starting from “patient zero” until the system evolved to an absorbing state, growing 106 clusters, and calculated an ensemble composed of 106 measurements of order parameter, mean cluster size and Binder cumulant, one for each cluster generated, in order to obtain the relevant ensemble averages. We calculated statistical errors by using the “jackknife” resampling [47]. In the following section, we discuss our numerical results.

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3. Results and discussion

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In this section, we show our numerical results of Asynchronous SIR model coupled to Penrose and Ammann-Beenker quasiperiodic lattices. First, we show results for the Binder cumulant in Figs. 2a and 2b. From Binder cumulant crossings, we estimated the epidemic thresholds for Penrose and Ammann-Beenker lattices, shown in Table 1. Binder cumulant expressed in Eq. (17) should not depend on the lattice size at the critical point, but it should depend on the dimensionality and boundary conditions as seen from comparing Figs. 2a and 2b. Note that the Binder cumulant values at the critical threshold are not the same for Penrose and Ammann-Beenker lattices because of different boundary conditions we adopted to identify spanning clusters, as explained in the previous section. Following Binder cumulant discussion, we show order parameter results in Figs. 2c and 2d. We see that the curves for both Penrose and Ammann-Beenker lattices have the typical sigmoidal shape, indicating a continuous phase transition from the epidemic phase to the endemic phase, where the order parameter vanishes, i.e., where we do not have any spanning cluster. By using NewmanZiff algorithm, we were able to numerically calculate the mean cluster size. In Figs. 2e and 2f, we show the susceptibilities for Penrose and Ammann-Beenker lattices, respectively. They diverge at the epidemic threshold λc in the infinite size lattice limit, according to the finite size scaling relation

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λmax = λc + bL

1/ν

,

(19)

which means that the mean cluster size maxima λmax gets closer to the epidemic threshold λc when increasing N, as expected for a continuous phase transition and as observed from our data.

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Fig. 2. In panels (a), (c), and (e), we show our numerical data for the Binder cumulant U , order parameter P and susceptibility χ as functions of recovery rate λ for Penrose lattice, respectively. The same in panels (b), (d), and (f) for AmmannBeenker lattice. Both lattices display a continuous phase transition from an epidemic phase to an endemic phase by increasing the recover rate λ. Note that the Binder cumulant crossings allows us to estimate the epidemic thresholds at λc = 0.1713(2) and λc = 0.1732(5) for Penrose and Ammann-Beenker lattices, respectively. The order parameter, defined as the probability of a single seed start an epidemics, goes to zero at the epidemic threshold in the infinite lattice limit. The susceptibility, defined as the size of the finite cluster, diverges at the epidemic threshold. Statistical errors are smaller than symbols.

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Now, we discuss the critical exponent values. We estimated the critical exponent ratios 1 /ν , β/ ν , and γ /ν by using finite

d(ln P ) , ln P , and ln χ , evaluated at dλ d(ln P ) the critical threshold λc , respectively. In order to extract dλ size scaling regressions of ln

data, we made linear regressions of the logarithm of order parameter in the vicinity of the epidemic threshold. Error bars are extracted in this case from the least method. In Figs. 3a squares

d(ln P ) as function of (ln N ) /2, dλ allowing us to estimate the numerical values for 1/ν shown in and Figs. 3b we show data for ln

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Fig. 3. In panels (a), (c), and (e), we show estimates of critical exponent ratios 1/ν , P) β/ν , and γ /ν from finite size scaling regressions of ln d(ln , ln P , and ln χ , redλ spectively, at the epidemic threshold λc = 0.1713(2) as functions of (ln N ) /2 for Penrose lattice. The same for panels (b), (d), and (f) at the epidemic threshold λc = 0.1732(5) for Ammann-Beenker lattice. All estimated exponent ratios, summarized in Table 1 are all close to the respective 2D dynamic percolation exponents 1/ν = 3/4, β/ν = 5/48, and γ /ν = 43/24. Note the logarithmic scale for the data and statistical errors.

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Table 1. The numerical values for 1/ν deviate 5% and 4% for Penrose and Ammann-Beenker lattices, respectively, from the exact critical exponent ratio 1/ν = 3/4 of 2D dynamical percolation. Continuing the discussion of critical exponent ratios, in Figs. 3c and 3d we show the order parameter logarithm at the epidemic threshold as function of (ln N ) /2. In this case, finite size regressions should give the estimates of β/ν shown in Table 1. Our numerical values for β/ν deviate 3% and 1% from exact β/ν value of 2D dynamical percolation universality class for Penrose and Ammann-Beenker lattices, respectively. We show the analogous for susceptibility in Figs. 3e and 3f at the epidemic threshold, and summarized our obtained numerical values of γ /ν in Table 1, deviating 1% and 0.5% from the exact critical exponent ratio γ /ν = 43/24 for Penrose and Ammann-Beenker lattices, respectively. By noting that the numerical values obtained from the finite size scaling regressions are all close from exact 2D dynamical percolation values, we can expect that the system falls into 2D percolation universality class. To confirm that, we collapsed all our numerical data shown in Fig. 2 by using the finite size scaling relations written in Eqs. (13), (14), and (18) with the known exact

Fig. 4. In panels (a), (c), and (e), we show our numerical data for the Binder cumulant U , order parameter P and susceptibility χ rescaled by using the finite size scaling relations written in Eqs. (13), (14), and (18) with the 2D dynamic percolation exponent ratios 1/ν = 3/4, β/ν = 5/48, and γ /ν = 43/24 at the vicinity of epidemic threshold λc = 0.1713(2) for Penrose lattice, respectively. The same is shown in panels (b), (d), and (f) at the vicinity of the epidemic threshold λc = 0.1732(5) for Ammann-Beenker lattice. Data collapses suggest that the system falls in 2D percolation universality class, irrespective of quasiperiodicity. Statistical errors are smaller than symbols.

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values of 2D dynamic percolation exponent ratios, summarized in Table 1. All data collapse panels shown in Fig. 4 are respective to the panels shown in Fig. 2. The data collapses for both Penrose and Ammann-Beenker lattices are a strong evidence that the system falls into dynamic percolation universality class in the same way of 2D square lattice [11,12]. Therefore, quasiperiodic order is irrelevant, as seen from results for classical percolation in the same quasiperiodic lattices [48–50].

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4. Conclusions

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We proposed a new way to simulate the Asynchronous SIR model on non-periodic lattices by using Newman-Ziff algorithm to numerically determine the cluster size distribution and calculate any observable related to percolation. We applied NewmanZiff algorithm to identify the epidemic phase, characterized by a spanning cluster and used this technique to investigate a redefined version of SIR model, coupled to Penrose and Ammann-Beenker lattices. Our numerical results suggest that quasiperiodic order is irrelevant and the system falls into the 2D dynamical percolation

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universality class in the same way of the Asynchronous SIR model on periodic lattices, in agreement with classical percolation results, in a way that our redefined model has the same critical behavior. We obtained the order parameter, defined as the ensemble average of spanning cluster density, the susceptibility, defined as the ensemble average of mean cluster size and Binder cumulant ratio defined for percolation, in order to characterize the critical behavior. We estimated the epidemic thresholds: λc = 0.1713(2) and λc = 0.1732(5) for Penrose and Ammann-Beenker lattices, respectively. We would like to stress that this way to simulate a lattice dynamics controlled by stoichiometric equations with removed states who led a system always to an absorbing state, can be applied to some generalizations of SIR model. Another possibility of an application can be biological models based on Lotka-Volterra differential equations [51]. By using standard percolation methods, all these dynamics can be straightforwardly generalized to networks and non-sedentary populations, i.e., models with diffusing particles where we can consider a possibility of a particle being removed [52–61].

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Acknowledgements

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We would like to thank CNPq (Conselho Nacional de Desenvolvimento Científico e tecnológico), FUNCAP (Fundação Cearense de Apoio ao Desenvolvimento Científico e Tecnológico) and FAPEPI (Fundação de Amparo a Pesquisa do Estado do Piauí) for the financial support. We acknowledge the use of Dietrich Stauffer Computational Physics Lab, Teresina, Brazil, and Laboratório de Física Teórica e Modelagem Computacional - LFTMC, Piripiri, Brazil, where the numerical simulations were performed.

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