Modified Epidemic Diffusive Process on the Apollonian Network

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 scienceZaher et al. (2020); Sun et al. (2020); Lu et al. (2020); Zhang and Liu (2020). 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 Bartlett (1956); Castellano and Pastor-Satorras (2020); Mota et al. (2018); Macedo-Filho et al. (2018), Susceptible-Infected-Recovered (SIR) model Bailey (1964); Santos et al. (2020); Alencar et al. (2020), The Contact Process (CP) model Harris (1974); da Silva et al. (2013); Marro and Dickman (1999); Almeida et al. (2016), Diffusive Epidemic Process (DEP) model Kree et al. (1989); van Wijland et al. (1998); Fulco et al. (2001); Maia and Dickman (2007); Filho et al. (2010), 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 wide theoretical field based on analytical techniques such as Mean Field (MF) theory and Renormalization Group (RG) theory, besides computational methods that rely on Monte Carlo (MC) method simulations that were developed in the last years Hinrichsen (2000). In this last case, major support has been coming from technological advances. Thereby, through the use of computational tools, we can model real situations and make predictions, in a short time, that are useful to human society.

Over the last years, the DEP model has been useful for modeling systems closer to reality by inclusion of a non-sedentary population in order to describe a reaction-diffusion process. This model consists of two classes of individuals, susceptible (A) and infected (B), diffusing on a lattice obeying three diffusive regimes, i.e., $D_{A}<D_{B}$, $D_{A}=D_{B}$, and $D_{A}>D_{B}$, where $D_{A}$ and $D_{B}$ are constant probabilities in the [$0,1$] interval. It is worth mentioning that Kree et al. analyzed the DEP model for the first time investigating an ecological system, describing the effect of pollution on a population that is on the brink of extinction, aiming to understand ecological catastrophes, by using RG theory.

Thus, according to the RG analysis, the system presented a continuous phase transition to an absorbing state with critical exponents $\eta=-\epsilon/8$ to first order of $\epsilon=4-d$, $\nu_{\perp}=2/d$ and $z=2$ in all orders in $\epsilon$. Next, the Kree et al. modelKree et al. (1989) was extended by van Wijland et al.van Wijland et al. (1998), considering the $D_{A}>D_{B}$, $D_{A}=D_{B}$, and $D_{A}<D_{B}$ regimes. In this context, the RG theory predicted a continuous phase transition for the $D_{A}<D_{B}$ regime, defining a new universality class (WOH) with critical exponents given by $\nu_{\perp}=2/d$, $\eta=0$ and $z=2$ in all orders in $\epsilon$, 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 regimesFulco et al. (2001); Maia and Dickman (2007); Filho et al. (2010).

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.Jr. et al. (2005) in 2005, which is characterized by a degree distribution given by a power-law $P(k)\propto k^{-\gamma}$, where $k$ is the number of connected neighbors (degree), and $\gamma=1+\ln{3}/\ln{2}\approx 2.585$ is the power-law exponent. In this context, according to the MF theory, the critical behavior of the system depends on the distribution degree $\gamma$. Thus, we can classify three $\gamma$ regionsda Silva et al. (2013); Ferreira et al. (2011a); Noh and Park (2009); Ferreira et al. (2011b), i.e.:

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  $\gamma<2$ there is no existing phase transition;

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  $2<\gamma\leq 3$ with phase transition characterized by $\beta=1/(\gamma-2)$ and $\nu=(\gamma-1)/(\gamma-2)$ exponents, and

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  $\gamma>3$ with critical exponents $\beta=1$ and $\nu=2$.

Here, our main interest in networks with power-law distribution is the scale-free property, specifically the power-law exponent satisfying $2<\gamma\leq 3$ interval, yielding a finite average degree scaling as $\langle k\rangle\approx k_{c}^{2-\gamma}$ (first moment) while the second (and higher) moment diverges as $k_{c}\approx k_{c}^{3-\gamma}$, where $k_{c}$ is the maximum degree present in the network. Besides that, scale-free networks are ubiquitous in nature and societyCohen and Havlin (2010); Barabási and Pósfai (2016); Liljeros et al. (2001); Mendes and da Silva (2009); Almeida et al. (2013); de Solla Price (1965).

In this context, our investigation consists in a modified DEP (MDEP) model study that aims to estimate the existence of phase transition in each $D_{A}>D_{B}$, $D_{A}=D_{B}$ and $D_{A}<D_{B}$ regimes, in addition to its critical exponents $\beta/\nu$, $1/\nu$, and $\gamma/\nu$. 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 algorithmGillespie and Petzold (2013); Gillespie (1976, 1977), in order to introduce a finite threshold in the scale-free networksAlves et al. (2020a). This algorithm allows to stochastically solve the differential equations like SIS modelPastor-Satorras et al. (2015) 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 $\mu_{c}$ is the infection rate, $\mu_{r}$ is the recovery rate and $\rho=\rho_{A}+\rho_{B}$ is the total density population. When coupling a population of random walkers to a network, one can make the density population $\rho_{A}(i)$ and $\rho_{B}(i)$ of node $i$ obeying the Eq.(1), where the reactions take place in a time $t_{\mathrm{max}}$ while maintaining the discrete time diffusion. More details are shown in section II.

This paper is organized as follows: in section II we describe the Apollonian network and MDEP model and its dynamics. In section III we present our simulation results and discussions about our proposed model. And finally, in section IV we present our conclusions.

In this section, we focus the attention on our main MC simulation results. Thus, we present the results for the MDEP model on the Apollonian network. First, we employ a finite-size scaling analysis (FSS) to estimate the critical point, and to directly obtain the critical exponents $1/\nu$, $\beta/\nu$, and $\gamma/\nu$ for each $D_{A}<D_{B}$, $D_{A}=D_{B}$, and $D_{A}>D_{B}$ regime, 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 $\rho_{B}$ on the stationary state can be obtained as functions of the density of individuals $\rho$:

Here, $P$ is the order parameter, $U$ is the $5$-order cumulant ratio for directed percolation, and $\Delta$ 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 thresholdLübeck and Willmann (2002); Janssen et al. (2007); Henkel et al. (2008). Analogous averages $P_{\mathrm{node}}$, $U_{\mathrm{node}}$, and $\Delta_{\mathrm{node}}$ can be constructed from the fraction of active nodes, which have the same critical behavior of $P$, $U$, and $\Delta$, respectively.

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

close to the critical threshold $\rho_{c}$, where $1/\nu$, $\beta/\nu$, and $\gamma/\nu$ are the critical exponent ratios, and $f_{\rho_{B},U,\Delta}$ 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 $\mu_{c}=\mu_{r}=D_{A}=D_{B}=0.5$. 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, the order parameter and its fluctuations are compatible with monotonic increasing functions in the infinite lattice size limit. In particular, the inflection points of the order parameter curves, and the peaks of the order parameter fluctuations goes closer to zero when increasing the network size.

Continuing the discussion, we show results for the MDEP in Figs.(3), (4), and (5) for $D_{A}>D_{B}$, $D_{A}=D_{B}$, and $D_{A}<D_{B}$ regimes, respectively. In all cases, we considered contamination and recovering taxes $\mu_{c}=2$, and $\mu_{r}=1$, respectively. In panel (a) of Figs.(3), (4), and (5), we show the $5$-order cumulant curves and by the cumulant crossings, we estimate the collective thresholds $\rho_{c}=3.038(1)$, $\rho_{c}=2.957(1)$, and $\rho_{c}=2.709(3)$, respectively. In panels (c) of Figs.(3), (4), and (5), we show the average of the fraction of active nodes where we can see the typical sigmoidal shape where the curve inflection separates the absorbing and active phases. Particularly interesting is the behavior of the order parameter fluctuations, which is compatible with a finite jump, instead of a typical peak that increases with the network size.

Regarding the critical behavior results, in panels (b), (d), and (f) of Figs.(3), (4), and (5), 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 $\rho_{c}$ computed and estimated the following critical exponent ratios: $\beta/\nu=0.66(2)$, $1/\nu=0.46(2)$, and $\gamma/\nu=-0.24(2)$ in all the three $D_{A}>D_{B}$, $D_{A}=D_{B}$, and $D_{A}<D_{B}$ regimes. The $\gamma$ exponent is negative, corresponding to a finite jump, instead of a divergence with the network size.

Additionally, reporting the literature about simulation studies on low dimensional regular lattices for DEP model, all investigations exhibit a continuous phase transition to an absorbing state with well defined critical properties, characterizing distinct universality classes. This is in contrast to van Wijland et al. to $D_{A}>D_{B}$ regimeFulco et al. (2001); Maia and Dickman (2007); Filho et al. (2010) which predict a discontinuous phase transition. Regarding complex networks, CP model on the Apollonian network presented a continuous phase transition with well defined critical point and exponents da Silva et al. (2013). They concluded that the Apollonian network topology, with $2<\gamma\leq 3$, cannot affect the overall epidemic spreading, and that the CP model on this network presents a phase transition with critical exponents $\beta/\nu=0.54(2)$ and $1/\nu=0.51(2)$, closer to the MF exponent for the $\gamma>3$, corroborating with the universality class of the MF theory of regular lattice.

Recently, the MDEP model was implemented on Barabasi-Albert network Alves et al. (2020a). Here, was analyzed a MDEP furnishing a continuous phase transition obeying the MF critical exponents $\beta=1$, $\nu=0.5$, and $\gamma=0$, but with logarithmic corrections to order parameter and its fluctuations to all $D_{A}<D_{B}$, $D_{A}=D_{B}$, and $D_{A}>D_{B}$ studied regimes. In this context, although the MDEP model on the Apollonian network presents continuous phase transition to all diffusion regimes, it does not agree with the literature, exhibiting a new set of critical exponents which are different from MF theory, both characterizing the different diffusive regimes investigated in this work.

In this work, we study the DEP model on the Apollonian network by implementing a new approach, based on Gillespie algorithm which introduces a modification in the reaction process, limiting the contacts among individuals into a same site and, consequently, introducing a phase transition that is not present if the contacts are unrestricted. In this model, each individual may diffuse on the network with constant diffusion rates $D_{A}$ and $D_{B}$. Here, for the first time, the DEP model displays a phase transition on the Apollonian network.

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 $\beta/\nu=0.66(2)$, $1/\nu=0.46(2)$, and $\gamma/\nu=-0.24(2)$. This set of critical exponents does not belong to MF universality class in both regular lattices and complex networks.