Fast variables determine the epidemic threshold in the pairwise model with an improved closure

Pairwise models are used widely to model epidemic spread on networks. These include the modelling of susceptible-infected-removed (SIR) epidemics on regular networks and extensions to SIS dynamics and contact tracing on more exotic networks exhibiting degree heterogeneity, directed and/or weighted links and clustering. However, extra features of the disease dynamics or of the network lead to an increase in system size and analytical tractability becomes problematic. Various `closures' can be used to keep the system tractable. Focusing on SIR epidemics on regular but clustered networks, we show that even for the most complex closure we can determine the epidemic threshold as an asymptotic expansion in terms of the clustering coefficient.We do this by exploiting the presence of a system of fast variables, specified by the correlation structure of the epidemic, whose steady state determines the epidemic threshold. While we do not find the steady state analytically, we create an elegant asymptotic expansion of it. We validate this new threshold by comparing it to the numerical solution of the full system and find excellent agreement over a wide range of values of the clustering coefficient, transmission rate and average degree of the network. The technique carries over to pairwise models with other closures [1] and we note that the epidemic threshold will be model dependent. This emphasises the importance of model choice when dealing with realistic outbreaks

Simulation of an SEIR infectious disease model on the dynamic contact network of conference attendees

The spread of infectious diseases crucially depends on the pattern of contacts among individuals. Knowledge of these patterns is thus essential to inform models and computational efforts. Few empirical studies are however available that provide estimates of the number and duration of contacts among social groups. Moreover, their space and time resolution are limited, so that data is not explicit at the person-to-person level, and the dynamical aspect of the contacts is disregarded. Here, we want to assess the role of data-driven dynamic contact patterns among individuals, and in particular of their temporal aspects, in shaping the spread of a simulated epidemic in the population. We consider high resolution data of face-to-face interactions between the attendees of a conference, obtained from the deployment of an infrastructure based on Radio Frequency Identification (RFID) devices that assess mutual face-to-face proximity. The spread of epidemics along these interactions is simulated through an SEIR model, using both the dynamical network of contacts defined by the collected data, and two aggregated versions of such network, in order to assess the role of the data temporal aspects. We show that, on the timescales considered, an aggregated network taking into account the daily duration of contacts is a good approximation to the full resolution network, whereas a homogeneous representation which retains only the topology of the contact network fails in reproducing the size of the epidemic. These results have important implications in understanding the level of detail needed to correctly inform computational models for the study and management of real epidemics

Mapping structural diversity in networks sharing a given degree distribution and global clustering: Adaptive resolution grid search evolution with Diophantine equation-based mutations

Methods that generate networks sharing a given degree distribution and global clustering can induce changes in structural properties other than that controlled for. Diversity in structural properties, in turn, can affect the outcomes of dynamical processes operating on those networks. Since exhaustive sampling is not possible, we propose a novel evolutionary framework for mapping this structural diversity. The three main features of this framework are: (a) subgraph-based encoding of networks, (b) exact mutations based on solving systems of Diophantine equations, and (c) heuristic diversity-driven mechanism to drive resolution changes in the MapElite algorithm.We show that our framework can elicit networks with diversity in their higher-order structure and that this diversity affects the behaviour of the complex contagion model. Through a comparison with state of the art clustered network generation methods, we demonstrate that our approach can uncover a comparably diverse range of networks without needing computationally unfeasible mixing times. Further, we suggest that the subgraph-based encoding provides greater confidence in the diversity of higher-order network structure for low numbers of samples and is the basis for explaining our results with complex contagion model. We believe that this framework could be applied to other complex landscapes that cannot be practically mapped via exhaustive sampling

Recommendations for and compliance with social restrictions during implementation of school closures in the early phase of the influenza A (H1N1) 2009 outbreak in Melbourne, Australia

Background Localized reactive school and classroom closures were implemented as part of a suite of pandemic containment measures during the initial response to influenza A (H1N1) 2009 in Melbourne, Australia. Infected individuals, and those who had been in close contact with a case, were asked to stay in voluntary home quarantine and refrain from contact with visitors for seven days from the date of symptom onset or exposure to an infected person. Oseltamivir (Tamiflu®) was available for treatment or prophylaxis. Methods We surveyed affected families through schools involved in the closures. Analyses of responses were descriptive. We characterized recommendations made to case and contact households and quantified adherence to guidelines and antiviral therapy. Results Of the 314 respondent households, 51 contained a confirmed case. The prescribed quarantine period ranged from 1-14 days, reflecting logistic difficulties in reactive implementation relative to the stated guidelines. Household-level compliance with the requirement to stay at home was high (84.5%, 95% CI 79.3,88.5) and contact with children outside the immediate family infrequent. Conclusions Levels of compliance with recommendations in our sample were high compared with other studies, likely due to heightened public awareness of a newly introduced virus of uncertain severity. The variability of reported recommendations highlighted the difficulties inherent in implementing a targeted reactive strategy, such as that employed in Melbourne, on a large scale during a public health emergency. This study emphasizes the need to understand how public health measures are implemented when seeking to evaluate their effectiveness

Mean-field models for non-Markovian epidemics on networks

This paper introduces a novel extension of the edge-based compartmental model to epidemics where the transmission and recovery processes are driven by general independent probability distributions. Edge-based compartmental modelling is just one of many different approaches used to model the spread of an infectious disease on a network; the major result of this paper is the rigorous proof that the edge-based compartmental model and the message passing models are equivalent for general independent transmission and recovery processes. This implies that the new model is exact on the ensemble of configuration model networks of infinite size. For the case of Markovian transmission themessage passing model is re-parametrised into a pairwise-like model which is then used to derive many well-known pairwise models for regular networks, or when the infectious period is exponentially distributed or is of a fixed length

The spatial resolution of epidemic peaks

The emergence of novel respiratory pathogens can challenge the capacity of key health care resources, such as intensive care units, that are constrained to serve only specific geographical populations. An ability to predict the magnitude and timing of peak incidence at the scale of a single large population would help to accurately assess the value of interventions designed to reduce that peak. However, current disease-dynamic theory does not provide a clear understanding of the relationship between: epidemic trajectories at the scale of interest (e.g. city); population mobility; and higher resolution spatial effects (e.g. transmission within small neighbourhoods). Here, we used a spatially-explicit stochastic meta-population model of arbitrary spatial resolution to determine the effect of resolution on model-derived epidemic trajectories. We simulated an influenza-like pathogen spreading across theoretical and actual population densities and varied our assumptions about mobility using Latin-Hypercube sampling. Even though, by design, cumulative attack rates were the same for all resolutions and mobilities, peak incidences were different. Clear thresholds existed for all tested populations, such that models with resolutions lower than the threshold substantially overestimated population-wide peak incidence. The effect of resolution was most important in populations which were of lower density and lower mobility. With the expectation of accurate spatial incidence datasets in the near future, our objective was to provide a framework for how to use these data correctly in a spatial meta-population model. Our results suggest that there is a fundamental spatial resolution for any pathogen-population pair. If underlying interactions between pathogens and spatially heterogeneous populations are represented at this resolution or higher, accurate predictions of peak incidence for city-scale epidemics are feasible

Epidemic Contact Tracing via Communication Traces

Traditional contact tracing relies on knowledge of the interpersonal network of physical interactions, where contagious outbreaks propagate. However, due to privacy constraints and noisy data assimilation, this network is generally difficult to reconstruct accurately. Communication traces obtained by mobile phones are known to be good proxies for the physical interaction network, and they may provide a valuable tool for contact tracing. Motivated by this assumption, we propose a model for contact tracing, where an infection is spreading in the physical interpersonal network, which can never be fully recovered; and contact tracing is occurring in a communication network which acts as a proxy for the first. We apply this dual model to a dataset covering 72 students over a 9 month period, for which both the physical interactions as well as the mobile communication traces are known. Our results suggest that a wide range of contact tracing strategies may significantly reduce the final size of the epidemic, by mainly affecting its peak of incidence. However, we find that for low overlap between the face-to-face and communication interaction network, contact tracing is only efficient at the beginning of the outbreak, due to rapidly increasing costs as the epidemic evolves. Overall, contact tracing via mobile phone communication traces may be a viable option to arrest contagious outbreaks

PLoS Negl Trop Dis

BACKGROUND: During the Ebola virus disease (EVD) epidemic in Liberia, contact tracing was implemented to rapidly detect new cases and prevent further transmission. We describe the scope and characteristics of contact tracing in Liberia and assess its performance during the 2014-2015 EVD epidemic. METHODOLOGY/PRINCIPAL FINDINGS: We performed a retrospective descriptive analysis of data collection forms for contact tracing conducted in six counties during June 2014-July 2015. EVD case counts from situation reports in the same counties were used to assess contact tracing coverage and sensitivity. Contacts who presented with symptoms and/or died, and monitoring was stopped, were classified as "potential cases". Positive predictive value (PPV) was defined as the proportion of traced contacts who were identified as potential cases. Bivariate and multivariate logistic regression models were used to identify characteristics among potential cases. We analyzed 25,830 contact tracing records for contacts who had monitoring initiated or were last exposed between June 4, 2014 and July 13, 2015. Contact tracing was initiated for 26.7% of total EVD cases and detected 3.6% of all new cases during this period. Eighty-eight percent of contacts completed monitoring, and 334 contacts were identified as potential cases (PPV = 1.4%). Potential cases were more likely to be detected early in the outbreak; hail from rural areas; report multiple exposures and symptoms; have household contact or direct bodily or fluid contact; and report nausea, fever, or weakness compared to contacts who completed monitoring. CONCLUSIONS/SIGNIFICANCE: Contact tracing was a critical intervention in Liberia and represented one of the largest contact tracing efforts during an epidemic in history. While there were notable improvements in implementation over time, these data suggest there were limitations to its performance-particularly in urban districts and during peak transmission. Recommendations for improving performance include integrated surveillance, decentralized management of multidisciplinary teams, comprehensive protocols, and community-led strategies

School closures during the 2009 influenza pandemic: national and local experiences

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