key: cord-0537222-3xw4y0j1
authors: Alencar, D. S. M.; Macedo-Filho, A.; Alves, T. F. A.; Alves, G. A.; Ferreira, R. S.; Lima, F. W. S.
title: Modified Epidemic Diffusive Process on the Apollonian Network
date: 2021-10-27
journal: nan
DOI: nan
sha: a88368628e6861fa4a8270f4340e412abe8bb66f
doc_id: 537222
cord_uid: 3xw4y0j1

We present an analysis of an epidemic spreading process on the Apollonian network that can describe an epidemic spreading in a non-sedentary population. The modified diffusive epidemic process was employed in this analysis in a computational context by means of the Monte Carlo method. Our model has been useful for modeling systems closer to reality consisting of two classes of individuals: susceptible (A) and infected (B). The individuals can diffuse in a network according to constant diffusion rates $D_{A}$ and $D_{B}$, for the classes A and B, respectively, and obeying three diffusive regimes, i.e., $D_{A}D_{B}$. Into the same site $i$, the reaction occurs according to the dynamical rule based on Gillespie's algorithm. Finite-size scaling analysis has shown that our model exhibit continuous phase transition to an absorbing state with a set of critical exponents given by $beta/nu=0.66(1)$, $1/nu=0.46(2)$, and $gamma/nu=-0.24(2)$ common to every investigated regime. In summary, the continuous phase transition, characterized by this set of critical exponents, does not have the same exponents of the Mean-Field universality class in both regular lattices and complex networks.

Nowadays, we have passed by an infectious disease named COVID-19, which was considered a threat to global public health. This epidemic process caused acute pneumonia in patients around the world, being widely investigated in many areas of the science [1] [2] [3] [4] .

Based on this, we report that epidemic processes have been widely studied over the last years, for instance, by the physicists' community. Thus, many models were created and applied to mimic and to understand such epidemic processes, including the Susceptible-Infected-Susceptible (SIS) model [5] [6] [7] [8] , Susceptible-Infected-Recovered (SIR) model [9] [10] [11] ,

The Contact Process (CP) model [12] [13] [14] [15] , Diffusive Epidemic Process (DEP) model [16] [17] [18] [19] [20] , among others.

These processes belong to the class of non-equilibrium systems and usually display a continuous phase transition to an absorbing state. To investigate these processes, we have a new universality class (WOH) with critical exponents given by ν ⊥ = 2/d, η = 0 and z = 2 in all orders in , and recovering the Kree et al. universality class (KSS) studied to D A = D B regime. In addition, to the D A > D B regime, they conjectured a discontinuous phase transition, however, MC simulations analysis with distinct algorithms are consistent with a continuous phase transition to all three regimes [18] [19] [20] .

In this work, we performed a study about the DEP model coupled to Apollonian networks, a particular complex network model introduced by J. S. Andrade Jr. et al. [22] in 2005, which is characterized by a degree distribution given by a power-law P (k) ∝ k −γ , where k is the number of connected neighbors (degree), and γ = 1 + ln 3/ ln 2 ≈ 2.585 is the power-law exponent. In this context, according to the MF theory, the critical behavior of the system depends on the distribution degree γ. Thus, we can classify three γ regions [13, [23] [24] [25] , i.e.:

• γ < 2 there is no existing phase transition;

• 2 < γ ≤ 3 with phase transition characterized by β = 1/(γ −2) and ν = (γ −1)/(γ −2) exponents, and • γ > 3 with critical exponents β = 1 and ν = 2.

Here, our main interest in networks with power-law distribution is the scale-free property, specifically the power-law exponent satisfying 2 < γ ≤ 3 interval, yielding a finite average degree scaling as k ≈ k 2−γ c (first moment) while the second (and higher) moment diverges as k c ≈ k 3−γ c , where k c is the maximum degree present in the network. Besides that, scale-free networks are ubiquitous in nature and society [26] [27] [28] [29] [30] [31] .

In this context, our investigation consists in a modified DEP (MDEP) model study that aims to estimate the existence of phase transition in each

regimes, in addition to its critical exponents β/ν, 1/ν, and γ/ν. Thus, the main modification to the DEP usual definition was done on the reaction stage, by simulating the reaction process as a chemical reaction by using Gillespie algorithm [32] [33] [34] , in order to introduce a finite threshold in the scale-free networks [35] . This algorithm allows to stochastically solve the differential equations like SIS model [36] of a homogeneous population, not coupled to a network, and find the time evolution of the infected and susceptible compartments A and B, respectively

where µ c is the infection rate, µ r is the recovery rate and ρ = ρ A + ρ B is the total density population. When coupling a population of random walkers to a network, one can make the density population ρ A (i) and ρ B (i) of node i obeying the Eq. The Apollonian network belongs to the complex network family, having its origin based on the problem of a space-filling packing of disks first proposed by the Greek mathematician Apollonius of Perga [37] . To build it, we consider three equal radius disks touching each other and the space among them is filled by another disk that touches all the previous three disks. This procedure can be iterated by inserting smaller disks inside the space among any three touching disks. The network is formed by connecting the centers of the touching disks, obtaining a network that gives a triangulation that physically corresponds to the force network of a dense granular packing. The number of nodes N at each generation n = 0, 1, 2, 3, . . . can be found according to the relation

As an example, Fig. (1) displays an Apollonian network of fourth-generation with N = 43

nodes. Next, it is worth mentioning that a detailed description of the Apollonian network building can be found in Ref. [38] . In comparison to other complex network models, the Apollonian network is a particular case of a deterministic complex scale-free network embedded in a Euclidean lattice with space-filling and matching graph properties. Also, it displays small-world effect, which means that the average length of the shortest paths l between two any nodes grows up slower than any positive power of the system size N . In addition, it presents a large clustering coefficient C. Thus, since l grows up logarithmically and C tends to unity, the Apollonian network indeed exhibits a small-world effect. In this sense, the literature shows that the average length of the shortest paths is l ∝ [ln(N )] 3/4 and clustering coefficient C = 0.828 in the limit of large N [13, 22, 39, 40] .

Here, we introduce the DEP simulation based on kinetic Monte-Carlo dynamic rules.

The main modification introduced is relative to the reaction process. According to the usual definition of the DEP, the reaction process is modeled by using rejection sampling. Thus 

in the network at t = 0 with half infected and half susceptible. However, the original definition does not have a finite threshold, and this means that any infection can survive in the Apollonian network if we allow unrestricted contacts between the individuals in the same node.

We introduce a modification to the original definition that transforms the x ≤ µ c and

x ≤ µ r probabilities into taxes and the contamination and cure processes inside a node are simulated by the Gillespie Algorithm. Next, we enumerate the following rules which define the MDEP model applied to Apollonian networks [35] :

1. 

and if a random number uniformly distributed on [0, 1) interval is less or equal to D B (where D B ∈ [0, 1]) the infected particle jumps from the node i to a randomly chosen neighboring node j. Then, the infected populations are updated as follows

• Reaction stage: The time evolution of the populations in each node i is stochastically simulated by using Gillespie algorithm in a time t max , exponentially distributed with mean 1/P (i), i.e. to select if a contamination or a spontaneous recover will take place, with probabilities proportional to its propensities. The contamination channel

have a propensity κ(i, 1) given by

and the spontaneous recover channel A(i, t q + ∆t q ) = A(i, t q ) + 1,

have a propensity κ(i, 2) given by

The reaction time t q is then updated by adding it with an exponentially distributed time interval ∆t q with mean given by 1/(κ(i, 1) + κ(i, 2)); Our approach is a mixed one, that can be interpreted as a diffusion coupled agent model.

The propagation is modeled with diffusion probabilities by rejection sampling and the reac- respectively. In order to investigate the MDEP critical behavior, one can obtain a time series of the following observables at the stationary state

which are the infection density and the fraction of active nodes, i.e., nodes with at least one infected individual. The following averages from the time series of the infection density ρ B on the stationary state can be obtained as functions of the density of individuals ρ:

Here, P is the order parameter, U is the 5-order cumulant ratio for directed percolation, and ∆ is the order parameter fluctuation. The 5-order cumulant ratio is finite at the absorbing phase, and crosses on distinct network size data at the collective critical threshold [42] [43] [44] .

Analogous averages P node , U node , and ∆ node can be constructed from the fraction of active nodes, which have the same critical behavior of P , U , and ∆, respectively.

We conjecture that the averages shown on Eq.(12) obey the following FSS relations

close to the critical threshold ρ c , where 1/ν, β/ν, and γ/ν are the critical exponent ratios, and f ρ B ,U,∆ are, respectively, the FSS functions.

Now, we turn to the simulation results. First, we show an example of DEP dynamics on Apollonian networks in Fig.(2) for µ c = µ r = D A = D B = 0.5. Note that the crossings for increasing network sizes are closer to zero concentration which allows concluding that the system is active for any finite concentration in the infinite network limit. In addition, we show the best data collapses of the cumulant, the active node fraction average and the order parameter fluctuations by the use of each previously critical thresholds ρ c computed nents [13] . They concluded that the Apollonian network topology, with 2 < γ ≤ 3, cannot affect the overall epidemic spreading, and that the CP model on this network presents a phase transition with critical exponents β/ν = 0.54(2) and 1/ν = 0.51 (2) , closer to the MF exponent for the γ > 3, corroborating with the universality class of the MF theory of regular lattice.

Recently, the MDEP model was implemented on Barabasi-Albert network [35] . Regarding the epidemic spreading, the model is compatible with the fact that restricting the contacts between individuals can allow for epidemic control in a way that the system can present a critical threshold between the absorbing phase and the active phase by increasing the populational concentration. In this way, the model favors a social distancing behavior in order to allow epidemic control. In the converse, unrestricted contacts lead to epidemic survival for any non-zero concentration in the infinite network limit.

In contrast to van Wijland et al. to D A > D B regime, the MDEP model presents a continuous phase transition to an absorbing state being characterized by the same set critical exponents, for every investigated diffusive regime. The critical exponent ratios are given by β/ν = 0.66(2), 1/ν = 0.46 (2) , and γ/ν = −0.24 (2) . This set of critical exponents does not belong to MF universality class in both regular lattices and complex networks.

Proc. Third Berkeley Symp

The Elements of Stochastic Processes with Applications to the Natural Sciences

Nonequilibrium Phase Transitions in Lattice Models

Advances in Physics

Complex Networks: Structure, Robustness and Function

Network science

We would like to thank CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível Su-