Recently, hypergraphs have attracted considerable interest from the research community as a generalization of networks capable of encoding higher-order interactions, which commonly appear in both natural and social systems. Epidemic dynamics in hypergraphs has been studied by using the simplicial susceptible-infected-susceptible ($s$-SIS) model; however, the efficient immunization strategy for epidemics in hypergraphs is not studied despite the importance of the topic in mathematical epidemiology. Here, we propose an immunization strategy that immunizes hyperedges with high simultaneous infection probability (SIP). This strategy can be implemented in general hypergraphs. We also generalize the edge epidemic importance (EI)-based immunization strategy, which is the state of the art in complex networks. However, it does not perform as well as the SIP-based method in hypergraphs despite its high computational cost. We also show that immunizing hyperedges with high H-eigenscore effectively contains the epidemics in uniform hypergraphs. A high SIP of a hyperedge suggests that the hyperedge is a "hotspot" of the epidemic process. Therefore, SIP can be used as a centrality measure to quantify a hyperedge's influence on higher-order dynamics in general hypergraphs. The effectiveness of the immunization strategies suggests the necessity of scientific, data-driven, systematic policy-making for epidemic containment.
We propose a tractable epidemic model that includes containment measures. In the absence of containment measures, the epidemics spread exponentially fast whenever the infectivity rate is positive, $λ>0$. The containment measures are modeled by considering a time-dependent modulation of the bare infectivity $λ$ leading to effective infectivity that decays in time for each infected individual, mimicking for instance the combined effect of the asymptomatic onset of the disease, testing policies and quarantine. We consider a wide range of temporal kernels for effective infectivity and we investiga…
Epidemic containment is a major concern when confronting large-scale infections in complex networks. Many works have been devoted to analytically understand how to restructure the network to minimize the impact of major outbreaks of infections at large scale. In many cases, the strategies consist in the isolation of certain nodes, while less attention has been paid to the intervention on links. In epidemic spreading, links inform about the probability of carrying the contagion of the disease from infected to susceptible individuals. Note that these states depend on the full structure of the ne…
In this chapter, we focus on the problem of containing the spread of diseases taking place in both temporal and adaptive networks (i.e., networks whose structure `adapts' to the state of the disease). We specifically focus on the problem of finding the optimal allocation of containment resources (e.g., vaccines, medical personnel, traffic control resources, etc.) to eradicate epidemic outbreaks over the following three models of temporal and adaptive networks: (i) Markovian temporal networks, (ii) aggregated-Markovian temporal networks, and (iii) stochastically adaptive models. For each model,…
In this paper, we study the dynamics of epidemic processes taking place in temporal and adaptive networks. Building on the activity-driven network model, we propose an adaptive model of epidemic processes, where the network topology dynamically changes due to both exogenous factors independent of the epidemic dynamics as well as endogenous preventive measures adopted by individuals in response to the state of the infection. A direct analysis of the model using Markov processes involves the spectral analysis of a transition probability matrix whose size grows exponentially with the number of no…
Mathematical modeling of epidemic propagation on networks is extended to hypergraphs in order to account for both the community structure and the nonlinear dependence of the infection pressure on the number of infected neighbours. The exact master equations of the propagation process are derived for an arbitrary hypergraph given by its incidence matrix. Based on these, moment closure approximation and mean-field models are introduced and compared to individual-based stochastic simulations. The simulation algorithm, developed for networks, is extended to hypergraphs. The effects of hypergraph s…
University of Gothenburg（哥德堡大学）；CNR-IPCF, Istituto per i Processi Chimico-Fisici（意大利国家研究委员会化学物理过程研究所）
Containment of epidemic outbreaks entails great societal and economic costs. Cost-effective containment strategies rely on efficiently identifying infected individuals, making the best possible use of the available testing resources. Therefore, quickly identifying the optimal testing strategy is of critical importance. Here, we demonstrate that machine learning can be used to identify which individuals are most beneficial to test, automatically and dynamically adapting the testing strategy to the characteristics of the disease outbreak. Specifically, we simulate an outbreak using the archetypa…
Human mobility is one of the key factors at the basis of the spreading of diseases in a population. Containment strategies are usually devised on movement scenarios based on coarse-grained assumptions. Mobility phone data provide a unique opportunity for building models and defining strategies based on very precise information about the movement of people in a region or in a country. Another very important aspect is the underlying social structure of a population, which might play a fundamental role in devising information campaigns to promote vaccination and preventive measures, especially in…
Resource support between individuals is of particular importance in controlling or mitigating epidemic spreading, especially during pandemics. Whereas there remains the question of how we can protect ourselves from being infected while helping others by donating resources in fighting against the epidemic. To answer the question, we propose a novel resource allocation model by considering the awareness of self-protection of individuals. In the model, a tuning parameter is introduced to quantify the reaction strength of individuals when they are aware of the disease. And then, a coupled model of…
Here, we focus on the data analysis of the growth of epidemic spread of Covid-19 in countries where different policies of containment were activated. It is known that the growth of pandemic spread at its threshold is exponential, but it is not known how to quantify the success of different containment policies. We identify that a successful approach gives an arrested phase regime following the Ostwald growth, where, over the course of time, one phase transforms into another metastable phase with a similar free energy as observed in oxygen interstitial diffusion in quantum complex matter and in…
During the COVID pandemic, periods of exponential growth of the disease have been mitigated by containment measures that in different occasions have resulted in a power-law growth of the number of cases. The first observation of such behaviour has been obtained from 2020 late spring data coming from China by Ziff and Ziff in Ref. [1]. After this important observation the power-law scaling (albeit with different exponents) has also been observed in other countries during periods of containment of the spread. Early interpretations of these results suggest that this phenomenon might be due to spa…
The Covid-19 epidemic of the novel coronavirus (severe acute respiratory syndrome SARS - CoV-2) has been spreading around the world. While different containment policies using non-pharmaceutical interventions have been applied, their efficiency are not known quantitatively. We show that the doubling time Td(t) with the success s factor, the characteristic time of the exponential growth of Td(t) in the arrested regime, is a reliable tool for early predictions of epidemic spread time evolution and it provides a quantitative measure of the success of different containment measures. The efficiency…
Epidemic outbreaks are an important healthcare challenge, especially in developing countries where they represent one of the major causes of mortality. Approaches that can rapidly target subpopulations for surveillance and control are critical for enhancing containment processes during epidemics. Using a real-world dataset from Ivory Coast, this work presents an attempt to unveil the socio-geographical heterogeneity of disease transmission dynamics. By employing a spatially explicit meta-population epidemic model derived from mobile phone Call Detail Records (CDRs), we investigate how the diff…
Information-Epidemic Dynamics in Cyber-Physical Systems: A Hypergraph Framework with Interpersonal Relationships
Southwest University（西南大学）；University of Electronic Science and Technology of China（电子科技大学）；University of Maribor（马里博尔大学）；Community Healthcare Center Dr. Adolf Drolc Maribor（马里博尔市阿道夫·德罗尔克社区医疗中心）；Kyung Hee University（庆熙大学）；Korea University（高丽大学）；Potsdam Institute for Climate Impact Research（波茨坦气候影响研究所）；Humboldt University of Berlin（柏林洪堡大学）
We analyse 'stop-and-go' containment policies that produce infection cycles as periods of tight lockdowns are followed by periods of falling infection rates. The subsequent relaxation of containment measures allows cases to increase again until another lockdown is imposed and the cycle repeats. The policies followed by several European countries during the Covid-19 pandemic seem to fit this pattern. We show that 'stop-and-go' should lead to lower medical costs than keeping infections at the midpoint between the highs and lows produced by 'stop-and-go'. Increasing the upper and reducing the low…
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…
Until a vaccine or therapy is found against the SARS-CoV-2 coronavirus, reaching herd immunity appears to be the only mid-term option. However, if the number of infected individuals decreases and eventually fades only beyond this threshold, a significant proportion of susceptible may still be infected until the epidemic is over. A containment strategy is likely the best policy in the worst case where no vaccine or therapy is found. In order to keep the number of newly infected persons to a minimum, a possible strategy is to apply strict containment measures, so that the number of susceptible i…
Epidemic spreading can be suppressed by the introduction of containment measures such as social distancing and lock downs. Yet, when such measures are relaxed, new epidemic waves and infection cycles may occur. Here we explore this issue in compartmentalized epidemic models on graphs in presence of a feedback between the infection state of the population and the structure of its social network for the case of discontinuous control. We show that in random graphs the effect of containment measures is simply captured by a renormalization of the effective infection rate that accounts for the chang…
The disease spreading on complex networks is studied in SIR model. Simulations on empirical complex networks reveal two specific regimes of disease spreading: local containment and epidemic outbreak. The variables measuring the extent of disease spreading are in general characterized by a bimodal probability distribution. Phase diagrams of disease spreading for empirical complex networks are introduced. A theoretical model of disease spreading on m-ary tree is investigated both analytically and in simulations. It is shown that the model reproduces qualitative features of phase diagrams of dise…
The rapid spread of the Coronavirus (COVID-19) confronts policy makers with the problem of measuring the effectiveness of containment strategies, balancing public health considerations with the economic costs of social distancing measures. We introduce a modified epidemic model that we name the controlled-SIR model, in which the disease reproduction rate evolves dynamically in response to political and societal reactions. An analytic solution is presented. The model reproduces official COVID-19 cases counts of a large number of regions and countries that surpassed the first peak of the outbrea…
Mumbai, amongst the most densely populated cities in the world, has witnessed the fourth largest number of cases and the largest number of deaths among all the cities in India (as of 28th October 2020). Along with the rest of India, lockdowns (of varying degrees) have been in effect in Mumbai since March 25, 2020. Given the large economic toll on the country from the lockdown and the related restrictions on mobility of people and goods, swift opening of the economy especially in a financial hub such as Mumbai becomes critical. In this report, we use the IISc-TIFR agent based simulator to devel…
The SARS-CoV-2 outbreak changed the everyday life of almost all the people over the world.Currently, we are facing with the problem of containing the spread of the virus both using the more effective forced lockdown, which has the drawback of slowing down the economy of the involved countries, and by identifying and isolating the positive individuals, which, instead, is an hard task in general due to the lack of information. For this specific disease, the identificato of the infected is particularly challenging since there exists cathegories, namely the asymptomatics, who are positive and pote…
During an epidemic control, the containment of the disease is usually achieved through increasing devoted resource to shorten the duration of infectiousness. However, the impact of this resource expenditure has not been studied quantitatively. Using the well-documented cholera data, we observe empirically that the recovery rate which is related to the duration of infectiousness has a strong positive correlation with the average resource devoted to the infected individuals. By incorporating this relation we build a novel model and find that insufficient resource leads to an abrupt increase in t…
We analyze an epidemic model on a network consisting of susceptible-infected-recovered equations at the nodes coupled by diffusion using a graph Laplacian. We introduce an epidemic criterion and examine different vaccination/containment strategies: we prove that it is most effective to vaccinate a node of highest degree. The model is also useful to evaluate deconfinement scenarios and prevent a so-called second wave. The model has few parameters enabling fitting to the data and the essential ingredient of importation of infected; these features are particularly important for the current COVID-…
Effectiveness of probabilistic contact tracing in epidemic containment: the role of super-spreaders and transmission path reconstruction
Politecnico di Torino（都灵理工大学）；Italian Institute for Genomic Medicine（意大利基因组医学研究所）；Politecnico di Milano（米兰理工大学）；Universidad Complutense（康普顿斯大学）；Collegio Carlo Alberto（卡洛·阿尔贝托学院）
Adaptive Epidemic Dynamics on Hypergraphs with Group-Level Immunization and Rewiring
College of Artificial Intelligence, Southwest University（西南大学人工智能学院）；School of Systems Science/Institute of Nonequilibrium Systems, Beijing Normal University（北京师范大学系统科学学院/非平衡系统研究所）；Business College, Southwest University（西南大学商学院）；Faculty of Natural Sciences and Mathematics, University of Maribor（马里博尔大学自然科学与数学学院）；Community Healthcare Center Dr. Adolf Drolc Maribor（马里博尔阿道夫·德罗尔博士社区健康中心）；Department of Physics, Kyung Hee University（庆熙大学物理系）；University College, Korea University（高丽大学大学院）
So far most of the analysis of coronavirus 2020 epidemic data has been focusing on a short-time window and consequently a quantitative test of statistical physical laws of Coronavirus Epidemics with Containment Measures (CEwCM) is currently lacking. Here we report a quantitative analysis of CEwCM over 230 days, covering the full-time lapse of the first epidemic wave. We use a 3D phase diagram tracking the simultaneous evolution of the doubling time Td(t) and reproductive number Rt(t) showing that this expanded parameter space is needed for biological physics of CEwCP. We have verified that in …
Wireless networks are commonly used in public spaces, universities and public institutions and provide accurate and easily accessible information to monitor the mobility and behavior of users. Following the application of containment measures during the recent pandemic, we analyse extensive data from the WiFi network in a University Campus in Italy during three periods, corresponding to partial lockdown, partial opening, and almost complete opening. We measure the probability distributions of groups and link activation at Wi-Fi Access Points, investigating how different areas are used in the p…
Metapopulation epidemic models describe epidemic dynamics in networks of spatially distant patches connected with pathways for migration of individuals. In the present study, we deal with a susceptible-infected-recovered (SIR) metapopulation model where the epidemic process in each patch is represented by an SIR model and the mobility of individuals is assumed to be a homogeneous diffusion. Our study focuses on two types of patches including high-risk and low-risk ones, in order to evaluate intervention strategies for epidemic control. We theoretically analyze the intervention threshold, indic…
High impact epidemics constitute one of the largest threats humanity is facing in the 21st century. Testing, contact tracing and quarantining are critical in slowing down epidemic dynamics, but may prove insufficient for highly contagious diseases. In the absence of pharmaceutical interventions, physical distancing measures remain as the last resort to avoid a widespread outbreak. Here we show that such combined countermeasures drastically change the rules of the epidemic transition if testing capacities are limited: Instead of continuous the response to countermeasures becomes discontinuous a…
We study the dynamics of epidemic and reaction-diffusion processes in metapopulation models with heterogeneous connectivity pattern. In SIR-like processes, along with the standard local epidemic threshold, the system exhibits a global invasion threshold. We provide an explicit expression of the threshold that sets a critical value of the diffusion/mobility rate below which the epidemic is not able to spread to a macroscopic fraction of subpopulations. The invasion threshold is found to be affected by the topological fluctuations of the metapopulation network. The presented results provide a ge…
The cataclysmic contagion based calamity -- Covid-19 has shown us a clear need for a comprehensive community based strategy that overcomes the sheer complexity of controlling it and the caveats of current methods. In this regard, as seen in earlier epidemics, testing has always been an integral part of containment policy. But one has to consider the optimality of a testing scheme based on the resultant disease spread in the community and not based on purely increasing testing efficiency. Therefore, taking a decision is no easy feat and must consider the community utility constrained by its pri…
The ongoing Coronavirus Disease 2019 (COVID-19) pandemic threatens the health of humans and causes great economic losses. Predictive modelling and forecasting the epidemic trends are essential for developing countermeasures to mitigate this pandemic. We develop a network model, where each node represents an individual and the edges represent contacts between individuals where the infection can spread. The individuals are classified based on the number of contacts they have each day (their node degrees) and their infection status. The transmission network model was respectively fitted to the re…
We consider an epidemic process on adaptive activity-driven temporal networks, with adaptive behaviour modelled as a change in activity and attractiveness due to infection. By using a mean-field approach, we derive an analytical estimate of the epidemic threshold for SIS and SIR epidemic models for a general adaptive strategy, which strongly depends on the correlations between activity and attractiveness in the susceptible and infected states. We focus on strong social distancing, implementing two types of quarantine inspired by recent real case studies: an active quarantine, in which the popu…
The containment of epidemic spreading is a major challenge in science. Vaccination, whenever available, is the best way to prevent the spreading, because it eventually immunizes individuals. However, vaccines are not perfect, and total immunization is not guaranteed. Imperfect immunization has driven the emergence of anti-vaccine movements that totally alter the predictions about the epidemic incidence. Here, we propose a mathematically solvable mean-field vaccination model to mimic the spontaneous adoption of vaccines against influenza-like diseases, and the expected epidemic incidence. The r…
The effectiveness of vaccination highly depends on the choice of individuals to vaccinate, even if the same number of individuals are vaccinated. Vaccinating individuals with high centrality measures such as betweenness centrality (BC) and eigenvector centrality (EC) are effective in containing epidemics. However, in many real-world cases, each individual has distinct epidemic characteristics such as contagion, recovery, fatality rate, efficacy, and probability of severe reaction to a vaccine. Moreover, the relative effectiveness of vaccination strategies depends on the number of available vac…
Infectious diseases that incorporate pre-symptomatic transmission are challenging to monitor, model, predict and contain. We address this scenario by studying a variant of a stochastic susceptible-exposed-infected-recovered model on arbitrary network instances using an analytical framework based on the method of dynamic message-passing. This framework provides a good estimate of the probabilistic evolution of the spread on both static and contact networks, offering a significantly improved accuracy with respect to individual-based mean-field approaches while requiring a much lower computationa…
The spread of rumors through social media and online social networks can not only disrupt the daily lives of citizens but also result in loss of life and property. A rumor spreads when individuals, who are unable decide the authenticity of the information, mistake the rumor as genuine information and pass it on to their acquaintances. We propose a solution where a set of individuals (based on their degree) in the social network are trained and provided resources to help them distinguish a rumor from genuine information. By formulating an optimization problem we calculate the optimum set of ind…
Network-based epidemic models have been extensively employed to understand the spread of infectious diseases, but have generally overlooked the fact that most realistic networks are dynamical rather than static. In this paper, we study a susceptible-infected-susceptible epidemic model with vaccination in a dynamical contact network of moving individuals, where we regard mobile individuals as random walkers that are allowed to perform long-range jumps. Different from previous studies of epidemics in a random walk network with a constant interaction radius, we consider the scenario where the ind…
Unveiling the impact of cross-order hyperdegree correlations in contagion processes on hypergraphs
Network Science Institute, Northeastern University London（东北大学伦敦网络科学研究所）；Department of Network and Data Science, Central European University（中欧大学网络与数据科学系）；Department of Mathematics, Northeastern University（东北大学数学系）
While several non-pharmacological measures have been implemented for a few months in an effort to slow the coronavirus disease (COVID-19) pandemic in the United States, the disease remains a danger in a number of counties as restrictions are lifted to revive the economy. Making a trade-off between economic recovery and infection control is a major challenge confronting many hard-hit counties. Understanding the transmission process and quantifying the costs of local policies are essential to the task of tackling this challenge. Here, we investigate the dynamic contact patterns of the population…
The possibility to analyze, quantify and forecast epidemic outbreaks is fundamental when devising effective disease containment strategies. Policy makers are faced with the intricate task of drafting realistically implementable policies that strike a balance between risk management and cost. Two major techniques policy makers have at their disposal are: epidemic modeling and contact tracing. Models are used to forecast the evolution of the epidemic both globally and regionally, while contact tracing is used to reconstruct the chain of people who have been potentially infected, so that they can…
Higher order interactions are increasingly recognised as a fundamental aspect of complex systems ranging from the brain to social contact networks. Hypergraph as well as simplicial complexes capture the higher-order interactions of complex systems and allow to investigate the relation between their higher-order structure and their function. Here we establish a general framework for assessing hypergraph robustness and we characterize the critical properties of simple and higher-order percolation processes. This general framework builds on the formulation of the random multiplex hypergraph ensem…
National stay-at-home orders, or lockdowns, were imposed in several countries to drastically reduce the social interactions mainly responsible for the transmission of the SARS-CoV-2 virus. Despite being essential to slow down the COVID-19 pandemic, these containment measures are associated with an economic burden. In this work, we propose a network approach to model the implementation of a partial lockdown, breaking the society into disconnected components, or partitions. Our model is composed by two main ingredients: a multiplex network representing human contacts within different contexts, f…
The spatial structure of populations is a key element in the understanding of the large scale spreading of epidemics. Motivated by the recent empirical evidence on the heterogeneous properties of transportation and commuting patterns among urban areas, we present a thorough analysis of the behavior of infectious diseases in metapopulation models characterized by heterogeneous connectivity and mobility patterns. We derive the basic reaction-diffusion equation describing the metapopulation system at the mechanistic level and derive an early stage dynamics approximation for the subpopulation inva…
The integration of empirical data in computational frameworks to model the spread of infectious diseases poses challenges that are becoming pressing with the increasing availability of high-resolution information on human mobility and contacts. This deluge of data has the potential to revolutionize the computational efforts aimed at simulating scenarios and designing containment strategies. However, the integration of detailed data sources yields models that are less transparent and general. Hence, given a specific disease model, it is crucial to assess which representations of the raw data st…
When effective medical treatment and vaccination are not available, non-pharmaceutical interventions such as social distancing, home quarantine and far-reaching shutdown of public life are the only available strategies to prevent the spread of epidemics. Based on an extended SEIR (susceptible-exposed-infectious-recovered) model and continuous-time optimal control theory, we compute the optimal non-pharmaceutical intervention strategy for the case that a vaccine is never found and complete containment (eradication of the epidemic) is impossible. In this case, the optimal control must meet compe…
We have two main aims in this paper. First we use theories of disease spreading on networks to look at the COVID-19 epidemic on the basis of individual contacts -- these give rise to predictions which are often rather different from the homogeneous mixing approaches usually used. Our second aim is to look at the role of social deprivation, again using networks as our basis, in the spread of this epidemic. We choose the city of Kolkata as a case study, but assert that the insights so obtained are applicable to a wide variety of urban environments which are densely populated and where social ine…
