Simple quasistationary method for simulations of epidemic processes with localized states☆
Introduction
Many phenomena can be suited within the framework of dynamical processes on complex networks, in which individuals represented by nodes (or vertices) interact with each other and the interactions are mediated by links (or edges) [1], as illustrated in Fig. 1. Nodes and links are more frequently used in the context of physical systems while vertices and edges for their mathematical representation as graphs. We use both terminologies interchangeably. Examples range from rumor and epidemic spreading [2], [3], [4], [5], [6] to urban mobility [7], [8]. In particular, those with absorbing states, in which the dynamics is trapped forever when visited [9], [10], represent an important class of processes on networks [1].
Epidemics can be modeled as reaction-diffusion systems, in which the states of agents and the corresponding transitions are related to epidemiological stages (susceptible, infectious, immunized, etc.) [11]. One of the most fundamental epidemic models is the susceptible-infected-susceptible (SIS), in which individuals can be infectious or susceptible. The former heals spontaneously with rate μ and the latter is infected with rate λ per infectious contact [12]. On networks, the contagion is mediated by edges connecting infected and susceptible nodes. Since the infection cannot arise spontaneously, a configuration with only susceptible individuals is an absorbing state.
The SIS model on power-law networks with degree distribution , defined as the probability that a randomly selected vertex has k connections (Fig. 1), presents remarkable properties. For example, the epidemic threshold, which separates an inactive (absorbing) from an active (fluctuating) phase, is asymptotically null [13], [14] that implies a nonzero prevalence (fraction of infected nodes) for any finite infection rate in an infinite network. For finite systems, the dynamics near to the transition is featured by a localized activation [15], [16], [17], [18], [19]. In the particular case of random power-law networks, the competition of two basic localized structures, stars and maximum k-core subgraphs [20], [21], plays a central role on the activity distribution. A star subgraph is composed of a vertex of degree k plus its nearest-neighbors, see Fig. 1, and can sustain a localized metastable activity of a SIS dynamics [13], permitting mutual activation of hubs even if they are far from each other [13], [14], [22]. The maximum k-core [20], which is the minimal connected subset containing only elements of degree after the removal of all vertices of degree and the edges connected to them, may undergo an activation before the star graphs being responsible by the onset of the endemic phase [21]. However, more complex scenarios can emerge on real networks, in which other motifs, such as cliques of fully connected vertices (Fig. 1), rule the epidemic activation [23].
Analyses of absorbing-state phase transitions by means of simulations demand approaches to prevent trapping into absorbing states since, strictly, they are the only actual steady-state for finite sizes [9]; the only ones accessible in simulations. A fundamental approach is the quasistationary (QS) analysis [9], in which averages at a time t are performed only over a subspace of configurations that did not visit the absorbing state until that time. Alternatively, one can consider some perturbation of the dynamics in such a way that the absorbing state is no longer accessible. However, these perturbations must be negligible for intensive and extensive quantities in the thermodynamic limit [24]. There are some methods conceived within this approach [25], [26], [27].
The standard quasistationary method which is hereafter generally called QS method (SQS), can be implemented by storing configurations picked up at random along simulations with a certain rate (a parameter of the method) and using them to form a new active state whenever the dynamics is trapped into an absorbing state [28]. The method has frequently been used to investigate localization of epidemic processes on networks [15], [16], [29], [30], [31], [32]. Despite being general, the method has some drawbacks for simulations on large networks such as the high RAM memory load to store a large number of configurations, metastability problems due to finite sampling and averaging time on critical and subcritical simulations, and parameters which do not have a systematic calibration and are model dependent. Simpler methods, such as reflecting boundary condition (RBC), in which the dynamics returns to the previously visited configuration if it has fallen into an absorbing state, can circumvent these problems [27]. A systematic comparison between SQS and RBC [24] for the SIS model on power-law networks shows that the latter is unable to capture some metastable dynamics related to localized states as the former does, leading, for example, to different finite-size scaling (FSS) exponents at the transition. A variant of the RBC, called hub reactivation [24], [33], consists of reinfecting the most connected vertex of the network every time the absorbing state is visited. This method outperformed RBC on the identification of localized phases, equating to the SQS method. However, this method is biased for dynamics in which a single hub plays a major role in the localization and calls for a generalization towards other dynamical processes and relevant network structures such as the cliques shown in Fig. 1.
In this paper, we investigate a QS method without tuning parameters and metastability issues, demanding reduced amount of RAM, and able to capture the same localization patterns on heterogeneous networks as the SQS does. It consists of reactivating a given number of vertices, chosen proportionally to their activity times along the whole history of the simulation, being hereafter called reactivation per activity time (RAT). We compare RAT with the SQS and RBC methods using the challenging SIS dynamics on a variety of synthetic and real networks with high heterogeneity, including scale-free and multiplex structures [34]. Among other findings, we report that all methods provide the same epidemic thresholds in scale-networks with degree exponents , but RBC differs on the FSS of the epidemic prevalence evaluated at the epidemic threshold. Using dynamical susceptibility curves [15], we report that the SQS and RAT methods capture the same localization patterns, even in the cases where RBC and SQS mismatch, as for example, in synthetic networks with degree exponent and real networks with degree correlations and more complicated structures. Metastability problems due to insufficient time averaging were not observed for RAT in the networks where they were for SQS. Last but not least, the RAM load in RAT is greatly reduced with respect to SQS while CPU times are slightly smaller.
The remainder of this paper is organized as follows. In Section 2 we describe the investigated QS methods and the algorithms to simulate the SIS model. We present simulations comparing the QS methods on synthetic and real networks in section 3. The computer performance of the QS methods is presented in section 4. Finally, we draw our remarks and prospects for potential applications of the RAT method in section 5.
Section snippets
Methods
Consider systems with N elements labeled by which can either be active or inactive. In the case of epidemic processes we have infected and susceptible individuals, but we keep the generic active and inactive terms allowing extension to other dynamical processes. We follow the approach of QS method as a perturbation of the dynamical rules that prevents the trapping into absorbing states in such a way that this disturbance is negligible and does not alter any system's intensive or
Comparison of QS methods
We performed QS simulations of the SIS model on complex networks with broad levels of localization to compare the different methods. For simplicity we adopt , fixing therefore the time unit without loss of generality. We start with simple graphs where the localization is highly controlled in subsection 3.1. Then, we consider increasing complexity with synthetic power-law, multiplex, and real networks in subsections 3.2, 3.3, and 3.4, respectively.
Comparison on the computer efficiencies of QS methods
We now discuss the computational complexity of the QS methods. The RAT computational complexity is expected to lie between the other methods. We analyze two key ingredients of computational complexity: RAM load and CPU time. Simulations of the SIS model on UCM networks for (the position of the maximum value of the susceptibility curves) were run on a workstation with an Intel i5 8400 - 2.80 GHz processor and 8 GB of RAM memory. Besides its physical relevance, the choice of the transition
Conclusions and prospects
Dynamical processes with absorbing configurations, which once visited trap the dynamics indefinitely [9], are relevant for many biological, chemical and physical systems. In particular, the study of these processes on non-regular structures, such as complex networks [1], constitutes an interdisciplinary issue with applications to real cases [7], [49], [50], [51]. Considering spreading on networks [11], computational simulation is a central actor in the validation of approximated theories and in
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
We thank M. M de Oliveira, D. H. Silva and W. Cota for comments and suggestions. This work was partially supported by the Brazilian agencies Conselho Nacional de Desenvolvimento Científico e Tecnológico- CNPq (Grants no. 430768/2018-4 and 311183/2019-0) and Fundação de Amparo à Pesquisa do Estado de Minas Gerais - FAPEMIG (Grant no. APQ-02393-18). This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) - Brasil - Finance Code 001.
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The review of this paper was arranged by Prof. Blum Volker.