Accuracy of discrete- and continuous-time mean-field theories for epidemic processes on complex networks.

Discrete- and continuous-time approaches are frequently used to model the role of heterogeneity on dynamical interacting agents on the top of complex networks. While, on the one hand, one does not expect drastic differences between these approaches, and the choice is usually based on one's expertise or methodological convenience, on the other hand, a detailed analysis of the differences is necessary to guide the proper choice of one or another approach. We tackle this problem by investigating both discrete- and continuous-time mean-field theories for the susceptible-infected-susceptible (SIS) epidemic model on random networks with power-law degree distributions. We compare the discrete epidemic link equations (ELE) and continuous pair quenched mean-field (PQMF) theories with the corresponding stochastic simulations, both theories that reckon pairwise interactions explicitly. We show that ELE converges to the PQMF theory when the time step goes to zero. We performed an epidemic localization analysis considering the inverse participation ratio (IPR). Both theories present the same localization dependence on network degree exponent γ: for γ<5/2 the epidemics are localized on the maximum k-core of networks with a vanishing IPR in the infinite-size limit while, for γ>5/2, the localization happens on hubs that do not form a densely connected set and leads to a finite value of the IPR. However, the IPR and epidemic threshold of ELE depend on the time-step discretization such that a larger time step leads to more localized epidemics. A remarkable difference between discrete- and continuous-time approaches is revealed in the epidemic prevalence near the epidemic threshold, in which the discrete-time stochastic simulations indicate a mean-field critical exponent θ=1 instead of the value θ=1/(3-γ) obtained rigorously and verified numerically for the continuous-time SIS on the same networks.

Bifractality in the one-dimensional Wolf-Villain model.

We introduce a multifractal optimal detrended fluctuation analysis to study the scaling properties of the one-dimensional Wolf-Villain (WV) model for surface growth. This model produces coarsened surface morphologies for long timescales (up to 10^{9} monolayers) and its universality class remains an open problem. Our results for the multifractal exponent τ(q) reveal an effective local roughness exponent consistent with a transient given by the molecular beam epitaxy (MBE) growth regime and Edwards-Wilkinson (EW) universality class for negative and positive q values, respectively. Therefore, although the results corroborate that long-wavelength fluctuations belong to the EW class in the hydrodynamic limit, as conjectured in the recent literature, a bifractal signature of the WV model with an MBE regime at short wavelengths was observed.

Visibility graphs of critical and off-critical time series for absorbing state phase transitions.

It is possible to investigate emergence in many real systems using time-ordered data. However, classical time series analysis is usually conditioned by data accuracy and quantity. A modern method is to map time series onto graphs and study these structures using the toolbox available in complex network analysis. An important practical problem to investigate the criticality in experimental systems is to determine whether an observed time series is associated with a critical regime or not. We contribute to this problem by investigating the mapping called visibility graph (VG) of a time series generated in dynamical processes with absorbing-state phase transitions. Analyzing degree correlation patterns of the VGs, we are able to distinguish between critical and off-critical regimes. One central hallmark is an asymptotic disassortative correlation on the degree for series near the critical regime in contrast with a pure assortative correlation observed for noncritical dynamics. We are also able to distinguish between continuous (critical) and discontinuous (noncritical) absorbing state phase transitions, the second of which is commonly involved in catastrophic phenomena. The determination of critical behavior converges very quickly in higher dimensions, where many complex system dynamics are relevant.

Self-tuned criticality: Controlling a neuron near its bifurcation point via temporal correlations.

Previous work showed that the collective activity of large neuronal networks can be tamed to remain near its critical point by a feedback control that maximizes the temporal correlations of the mean-field fluctuations. Since such correlations behave similarly near instabilities across nonlinear dynamical systems, it is expected that the principle should control also low-dimensional dynamical systems exhibiting continuous or discontinuous bifurcations from fixed points to limit cycles. Here we present numerical evidence that the dynamics of a single neuron can be controlled in the vicinity of its bifurcation point. The approach is tested in two models: a two-dimensional generic excitable map and the paradigmatic FitzHugh-Nagumo neuron model. The results show that in both cases, the system can be self-tuned to its bifurcation point by modifying the control parameter according to the first coefficient of the autocorrelation function.

Effects of infection fatality ratio and social contact matrices on vaccine prioritization strategies.

Effective strategies of vaccine prioritization are essential to mitigate the impacts of severe infectious diseases. We investigate the role of infection fatality ratio (IFR) and social contact matrices on vaccination prioritization using a compartmental epidemic model fueled by real-world data of different diseases and countries. Our study confirms that massive and early vaccination is extremely effective to reduce the disease fatality if the contagion is mitigated, but the effectiveness is increasingly reduced as vaccination beginning delays in an uncontrolled epidemiological scenario. The optimal and least effective prioritization strategies depend non-linearly on epidemiological variables. Regions of the epidemiological parameter space, in which prioritizing the most vulnerable population is more effective than the most contagious individuals, depend strongly on the IFR age profile being, for example, substantially broader for COVID-19 in comparison with seasonal influenza. Demographics and social contact matrices deform the phase diagrams but do not alter their qualitative shapes.

Comparison of theoretical approaches for epidemic processes with waning immunity in complex networks.

The role of waning immunity in basic epidemic models on networks has been undervalued while being noticeably fundamental for real epidemic outbreaks. One central question is which mean-field approach is more accurate in describing the epidemic dynamics. We tackled this problem considering the susceptible-infected-recovered-susceptible (SIRS) epidemic model on networks. Two pairwise mean-field theories, one based on recurrent dynamical message-passing (rDMP) theory and the other on the pair quenched mean-field (PQMF) theory, are compared with extensive stochastic simulations on large networks of different levels of heterogeneity. For waning immunity times longer than or comparable with the recovering time, rDMP outperforms PQMF theory on power-law networks with degree distribution P(k)∼k^{-γ}. In particular, for γ>3, the epidemic threshold observed in simulations is finite, in qualitative agreement with rDMP, while PQMF leads to an asymptotically null threshold. The critical epidemic prevalence for γ>3 is localized in a finite set of vertices in the case of the PQMF theory. In contrast, the localization happens in a subextensive fraction of the network in rDMP theory. Simulations, however, indicate that localization patterns of the actual epidemic lay between the two mean-field theories, and improved theoretical approaches are necessary to understanding the SIRS dynamics.

Heterogeneous mean-field theory for two-species symbiotic processes on networks.

A simple model to study cooperation is the two-species symbiotic contact process (2SCP), in which two different species spread on a graph and interact by a reduced death rate if both occupy the same vertex, representing a symbiotic interaction. The 2SCP is known to exhibit a complex behavior with a rich phase diagram, including continuous and discontinuous transitions between the active phase and extinction. In this work, we advance the understanding of the phase transition of the 2SCP on uncorrelated networks by developing a heterogeneous mean-field (HMF) theory, in which the heterogeneity of contacts is explicitly reckoned. The HMF theory for networks with power-law degree distribution shows that the region of bistability (active and inactive phases) in the phase diagram shrinks as the heterogeneity level is increased by reducing the degree exponent. Finite-size analysis reveals a complex behavior where a pseudodiscontinuous transition at a finite size can be converted into a continuous one in the thermodynamic limit, depending on degree exponent and symbiotic coupling. The theoretical results are supported by extensive numerical simulations.

Data-driven approach in a compartmental epidemic model to assess undocumented infections.

Nowcasting and forecasting of epidemic spreading rely on incidence series of reported cases to derive the fundamental epidemiological parameters for a given pathogen. Two relevant drawbacks for predictions are the unknown fractions of undocumented cases and levels of nonpharmacological interventions, which span highly heterogeneously across different places and times. We describe a simple data-driven approach using a compartmental model including asymptomatic and pre-symptomatic contagions that allows to estimate both the level of undocumented infections and the value of effective reproductive number R t from time series of reported cases, deaths, and epidemiological parameters. The method was applied to epidemic series for COVID-19 across different municipalities in Brazil allowing to estimate the heterogeneity level of under-reporting across different places. The reproductive number derived within the current framework is little sensitive to both diagnosis and infection rates during the asymptomatic states. The methods described here can be extended to more general cases if data is available and adapted to other epidemiological approaches and surveillance data.

High prevalence regimes in the pair-quenched mean-field theory for the susceptible-infected-susceptible model on networks.

Reckoning of pairwise dynamical correlations significantly improves the accuracy of mean-field theories and plays an important role in the investigation of dynamical processes in complex networks. In this work, we perform a nonperturbative numerical analysis of the quenched mean-field theory (QMF) and the inclusion of dynamical correlations by means of the pair quenched mean-field (PQMF) theory for the susceptible-infected-susceptible model on synthetic and real networks. We show that the PQMF considerably outperforms the standard QMF theory on synthetic networks of distinct levels of heterogeneity and degree correlations, providing extremely accurate predictions when the system is not too close to the epidemic threshold, while the QMF theory deviates substantially from simulations for networks with a degree exponent γ>2.5. The scenario for real networks is more complicated, still with PQMF significantly outperforming the QMF theory. However, despite its high accuracy for most investigated networks, in a few cases PQMF deviations from simulations are not negligible. We found correlations between accuracy and average shortest path, while other basic network metrics seem to be uncorrelated with the theory accuracy. Our results show the viability of the PQMF theory to investigate the high-prevalence regimes of recurrent-state epidemic processes in networks, a regime of high applicability.

Nonmassive immunization to contain spreading on complex networks.

Optimal strategies for epidemic containment are focused on dismantling the contact network through effective immunization with minimal costs. However, network fragmentation is seldom accessible in practice and may present extreme side effects. In this work, we investigate the epidemic containment immunizing population fractions far below the percolation threshold. We report that moderate and weakly supervised immunizations can lead to finite epidemic thresholds of the susceptible-infected-susceptible model on scale-free networks by changing the nature of the transition from a specific motif to a collectively driven process. Both pruning of efficient spreaders and increasing of their mutual separation are necessary for a collective activation. Fractions of immunized vertices needed to eradicate the epidemics which are much smaller than the percolation thresholds were observed for a broad spectrum of real networks considering targeted or acquaintance immunization strategies. Our work contributes for the construction of optimal containment, preserving network functionality through nonmassive and viable immunization strategies.

Dynamical correlations and pairwise theory for the symbiotic contact process on networks.

The two-species symbiotic contact process (2SCP) is a stochastic process in which each vertex of a graph may be vacant or host at most one individual of each species. Vertices with both species have a reduced death rate, representing a symbiotic interaction, while the dynamics evolves according to the standard (single species) contact process rules otherwise. We investigate the role of dynamical correlations on the 2SCP on homogeneous and heterogeneous networks using pairwise mean-field theory. This approach is compared with the ordinary one-site theory and stochastic simulations. We show that our approach significantly outperforms the one-site theory. In particular, the stationary state of the 2SCP model on random regular networks is very accurately reproduced by the pairwise mean-field, even for relatively small values of vertex degree, where expressive deviations of the standard mean-field are observed. The pairwise approach is also able to capture the transition points accurately for heterogeneous networks and provides rich phase diagrams with transitions not predicted by the one-site method. Our theoretical results are corroborated by extensive numerical simulations.

Effects of a kinetic barrier on limited-mobility interface growth models.

The role played by a kinetic barrier originated by out-of-plane step edge diffusion, introduced by Leal et al. [J. Phys.: Condens. Matter 23, 292201 (2011)JCOMEL0953-898410.1088/0953-8984/23/29/292201], is investigated in the Wolf-Villain and Das Sarma-Tamborenea models with short-range diffusion. Using large-scale simulations, we observe that this barrier is sufficient to produce growth instability, forming quasiregular mounds in one and two dimensions. The characteristic surface length saturates quickly indicating a uncorrelated growth of the three-dimensional structures, which is also confirmed by a growth exponent ß=1/2. The out-of-plane particle current shows a large reduction of the downward flux in the presence of the kinetic barrier enhancing, consequently, the net upward diffusion and the formation of three-dimensional self-assembled structures.

Local roughness exponent in the nonlinear molecular-beam-epitaxy universality class in one dimension.

We report local roughness exponents, α_{loc}, for three interface growth models in one dimension which are believed to belong to the nonlinear molecular-beam-epitaxy (nMBE) universality class represented by the Villain-Lais-Das Sarma (VLDS) stochastic equation. We applied an optimum detrended fluctuation analysis (ODFA) [Luis et al., Phys. Rev. E 95, 042801 (2017)2470-004510.1103/PhysRevE.95.042801] and compared the outcomes with standard detrending methods. We observe in all investigated models that ODFA outperforms the standard methods providing exponents in the narrow interval α_{loc}^{}∈[0.96,0.98] quantitatively consistent with two-loop renormalization group predictions for the VLDS equation. In particular, these exponent values are calculated for the Clarke-Vvdensky and Das Sarma-Tamborenea models characterized by very strong corrections to the scaling, for which large deviations of these values had been reported. Our results strongly support the absence of anomalous scaling in the nMBE universality class and the existence of corrections in the form α_{loc}^{}=1-ε of the one-loop renormalization group analysis of the VLDS equation.

Eden model with nonlocal growth rules and kinetic roughening in biological systems.

We investigate an off-lattice Eden model where the growth of new cells is performed with a probability dependent on the availability of resources coming externally towards the growing aggregate. The concentration of nutrients necessary for replication is assumed to be proportional to the voids connecting the replicating cells to the outer region, introducing therefore a nonlocal dependence on the replication rule. Our simulations point out that the Kadar-Parisi-Zhang (KPZ) universality class is a transient that can last for long periods in plentiful environments. For conditions of nutrient scarcity, we observe a crossover from regular KPZ to unstable growth, passing by a transient consistent with the quenched KPZ class at the pinning transition. Our analysis sheds light on results reporting on the universality class of kinetic roughening in akin experiments of biological growth.

Robustness and fragility of the susceptible-infected-susceptible epidemic models on complex networks.

We analyze two alterations of the standard susceptible-infected-susceptible (SIS) dynamics that preserve the central properties of spontaneous healing and infection capacity of a vertex increasing unlimitedly with its degree. All models have the same epidemic thresholds in mean-field theories but depending on the network properties, simulations yield a dual scenario, in which the epidemic thresholds of the modified SIS models can be either dramatically altered or remain unchanged in comparison with the standard dynamics. For uncorrelated synthetic networks having a power-law degree distribution with exponent γ<5/2, the SIS dynamics are robust exhibiting essentially the same outcomes for all investigated models. A threshold in better agreement with the heterogeneous rather than quenched mean-field theory is observed in the modified dynamics for exponent γ>5/2. Differences are more remarkable for γ>3, where a finite threshold is found in the modified models in contrast with the vanishing threshold of the original one. This duality is elucidated in terms of epidemic lifespan on star graphs. We verify that the activation of the modified SIS models is triggered in the innermost component of the network given by a k-core decomposition for γ<3 while it happens only for γ<5/2 in the standard model. For γ>3, the activation in the modified dynamics is collective involving essentially the whole network while it is triggered by hubs in the standard SIS. The duality also appears in the finite-size scaling of the critical quantities where mean-field behaviors are observed for the modified but not for the original dynamics. Our results feed the discussions about the most proper conceptions of epidemic models to describe real systems and the choices of the most suitable theoretical approaches to deal with these models.

Griffiths phases in infinite-dimensional, non-hierarchical modular networks.

Griffiths phases (GPs), generated by the heterogeneities on modular networks, have recently been suggested to provide a mechanism, rid of fine parameter tuning, to explain the critical behavior of complex systems. One conjectured requirement for systems with modular structures was that the network of modules must be hierarchically organized and possess finite dimension. We investigate the dynamical behavior of an activity spreading model, evolving on heterogeneous random networks with highly modular structure and organized non-hierarchically. We observe that loosely coupled modules act as effective rare-regions, slowing down the extinction of activation. As a consequence, we find extended control parameter regions with continuously changing dynamical exponents for single network realizations, preserved after finite size analyses, as in a real GP. The avalanche size distributions of spreading events exhibit robust power-law tails. Our findings relax the requirement of hierarchical organization of the modular structure, which can help to rationalize the criticality of modular systems in the framework of GPs.

Activation thresholds in epidemic spreading with motile infectious agents on scale-free networks.

We investigate a fermionic susceptible-infected-susceptible model with the mobility of infected individuals on uncorrelated scale-free networks with power-law degree distributions P ( k ) ∼ k - γ of exponents 2 < γ < 3 . Two diffusive processes with diffusion rate D of an infected vertex are considered. In the standard diffusion, one of the nearest-neighbors is chosen with equal chance, while in the biased diffusion, this choice happens with probability proportional to the neighbor's degree. A non-monotonic dependence of the epidemic threshold on D with an optimum diffusion rate D ∗ , for which the epidemic spreading is more efficient, is found for standard diffusion while monotonic decays are observed in the biased case. The epidemic thresholds go to zero as the network size is increased and the form that this happens depends on the diffusion rule and the degree exponent. We analytically investigated the dynamics using quenched and heterogeneous mean-field theories. The former presents, in general, a better performance for standard and the latter for biased diffusion models, indicating different activation mechanisms of the epidemic phases that are rationalized in terms of hubs or max k -core subgraphs.

Optimal detrended fluctuation analysis as a tool for the determination of the roughness exponent of the mounded surfaces.

We present an optimal detrended fluctuation analysis (DFA) and apply it to evaluate the local roughness exponent in nonequilibrium surface growth models with mounded morphology. Our method consists in analyzing the height fluctuations computing the shortest distance of each point of the profile to a detrending curve that fits the surface within the investigated interval. We compare the optimal DFA (ODFA) with both the standard DFA and nondetrended analysis. We validate the ODFA method considering a one-dimensional model in the Kardar-Parisi-Zhang universality class starting from a mounded initial condition. We applied the methods to the Clarke-Vvedensky (CV) model in 2+1 dimensions with thermally activated surface diffusion and absence of step barriers. It is expected that this model belongs to the nonlinear molecular beam epitaxy (nMBE) universality class. However, an explicit observation of the roughness exponent in agreement with the nMBE class was still missing. The effective roughness exponent obtained with ODFA agrees with the value expected for the nMBE class, whereas using the other methods it does not agree. We also characterize the transient anomalous scaling of the CV model and obtained that the corresponding exponent is in agreement with the value reported for other nMBE models with weaker corrections to the scaling.

Sampling methods for the quasistationary regime of epidemic processes on regular and complex networks.

A major hurdle in the simulation of the steady state of epidemic processes is that the system will unavoidably visit an absorbing, disease-free state at sufficiently long times due to the finite size of the networks where epidemics evolves. In the present work, we compare different quasistationary (QS) simulation methods where the absorbing states are suitably handled and the thermodynamical limit of the original dynamics can be achieved. We analyze the standard QS (SQS) method, where the sampling is constrained to active configurations, the reflecting boundary condition (RBC), where the dynamics returns to the pre-absorbing configuration, and hub reactivation (HR), where the most connected vertex of the network is reactivated after a visit to an absorbing state. We apply the methods to the contact process (CP) and susceptible-infected-susceptible (SIS) models on regular and scale free networks. The investigated methods yield the same epidemic threshold for both models. For CP, that undergoes a standard collective phase transition, the methods are equivalent. For SIS, whose phase transition is ruled by the hub mutual reactivation, the SQS and HR methods are able to capture localized epidemic phases while RBC is not. We also apply the autocorrelation time as a tool to characterize the phase transition and observe that this analysis provides the same finite-size scaling exponents for the critical relaxation time for the investigated methods. Finally, we verify the equivalence between RBC method and a weak external field for epidemics on networks.

Scaling, cumulant ratios, and height distribution of ballistic deposition in 3+1 and 4+1 dimensions.

We investigate the origin of the scaling corrections in ballistic deposition models in high dimensions using the method proposed by Alves et al. [Phys. Rev. E 90, 052405 (2014)PLEEE81539-375510.1103/PhysRevE.90.052405] in d=2+1 dimensions, where the intrinsic width associated with the fluctuations of the height increments during the deposition processes is explicitly taken into account. In the present work, we show that this concept holds for d=3+1 and 4+1 dimensions. We have found that growth and roughness exponents and dimensionless cumulant ratios are in agreement with other models, presenting small finite-time corrections to the scaling, that in principle belong to the Kardar-Parisi-Zhang (KPZ) universality class in both d=3+1 and 4+1. Our results constitute further evidence that the upper critical dimension of the KPZ class, if it exists, is larger than 4.