Predicting and controlling infectious disease epidemics using temporal networks

Adaption of the temporal correlation coefficient calculation for temporal networks (applied to a real-world pig trade network)

The average topological overlap of two graphs of two consecutive time steps measures the amount of changes in the edge configuration between the two snapshots. This value has to be zero if the edge configuration changes completely and one if the two consecutive graphs are identical. Current methods depend on the number of nodes in the network or on the maximal number of connected nodes in the consecutive time steps. In the first case, this methodology breaks down if there are nodes with no edges. In the second case, it fails if the maximal number of active nodes is larger than the maximal number of connected nodes. In the following, an adaption of the calculation of the temporal correlation coefficient and of the topological overlap of the graph between two consecutive time steps is presented, which shows the expected behaviour mentioned above. The newly proposed adaption uses the maximal number of active nodes, i.e. the number of nodes with at least one edge, for the calculation of the topological overlap. The three methods were compared with the help of vivid example networks to reveal the differences between the proposed notations. Furthermore, these three calculation methods were applied to a real-world network of animal movements in order to detect influences of the network structure on the outcome of the different methods.

The role of heterogeneity in contact timing and duration in network models of influenza spread in schools

Influenza poses a significant health threat to children, and schools may play a critical role in community outbreaks. Mathematical outbreak models require assumptions about contact rates and patterns among students, but the level of temporal granularity required to produce reliable results is unclear. We collected objective contact data from students aged 5–14 at an elementary school and middle school in the state of Utah, USA, and paired those data with a novel, data-based model of influenza transmission in schools. Our simulations produced within-school transmission averages consistent with published estimates. We compared simulated outbreaks over the full resolution dynamic network with simulations on networks with averaged representations of contact timing and duration. For both schools, averaging the timing of contacts over one or two school days caused average outbreak sizes to increase by 1–8%. Averaging both contact timing and pairwise contact durations caused average outbreak sizes to increase by 10% at the middle school and 72% at the elementary school. Averaging contact durations separately across within-class and between-class contacts reduced the increase for the elementary school to 5%. Thus, the effect of ignoring details about contact timing and duration in school contact networks on outbreak size modelling can vary across different schools.

Accelerating coordination in temporal networks by engineering the link order

Social dynamics on a network may be accelerated or decelerated depending on which pairs of individuals in the network communicate early and which pairs do later. The order with which the links in a given network are sequentially used, which we call the link order, may be a strong determinant of dynamical behaviour on networks, potentially adding a new dimension to effects of temporal networks relative to static networks. Here we study the effect of the link order on linear coordination (i.e., synchronisation) dynamics. We show that the coordination speed considerably depends on specific orders of links. In addition, applying each single link for a long time to ensure strong pairwise coordination before moving to a next pair of individuals does not often enhance coordination of the entire network. We also implement a simple greedy algorithm to optimise the link order in favour of fast coordination.

Temporal dynamics of connectivity and epidemic properties of growing networks

Traditional mathematical models of epidemic disease had for decades conventionally considered static structure for contacts. Recently, an upsurge of theoretical inquiry has strived towards rendering the models more realistic by incorporating the temporal aspects of networks of contacts, societal and online, that are of interest in the study of epidemics (and other similar diffusion processes). However, temporal dynamics have predominantly focused on link fluctuations and nodal activities, and less attention has been paid to the growth of the underlying network. Many real networks grow: Online networks are evidently in constant growth, and societal networks can grow due to migration flux and reproduction. The effect of network growth on the epidemic properties of networks is hitherto unknown, mainly due to the predominant focus of the network growth literature on the so-called steady state. This paper takes a step towards alleviating this gap. We analytically study the degree dynamics of a given arbitrary network that is subject to growth. We use the theoretical findings to predict the epidemic properties of the network as a function of time. We observe that the introduction of new individuals into the network can enhance or diminish its resilience against endemic outbreaks and investigate how this regime shift depends upon the connectivity of newcomers and on how they establish connections to existing nodes. Throughout, theoretical findings are corroborated with Monte Carlo simulations over synthetic and real networks. The results shed light on the effects of network growth on the future epidemic properties of networks and offers insights for devising a priori immunization strategies.

Compensating for population sampling in simulations of epidemic spread on temporal contact networks

Data describing human interactions often suffer from incomplete sampling of the underlying population. As a consequence, the study of contagion processes using data-driven models can lead to a severe underestimation of the epidemic risk. Here we present a systematic method to alleviate this issue and obtain a better estimation of the risk in the context of epidemic models informed by high-resolution time-resolved contact data. We consider several such data sets collected in various contexts and perform controlled resampling experiments. We show how the statistical information contained in the resampled data can be used to build a series of surrogate versions of the unknown contacts. We simulate epidemic processes on the resulting reconstructed data sets and show that it is possible to obtain good estimates of the outcome of simulations performed using the complete data set. We discuss limitations and potential improvements of our method.

Information content of contact-pattern representations and predictability of epidemic outbreaks

To understand the contact patterns of a population--who is in contact with whom, and when the contacts happen--is crucial for modeling outbreaks of infectious disease. Traditional theoretical epidemiology assumes that any individual can meet any with equal probability. A more modern approach, network epidemiology, assumes people are connected into a static network over which the disease spreads. Newer yet, temporal network epidemiology, includes the time in the contact representations. In this paper, we investigate the effect of these successive inclusions of more information. Using empirical proximity data, we study both outbreak sizes from unknown sources, and from known states of ongoing outbreaks. In the first case, there are large differences going from a fully mixed simulation to a network, and from a network to a temporal network. In the second case, differences are smaller. We interpret these observations in terms of the temporal network structure of the data sets. For example, a fast overturn of nodes and links seem to make the temporal information more important.

Dynamics of history-dependent epidemics in temporal networks

The structural properties of temporal networks often influence the dynamical processes that occur on these networks, e.g., bursty interaction patterns have been shown to slow down epidemics. In this paper, we investigate the effect of link lifetimes on the spread of history-dependent epidemics. We formulate an analytically tractable activity-driven temporal network model that explicitly incorporates link lifetimes. For Markovian link lifetimes, we use mean-field analysis for computing the epidemic threshold, while the effect of non-Markovian link lifetimes is studied using simulations. Furthermore, we also study the effect of negative correlation between the number of links spawned by an individual and the lifetimes of those links. Such negative correlations may arise due to the finite cognitive capacity of the individuals. Our investigations reveal that heavy-tailed link lifetimes slow down the epidemic, while negative correlations can reduce epidemic prevalence. We believe that our results help shed light on the role of link lifetimes in modulating diffusion processes on temporal networks.

Most probable paths in temporal weighted networks: An application to ocean transport

We consider paths in weighted and directed temporal networks, introducing tools to compute sets of paths of high probability. We quantify the relative importance of the most probable path between two nodes with respect to the whole set of paths and to a subset of highly probable paths that incorporate most of the connection probability. These concepts are used to provide alternative definitions of betweenness centrality. We apply our formalism to a transport network describing surface flow in the Mediterranean sea. Despite the full transport dynamics is described by a very large number of paths we find that, for realistic time scales, only a very small subset of high probability paths (or even a single most probable one) is enough to characterize global connectivity properties of the network.

The basic reproduction number as a predictor for epidemic outbreaks in temporal networks

The basic reproduction number R₀—the number of individuals directly infected by an infectious person in an otherwise susceptible population—is arguably the most widely used estimator of how severe an epidemic outbreak can be. This severity can be more directly measured as the fraction of people infected once the outbreak is over, Ω. In traditional mathematical epidemiology and common formulations of static network epidemiology, there is a deterministic relationship between R₀ and Ω. However, if one considers disease spreading on a temporal contact network—where one knows when contacts happen, not only between whom—then larger R₀ does not necessarily imply larger Ω. In this paper, we numerically investigate the relationship between R₀ and Ω for a set of empirical temporal networks of human contacts. Among 31 explanatory descriptors of temporal network structure, we identify those that make R₀ an imperfect predictor of Ω. We find that descriptors related to both temporal and topological aspects affect the relationship between R₀ and Ω, but in different ways.

Efficient allocation of heterogeneous response times in information spreading process

Recently, the impacts of spatiotemporal heterogeneities of human activities on spreading dynamics have attracted extensive attention. In this paper, we intend to understand how the heterogeneous distribution of response times at the individual level influences information spreading. Based on the uncorrelated scale-free networks without degree-degree correlation, we study the susceptible-infected spreading dynamics with adjustable power-law response time distribution, and find that the stronger the heterogeneity of response times is, the faster the information spreading is in the early and middle stages. Following a given heterogeneity, the procedure of reducing the correlation between the response times and degrees of individuals can also accelerate the spreading dynamics in the early and middle stages. However, the dynamics in the late stage is slightly more complicated, and there is an optimal value of the full prevalence time (i.e., the time for full infection on a network) changing with the heterogeneity of response times and the response time-degree correlation, respectively. The optimal phenomena result from the efficient allocation of heterogeneous response times.

Birth and death of links control disease spreading in empirical contact networks

We investigate what structural aspects of a collection of twelve empirical temporal networks of human contacts are important to disease spreading. We scan the entire parameter spaces of the two canonical models of infectious disease epidemiology—the Susceptible-Infectious-Susceptible (SIS) and Susceptible-Infectious-Removed (SIR) models. The results from these simulations are compared to reference data where we eliminate structures in the interevent intervals, the time to the first contact in the data, or the time from the last contact to the end of the sampling. The picture we find is that the birth and death of links, and the total number of contacts over a link, are essential to predict outbreaks. On the other hand, the exact times of contacts between the beginning and end, or the interevent interval distribution, do not matter much. In other words, a simplified picture of these empirical data sets that suffices for epidemiological purposes is that links are born, is active with some intensity, and die.

Extinction times of epidemic outbreaks in networks

In the Susceptible-Infectious-Recovered (SIR) model of disease spreading, the time to extinction of the epidemics happens at an intermediate value of the per-contact transmission probability. Too contagious infections burn out fast in the population. Infections that are not contagious enough die out before they spread to a large fraction of people. We characterize how the maximal extinction time in SIR simulations on networks depend on the network structure. For example we find that the average distances in isolated components, weighted by the component size, is a good predictor of the maximal time to extinction. Furthermore, the transmission probability giving the longest outbreaks is larger than, but otherwise seemingly independent of, the epidemic threshold.

Temporal networks: slowing down diffusion by long lasting interactions

Interactions among units in complex systems occur in a specific sequential order, thus affecting the flow of information, the propagation of diseases, and general dynamical processes. We investigate the Laplacian spectrum of temporal networks and compare it with that of the corresponding aggregate network. First, we show that the spectrum of the ensemble average of a temporal network has identical eigenmodes but smaller eigenvalues than the aggregate networks. In large networks without edge condensation, the expected temporal dynamics is a time-rescaled version of the aggregate dynamics. Even for single sequential realizations, diffusive dynamics is slower in temporal networks. These discrepancies are due to the noncommutability of interactions. We illustrate our analytical findings using a simple temporal motif, larger network models, and real temporal networks.

Efficient interruption of infection chains by targeted removal of central holdings in an animal trade network

Centrality parameters in animal trade networks typically have right-skewed distributions, implying that these networks are highly resistant against the random removal of holdings, but vulnerable to the targeted removal of the most central holdings. In the present study, we analysed the structural changes of an animal trade network topology based on the targeted removal of holdings using specific centrality parameters in comparison to the random removal of holdings. Three different time periods were analysed: the three-year network, the yearly and the monthly networks. The aim of this study was to identify appropriate measures for the targeted removal, which lead to a rapid fragmentation of the network. Furthermore, the optimal combination of the removal of three holdings regardless of their centrality was identified. The results showed that centrality parameters based on ingoing trade contacts, e.g. in-degree, ingoing infection chain and ingoing closeness, were not suitable for a rapid fragmentation in all three time periods. More efficient was the removal based on parameters considering the outgoing trade contacts. In all networks, a maximum percentage of 7.0% (on average 5.2%) of the holdings had to be removed to reduce the size of the largest component by more than 75%. The smallest difference from the optimal combination for all three time periods was obtained by the removal based on out-degree with on average 1.4% removed holdings, followed by outgoing infection chain and outgoing closeness. The targeted removal using the betweenness centrality differed the most from the optimal combination in comparison to the other parameters which consider the outgoing trade contacts. Due to the pyramidal structure and the directed nature of the pork supply chain the most efficient interruption of the infection chain for all three time periods was obtained by using the targeted removal based on out-degree.

Epidemiologically optimal static networks from temporal network data

One of network epidemiology's central assumptions is that the contact structure over which infectious diseases propagate can be represented as a static network. However, contacts are highly dynamic, changing at many time scales. In this paper, we investigate conceptually simple methods to construct static graphs for network epidemiology from temporal contact data. We evaluate these methods on empirical and synthetic model data. For almost all our cases, the network representation that captures most relevant information is a so-called exponential-threshold network. In these, each contact contributes with a weight decreasing exponentially with time, and there is an edge between a pair of vertices if the weight between them exceeds a threshold. Networks of aggregated contacts over an optimally chosen time window perform almost as good as the exponential-threshold networks. On the other hand, networks of accumulated contacts over the entire sampling time, and networks of concurrent partnerships, perform worse. We discuss these observations in the context of the temporal and topological structure of the data sets.

To understand how diseases spread in a population, it is important to study the network of people in contact. Many methods to model epidemic outbreaks make the assumption that one can treat this network as static. In reality, we know that contact patterns between people change in time, and old contacts are soon irrelevant—it does not matter that we know Marie Antoinette's lovers to understand the HIV epidemic. This paper investigates methods for constructing networks of people that are as relevant as possible for disease spreading. The most promising method we call exponential-threshold network works by letting contacts contribute less, the further from the beginning of an outbreak they take place. We investigate the methods both on artificial models of the contact patterns and empirical data. Except searching for the optimal network representation, we also investigate how the structure of the original data set affects the performance of the representations.

An infectious disease model on empirical networks of human contact: bridging the gap between dynamic network data and contact matrices

BACKGROUND

The integration of empirical data in computational frameworks designed to model the spread of infectious diseases poses a number of challenges that are becoming more 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, designing containment strategies, and evaluating outcomes. However, the integration of highly detailed data sources yields models that are less transparent and general in their applicability. Hence, given a specific disease model, it is crucial to assess which representations of the raw data work best to inform the model, striking a balance between simplicity and detail.

METHODS

We consider high-resolution data on the face-to-face interactions of individuals in a pediatric hospital ward, obtained by using wearable proximity sensors. We simulate the spread of a disease in this community by using an SEIR model on top of different mathematical representations of the empirical contact patterns. At the most detailed level, we take into account all contacts between individuals and their exact timing and order. Then, we build a hierarchy of coarse-grained representations of the contact patterns that preserve only partially the temporal and structural information available in the data. We compare the dynamics of the SEIR model across these representations.

RESULTS

We show that a contact matrix that only contains average contact durations between role classes fails to reproduce the size of the epidemic obtained using the high-resolution contact data and also fails to identify the most at-risk classes. We introduce a contact matrix of probability distributions that takes into account the heterogeneity of contact durations between (and within) classes of individuals, and we show that, in the case study presented, this representation yields a good approximation of the epidemic spreading properties obtained by using the high-resolution data.

CONCLUSIONS

Our results mark a first step towards the definition of synopses of high-resolution dynamic contact networks, providing a compact representation of contact patterns that can correctly inform computational models designed to discover risk groups and evaluate containment policies. We show in a typical case of a structured population that this novel kind of representation can preserve in simulation quantitative features of the epidemics that are crucial for their study and management.