Susceptible-infected-recovered epidemics in dynamic contact networks

Contact patterns in populations fundamentally influence the spread of infectious diseases. Current mathematical methods for epidemiological forecasting on networks largely assume that contacts between individuals are fixed, at least for the duration of an outbreak. In reality, contact patterns may be quite fluid, with individuals frequently making and breaking social or sexual relationships. Here, we develop a mathematical approach to predicting disease transmission on dynamic networks in which each individual has a characteristic behaviour (typical contact number), but the identities of their contacts change in time. We show that dynamic contact patterns shape epidemiological dynamics in ways that cannot be adequately captured in static network models or mass-action models. Our new model interpolates smoothly between static network models and mass-action models using a mixing parameter, thereby providing a bridge between disparate classes of epidemiological models. Using epidemiological and sexual contact data from an Atlanta high school, we demonstrate the application of this method for forecasting and controlling sexually transmitted disease outbreaks.     

Contact tracing and disease control

Contact tracing, followed by treatment or isolation, is a key control measure in the battle against infectious diseases. It is an extreme form of locally targeted control, and as such has the potential to be highly efficient when dealing with low numbers of cases. For this reason it is frequently used to combat sexually transmitted diseases and new invading pathogens. Accurate modelling of contact tracing requires explicit information about the disease-transmission pathways from each individual, and hence the network of contacts. Here, pairwise-approximation methods and full stochastic simulations are used to investigate the utility of contact tracing. A simple relationship is found between the efficiency of contact tracing necessary for eradication and the basic reproductive ratio of the disease. This holds for a wide variety of realistic situations including heterogeneous networks containing core-groups or super-spreaders, and asymptomatic individuals. Clustering (transitivity) within the transmission network is found to destroy the relationship, requiring lower efficiency than predicted.     

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.     

Epidemics Scenarios in the “Romantic Network”

The networks of sexual contacts together with temporal interactions play key roles in the spread of sexually transmitted infections. Unfortunately, data for this kind of network is scarce. One of the few exceptions, the “Romantic network”, is a complete structure of a real sexual network in a high school. Based on many network measurements the authors of the work have concluded that it does not correspond to any other model network. Regarding the temporal structure, several studies indicate that relationship timing can have an effect on the diffusion throughout networks, as relationship order determines transmission routes. The aim is to check if the particular structure, static and dynamic, of the Romantic network is determinant for the propagation of an STI. We performed simulations in two scenarios: the static network where all contacts are available and the dynamic case where contacts evolve over time. In the static case, we compared the epidemic results in the Romantic network with some paradigmatic topologies. In the dynamic scenario, we considered the dynamics of formation of pairs in the Romantic network and we studied the propagation of the diseases. Our results suggest that although this real network cannot be labeled as a Watts-Strogatz network, it is, in regard to the propagation of an STI, very similar to a high disorder network. Additionally, we found that: the effect that any individual contacting an externally infected subject is to make the network closer to a fully connected one, the higher the contact degree of patient zero the faster the spread of the outbreaks, and the epidemic impact is proportional to the numbers of contacts per unit time. Finally, our simulations confirm that relationship timing severely reduced the final outbreak size, and also, show a clear correlation between the average degree and the outbreak size over time.     

Effects of heterogeneous and clustered contact patterns on infectious disease dynamics

The spread of infectious diseases fundamentally depends on the pattern of contacts between individuals. Although studies of contact networks have shown that heterogeneity in the number of contacts and the duration of contacts can have far-reaching epidemiological consequences, models often assume that contacts are chosen at random and thereby ignore the sociological, temporal and/or spatial clustering of contacts. Here we investigate the simultaneous effects of heterogeneous and clustered contact patterns on epidemic dynamics. To model population structure, we generalize the configuration model which has a tunable degree distribution (number of contacts per node) and level of clustering (number of three cliques). To model epidemic dynamics for this class of random graph, we derive a tractable, low-dimensional system of ordinary differential equations that accounts for the effects of network structure on the course of the epidemic. We find that the interaction between clustering and the degree distribution is complex. Clustering always slows an epidemic, but simultaneously increasing clustering and the variance of the degree distribution can increase final epidemic size. We also show that bond percolation-based approximations can be highly biased if one incorrectly assumes that infectious periods are homogeneous, and the magnitude of this bias increases with the amount of clustering in the network. We apply this approach to model the high clustering of contacts within households, using contact parameters estimated from survey data of social interactions, and we identify conditions under which network models that do not account for household structure will be biased.     

Social mixing patterns in rural and urban areas of southern China

A dense population, global connectivity and frequent human–animal interaction give southern China an important role in the spread and emergence of infectious disease. However, patterns of person-to-person contact relevant to the spread of directly transmitted infections such as influenza remain poorly quantified in the region. We conducted a household-based survey of travel and contact patterns among urban and rural populations of Guangdong, China. We measured the character and distance from home of social encounters made by 1821 individuals. Most individuals reported 5–10 h of contact with around 10 individuals each day; however, both distributions have long tails. The distribution of distance from home at which contacts were made is similar: most were within a kilometre of the participant''s home, while some occurred further than 500 km away. Compared with younger individuals, older individuals made fewer contacts which tended to be closer to home. There was strong assortativity in age-based contact rates. We found no difference between the total number or duration of contacts between urban and rural participants, but urban participants tended to make contacts closer to home. These results can improve mathematical models of infectious disease emergence, spread and control in southern China and throughout the region.     

Models of epidemics: when contact repetition and clustering should be included

Background  

The spread of infectious disease is determined by biological factors, e.g. the duration of the infectious period, and social factors, e.g. the arrangement of potentially contagious contacts. Repetitiveness and clustering of contacts are known to be relevant factors influencing the transmission of droplet or contact transmitted diseases. However, we do not yet completely know under what conditions repetitiveness and clustering should be included for realistically modelling disease spread.     

Predicting epidemics on directed contact networks

Contact network epidemiology is an approach to modeling the spread of infectious diseases that explicitly considers patterns of person-to-person contacts within a community. Contacts can be asymmetric, with a person more likely to infect one of their contacts than to become infected by that contact. This is true for some sexually transmitted diseases that are more easily caught by women than men during heterosexual encounters; and for severe infectious diseases that cause an average person to seek medical attention and thereby potentially infect health care workers (HCWs) who would not, in turn, have an opportunity to infect that average person. Here we use methods from percolation theory to develop a mathematical framework for predicting disease transmission through semi-directed contact networks in which some contacts are undirected-the probability of transmission is symmetric between individuals-and others are directed-transmission is possible only in one direction. We find that the probability of an epidemic and the expected fraction of a population infected during an epidemic can be different in semi-directed networks, in contrast to the routine assumption that these two quantities are equal. We furthermore demonstrate that these methods more accurately predict the vulnerability of HCWs and the efficacy of various hospital-based containment strategies during outbreaks of severe respiratory diseases.     

Pooled testing of traced contacts under superspreading dynamics

Testing is recommended for all close contacts of confirmed COVID-19 patients. However, existing pooled testing methods are oblivious to the circumstances of contagion provided by contact tracing. Here, we build upon a well-known semi-adaptive pooled testing method, Dorfman’s method with imperfect tests, and derive a simple pooled testing method based on dynamic programming that is specifically designed to use information provided by contact tracing. Experiments using a variety of reproduction numbers and dispersion levels, including those estimated in the context of the COVID-19 pandemic, show that the pools found using our method result in a significantly lower number of tests than those found using Dorfman’s method. Our method provides the greatest competitive advantage when the number of contacts of an infected individual is small, or the distribution of secondary infections is highly overdispersed. Moreover, it maintains this competitive advantage under imperfect contact tracing and significant levels of dilution.     

Social encounter networks: characterizing Great Britain

A major goal of infectious disease epidemiology is to understand and predict the spread of infections within human populations, with the intention of better informing decisions regarding control and intervention. However, the development of fully mechanistic models of transmission requires a quantitative understanding of social interactions and collective properties of social networks. We performed a cross-sectional study of the social contacts on given days for more than 5000 respondents in England, Scotland and Wales, through postal and online survey methods. The survey was designed to elicit detailed and previously unreported measures of the immediate social network of participants relevant to infection spread. Here, we describe individual-level contact patterns, focusing on the range of heterogeneity observed and discuss the correlations between contact patterns and other socio-demographic factors. We find that the distribution of the number of contacts approximates a power-law distribution, but postulate that total contact time (which has a shorter-tailed distribution) is more epidemiologically relevant. We observe that children, public-sector and healthcare workers have the highest number of total contact hours and are therefore most likely to catch and transmit infectious disease. Our study also quantifies the transitive connections made between an individual''s contacts (or clustering); this is a key structural characteristic of social networks with important implications for disease transmission and control efficacy. Respondents'' networks exhibit high levels of clustering, which varies across social settings and increases with duration, frequency of contact and distance from home. Finally, we discuss the implications of these findings for the transmission and control of pathogens spread through close contact.     

Estimates of Social Contact in a Middle School Based on Self-Report and Wireless Sensor Data

Estimates of contact among children, used for infectious disease transmission models and understanding social patterns, historically rely on self-report logs. Recently, wireless sensor technology has enabled objective measurement of proximal contact and comparison of data from the two methods. These are mostly small-scale studies, and knowledge gaps remain in understanding contact and mixing patterns and also in the advantages and disadvantages of data collection methods. We collected contact data from a middle school, with 7th and 8th grades, for one day using self-report contact logs and wireless sensors. The data were linked for students with unique initials, gender, and grade within the school. This paper presents the results of a comparison of two approaches to characterize school contact networks, wireless proximity sensors and self-report logs. Accounting for incomplete capture and lack of participation, we estimate that “sensor-detectable”, proximal contacts longer than 20 seconds during lunch and class-time occurred at 2 fold higher frequency than “self-reportable” talk/touch contacts. Overall, 55% of estimated talk-touch contacts were also sensor-detectable whereas only 15% of estimated sensor-detectable contacts were also talk-touch. Contacts detected by sensors and also in self-report logs had longer mean duration than contacts detected only by sensors (6.3 vs 2.4 minutes). During both lunch and class-time, sensor-detectable contacts demonstrated substantially less gender and grade assortativity than talk-touch contacts. Hallway contacts, which were ascertainable only by proximity sensors, were characterized by extremely high degree and short duration. We conclude that the use of wireless sensors and self-report logs provide complementary insight on in-school mixing patterns and contact frequency.     

The Sociospatial Network: Risk and the Role of Place in the Transmission of Infectious Diseases

Control of sexually transmitted infections and blood-borne pathogens is challenging due to their presence in groups exhibiting complex social interactions. In particular, sharing injection drug use equipment and selling sex (prostitution) puts people at high risk. Previous work examining the involvement of risk behaviours in social networks has suggested that social and geographic distance of persons within a group contributes to these pathogens’ endemicity. In this study, we examine the role of place in the connectedness of street people, selected by respondent driven sampling, in the transmission of blood-borne and sexually transmitted pathogens. A sample of 600 injection drug users, men who have sex with men, street youth and homeless people were recruited in Winnipeg, Canada from January to December, 2009. The residences of participants and those of their social connections were linked to each other and to locations where they engaged in risk activity. Survey responses identified 101 unique sites where respondents participated in injection drug use or sex transactions. Risk sites and respondents’ residences were geocoded, with residence representing the individuals. The sociospatial network and estimations of geographic areas most likely to be frequented were mapped with network graphs and spatially using a Geographic Information System (GIS). The network with the most nodes connected 7.7% of respondents; consideration of the sociospatial network increased this to 49.7%. The mean distance between any two locations in the network was within 3.5 kilometres. Kernel density estimation revealed key activity spaces where the five largest networks overlapped. Here, the combination of spatial and social entities in network analysis defines the overlap of vulnerable populations in risk space, over and above the person to person links. Implications of this work are far reaching, not just for understanding transmission dynamics of sexually transmitted infections by identifying activity “hotspots” and their intersection with each social network, but also for the spread of other diseases (e.g. tuberculosis) and targeting prevention services.     

Simulated epidemics in an empirical spatiotemporal network of 50,185 sexual contacts

Sexual contact patterns, both in their temporal and network structure, can influence the spread of sexually transmitted infections (STI). Most previous literature has focused on effects of network topology; few studies have addressed the role of temporal structure. We simulate disease spread using SI and SIR models on an empirical temporal network of sexual contacts in high-end prostitution. We compare these results with several other approaches, including randomization of the data, classic mean-field approaches, and static network simulations. We observe that epidemic dynamics in this contact structure have well-defined, rather high epidemic thresholds. Temporal effects create a broad distribution of outbreak sizes, even if the per-contact transmission probability is taken to its hypothetical maximum of 100%. In general, we conclude that the temporal correlations of our network accelerate outbreaks, especially in the early phase of the epidemics, while the network topology (apart from the contact-rate distribution) slows them down. We find that the temporal correlations of sexual contacts can significantly change simulated outbreaks in a large empirical sexual network. Thus, temporal structures are needed alongside network topology to fully understand the spread of STIs. On a side note, our simulations further suggest that the specific type of commercial sex we investigate is not a reservoir of major importance for HIV.     

Sexual networks: implications for the transmission of sexually transmitted infections

The structures of sexual networks are essential for understanding the dynamics of sexually transmitted infections. Standard epidemiological models largely disregard the complex patterns of intimate contacts. Social network analysis offers important insight into how to conceptualize and model social interaction and has the potential to greatly enhance the understanding of disease epidemics.     

Hepatitis C Transmission and Treatment in Contact Networks of People Who Inject Drugs

Hepatitis C virus (HCV) chronically infects over 180 million people worldwide, with over 350,000 estimated deaths attributed yearly to HCV-related liver diseases. It disproportionally affects people who inject drugs (PWID). Currently there is no preventative vaccine and interventions feature long treatment durations with severe side-effects. Upcoming treatments will improve this situation, making possible large-scale treatment interventions. How these strategies should target HCV-infected PWID remains an important unanswered question. Previous models of HCV have lacked empirically grounded contact models of PWID. Here we report results on HCV transmission and treatment using simulated contact networks generated from an empirically grounded network model using recently developed statistical approaches in social network analysis. Our HCV transmission model is a detailed, stochastic, individual-based model including spontaneously clearing nodes. On transmission we investigate the role of number of contacts and injecting frequency on time to primary infection and the role of spontaneously clearing nodes on incidence rates. On treatment we investigate the effect of nine network-based treatment strategies on chronic prevalence and incidence rates of primary infection and re-infection. Both numbers of contacts and injecting frequency play key roles in reducing time to primary infection. The change from “less-” to “more-frequent” injector is roughly similar to having one additional network contact. Nodes that spontaneously clear their HCV infection have a local effect on infection risk and the total number of such nodes (but not their locations) has a network wide effect on the incidence of both primary and re-infection with HCV. Re-infection plays a large role in the effectiveness of treatment interventions. Strategies that choose PWID and treat all their contacts (analogous to ring vaccination) are most effective in reducing the incidence rates of re-infection and combined infection. A strategy targeting infected PWID with the most contacts (analogous to targeted vaccination) is the least effective.     

Epidemic management and control through risk-dependent individual contact interventions

Testing, contact tracing, and isolation (TTI) is an epidemic management and control approach that is difficult to implement at scale because it relies on manual tracing of contacts. Exposure notification apps have been developed to digitally scale up TTI by harnessing contact data obtained from mobile devices; however, exposure notification apps provide users only with limited binary information when they have been directly exposed to a known infection source. Here we demonstrate a scalable improvement to TTI and exposure notification apps that uses data assimilation (DA) on a contact network. Network DA exploits diverse sources of health data together with the proximity data from mobile devices that exposure notification apps rely upon. It provides users with continuously assessed individual risks of exposure and infection, which can form the basis for targeting individual contact interventions. Simulations of the early COVID-19 epidemic in New York City are used to establish proof-of-concept. In the simulations, network DA identifies up to a factor 2 more infections than contact tracing when both harness the same contact data and diagnostic test data. This remains true even when only a relatively small fraction of the population uses network DA. When a sufficiently large fraction of the population (≳ 75%) uses network DA and complies with individual contact interventions, targeting contact interventions with network DA reduces deaths by up to a factor 4 relative to TTI. Network DA can be implemented by expanding the computational backend of existing exposure notification apps, thus greatly enhancing their capabilities. Implemented at scale, it has the potential to precisely and effectively control future epidemics while minimizing economic disruption.     

Modeling the impact of random screening and contact tracing in reducing the spread of HIV

Mathematical models can help predict the effectiveness of control measures on the spread of HIV and other sexually transmitted diseases (STDs) by reducing the uncertainty in assessing the impact of intervention strategies such as random screening and contact tracing. Even though contact tracing is one of the most effective methods used for controlling treatable STDs, it is still a controversial strategy for controlling HIV because of cost and confidentiality issues. To help estimate the effectiveness of these control measures, we formulate two models with random screening and contact tracing based on the differential infectivity (DI) model and the staged-progression (SP) model. We derive formulas for the reproductive numbers and the endemic equilibria and compare the impact that random screening and contact tracing have in slowing the epidemic in the two models. In the DI model the infected population is divided into groups according to their infectiousness, and HIV is largely spread by a small, highly infectious, group of superspreaders. In this model contact tracing is an effective approach to identifying the superspreaders and has a large effect in slowing the epidemic. In the SP model every infected individual goes through a series of infection stages and the virus is primarily spread by individuals in an initial highly infectious stage or in the late stages of the disease. In this model random screening is more effective than for the DI model, and contact tracing is less effective. Thus the effectiveness of the intervention strategy strongly depends on the underlying etiology of the disease transmission.     

The long-term effect of current and new interventions on the new case detection of leprosy: a modeling study

Background

Although the number of newly detected leprosy cases has decreased globally, a quarter of a million new cases are detected annually and eradication remains far away. Current options for leprosy prevention are contact tracing and BCG vaccination of infants. Future options may include chemoprophylaxis and early diagnosis of subclinical infections. This study compared the predicted trends in leprosy case detection of future intervention strategies.

Methods

Seven leprosy intervention scenarios were investigated with a microsimulation model (SIMCOLEP) to predict future leprosy trends. The baseline scenario consisted of passive case detection, multidrug therapy, contact tracing, and BCG vaccination of infants. The other six scenarios were modifications of the baseline, as follows: no contact tracing; with chemoprophylaxis; with early diagnosis of subclinical infections; replacement of the BCG vaccine with a new tuberculosis vaccine ineffective against Mycobacterium leprae (“no BCG”); no BCG with chemoprophylaxis; and no BCG with early diagnosis.

Findings

Without contact tracing, the model predicted an initial drop in the new case detection rate due to a delay in detecting clinical cases among contacts. Eventually, this scenario would lead to new case detection rates higher than the baseline program. Both chemoprophylaxis and early diagnosis would prevent new cases due to a reduction of the infectious period of subclinical cases by detection and cure of these cases. Also, replacing BCG would increase the new case detection rate of leprosy, but this effect could be offset with either chemoprophylaxis or early diagnosis.

Conclusions

This study showed that the leprosy incidence would be reduced substantially by good BCG vaccine coverage and the combined strategies of contact tracing, early diagnosis, and treatment of infection and/or chemoprophylaxis among household contacts. To effectively interrupt the transmission of M. leprae, it is crucial to continue developing immuno- and chemoprophylaxis strategies and an effective test for diagnosing subclinical infections.     

A Contact-Network-Based Formulation of a Preferential Mixing Model

Heterogeneity in the number of potentially infectious contacts and connectivity correlations (“like attaches to like” i.e., assortatively mixed or “opposites attract” i.e., disassortatively mixed) have important implications for the value of the basic reproduction ratio R ₀ and final epidemic size. In this paper, we present a contact-network-based derivation of a simple differential equation model that accounts for preferential mixing based on the number of contacts. We show that results based on this model are in good qualitative agreement with results obtained from preferential mixing models used in the context of sexually transmitted diseases (STDs). This simple model can accommodate any mixing pattern ranging from completely disassortative to completely assortative and allows the derivation of a series of analytical results.     

A Simulation Study Comparing Epidemic Dynamics on Exponential Random Graph and Edge-Triangle Configuration Type Contact Network Models

We compare two broad types of empirically grounded random network models in terms of their abilities to capture both network features and simulated Susceptible-Infected-Recovered (SIR) epidemic dynamics. The types of network models are exponential random graph models (ERGMs) and extensions of the configuration model. We use three kinds of empirical contact networks, chosen to provide both variety and realistic patterns of human contact: a highly clustered network, a bipartite network and a snowball sampled network of a “hidden population”. In the case of the snowball sampled network we present a novel method for fitting an edge-triangle model. In our results, ERGMs consistently capture clustering as well or better than configuration-type models, but the latter models better capture the node degree distribution. Despite the additional computational requirements to fit ERGMs to empirical networks, the use of ERGMs provides only a slight improvement in the ability of the models to recreate epidemic features of the empirical network in simulated SIR epidemics. Generally, SIR epidemic results from using configuration-type models fall between those from a random network model (i.e., an Erdős-Rényi model) and an ERGM. The addition of subgraphs of size four to edge-triangle type models does improve agreement with the empirical network for smaller densities in clustered networks. Additional subgraphs do not make a noticeable difference in our example, although we would expect the ability to model cliques to be helpful for contact networks exhibiting household structure.