Epidemic threshold and network structure: The interplay of probability of transmission and of persistence in small-size directed networks

A growing number of studies are investigating the effect of contact structure on the dynamics of epidemics in large-scale complex networks. Whether findings thus obtained apply also to networks of small size, and thus to many real-world biological applications, is still an open question. We use numerical simulations of disease spread in directed networks of 100 individual nodes with a constant number of links. We show that, no matter the type of network structure (local, small-world, random and scale-free), there is a linear threshold determined by the probability of infection transmission between connected nodes and the probability of infection persistence in an infected node. The threshold is significantly lower for scale-free networks compared to local, random and small-world ones only if super-connected nodes have a higher number of links both to and from other nodes. The starting point, the node at which the epidemic starts, does not affect the threshold conditions, but has a marked influence on the final size of the epidemic in all kinds of network. There is evidence that contact structure has an influence on the average final size of an epidemic across all starting nodes, with significantly lower values in scale-free networks at equilibrium. Simulations in scale-free networks show a distinctive time-series pattern, which, if found in a real epidemic, can be used to infer the underlying network structure. The findings have relevance also for meta-population ecology and species conservation.     

Contact networks and transmission of an intestinal pathogen in bumble bee (Bombus impatiens) colonies

In socially living animals, individuals interact through complex networks of contact that may influence the spread of disease. Whereas traditional epidemiological models typically assume no social structure, network theory suggests that an individual’s location in the network determines its risk of infection. Empirical, especially experimental, studies of disease spread on networks are lacking, however, largely due to a shortage of amenable study systems. We used automated video-tracking to quantify networks of physical contact among individuals within colonies of the social bumble bee Bombus impatiens. We explored the effects of network structure on pathogen transmission in naturally and artificially infected hives. We show for the first time that contact network structure determines the spread of a contagious pathogen (Crithidia bombi) in social insect colonies. Differences in rates of infection among colonies resulted largely from differences in network density among hives. Within colonies, a bee’s rate of contact with infected nestmates emerged as the only significant predictor of infection risk. The activity of bees, in terms of their movement rates and division of labour (e.g., brood care, nest care, foraging), did not influence risk of infection. Our results suggest that contact networks may have an important influence on the transmission of pathogens in social insects and, possibly, other social animals.     

The number of links to and from the starting node as a predictor of epidemic size in small-size directed networks

Much recent modelling is focusing on epidemics in large-scale complex networks. Whether or not findings of these investigations also apply to networks of small size is still an open question. This is an important gap for many biological applications, including the spread of the oomycete pathogen Phytophthora ramorum in networks of plant nurseries. We use numerical simulations of disease spread and establishment in directed networks of 100 individual nodes at four levels of connectivity. Factors governing epidemic spread are network structure (local, small-world, random, scale-free) and the probabilities of infection persistence in a node and of infection transmission between connected nodes. Epidemic final size at equilibrium varies widely depending on the starting node of infection, although the latter does not affect the threshold condition for spread. The number of links from (out-degree) but not the number of links to (in-degree) the starting node of the epidemic explains a substantial amount of variation in final epidemic size at equilibrium irrespective of the structure of the network. The proportion of variance in epidemic size explained by the out-degree of the starting node increases with the level of connectivity. Targeting highly connected nodes is thus likely to make disease control more effective also in case of small-size populations, a result of relevance not just for the horticultural trade, but for epidemiology in general.     

Optimal link removal for epidemic mitigation: a two-way partitioning approach

The structure of the contact network through which a disease spreads may influence the optimal use of resources for epidemic control. In this work, we explore how to minimize the spread of infection via quarantining with limited resources. In particular, we examine which links should be removed from the contact network, given a constraint on the number of removable links, such that the number of nodes which are no longer at risk for infection is maximized. We show how this problem can be posed as a non-convex quadratically constrained quadratic program (QCQP), and we use this formulation to derive a link removal algorithm. The performance of our QCQP-based algorithm is validated on small Erd?s–Renyi and small-world random graphs, and then tested on larger, more realistic networks, including a real-world network of injection drug use. We show that our approach achieves near optimal performance and out-performs other intuitive link removal algorithms, such as removing links in order of edge centrality.     

Role of Cattle Movements in Bovine Tuberculosis Spread in France between 2005 and 2014

Live animal movements are a major transmission route for the spread of infectious agents such as Mycobacterium bovis, the main agent of bovine Tuberculosis (bTB). France became officially bTB-free in 2001, but M. bovis is still circulating in the cattle population, with about a hundred of outbreaks per year, most located in a few geographic areas. The aim of this study was to analyse the role of cattle movements in bTB spread in France between 2005 and 2014, using social network analysis and logistic regression models. At a global scale, the trade network was studied to assess the association between several centrality measures and bTB infection though a case-control analysis. The bTB infection status was associated with a higher in-degree (odds-ratio [OR] = 2.4 [1.1–5.4]) and with a higher ingoing contact chain (OR = 2.2 [1.0–4.7]). At a more local scale, a second case-control analysis was conducted to estimate the relative importance of cattle movements and spatial neighbourhood. Only direct purchase from infected herds was shown to be associated with bTB infection (OR = 2.9 [1.7–5.2]), spatial proximity to infected herds being the predominant risk factor, with decreasing ORs when distance increases. Indeed, the population attributable fraction was 12% [5%–18%] for cattle movements and 73% [68%–78%] for spatial neighbourhood. Based on these results, networks of potential effective contacts between herds were built and analysed for the three major spoligotypes reported in France. In these networks, the links representing cattle movements were associated with higher edge betweenness than those representing the spatial proximity between infected herds. They were often links connecting distinct communities and sometimes distinct geographical areas. Therefore, although their role was quantitatively lower than the one of spatial neighbourhood, cattle movements appear to have been essential in the French bTB dynamics between 2005 and 2014.     

Equal graph partitioning on estimated infection network as an effective epidemic mitigation measure

Controlling severe outbreaks remains the most important problem in infectious disease area. With time, this problem will only become more severe as population density in urban centers grows. Social interactions play a very important role in determining how infectious diseases spread, and organization of people along social lines gives rise to non-spatial networks in which the infections spread. Infection networks are different for diseases with different transmission modes, but are likely to be identical or highly similar for diseases that spread the same way. Hence, infection networks estimated from common infections can be useful to contain epidemics of a more severe disease with the same transmission mode. Here we present a proof-of-concept study demonstrating the effectiveness of epidemic mitigation based on such estimated infection networks. We first generate artificial social networks of different sizes and average degrees, but with roughly the same clustering characteristic. We then start SIR epidemics on these networks, censor the simulated incidences, and use them to reconstruct the infection network. We then efficiently fragment the estimated network by removing the smallest number of nodes identified by a graph partitioning algorithm. Finally, we demonstrate the effectiveness of this targeted strategy, by comparing it against traditional untargeted strategies, in slowing down and reducing the size of advancing epidemics.     

Transmission pathways of foot-and-mouth disease virus in the United Kingdom in 2007

Foot-and-mouth disease (FMD) virus causes an acute vesicular disease of domesticated and wild ruminants and pigs. Identifying sources of FMD outbreaks is often confounded by incomplete epidemiological evidence and the numerous routes by which virus can spread (movements of infected animals or their products, contaminated persons, objects, and aerosols). Here, we show that the outbreaks of FMD in the United Kingdom in August 2007 were caused by a derivative of FMDV O(1) BFS 1860, a virus strain handled at two FMD laboratories located on a single site at Pirbright in Surrey. Genetic analysis of complete viral genomes generated in real-time reveals a probable chain of transmission events, predicting undisclosed infected premises, and connecting the second cluster of outbreaks in September to those in August. Complete genome sequence analysis of FMD viruses conducted in real-time have identified the initial and intermediate sources of these outbreaks and demonstrate the value of such techniques in providing information useful to contemporary disease control programmes.     

Tracking Socioeconomic Vulnerability Using Network Analysis: Insights from an Avian Influenza Outbreak in an Ostrich Production Network

Background

The focus of management in many complex systems is shifting towards facilitation, adaptation, building resilience, and reducing vulnerability. Resilience management requires the development and application of general heuristics and methods for tracking changes in both resilience and vulnerability. We explored the emergence of vulnerability in the South African domestic ostrich industry, an animal production system which typically involves 3–4 movements of each bird during its lifetime. This system has experienced several disease outbreaks, and the aim of this study was to investigate whether these movements have contributed to the vulnerability of this system to large disease outbreaks.

Methodology/Principal Findings

The ostrich production system requires numerous movements of birds between different farm types associated with growth (i.e. Hatchery to juvenile rearing farm to adult rearing farm). We used 5 years of movement records between 2005 and 2011 prior to an outbreak of Highly Pathogenic Avian Influenza (H5N2). These data were analyzed using a network analysis in which the farms were represented as nodes and the movements of birds as links. We tested the hypothesis that increasing economic efficiency in the domestic ostrich industry in South Africa made the system more vulnerable to outbreak of Highly Pathogenic Avian Influenza (H5N2). Our results indicated that as time progressed, the network became increasingly vulnerable to pathogen outbreaks. The farms that became infected during the outbreak displayed network qualities, such as significantly higher connectivity and centrality, which predisposed them to be more vulnerable to disease outbreak.

Conclusions/Significance

Taken in the context of previous research, our results provide strong support for the application of network analysis to track vulnerability, while also providing useful practical implications for system monitoring and management.     

SIS along a continuum (SIS(c)) epidemiological modelling and control of diseases on directed trade networks

Network theory has been applied to many aspects of biosciences, including epidemiology. Most epidemiological models in networks, however, have used the standard assumption of either susceptible or infected individuals. In some cases (e.g. the spread of Phytophthora ramorum in plant trade networks), a continuum in the infection status of nodes can better capture the reality of epidemics in networks. In this paper, a Susceptible-Infected-Susceptible model along a continuum in the infection status (SIS(c)) is presented, using as a case study directed networks and two parameters governing the epidemic process (probability of infection persistence (p(p)) and of infection transmission (p(t)). The previously empirically reported linear epidemic threshold in a plot of p(p) as a function of p(t) (Pautasso and Jeger, 2008) is derived analytically. Also the previously observed negative correlation between the epidemic threshold and the correlation between links in and out of nodes (Moslonka-Lefebvre et al., 2009) is justified analytically. A simple algorithm to calculate the threshold conditions is introduced. Additionally, a control strategy based on targeting market hierarchical categories such as producers, wholesalers and retailers is presented and applied to a realistic reconstruction of the UK horticultural trade network. Finally, various applications (e.g., seed exchange networks, food trade, spread of ideas) and potential refinements of the SIS(c) model are discussed.     

Disease contact tracing in random and clustered networks

The efficacy of contact tracing, be it between individuals (e.g. sexually transmitted diseases or severe acute respiratory syndrome) or between groups of individuals (e.g. foot-and-mouth disease; FMD), is difficult to evaluate without precise knowledge of the underlying contact structure; i.e. who is connected to whom? Motivated by the 2001 FMD epidemic in the UK, we determine, using stochastic simulations and deterministic 'moment closure' models of disease transmission on networks of premises (nodes), network and disease properties that are important for contact tracing efficiency. For random networks with a high average number of connections per node, little clustering of connections and short latency periods, contact tracing is typically ineffective. In this case, isolation of infected nodes is the dominant factor in determining disease epidemic size and duration. If the latency period is longer and the average number of connections per node small, or if the network is spatially clustered, then the contact tracing performs better and an overall reduction in the proportion of nodes that are removed during an epidemic is observed.     

The impacts of network topology on disease spread

Individuals in a population susceptible to a disease may be represented as vertices in a network, with the edges that connect vertices representing social and/or spatial contact between individuals. Networks, which explicitly included six different patterns of connection between vertices, were created. Both scale-free networks and random graphs showed a different response in path level to increasing levels of clustering than regular lattices. Clustering promoted short path lengths in all network types, but randomly assembled networks displayed a logarithmic relationship between degree and path length; whereas this response was linear in regular lattices. In all cases, small-world models, generated by rewiring the connections of a regular lattice, displayed properties, which spanned the gap between random and regular networks.Simulation of a disease in these networks showed a strong response to connectance pattern, even when the number of edges and vertices were approximately equal. Epidemic spread was fastest, and reached the largest size, in scale-free networks, then in random graphs. Regular lattices were the slowest to be infected, and rewired lattices were intermediate between these two extremes. Scale-free networks displayed the capacity to produce an epidemic even at a likelihood of infection, which was too low to produce an epidemic for the other network types. The interaction between the statistical properties of the network and the results of epidemic spread provides a useful tool for assessing the risk of disease spread in more realistic networks.     

Disease spread in small-size directed networks: Epidemic threshold, correlation between links to and from nodes, and clustering

Network epidemiology has mainly focused on large-scale complex networks. It is unclear whether findings of these investigations also apply to networks of small size. This knowledge gap is of relevance for many biological applications, including meta-communities, plant–pollinator interactions and the spread of the oomycete pathogen Phytophthora ramorum in networks of plant nurseries. Moreover, many small-size biological networks are inherently asymmetrical and thus cannot be realistically modelled with undirected networks. We modelled disease spread and establishment in directed networks of 100 and 500 nodes at four levels of connectance in six network structures (local, small-world, random, one-way, uncorrelated, and two-way scale-free networks). The model was based on the probability of infection persistence in a node and of infection transmission between connected nodes. Regardless of the size of the network, the epidemic threshold did not depend on the starting node of infection but was negatively related to the correlation coefficient between in- and out-degree for all structures, unless networks were sparsely connected. In this case clustering played a significant role. For small-size scale-free directed networks to have a lower epidemic threshold than other network structures, there needs to be a positive correlation between number of links to and from nodes. When this correlation is negative (one-way scale-free networks), the epidemic threshold for small-size networks can be higher than in non-scale-free networks. Clustering does not necessarily have an influence on the epidemic threshold if connectance is kept constant. Analyses of the influence of the clustering on the epidemic threshold in directed networks can also be spurious if they do not consider simultaneously the effect of the correlation coefficient between in- and out-degree.     

Effectiveness of dynamic quarantines against pathogen spread in models of the horticultural trade network

The live plant nursery trade is a potential vector for pests and pathogens, which can spread to natural and developed environments with unintended ecosystem consequences. Simulated, approximately scale-free, tiered horticultural trade networks consisting of growers, wholesalers, and retailers were used to study the efficacy of quarantine inspection and isolation procedures for reducing the spread of infected materials to consumers. The quarantine algorithm temporarily isolated infected nurseries from the rest of the trade network, rewiring the affected trade connections to unquarantined nodes, until the infection was reduced below the detection threshold, at which time the formerly infected nursery was reincorporated into the trade network.Nodes were inspected for infection at regular intervals. Increasing the inspection interval resulted in higher levels of infection with large, system-wide oscillations whose period that matched the inspection interval. The timing of quarantine inspections of the largest hub in the grower tier drove the dynamics of the entire network. Increasing the proportion of growers or wholesalers increased infection level in most networks. Increasing the connectivity within the grower and wholesaler tiers led to large increases in mean infection levels. Focusing quarantine inspection efforts on hubs in the grower and wholesaler tiers may be the most efficient method for reducing the level of infected plant material sold by retailers in real plant trade networks.     

Epidemic progression on networks based on disease generation time

We investigate the time evolution of disease spread on a network and present an analytical framework using the concept of disease generation time. Assuming a susceptible–infected–recovered epidemic process, this network-based framework enables us to calculate in detail the number of links (edges) within the network that are capable of producing new infectious nodes (individuals), the number of links that are not transmitting the infection further (non-transmitting links), as well as the number of contacts that individuals have with their neighbours (also known as degree distribution) within each epidemiological class, for each generation period. Using several examples, we demonstrate very good agreement between our analytical calculations and the results of computer simulations.     

Epidemiological Investigation of Bovine Tuberculosis Herd Breakdowns in Spain 2009/2011

We analyzed the most likely cause of 687 bovine tuberculosis (bTB) breakdowns detected in Spain between 2009 and 2011 (i.e., 22% of the total number of breakdowns detected during this period). Seven possible causes were considered: i) residual infection; ii) introduction of infected cattle from other herds; iii) sharing of pastures with infected herds; iv) contiguous spread from infected neighbor herds; v) presence of infected goats in the farm; vi) interaction with wildlife reservoirs and vii) contact with an infected human. For each possible cause a decision tree was developed and key questions were included in each of them. Answers to these key questions lead to different events within each decision tree. In order to assess the likelihood of occurrence of the different events a qualitative risk assessment approach was used. For this purpose, an expert opinion workshop was organized and ordinal values, ranging from 0 to 9 (i.e., null to very high likelihood of occurrence) were assigned. The analysis identified residual infection as the most frequent cause of bTB breakdowns (22.3%; 95%CI: 19.4–25.6), followed by interaction with wildlife reservoirs (13.1%; 95%CI: 10.8–15.8). The introduction of infected cattle, sharing of pastures and contiguous spread from infected neighbour herds were also identified as relevant causes. In 41.6% (95%CI: 38.0–45.4) of the breakdowns the origin of infection remained unknown.Veterinary officers conducting bTB breakdown investigations have to state their opinion about the possible cause of each breakdown. Comparison between the results of our analysis and the opinion from veterinary officers revealed a slight concordance. This slight agreement might reflect a lack of harmonized criteria to assess the most likely cause of bTB breakdowns as well as different perceptions about the importance of the possible causes. This is especially relevant in the case of the role of wildlife reservoirs.     

Potential for epidemic take-off from the primary outbreak farm via livestock movements

Background

We consider the potential for infection to spread in a farm population from the primary outbreak farm via livestock movements prior to disease detection. We analyse how this depends on the time of the year infection occurs, the species transmitting, the length of infectious period on the primary outbreak farm, location of the primary outbreak, and whether a livestock market becomes involved. We consider short infectious periods of 1 week, 2 weeks and 4 weeks, characteristic of acute contagious livestock diseases. The analysis is based on farms in Scotland from 1 January 2003 to 31 July 2007.

Results

The proportion of primary outbreaks from which an acute contagious disease would spread via movement of livestock is generally low, but exhibits distinct annual cyclicity with peaks in May and August. The distance that livestock are moved varies similarly: at the time of the year when the potential for spread via movements is highest, the geographical spread via movements is largest. The seasonal patterns for cattle differ from those for sheep whilst there is no obvious seasonality for pigs. When spread via movements does occur, there is a high risk of infection reaching a livestock market; infection of markets can amplify disease spread. The proportion of primary outbreaks that would spread infection via livestock movements varies significantly between geographical regions.

Conclusions

In this paper we introduce a set-up for analysis of movement data that allows for a generalized assessment of the risk associated with infection spreading from a primary outbreak farm via livestock movements, applying this to Scotland, we assess how this risk depends upon the time of the year, species transmitting, location of the farm and other factors.     

Risk of Foot-and-Mouth Disease spread due to sole occupancy authorities and linked cattle holdings

Livestock movements in Great Britain are well recorded, have been extensively analysed with respect to their role in disease spread, and have been used in real time to advise governments on the control of infectious diseases. Typically, livestock holdings are treated as distinct entities that must observe movement standstills upon receipt of livestock, and must report livestock movements. However, there are currently two dispensations that can exempt holdings from either observing standstills or reporting movements, namely the Sole Occupancy Authority (SOA) and Cattle Tracing System (CTS) Links, respectively. In this report we have used a combination of data analyses and computational modelling to investigate the usage and potential impact of such linked holdings on the size of a Foot-and-Mouth Disease (FMD) epidemic. Our analyses show that although SOAs are abundant, their dynamics appear relatively stagnant. The number of CTS Links is also abundant, and increasing rapidly. Although most linked holdings are only involved in a single CTS Link, some holdings are involved in numerous links that can be amalgamated to form "CTS Chains" which can be both large and geographically dispersed. Our model predicts that under a worst case scenario of "one infected - all infected", SOAs do pose a risk of increasing the size (in terms of number of infected holdings) of a FMD epidemic, but this increase is mainly due to intra-SOA infection spread events. Furthermore, although SOAs do increase the geographic spread of an epidemic, this increase is predominantly local. Whereas, CTS Chains pose a risk of increasing both the size and the geographical spread of the disease substantially, under a worse case scenario. Our results highlight the need for further investigations into whether CTS Chains are transmission chains, and also investigations into intra-SOA movements and livestock distributions due to the lack of current data.     

Epithelial expression of mRNA and protein for IL-6, IL-10 and TNF-α in endobronchial biopsies in horses with recurrent airway obstruction

Background  

Models of Foot and Mouth Disease (FMD) transmission have assumed a homogeneous landscape across which Euclidean distance is a suitable measure of the spatial dependency of transmission. This paper investigated features of the landscape and their impact on transmission during the period of predominantly local spread which followed the implementation of the national movement ban during the 2001 UK FMD epidemic. In this study 113 farms diagnosed with FMD which had a known source of infection within 3 km (cases) were matched to 188 control farms which were either uninfected or infected at a later timepoint. Cases were matched to controls by Euclidean distance to the source of infection and farm size. Intervening geographical features and connectivity between the source of infection and case and controls were compared.     

Reconstruction of the Transmission History of RNA Virus Outbreaks Using Full Genome Sequences: Foot-and-Mouth Disease Virus in Bulgaria in 2011

Improvements to sequencing protocols and the development of computational phylogenetics have opened up opportunities to study the rapid evolution of RNA viruses in real time. In practical terms, these results can be combined with field data in order to reconstruct spatiotemporal scenarios that describe the origin and transmission pathways of viruses during an epidemic. In the case of notifiable diseases, such as foot-and-mouth disease (FMD), these analyses provide important insights into the epidemiology of field outbreaks that can support disease control programmes. This study reconstructs the origin and transmission history of the FMD outbreaks which occurred during 2011 in Burgas Province, Bulgaria, a country that had been previously FMD-free-without-vaccination since 1996. Nineteen full genome sequences (FGS) of FMD virus (FMDV) were generated and analysed, including eight representative viruses from all of the virus-positive outbreaks of the disease in the country and 11 closely-related contemporary viruses from countries in the region where FMD is endemic (Turkey and Israel). All Bulgarian sequences shared a single putative common ancestor which was closely related to the index case identified in wild boar. The closest relative from outside of Bulgaria was a FMDV collected during 2010 in Bursa (Anatolia, Turkey). Within Bulgaria, two discrete genetic clusters were detected that corresponded to two episodes of outbreaks that occurred during January and March-April 2011. The number of nucleotide substitutions that were present between, and within, these separate clusters provided evidence that undetected FMDV infection had occurred. These conclusions are supported by laboratory data that subsequently identified three additional FMDV-infected livestock premises by serosurveillance, as well as a number of antibody positive wild boar on both sides of the border with Turkish Thrace. This study highlights how FGS analysis can be used as an effective on-the-spot tool to support and help direct epidemiological investigations of field outbreaks.     

Habitat loss exacerbates pathogen spread: An Agent-based model of avian influenza infection in migratory waterfowl

Habitat availability determines the distribution of migratory waterfowl along their flyway, which further influences the transmission and spatial spread of avian influenza viruses (AIVs). The extensive habitat loss in the East Asian-Australasian Flyway (EAAF) may have potentially altered the virus spread and transmission, but those consequences are rarely studied. We constructed 6 fall migration networks that differed in their level of habitat loss, wherein an increase in habitat loss resulted in smaller networks with fewer sites and links. We integrated an agent-based model and a susceptible-infected-recovered model to simulate waterfowl migration and AIV transmission. We found that extensive habitat loss in the EAAF can 1) relocate the outbreaks northwards, responding to the distribution changes of wintering waterfowl geese, 2) increase the outbreak risk in remaining sites due to larger goose congregations, and 3) facilitate AIV transmission in the migratory population. In addition, our modeling output was in line with the predictions from the concept of “migratory escape”, i.e., the migration allows the geese to “escape” from the location where infection risk is high, affecting the pattern of infection prevalence in the waterfowl population. Our modeling shed light on the potential consequences of habitat loss in spreading and transmitting AIV at the flyway scale and suggested the driving mechanisms behind these effects, indicating the importance of conservation in changing spatial and temporal patterns of AIV outbreaks.