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

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.

The impact of positive and negative information on SIR-like epidemics in delayed multiplex networks

In order to better study the interaction between epidemic propagation and information diffusion, a new coupling model on multiplex networks with time delay is put forward in this paper. One layer represents the information diffusion about epidemics. There is not only information about the positive prevention of infectious diseases but also negative preventive information. Meanwhile, the dissemination of information at this layer will be influenced by the mass media, which can convey positive and reliable preventive measures to help the public avoid exposure to contagion. The other layer represents the transmission of infectious diseases, and the public in this layer no longer only exchange information related to infectious diseases in the virtual social network like the information layer but spread infectious diseases through contact among people. The classical SIR model is used to model for epidemic propagation. Since each infected individual needs to spend enough time to recover, the infected one at one time does not necessarily change to the recovered one at the next time, so time delay is an essential factor to be considered in the model. Based on the microscopic Markov chain approach, this paper obtains an explicit expression for epidemic threshold in the two-layered multiplex networks with time delay, which reveals some main factors affecting epidemic threshold. In particular, the time delay has a noticeable effect on the epidemic threshold to some extent. Finally, the influence of these main factors on the epidemic threshold and their interaction are proved through numerical simulations.

Population persistence under two conservation measures: Paradox of habitat protection in a patchy environment

Anthropogenic modification of natural habitats is a growing threat to biodiversity and ecosystem services. The protection of biospecies has become increasingly important. Here, we pay attention to a single species as a conservation target. The species has three processes: reproduction, death and movement. Two different measures of habitat protection are introduced. One is partial protection in a single habitat (patch); the mortality rate of the species is reduced inside a rectangular area. The other is patch protection in a two-patch system, where only the mortality rate in a particular patch is reduced. For the one-patch system, we carry out computer simulations of a stochastic cellular automaton for a "contact process". Individual movements follow random walking. For the two-patch system, we assume an individual migrates into the empty cell in the destination patch. The reaction-diffusion equation (RDE) is derived, whereby the recently developed "swapping migration" is used. It is found that both measures are mostly effective for population persistence. However, comparing the results of the two measures revealed different behaviors. ⅰ) In the case of the one-patch system, the steady-state densities in protected areas are always higher than those in wild areas. However, in the two-patch system, we have found a paradox: the densities in protected areas can be lower than those in wild areas. ⅱ) In the two-patch system, we have found another paradox: the total density in both patches can be lower, even though the proportion of the protected area is larger. Both paradoxes clearly occur for the RDE with swapping migration.

Probabilistic predictions of SIS epidemics on networks based on population-level observations

We predict the future course of ongoing susceptible-infected-susceptible (SIS) epidemics on regular, Erdős-Rényi and Barabási-Albert networks. It is known that the contact network influences the spread of an epidemic within a population. Therefore, observations of an epidemic, in this case at the population-level, contain information about the underlying network. This information, in turn, is useful for predicting the future course of an ongoing epidemic. To exploit this in a prediction framework, the exact high-dimensional stochastic model of an SIS epidemic on a network is approximated by a lower-dimensional surrogate model. The surrogate model is based on a birth-and-death process; the effect of the underlying network is described by a parametric model for the birth rates. We demonstrate empirically that the surrogate model captures the intrinsic stochasticity of the epidemic once it reaches a point from which it will not die out. Bayesian parameter inference allows for uncertainty about the model parameters and the class of the underlying network to be incorporated directly into probabilistic predictions. An evaluation of a number of scenarios shows that in most cases the resulting prediction intervals adequately quantify the prediction uncertainty. As long as the population-level data is available over a long-enough period, even if not sampled frequently, the model leads to excellent predictions where the underlying network is correctly identified and prediction uncertainty mainly reflects the intrinsic stochasticity of the spreading epidemic. For predictions inferred from shorter observational periods, uncertainty about parameters and network class dominate prediction uncertainty. The proposed method relies on minimal data at population-level, which is always likely to be available. This, combined with its numerical efficiency, makes the proposed method attractive to be used either as a standalone inference and prediction scheme or in conjunction with other inference and/or predictive models.

From subcritical behavior to a correlation-induced transition in rumor models

Rumors and information spreading emerge naturally from human-to-human interactions and have a growing impact on our everyday life due to increasing and faster access to information, whether trustworthy or not. A popular mathematical model for spreading rumors, data, or news is the Maki–Thompson model. Mean-field approximations suggested that this model does not have a phase transition, with rumors always reaching a fraction of the population. Conversely, here, we show that a continuous phase transition is present in this model. Moreover, we explore a modified version of the Maki–Thompson model that includes a forgetting mechanism, changing the Markov chain’s nature and allowing us to use a plethora of analytic and numeric methods. Particularly, we characterize the subcritical behavior, where the lifespan of a rumor increases as the spreading rate drops, following a power-law relationship. Our findings show that the dynamic behavior of rumor models is much richer than shown in previous investigations.

Rumors and information spreading emerge naturally from human-to-human interaction and have a growing impact on people’s lives due to increasing and faster access to information, whether trustworthy or not. The authors study the Maki–Thompson rumor model and its variation, revealing a phase transition and providing insights into the nature of this transition.

Master equation analysis of mesoscopic localization in contagion dynamics on higher-order networks

Simple models of infectious diseases tend to assume random mixing of individuals, but real interactions are not random pairwise encounters: they occur within various types of gatherings such as workplaces, households, schools, and concerts, best described by a higher-order network structure. We model contagions on higher-order networks using group-based approximate master equations, in which we track all states and interactions within a group of nodes and assume a mean-field coupling between them. Using the susceptible-infected-susceptible dynamics, our approach reveals the existence of a mesoscopic localization regime, where a disease can concentrate and self-sustain only around large groups in the network overall organization. In this regime, the phase transition is smeared, characterized by an inhomogeneous activation of the groups. At the mesoscopic level, we observe that the distribution of infected nodes within groups of the same size can be very dispersed, even bimodal. When considering heterogeneous networks, both at the level of nodes and at the level of groups, we characterize analytically the region associated with mesoscopic localization in the structural parameter space. We put in perspective this phenomenon with eigenvector localization and discuss how a focus on higher-order structures is needed to discern the more subtle localization at the mesoscopic level. Finally, we discuss how mesoscopic localization affects the response to structural interventions and how this framework could provide important insights for a broad range of dynamics.

An epidemiological model with voluntary quarantine strategies governed by evolutionary game dynamics

During pandemic events, strategies such as social distancing can be fundamental to reduce simultaneous infections and mitigate the disease spreading, which is very relevant to the risk of a healthcare system collapse. Although these strategies can be recommended, or even imposed, their actual implementation may depend on the population perception of the risks associated with a potential infection. The current COVID-19 crisis, for instance, is showing that some individuals are much more prone than others to remain isolated. To better understand these dynamics, we propose an epidemiological SIR model that uses evolutionary game theory for combining in a single process social strategies, individual risk perception, and viral spreading. In particular, we consider a disease spreading through a population, whose agents can choose between self-isolation and a lifestyle careless of any epidemic risk. The strategy adoption is individual and depends on the perceived disease risk compared to the quarantine cost. The game payoff governs the strategy adoption, while the epidemic process governs the agent's health state. At the same time, the infection rate depends on the agent's strategy while the perceived disease risk depends on the fraction of infected agents. Our results show recurrent infection waves, which are usually seen in previous historic epidemic scenarios with voluntary quarantine. In particular, such waves re-occur as the population reduces disease awareness. Notably, the risk perception is found to be fundamental for controlling the magnitude of the infection peak, while the final infection size is mainly dictated by the infection rates. Low awareness leads to a single and strong infection peak, while a greater disease risk leads to shorter, although more frequent, peaks. The proposed model spontaneously captures relevant aspects of a pandemic event, highlighting the fundamental role of social strategies.

An overview of epidemic models with phase transitions to absorbing states running on top of complex networks

Dynamical systems running on the top of complex networks have been extensively investigated for decades. But this topic still remains among the most relevant issues in complex network theory due to its range of applicability. The contact process (CP) and the susceptible-infected-susceptible (SIS) model are used quite often to describe epidemic dynamics. Despite their simplicity, these models are robust to predict the kernel of real situations. In this work, we review concisely both processes that are well-known and very applied examples of models that exhibit absorbing-state phase transitions. In the epidemic scenario, individuals can be infected or susceptible. A phase transition between a disease-free (absorbing) state and an active stationary phase (where a fraction of the population is infected) are separated by an epidemic threshold. For the SIS model, the central issue is to determine this epidemic threshold on heterogeneous networks. For the CP model, the main interest is to relate critical exponents with statistical properties of the network.

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.

Impact of the distribution of recovery rates on disease spreading in complex networks

We study a general epidemic model with arbitrary recovery rate distributions. This simple deviation from the standard setup is sufficient to prove that heterogeneity in the dynamical parameters can be as important as the more studied structural heterogeneity. Our analytical solution is able to predict the shift in the critical properties induced by heterogeneous recovery rates. We find that the critical value of infectivity tends to be smaller than the one predicted by quenched mean-field approaches in the homogeneous case and that it can be linked to the variance of the recovery rates. Our findings also illustrate the role of dynamical-structural correlations, where we allow a power-law network to dynamically behave as a homogeneous structure by an appropriate tuning of its recovery rates. Overall, our results demonstrate that heterogeneity in the recovery rates, eventually in all dynamical parameters, is as important as the structural heterogeneity.

Complex Networks: a Mini-review

Network analysis is a powerful tool that provides us a fruitful framework to describe phenomena related to social, technological, and many other real-world complex systems. In this paper, we present a brief review about complex networks including fundamental quantities, examples of network models, and the essential role of network topology in the investigation of dynamical processes as epidemics, rumor spreading, and synchronization. A quite of advances have been provided in this field, and many other authors also review the main contributions in this area over the years. However, we show an overview from a different perspective. Our aim is to provide basic information to a broad audience and more detailed references for those who would like to learn deeper the topic.

Onset of synchronization of Kuramoto oscillators in scale-free networks

Despite the great attention devoted to the study of phase oscillators on complex networks in the last two decades, it remains unclear whether scale-free networks exhibit a nonzero critical coupling strength for the onset of synchronization in the thermodynamic limit. Here, we systematically compare predictions from the heterogeneous degree mean-field (HMF) and the quenched mean-field (QMF) approaches to extensive numerical simulations on large networks. We provide compelling evidence that the critical coupling vanishes as the number of oscillators increases for scale-free networks characterized by a power-law degree distribution with an exponent 2<γ≤3, in line with what has been observed for other dynamical processes in such networks. For γ>3, we show that the critical coupling remains finite, in agreement with HMF calculations and highlight phenomenological differences between critical properties of phase oscillators and epidemic models on scale-free networks. Finally, we also discuss at length a key choice when studying synchronization phenomena in complex networks, namely, how to normalize the coupling between oscillators.

Metapopulation model for a prey-predator system: Nonlinear migration due to the finite capacities of patches

Many species live in spatially separated patches, and individuals can migrate between patches through paths. In real ecosystems, the capacities of patches are finite. If a patch is already occupied by the individuals of some species, then the migration into the patch is impossible. In the present paper, we deal with prey-predator system composed of two patches. Each patch contains a limited number of cells, where the cell is either empty or occupied by an individual of prey or predator. We introduce "swapping migration" defined by the exchange between occupied and empty cells. An individual can migrate, only when there are empty cells in the destination patch. Reaction-migration equations in prey-predator system are presented, where the migration term forms nonlinear function of densities. We numerically solve equilibrium densities, and find that the population dynamics are largely affected by nonlinear migration. Not only extinction points but also the responses to the environmental changes crucially depend on the patch capacities.

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.