Non-Markovian infection spread dramatically alters the susceptible-infected-susceptible epidemic threshold in networks

Impacts of memory-based and non-memory-based adoption in social contagion

In information diffusion within social networks, whether individuals adopt information often depends on the current and past information they receive. Some individuals adopt based on current information (i.e., no memory), while others rely on past information (i.e., with memory). Previous studies mainly focused on irreversible processes, such as the classic susceptible-infected and susceptible-infected-recovered threshold models, with less attention to reversible processes like the susceptible-infected-susceptible model. In this paper, we propose a susceptible-adopted-susceptible threshold model to study the competition between these two types of nodes and its impact on information diffusion. We also examine how memory length and differences in the adoption thresholds affect the diffusion process. First, we develop homogeneous and heterogeneous mean-field theories that accurately predict simulation results. Numerical simulations reveal that when node adoption thresholds are equal, increasing memory length raises the propagation threshold, thereby suppressing diffusion. When the adoption thresholds of the two node types differ, such as non-memory nodes having a lower threshold than memory-based nodes, increasing the memory length of the latter has little effect on the propagation threshold of the former. However, when the adoption threshold of the non-memory nodes is much higher than that of the memory-based nodes, increasing the memory length of the latter significantly suppresses the propagation threshold of the non-memory nodes. In heterogeneous networks, we find that as the degree of heterogeneity increases, the outbreak size of epidemic diffusion becomes smaller, while the propagation threshold also decreases. This work offers deeper insights into the impact of memory-based and non-memory-based adoption in social contagion.

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.

Higher-order correlations reveal complex memory in temporal hypergraphs

Many real-world complex systems are characterized by interactions in groups that change in time. Current temporal network approaches, however, are unable to describe group dynamics, as they are based on pairwise interactions only. Here, we use time-varying hypergraphs to describe such systems, and we introduce a framework based on higher-order correlations to characterize their temporal organization. The analysis of human interaction data reveals the existence of coherent and interdependent mesoscopic structures, thus capturing aggregation, fragmentation and nucleation processes in social systems. We introduce a model of temporal hypergraphs with non-Markovian group interactions, which reveals complex memory as a fundamental mechanism underlying the emerging pattern in the data.

Transition from time-variant to static networks: Timescale separation in N-intertwined mean-field approximation of susceptible-infectious-susceptible epidemics

We extend the N-intertwined mean-field approximation (NIMFA) for the susceptible-infectious-susceptible (SIS) epidemiological process to time-varying networks. Processes on time-varying networks are often analyzed under the assumption that the process and network evolution happen on different timescales. This approximation is called timescale separation. We investigate timescale separation between disease spreading and topology updates of the network. We introduce the transition times [under T]̲(r) and T[over ¯](r) as the boundaries between the intermediate regime and the annealed (fast changing network) and quenched (static network) regimes, respectively, for a fixed accuracy tolerance r. By analyzing the convergence of static NIMFA processes, we analytically derive upper and lower bounds for T[over ¯](r). Our results provide insights and bounds on the time of convergence to the steady state of the static NIMFA SIS process. We show that, under our assumptions, the upper-transition time T[over ¯](r) is almost entirely determined by the basic reproduction number R_{0} of the network. The value of the upper-transition time T[over ¯](r) around the epidemic threshold is large, which agrees with the current understanding that some real-world epidemics cannot be approximated with the aforementioned timescale separation.

Scalable parallel and distributed simulation of an epidemic on a graph

We propose an algorithm to simulate Markovian SIS epidemics with homogeneous rates and pairwise interactions on a fixed undirected graph, assuming a distributed memory model of parallel programming and limited bandwidth. This setup can represent a broad class of simulation tasks with compartmental models. Existing solutions for such tasks are sequential by nature. We provide an innovative solution that makes trade-offs between statistical faithfulness and parallelism possible. We offer an implementation of the algorithm in the form of pseudocode in the Appendix. Also, we analyze its algorithmic complexity and its induced dynamical system. Finally, we design experiments to show its scalability and faithfulness. In our experiments, we discover that graph structures that admit good partitioning schemes, such as the ones with clear community structures, together with the correct application of a graph partitioning method, can lead to better scalability and faithfulness. We believe this algorithm offers a way of scaling out, allowing researchers to run simulation tasks at a scale that was not accessible before. Furthermore, we believe this algorithm lays a solid foundation for extensions to more advanced epidemic simulations and graph dynamics in other fields.

Generalized epidemic model incorporating non-Markovian infection processes and waning immunity

The Markovian approach, which assumes exponentially distributed interinfection times, is dominant in epidemic modeling. However, this assumption is unrealistic as an individual's infectiousness depends on its viral load and varies over time. In this paper, we present a Susceptible-Infected-Recovered-Vaccinated-Susceptible epidemic model incorporating non-Markovian infection processes. The model can be easily adapted to accurately capture the generation time distributions of emerging infectious diseases, which is essential for accurate epidemic prediction. We observe noticeable variations in the transient behavior under different infectiousness profiles and the same basic reproduction number R_{0}. The theoretical analyses show that only R_{0} and the mean immunity period of the vaccinated individuals have an impact on the critical vaccination rate needed to achieve herd immunity. A vaccination level at the critical vaccination rate can ensure a very low incidence among the population in the case of future epidemics, regardless of the infectiousness profiles.

The coevolution of the spread of a disease and competing opinions in multiplex networks

The COVID-19 pandemic has resulted in a proliferation of conflicting opinions on physical distancing across various media platforms, which has had a significant impact on human behavior and the transmission dynamics of the disease. Inspired by this social phenomenon, we present a novel UAP-SIS model to study the interaction between conflicting opinions and epidemic spreading in multiplex networks, in which individual behavior is based on diverse opinions. We distinguish susceptibility and infectivity among individuals who are unaware, pro-physical distancing and anti-physical distancing, and we incorporate three kinds of mechanisms for generating individual awareness. The coupled dynamics are analyzed in terms of a microscopic Markov chain approach that encompasses the aforementioned elements. With this model, we derive the epidemic threshold which is related to the diffusion of competing opinions and their coupling configuration. Our findings demonstrate that the transmission of the disease is shaped in a significant manner by conflicting opinions, due to the complex interaction between such opinions and the disease itself. Furthermore, the implementation of awareness-generating mechanisms can help to mitigate the overall prevalence of the epidemic, and global awareness and self-awareness can be interchangeable in certain instances. To effectively curb the spread of epidemics, policymakers should take steps to regulate social media and promote physical distancing as the mainstream opinion.

Aging in binary-state models: The Threshold model for complex contagion

We study the non-Markovian effects associated with aging for binary-state dynamics in complex networks. Aging is considered as the property of the agents to be less prone to change their state the longer they have been in the current state, which gives rise to heterogeneous activity patterns. In particular, we analyze aging in the Threshold model, which has been proposed to explain the process of adoption of new technologies. Our analytical approximations give a good description of extensive Monte Carlo simulations in Erdős-Rényi, random-regular and Barabási-Albert networks. While aging does not modify the cascade condition, it slows down the cascade dynamics towards the full-adoption state: the exponential increase of adopters in time from the original model is replaced by a stretched exponential or power law, depending on the aging mechanism. Under several approximations, we give analytical expressions for the cascade condition and for the exponents of the adopters' density growth laws. Beyond random networks, we also describe by Monte Carlo simulations the effects of aging for the Threshold model in a two-dimensional lattice.

Exploring sexual contact networks by analyzing a nationwide commercial-sex review website

Understanding the structure of human sexual contact networks is vital in a broad range of disciplines, including sociology, biology, public health, and anthropology. However, sexual contact networks are yet to be understood because technical and privacy issues make it difficult to conduct accurate, large-scale surveys. In this study, we surveyed data openly available on one of the largest adult entertainment websites in Japan, where male clients (MCs) can write online customer reviews of female commercial sex workers (FCSWs). In particular, our investigation focused on a type of establishment called "soapland," the only type of sex industry in Japan where sexual intercourse is publicly permitted. Soaplands are scattered throughout Japan, and the study website covers approximately 66% of them. Using such a vast amount of data on a nationwide scale, we clarified the network structure of commercial sex, characterized by small-world, scale-free, and disassortative mating properties. To study geographical characteristics, we compared the resulting network with three different artificially generated networks via the random rewiring of links. Moreover, we considered a simple epidemic model on the resulting network, and investigated whether it would be more effective to provide infection control measures to FCSWs or MCs. We determined that active FCSWs constitute an important pathway of infection propagation in commercial sex networks, but MCs also play an essential role as weak ties.

A new buffering theory of social support and psychological stress

A dynamical model linking stress, social support, and health has been recently proposed and numerically analyzed from a classical point of view of integer-order calculus. Although interesting observations have been obtained in this way, the present work conducts a fractional-order analysis of that model. Under a periodic forcing of an environmental stress variable, the perceived stress has been analyzed through bifurcation diagrams and two well-known metrics of entropy and complexity, such as spectral entropy and C0 complexity. The results obtained by numerical simulations have shown novel insights into how stress evolves with frequency and amplitude of the perturbation, as well as with initial conditions for the system variables. More precisely, it has been observed that stress can alternate between chaos, periodic oscillations, and stable behaviors as the fractional order varies. Moreover, the perturbation frequency has revealed a narrow interval for the chaotic oscillations, while its amplitude may present different values indicating a low sensitivity regarding chaos generation. Also, the perceived stress has been noted to be highly sensitive to initial conditions for the symptoms of stress-related ill-health and for the social support received from family and friends. This work opens new directions of research whereby fractional calculus might offer more insight into psychology, life sciences, mental disorders, and stress-free well-being.

Estimation of the basic reproduction number of COVID-19 from the incubation period distribution

The estimates of the future course of spreading of the SARS-CoV-2 virus are frequently based on Markovian models in which the duration of residence in any compartment is exponentially distributed. Accordingly, the basic reproduction number R 0 is also determined from formulae where it is related to the parameters of such models. The observations show that the start of infectivity of an individual appears nearly at the same time as the onset of symptoms, while the distribution of the incubation period is not an exponential. Therefore, we propose a method for estimation of R 0 for COVID-19 based on the empirical incubation period distribution and assumed very short infectivity period that lasts only few days around the onset of symptoms. We illustrate this venerable approach to estimate R 0 for six major European countries in the first wave of the epidemic. The calculations show that even if the infectivity starts 2 days before the onset of symptoms and stops instantly when they appear (immediate isolation), the value of R 0 is larger than that from the classical, SIR model. For more realistic cases, when only individuals with mild symptoms spread the virus for few days after onset of symptoms, the respective values are even larger. This implies that calculations of R 0 and other characteristics of spreading of COVID-19 based on the classical, Markovian approaches should be taken very cautiously.

Endemic state equivalence between non-Markovian SEIS and Markovian SIS model in complex networks

In the light of several major epidemic events that emerged in the past two decades, and emphasized by the COVID-19 pandemics, the non-Markovian spreading models occurring on complex networks gained significant attention from the scientific community. Following this interest, in this article, we explore the relations that exist between the mean-field approximated non-Markovian SEIS (Susceptible-Exposed-Infectious-Susceptible) and the classical Markovian SIS, as basic reoccurring virus spreading models in complex networks. We investigate the similarities and seek for equivalences both for the discrete-time and the continuous-time forms. First, we formally introduce the continuous-time non-Markovian SEIS model, and derive the epidemic threshold in a strict mathematical procedure. Then we present the main result of the paper that, providing certain relations between process parameters hold, the stationary-state solutions of the status probabilities in the non-Markovian SEIS may be found from the stationary state probabilities of the Markovian SIS model. This result has a two-fold significance. First, it simplifies the computational complexity of the non-Markovian model in practical applications, where only the stationary distributions of the state probabilities are required. Next, it defines the epidemic threshold of the non-Markovian SEIS model, without the necessity of a thrall mathematical analysis. We present this result both in analytical form, and confirm the result through numerical simulations. Furthermore, as of secondary importance, in an analytical procedure we show that each Markovian SIS may be represented as non-Markovian SEIS model.

Non-Markovian SIR epidemic spreading model of COVID-19

We introduce non-Markovian SIR epidemic spreading model inspired by the characteristics of the COVID-19, by considering discrete- and continuous-time versions. The distributions of infection intensity and recovery period may take an arbitrary form. By taking corresponding choice of these functions, it is shown that the model reduces to the classical Markovian case. The epidemic threshold is analytically determined for arbitrary functions of infectivity and recovery and verified numerically. The relevance of the model is shown by modeling the first wave of the epidemic in Italy, Spain and the UK, in the spring, 2020.

Balancing Quarantine and Self-Distancing Measures in Adaptive Epidemic Networks

We study the relative importance of two key control measures for epidemic spreading: endogenous social self-distancing and exogenous imposed quarantine. We use the framework of adaptive networks, moment-closure, and ordinary differential equations to introduce new model types of susceptible-infected-recovered (SIR) dynamics. First, we compare computationally expensive, adaptive network simulations with their corresponding computationally efficient ODE equivalents and find excellent agreement. Second, we discover that there exists a critical curve in parameter space for the epidemic threshold, which suggests a mutual compensation effect between the two mitigation strategies: as long as social distancing and quarantine measures are both sufficiently strong, large outbreaks are prevented. Third, we study the total number of infected and the maximum peak during large outbreaks using a combination of analytical estimates and numerical simulations. Also for large outbreaks we find a similar compensation mechanism as for the epidemic threshold. This means that if there is little incentive for social distancing in a population, drastic quarantining is required, and vice versa. Both pure scenarios are unrealistic in practice. The new models show that only a combination of measures is likely to succeed to control epidemic spreading. Fourth, we analytically compute an upper bound for the total number of infected on adaptive networks, using integral estimates in combination with a moment-closure approximation on the level of an observable. Our method allows us to elegantly and quickly check and cross-validate various conjectures about the relevance of different network control measures. In this sense it becomes possible to adapt also other models rapidly to new epidemic challenges.

Dynamic survival analysis for non-Markovian epidemic models

We present a new method for analysing stochastic epidemic models under minimal assumptions. The method, dubbed dynamic survival analysis (DSA), is based on a simple yet powerful observation, namely that population-level mean-field trajectories described by a system of partial differential equations may also approximate individual-level times of infection and recovery. This idea gives rise to a certain non-Markovian agent-based model and provides an agent-level likelihood function for a random sample of infection and/or recovery times. Extensive numerical analyses on both synthetic and real epidemic data from foot-and-mouth disease in the UK (2001) and COVID-19 in India (2020) show good accuracy and confirm the method's versatility in likelihood-based parameter estimation. The accompanying software package gives prospective users a practical tool for modelling, analysing and interpreting epidemic data with the help of the DSA approach.

Analysis of continuous-time Markovian ɛ-SIS epidemics on networks

We analyze continuous-time Markovian ɛ-SIS epidemics with self-infections on the complete graph. The majority of the graphs are analytically intractable, but many physical features of the ɛ-SIS process observed in the complete graph can occur in any other graph. In this work, we illustrate that the timescales of the ɛ-SIS process are related to the eigenvalues of the tridiagonal matrix of the SIS Markov chain. We provide a detailed analysis of all eigenvalues and illustrate that the eigenvalues show staircases, which are caused by the nearly degenerate (but strictly distinct) pairs of eigenvalues. We also illustrate that the ratio between the second-largest and third-largest eigenvalue is a good indicator of metastability in the ɛ-SIS process. Additionally, we show that the epidemic threshold of the Markovian ɛ-SIS process can be accurately approximated by the effective infection rate for which the third-largest eigenvalue of the transition matrix is the smallest. Finally, we derive the exact mean-field solution for the ɛ-SIS process on the complete graph, and we show that the mean-field approximation does not correctly represent the metastable behavior of Markovian ɛ-SIS epidemics.

Effects of void nodes on epidemic spreads in networks

We present the pair approximation models for susceptible–infected–recovered (SIR) epidemic dynamics in a sparse network based on a regular network. Two processes are considered, namely, a Markovian process with a constant recovery rate and a non-Markovian process with a fixed recovery time. We derive the implicit analytical expression for the final epidemic size and explicitly show the epidemic threshold in both Markovian and non-Markovian processes. As the connection rate decreases from the original network connection, the epidemic threshold in which epidemic phase transits from disease-free to endemic increases, and the final epidemic size decreases. Additionally, for comparison with sparse and heterogeneous networks, the pair approximation models were applied to a heterogeneous network with a degree distribution. The obtained phase diagram reveals that, upon increasing the degree of the original random regular networks and decreasing the effective connections by introducing void nodes accordingly, the final epidemic size of the sparse network is close to that of the random network with average degree of 4. Thus, introducing the void nodes in the network leads to more heterogeneous network and reduces the final epidemic size.

Non-Markovian temporal networks with auto- and cross-correlated link dynamics

Many of the biological, social and man-made networks around us are inherently dynamic, with their links switching on and off over time. The evolution of these networks is often observed to be non-Markovian, and the dynamics of their links are often correlated. Hence, to accurately model these networks, predict their evolution, and understand how information and other relevant quantities propagate over them, the inclusion of both memory and dynamical dependencies between links is key. In this article we introduce a general class of models of temporal networks based on discrete autoregressive processes for link dynamics. As a concrete and useful case study, we then concentrate on a specific model within this class, which allows to generate temporal networks with a specified underlying structural backbone, and with precise control over the dynamical dependencies between links and the strength and length of their memories. In this network model the presence of each link is influenced not only by its past activity, but also by the past activities of other links, as specified by a coupling matrix, which directly controls the causal relations, and hence the correlations, among links. We propose a maximum likelihood method for estimating the model's parameters from data, showing how the model allows a more realistic description of real-world temporal networks and also to predict their evolution. Due to the flexibility of maximum likelihood inference, we illustrate how to deal with heterogeneity and time-varying patterns, possibly including also nonstationary network dynamics. We then use our network model to investigate the role that, both the features of memory and the type of correlations in the dynamics of links have on the properties of processes occurring over a temporal network. Namely, we study the speed of a spreading process, as measured by the time it takes for diffusion to reach equilibrium. Through both numerical simulations and analytical results, we are able to separate the roles of autocorrelations and neighborhood correlations in link dynamics, showing that not only is the speed of diffusion nonmonotonically dependent on the memory length, but also that correlations among neighboring links help to speed up the spreading process, while autocorrelations slow it back down. Our results have implications in the study of opinion formation, the modeling of social networks, and the spreading of epidemics through mobile populations.

Stability analysis of a nonlocal SIHRDP epidemic model with memory effects

The prediction and control of COVID-19 is critical for ending this pandemic. In this paper, a nonlocal SIHRDP (S-susceptible class, I-infective class (infected but not hospitalized), H-hospitalized class, R-recovered class, D-death class and P-isolated class) epidemic model with long memory is proposed to describe the multi-wave peaks for the spread of COVID-19. Based on the basic reproduction number R 0 , which is completely controlled by fractional order, the stability of the proposed system is studied. Furthermore, the numerical simulation is conducted to gauge the performance of the proposed model. The results on Hunan, China, reveal thatR 0 < 1 suggests that the disease-free equilibrium point is globally asymptotically stable. Likewise, the situation of the multi-peak case in China is presented, and it is clear that the nonlocal epidemic system has a superior fitting effect than the classical model. Finally an adaptive impulsive vaccination is introduced based on the proposed system. Then employing the real data of France, India, the USA and Argentina, parameters identification and short-term forecasts are carried out to verify the effectiveness of the proposed model in describing the case of multiple peaks. Moreover, the implementation of vaccine control is expected once the hospitalized population exceeds 20 % of the total population. Numerical results of France, Indian, the USA and Argentina shed light on the varied effect of vaccine control in different countries. According to the vaccine control imposed on France, no obvious effect is observed even consider reducing human contact. As for India, although there will be a temporary increase in hospitalized admissions after execution of vaccination control, COVID-19 will eventually disappear. Results on the USA have seen most significant effect of vaccine control, the number of hospitalized individuals drops off and the disease is eventually eradicated. In contrast to the USA, vaccine control in Argentina has also been very effective, but COVID-19 cannot be completely eradicated.

The shape of memory in temporal networks

How to best define, detect and characterize network memory, i.e. the dependence of a network’s structure on its past, is currently a matter of debate. Here we show that the memory of a temporal network is inherently multidimensional, and we introduce a mathematical framework for defining and efficiently estimating the microscopic shape of memory, which characterises how the activity of each link intertwines with the activities of all other links. We validate our methodology on a range of synthetic models, and we then study the memory shape of real-world temporal networks spanning social, technological and biological systems, finding that these networks display heterogeneous memory shapes. In particular, online and offline social networks are markedly different, with the latter showing richer memory and memory scales. Our theory also elucidates the phenomenon of emergent virtual loops and provides a novel methodology for exploring the dynamically rich structure of complex systems.

The evolution of networks with structure changing in time is dependent on their past states and relevant to diffusion and spreading processes. The authors show that temporal network’s memory is described by multidimensional patterns at a microscopic scale, and cannot be reduced to a scalar quantity.

Accuracy of predicting epidemic outbreaks

During the outbreak of a virus, perhaps the greatest concern is the future evolution of the epidemic: How many people will be infected and which regions will be affected the most? The accurate prediction of an epidemic enables targeted disease countermeasures (e.g., allocating medical staff and quarantining). But when can we trust the prediction of an epidemic to be accurate? In this work we consider susceptible-infected-susceptible (SIS) and susceptible-infected-removed (SIR) epidemics on networks with time-invariant spreading parameters. (For time-varying spreading parameters, our results correspond to an optimistic scenario for the predictability of epidemics.) Our contribution is twofold. First, accurate long-term predictions of epidemics are possible only after the peak rate of new infections. Hence, before the peak, only short-term predictions are reliable. Second, we define an exponential growth metric, which quantifies the predictability of an epidemic. In particular, even without knowing the future evolution of the epidemic, the growth metric allows us to compare the predictability of an epidemic at different points in time. Our results are an important step towards understanding when and why epidemics can be predicted reliably.

Self-initiated behavioral change and disease resurgence on activity-driven networks

We consider a population that experienced a first wave of infections, interrupted by strong, top-down, governmental restrictions and did not develop a significant immunity to prevent a second wave (i.e., resurgence). As restrictions are lifted, individuals adapt their social behavior to minimize the risk of infection. We explore two scenarios. In the first, individuals reduce their overall social activity towards the rest of the population. In the second scenario, they maintain normal social activity within a small community of peers (i.e., social bubble) while reducing social interactions with the rest of the population. In both cases, we investigate possible correlations between social activity and behavior change, reflecting, for example, the social dimension of certain occupations. We model these scenarios considering a susceptible-infected-recovered epidemic model unfolding on activity-driven networks. Extensive analytical and numerical results show that (i) a minority of very active individuals not changing behavior may nullify the efforts of the large majority of the population and (ii) imperfect social bubbles of normal social activity may be less effective than an overall reduction of social interactions.

Higher-order temporal network effects through triplet evolution

We study the evolution of networks through 'triplets'-three-node graphlets. We develop a method to compute a transition matrix to describe the evolution of triplets in temporal networks. To identify the importance of higher-order interactions in the evolution of networks, we compare both artificial and real-world data to a model based on pairwise interactions only. The significant differences between the computed matrix and the calculated matrix from the fitted parameters demonstrate that non-pairwise interactions exist for various real-world systems in space and time, such as our data sets. Furthermore, this also reveals that different patterns of higher-order interaction are involved in different real-world situations. To test our approach, we then use these transition matrices as the basis of a link prediction algorithm. We investigate our algorithm's performance on four temporal networks, comparing our approach against ten other link prediction methods. Our results show that higher-order interactions in both space and time play a crucial role in the evolution of networks as we find our method, along with two other methods based on non-local interactions, give the best overall performance. The results also confirm the concept that the higher-order interaction patterns, i.e., triplet dynamics, can help us understand and predict the evolution of different real-world systems.

Clustering for epidemics on networks: A geometric approach

Infectious diseases typically spread over a contact network with millions of individuals, whose sheer size is a tremendous challenge to analyzing and controlling an epidemic outbreak. For some contact networks, it is possible to group individuals into clusters. A high-level description of the epidemic between a few clusters is considerably simpler than on an individual level. However, to cluster individuals, most studies rely on equitable partitions, a rather restrictive structural property of the contact network. In this work, we focus on Susceptible-Infected-Susceptible (SIS) epidemics, and our contribution is threefold. First, we propose a geometric approach to specify all networks for which an epidemic outbreak simplifies to the interaction of only a few clusters. Second, for the complete graph and any initial viral state vectors, we derive the closed-form solution of the nonlinear differential equations of the N-intertwined mean-field approximation of the SIS process. Third, by relaxing the notion of equitable partitions, we derive low-complexity approximations and bounds for epidemics on arbitrary contact networks. Our results are an important step toward understanding and controlling epidemics on large networks.

A fractional-order SIRD model with time-dependent memory indexes for encompassing the multi-fractional characteristics of the COVID-19

COVID-19 is a novel coronavirus affecting all the world since December last year. Up to date, the spread of the outbreak continues to complicate our lives, and therefore, several research efforts from many scientific areas are proposed. Among them, mathematical models are an excellent way to understand and predict the epidemic outbreaks evolution to some extent. Due to the COVID-19 may be modeled as a non-Markovian process that follows power-law scaling features, we present a fractional-order SIRD (Susceptible-Infected-Recovered-Dead) model based on the Caputo derivative for incorporating the memory effects (long and short) in the outbreak progress. Additionally, we analyze the experimental time series of 23 countries using fractal formalism. Like previous works, we identify that the COVID-19 evolution shows various power-law exponents (no just a single one) and share some universality among geographical regions. Hence, we incorporate numerous memory indexes in the proposed model, i.e., distinct fractional-orders defined by a time-dependent function that permits us to set specific memory contributions during the evolution. This allows controlling the memory effects of more early states, e.g., before and after a quarantine decree, which could be less relevant than the contribution of more recent ones on the current state of the SIRD system. We also prove our model with Italy's real data from the Center for Systems Science and Engineering (CSSE) at Johns Hopkins University.

Non-Markovian majority-vote model

Non-Markovian dynamics pervades human activity and social networks and it induces memory effects and burstiness in a wide range of processes including interevent time distributions, duration of interactions in temporal networks, and human mobility. Here, we propose a non-Markovian majority-vote model (NMMV) that introduces non-Markovian effects in the standard (Markovian) majority-vote model (SMV). The SMV model is one of the simplest two-state stochastic models for studying opinion dynamics, and displays a continuous order-disorder phase transition at a critical noise. In the NMMV model we assume that the probability that an agent changes state is not only dependent on the majority state of his neighbors but it also depends on his age, i.e., how long the agent has been in his current state. The NMMV model has two regimes: the aging regime implies that the probability that an agent changes state is decreasing with his age, while in the antiaging regime the probability that an agent changes state is increasing with his age. Interestingly, we find that the critical noise at which we observe the order-disorder phase transition is a nonmonotonic function of the rate β of the aging (antiaging) process. In particular the critical noise in the aging regime displays a maximum as a function of β while in the antiaging regime displays a minimum. This implies that the aging/antiaging dynamics can retard/anticipate the transition and that there is an optimal rate β for maximally perturbing the value of the critical noise. The analytical results obtained in the framework of the heterogeneous mean-field approach are validated by extensive numerical simulations on a large variety of network topologies.

SEAIR Epidemic spreading model of COVID-19

We study Susceptible-Exposed-Asymptomatic-Infectious-Recovered (SEAIR) epidemic spreading model of COVID-19. It captures two important characteristics of the infectiousness of COVID-19: delayed start and its appearance before onset of symptoms, or even with total absence of them. The model is theoretically analyzed in continuous-time compartmental version and discrete-time version on random regular graphs and complex networks. We show analytically that there are relationships between the epidemic thresholds and the equations for the susceptible populations at the endemic equilibrium in all three versions, which hold when the epidemic is weak. We provide theoretical arguments that eigenvector centrality of a node approximately determines its risk to become infected.

Giant leaps and long excursions: Fluctuation mechanisms in systems with long-range memory

We analyze large deviations of time-averaged quantities in stochastic processes with long-range memory, where the dynamics at time t depends itself on the value q_{t} of the time-averaged quantity. First we consider the elephant random walk and a Gaussian variant of this model, identifying two mechanisms for unusual fluctuation behavior, which differ from the Markovian case. In particular, the memory can lead to large-deviation principles with reduced speeds and to nonanalytic rate functions. We then explain how the mechanisms operating in these two models are generic for memory-dependent dynamics and show other examples including a non-Markovian simple exclusion process.

Classification of link-breaking and link-creation updating rules in susceptible-infected-susceptible epidemics on adaptive networks

In the classical susceptible-infected-susceptible (SIS) model, a disease or infection spreads over a given, mostly fixed graph. However, in many real complex networks, the topology of the underlying graph can change due to the influence of the dynamical process. In this paper, besides the spreading process, the network adaptively changes its topology based on the states of the nodes in the network. An entire class of link-breaking and link-creation mechanisms, which we name Generalized Adaptive SIS (G-ASIS), is presented and analyzed. For each instance of G-ASIS using the complete graph as initial network, the relation between the epidemic threshold and the effective link-breaking rate is determined to be linear, constant, or unknown. Additionally, we show that there exist link-breaking and link-creation mechanisms for which the metastable state does not exist. We confirm our theoretical results with several numerical simulations.

Non-Markovian recovery makes complex networks more resilient against large-scale failures

Non-Markovian spontaneous recovery processes with a time delay (memory) are ubiquitous in the real world. How does the non-Markovian characteristic affect failure propagation in complex networks? We consider failures due to internal causes at the nodal level and external failures due to an adverse environment, and develop a pair approximation analysis taking into account the two-node correlation. In general, a high failure stationary state can arise, corresponding to large-scale failures that can significantly compromise the functioning of the network. We uncover a striking phenomenon: memory associated with nodal recovery can counter-intuitively make the network more resilient against large-scale failures. In natural systems, the intrinsic non-Markovian characteristic of nodal recovery may thus be one reason for their resilience. In engineering design, incorporating certain non-Markovian features into the network may be beneficial to equipping it with a strong resilient capability to resist catastrophic failures.

Time dependence of susceptible-infected-susceptible epidemics on networks with nodal self-infections

The average fraction of infected nodes, in short the prevalence, of the Markovian ɛ-SIS (susceptible-infected-susceptible) process with small self-infection rate ɛ>0 exhibits, as a function of time, a typical "two-plateau" behavior, which was first discovered in the complete graph K_{N}. Although the complete graph is often dismissed as an unacceptably simplistic approximation, its analytic tractability allows to unravel deeper details, that are surprisingly also observed in other graphs as demonstrated by simulations. The time-dependent mean-field approximation for K_{N} performs only reasonably well for relatively large self-infection rates, but completely fails to mimic the typical Markovian ɛ-SIS process with small self-infection rates. While self-infections, particularly when their rate is small, are usually ignored, the interplay of nodal self-infection and spread over links may explain why absorbing processes are hardly observed in reality, even over long time intervals.

Spectral Analysis of Epidemic Thresholds of Temporal Networks

Many complex systems can be modeled as temporal networks with time-evolving connections. The influence of their characteristics on epidemic spreading is analyzed in a susceptible-infected-susceptible epidemic model illustrated by the discrete-time Markov chain approach. We develop the analytical epidemic thresholds in terms of the spectral radius of weighted adjacency matrix by averaging temporal networks, e.g., periodic, nonperiodic Markovian networks, and a special nonperiodic non-Markovian network (the link activation network) in time. We discuss the impacts of statistical characteristics, e.g., bursts and duration heterogeneity, as well as time-reversed characteristic on epidemic thresholds. We confirm the tightness of the proposed epidemic thresholds with numerical simulations on seven artificial and empirical temporal networks and show that the epidemic threshold of our theory is more precise than those of previous studies.

Macroscopic patterns of interacting contagions are indistinguishable from social reinforcement

From fake news to innovative technologies, many contagions spread as complex contagions via a process of social reinforcement, where multiple exposures are distinct from prolonged exposure to a single source.1 Contrarily, biological agents such as Ebola or measles are typically thought to spread as simple contagions.2 Here, we demonstrate that these different spreading mechanisms can have indistinguishable population-level dynamics once multiple contagions interact. In the social context, our results highlight the challenge of identifying and quantifying spreading mechanisms, such as social reinforcement,3 in a world where an innumerable amount of ideas, memes and behaviors interact. In the biological context, this parallel allows the use of complex contagions to effectively quantify the non-trivial interactions of infectious diseases.

A multilayer temporal network model for STD spreading accounting for permanent and casual partners

Sexually transmitted diseases (STD) modeling has used contact networks to study the spreading of pathogens. Recent findings have stressed the increasing role of casual partners, often enabled by online dating applications. We study the Susceptible-Infected-Susceptible (SIS) epidemic model –appropriate for STDs– over a two-layer network aimed to account for the effect of casual partners in the spreading of STDs. In this novel model, individuals have a set of steady partnerships (links in layer 1). At certain rates, every individual can switch between active and inactive states and, while active, it establishes casual partnerships with some probability with active neighbors in layer 2 (whose links can be thought as potential casual partnerships). Individuals that are not engaged in casual partnerships are classified as inactive, and the transitions between active and inactive states are independent of their infectious state. We use mean-field equations as well as stochastic simulations to derive the epidemic threshold, which decreases substantially with the addition of the second layer. Interestingly, for a given expected number of casual partnerships, which depends on the probabilities of being active, this threshold turns out to depend on the duration of casual partnerships: the longer they are, the lower the threshold.

Explicit non-Markovian susceptible-infected-susceptible mean-field epidemic threshold for Weibull and Gamma infections but Poisson curings

Although non-Markovian processes are considerably more complicated to analyze, real-world epidemics are likely non-Markovian, because the infection time is not always exponentially distributed. Here, we present analytic expressions of the epidemic threshold in a Weibull and a Gamma SIS epidemic on any network, where the infection time is Weibull, respectively, Gamma, but the recovery time is exponential. The theory is compared with precise simulations. The mean-field non-Markovian epidemic thresholds, both for a Weibull and Gamma infection time, are physically similar and interpreted via the occurrence time of an infection during a healthy period of each node in the graph. Our theory couples the type of a viral item, specified by a shape parameter of the Weibull or Gamma distribution, to its corresponding network-wide endemic spreading power, which is specified by the mean-field non-Markovian epidemic threshold in any network.

Markovian approaches to modeling intracellular reaction processes with molecular memory

Many cellular processes are governed by stochastic reaction events. These events do not necessarily occur in single steps of individual molecules, and, conversely, each birth or death of a macromolecule (e.g., protein) could involve several small reaction steps, creating a memory between individual events and thus leading to nonmarkovian reaction kinetics. Characterizing this kinetics is challenging. Here, we develop a systematic approach for a general reaction network with arbitrary intrinsic waiting-time distributions, which includes the stationary generalized chemical-master equation (sgCME), the stationary generalized Fokker-Planck equation, and the generalized linear-noise approximation. The first formulation converts a nonmarkovian issue into a markovian one by introducing effective transition rates (that explicitly decode the effect of molecular memory) for the reactions in an equivalent reaction network with the same substrates but without molecular memory. Nonmarkovian features of the reaction kinetics can be revealed by solving the sgCME. The latter 2 formulations can be used in the fast evaluation of fluctuations. These formulations can have broad applications, and, in particular, they may help us discover new biological knowledge underlying memory effects. When they are applied to generalized stochastic models of gene-expression regulation, we find that molecular memory is in effect equivalent to a feedback and can induce bimodality, fine-tune the expression noise, and induce switch.

Equivalence and its invalidation between non-Markovian and Markovian spreading dynamics on complex networks

Epidemic spreading processes in the real world depend on human behaviors and, consequently, are typically non-Markovian in that the key events underlying the spreading dynamics cannot be described as a Poisson random process and the corresponding event time is not exponentially distributed. In contrast to Markovian type of spreading dynamics for which mathematical theories have been well developed, we lack a comprehensive framework to analyze and fully understand non-Markovian spreading processes. Here we develop a mean-field theory to address this challenge, and demonstrate that the theory enables accurate prediction of both the transient phase and the steady states of non-Markovian susceptible-infected-susceptible spreading dynamics on synthetic and empirical networks. We further find that the existence of equivalence between non-Markovian and Markovian spreading depends on a specific edge activation mechanism. In particular, when temporal correlations are absent on active edges, the equivalence can be expected; otherwise, an exact equivalence no longer holds.

Contact-Based Model for Epidemic Spreading on Temporal Networks

We present a contact-based model to study the spreading of epidemics by means of extending the dynamic message-passing approach to temporal networks. The shift in perspective from node- to edge-centric quantities enables accurate modeling of Markovian susceptible-infected-recovered outbreaks on time-varying trees, i.e., temporal networks with a loop-free underlying topology. On arbitrary graphs, the proposed contact-based model incorporates potential structural and temporal heterogeneities of the contact network and improves analytic estimations with respect to the individual-based (node-centric) approach at a low computational and conceptual cost. Within this new framework, we derive an analytical expression for the epidemic threshold on temporal networks and demonstrate the feasibility of this method on empirical data.

The spread of infection, information, computer malware, or any contagionlike process is often described by disease models on complex networks with a time-varying topology. Recurrent, or flulike, spreading can be modeled accurately by taking an “individual-based” approach that focuses on nodes in a network. Here, we instead focus on the interactions—the links in a network—and present a contact-based model that accurately describes a second group of contagion processes: those that lead to permanent immunization. Taking this new perspective, we derive a criterion that separates local outbreaks from global epidemics, a crucial tool for risk assessment and control of, for instance, viral marketing.

To develop our model, we integrate time-varying network topologies into dynamic message passing, a widely used approach to describe unidirectional contagion processes. Based on this generalized model, we derive a spectral criterion for the stability of the disease-free solution, which determines the critical disease parameters. Through numerous numerical studies, we provide evidence that the contact-based perspective improves the individual-based approach. Finally, we investigate the epidemic risk based on the German cattle-trade network with over 180 000 nodes. Results from the individual-based and contact-based approaches deviate considerably, and thus justify this paradigmatic shift.

Our contact-based model is conceptually similar to those that focus on individuals, so we expect that numerous individual-based findings as well as results from networks with a static topology can be transferred in the future. These may include general epidemic models with a non-Poissonian transition process that go beyond the assumption of treelike topologies, stochastic effects, and temporal networks that evolve continuously in time.

Contact Adaption During Epidemics: A Multilayer Network Formulation Approach

People change their physical contacts as a preventive response to infectious disease propagations. Yet, only a few mathematical models consider the coupled dynamics of the disease propagation and the contact adaptation process. This paper presents a model where each agent has a default contact neighborhood set, and switches to a different contact set once she becomes alert about infection among her default contacts. Since each agent can adopt either of two possible neighborhood sets, the overall contact network switches among [Formula: see text] possible configurations. Notably, a two-layer network representation can fully model the underlying adaptive, state-dependent contact network. Contact adaptation influences the size of the disease prevalence and the epidemic threshold-a characteristic measure of a contact network robustness against epidemics-in a nonlinear fashion. Particularly, the epidemic threshold for the presented adaptive contact network belongs to the solution of a nonlinear Perron-Frobenius (NPF) problem, which does not depend on the contact adaptation rate monotonically. Furthermore, the network adaptation model predicts a counter-intuitive scenario where adaptively changing contacts may adversely lead to lower network robustness against epidemic spreading if the contact adaptation is not fast enough. An original result for a class of NPF problems facilitate the analytical developments in this paper.

An SIR pairwise epidemic model with infection age and demography

The demography and infection age play an important role in the spread of slowly progressive diseases. To investigate their effects on the disease spreading, we propose a pairwise epidemic model with infection age and demography on dynamic networks. The basic reproduction number of this model is derived. It is proved that there is a disease-free equilibrium which is globally asymptotically stable if the basic reproduction number is less that unity. Besides, sensitivity analysis is performed and shows that increasing the variance in recovery time and decreasing the variance in infection time can effectively control the diseases. The complex interaction between the death rate and equilibrium prevalence suggests that it is imperative to correctly estimate the parameters of demography in order to assess the disease transmission dynamics accurately. Moreover, numerical simulations show that the endemic equilibrium is globally asymptotically stable.

Burst of virus infection and a possibly largest epidemic threshold of non-Markovian susceptible-infected-susceptible processes on networks

Since a real epidemic process is not necessarily Markovian, the epidemic threshold obtained under the Markovian assumption may be not realistic. To understand general non-Markovian epidemic processes on networks, we study the Weibullian susceptible-infected-susceptible (SIS) process in which the infection process is a renewal process with a Weibull time distribution. We find that, if the infection rate exceeds 1/ln(λ_{1}+1), where λ_{1} is the largest eigenvalue of the network's adjacency matrix, then the infection will persist on the network under the mean-field approximation. Thus, 1/ln(λ_{1}+1) is possibly the largest epidemic threshold for a general non-Markovian SIS process with a Poisson curing process under the mean-field approximation. Furthermore, non-Markovian SIS processes may result in a multimodal prevalence. As a byproduct, we show that a limiting Weibullian SIS process has the potential to model bursts of a synchronized infection.

Impact of the infectious period on epidemics

The duration of the infectious period is a crucial determinant of the ability of an infectious disease to spread. We consider an epidemic model that is network based and non-Markovian, containing classic Kermack-McKendrick, pairwise, message passing, and spatial models as special cases. For this model, we prove a monotonic relationship between the variability of the infectious period (with fixed mean) and the probability that the infection will reach any given subset of the population by any given time. For certain families of distributions, this result implies that epidemic severity is decreasing with respect to the variance of the infectious period. The striking importance of this relationship is demonstrated numerically. We then prove, with a fixed basic reproductive ratio (R_{0}), a monotonic relationship between the variability of the posterior transmission probability (which is a function of the infectious period) and the probability that the infection will reach any given subset of the population by any given time. Thus again, even when R_{0} is fixed, variability of the infectious period tends to dampen the epidemic. Numerical results illustrate this but indicate the relationship is weaker. We then show how our results apply to message passing, pairwise, and Kermack-McKendrick epidemic models, even when they are not exactly consistent with the stochastic dynamics. For Poissonian contact processes, and arbitrarily distributed infectious periods, we demonstrate how systems of delay differential equations and ordinary differential equations can provide upper and lower bounds, respectively, for the probability that any given individual has been infected by any given time.

Phase transition of the susceptible-infected-susceptible dynamics on time-varying configuration model networks

We present a degree-based theoretical framework to study the susceptible-infected-susceptible (SIS) dynamics on time-varying (rewired) configuration model networks. Using this framework on a given degree distribution, we provide a detailed analysis of the stationary state using the rewiring rate to explore the whole range of the time variation of the structure relative to that of the SIS process. This analysis is suitable for the characterization of the phase transition and leads to three main contributions: (1) We obtain a self-consistent expression for the absorbing-state threshold, able to capture both collective and hub activation. (2) We recover the predictions of a number of existing approaches as limiting cases of our analysis, providing thereby a unifying point of view for the SIS dynamics on random networks. (3) We obtain bounds for the critical exponents of a number of quantities in the stationary state. This allows us to reinterpret the concept of hub-dominated phase transition. Within our framework, it appears as a heterogeneous critical phenomenon: observables for different degree classes have a different scaling with the infection rate. This phenomenon is followed by the successive activation of the degree classes beyond the epidemic threshold.

Effects of communication burstiness on consensus formation and tipping points in social dynamics

Current models for opinion dynamics typically utilize a Poisson process for speaker selection, making the waiting time between events exponentially distributed. Human interaction tends to be bursty though, having higher probabilities of either extremely short waiting times or long periods of silence. To quantify the burstiness effects on the dynamics of social models, we place in competition two groups exhibiting different speakers' waiting-time distributions. These competitions are implemented in the binary naming game and show that the relevant aspect of the waiting-time distribution is the density of the head rather than that of the tail. We show that even with identical mean waiting times, a group with a higher density of short waiting times is favored in competition over the other group. This effect remains in the presence of nodes holding a single opinion that never changes, as the fraction of such committed individuals necessary for achieving consensus decreases dramatically when they have a higher head density than the holders of the competing opinion. Finally, to quantify differences in burstiness, we introduce the expected number of small-time activations and use it to characterize the early-time regime of the system.

Equivalence between Non-Markovian and Markovian Dynamics in Epidemic Spreading Processes

A general formalism is introduced to allow the steady state of non-Markovian processes on networks to be reduced to equivalent Markovian processes on the same substrates. The example of an epidemic spreading process is considered in detail, where all the non-Markovian aspects are shown to be captured within a single parameter, the effective infection rate. Remarkably, this result is independent of the topology of the underlying network, as demonstrated by numerical simulations on two-dimensional lattices and various types of random networks. Furthermore, an analytic approximation for the effective infection rate is introduced, which enables the calculation of the critical point and of the critical exponents for the non-Markovian dynamics.

Unification of theoretical approaches for epidemic spreading on complex networks

Models of epidemic spreading on complex networks have attracted great attention among researchers in physics, mathematics, and epidemiology due to their success in predicting and controlling scenarios of epidemic spreading in real-world scenarios. To understand the interplay between epidemic spreading and the topology of a contact network, several outstanding theoretical approaches have been developed. An accurate theoretical approach describing the spreading dynamics must take both the network topology and dynamical correlations into consideration at the expense of increasing the complexity of the equations. In this short survey we unify the most widely used theoretical approaches for epidemic spreading on complex networks in terms of increasing complexity, including the mean-field, the heterogeneous mean-field, the quench mean-field, dynamical message-passing, link percolation, and pairwise approximation. We build connections among these approaches to provide new insights into developing an accurate theoretical approach to spreading dynamics on complex networks.

Memory effects on epidemic evolution: The susceptible-infected-recovered epidemic model

Memory has a great impact on the evolution of every process related to human societies. Among them, the evolution of an epidemic is directly related to the individuals' experiences. Indeed, any real epidemic process is clearly sustained by a non-Markovian dynamics: memory effects play an essential role in the spreading of diseases. Including memory effects in the susceptible-infected-recovered (SIR) epidemic model seems very appropriate for such an investigation. Thus, the memory prone SIR model dynamics is investigated using fractional derivatives. The decay of long-range memory, taken as a power-law function, is directly controlled by the order of the fractional derivatives in the corresponding nonlinear fractional differential evolution equations. Here we assume "fully mixed" approximation and show that the epidemic threshold is shifted to higher values than those for the memoryless system, depending on this memory "length" decay exponent. We also consider the SIR model on structured networks and study the effect of topology on threshold points in a non-Markovian dynamics. Furthermore, the lack of access to the precise information about the initial conditions or the past events plays a very relevant role in the correct estimation or prediction of the epidemic evolution. Such a "constraint" is analyzed and discussed.

Measuring burstiness for finite event sequences

Characterizing inhomogeneous temporal patterns in natural and social phenomena is important to understand underlying mechanisms behind such complex systems and, hence, even to predict and control them. Temporal inhomogeneities in event sequences have been described in terms of bursts that are rapidly occurring events in short time periods alternating with long inactive periods. The bursts can be quantified by a simple measure, called the burstiness parameter, which was introduced by Goh and Barabási [Europhys. Lett. 81, 48002 (2008)EULEEJ0295-507510.1209/0295-5075/81/48002]. The burstiness parameter has been widely used due to its simplicity, which, however, turns out to be strongly affected by the finite number of events in the time series. As the finite-size effects on burstiness parameter have been largely ignored, we analytically investigate the finite-size effects of the burstiness parameter. Then we suggest an alternative definition of burstiness that is free from finite-size effects and yet simple. Using our alternative burstiness measure, one can distinguish the finite-size effects from the intrinsic bursty properties in the time series. We also demonstrate the advantages of our burstiness measure by analyzing empirical data sets.