Network frailty and the geometry of herd immunity

The spread of infectious disease through communities depends fundamentally on the underlying patterns of contacts between individuals. Generally, the more contacts one individual has, the more vulnerable they are to infection during an epidemic. Thus, outbreaks disproportionately impact the most highly connected demographics. Epidemics can then lead, through immunization or removal of individuals, to sparser networks that are more resistant to future transmission of a given disease. Using several classes of contact networks-Poisson, scale-free and small-world-we characterize the structural evolution of a network due to an epidemic in terms of frailty (the degree to which highly connected individuals are more vulnerable to infection) and interference (the extent to which the epidemic cuts off connectivity among the susceptible population that remains following an epidemic). The evolution of the susceptible network over the course of an epidemic differs among the classes of networks; frailty, relative to interference, accounts for an increasing component of network evolution on networks with greater variance in contacts. The result is that immunization due to prior epidemics can provide greater community protection than random vaccination on networks with heterogeneous contact patterns, while the reverse is true for highly structured populations.     

Analysis of a stochastic SIR epidemic on a random network incorporating household structure

This paper is concerned with a stochastic SIR (susceptible → infective → removed) model for the spread of an epidemic amongst a population of individuals, with a random network of social contacts, that is also partitioned into households. The behaviour of the model as the population size tends to infinity in an appropriate fashion is investigated. A threshold parameter which determines whether or not an epidemic with few initial infectives can become established and lead to a major outbreak is obtained, as are the probability that a major outbreak occurs and the expected proportion of the population that are ultimately infected by such an outbreak, together with methods for calculating these quantities. Monte Carlo simulations demonstrate that these asymptotic quantities accurately reflect the behaviour of finite populations, even for only moderately sized finite populations. The model is compared and contrasted with related models previously studied in the literature. The effects of the amount of clustering present in the overall population structure and the infectious period distribution on the outcomes of the model are also explored.     

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.     

Random Modelling of Contagious Diseases

Modelling contagious diseases needs to include a mechanistic knowledge about contacts between hosts and pathogens as specific as possible, e.g., by incorporating in the model information about social networks through which the disease spreads. The unknown part concerning the contact mechanism can be modelled using a stochastic approach. For that purpose, we revisit SIR models by introducing first a microscopic stochastic version of the contacts between individuals of different populations (namely Susceptible, Infective and Recovering), then by adding a random perturbation in the vicinity of the endemic fixed point of the SIR model and eventually by introducing the definition of various types of random social networks. We propose as example of application to contagious diseases the HIV, and we show that a micro-simulation of individual based modelling (IBM) type can reproduce the current stable incidence of the HIV epidemic in a population of HIV-positive men having sex with men (MSM).     

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.     

Network-based analysis of stochastic SIR epidemic models with random and proportionate mixing

In this paper, we outline the theory of epidemic percolation networks and their use in the analysis of stochastic susceptible-infectious-removed (SIR) epidemic models on undirected contact networks. We then show how the same theory can be used to analyze stochastic SIR models with random and proportionate mixing. The epidemic percolation networks for these models are purely directed because undirected edges disappear in the limit of a large population. In a series of simulations, we show that epidemic percolation networks accurately predict the mean outbreak size and probability and final size of an epidemic for a variety of epidemic models in homogeneous and heterogeneous populations. Finally, we show that epidemic percolation networks can be used to re-derive classical results from several different areas of infectious disease epidemiology. In an Appendix, we show that an epidemic percolation network can be defined for any time-homogeneous stochastic SIR model in a closed population and prove that the distribution of outbreak sizes given the infection of any given node in the SIR model is identical to the distribution of its out-component sizes in the corresponding probability space of epidemic percolation networks. We conclude that the theory of percolation on semi-directed networks provides a very general framework for the analysis of stochastic SIR models in closed populations.     

The large graph limit of a stochastic epidemic model on a dynamic multilayer network

We consider a Markovian SIR-type (Susceptible → Infected → Recovered) stochastic epidemic process with multiple modes of transmission on a contact network. The network is given by a random graph following a multilayer configuration model where edges in different layers correspond to potentially infectious contacts of different types. We assume that the graph structure evolves in response to the epidemic via activation or deactivation of edges of infectious nodes. We derive a large graph limit theorem that gives a system of ordinary differential equations (ODEs) describing the evolution of quantities of interest, such as the proportions of infected and susceptible vertices, as the number of nodes tends to infinity. Analysis of the limiting system elucidates how the coupling of edge activation and deactivation to infection status affects disease dynamics, as illustrated by a two-layer network example with edge types corresponding to community and healthcare contacts. Our theorem extends some earlier results describing the deterministic limit of stochastic SIR processes on static, single-layer configuration model graphs. We also describe precisely the conditions for equivalence between our limiting ODEs and the systems obtained via pair approximation, which are widely used in the epidemiological and ecological literature to approximate disease dynamics on networks. The flexible modeling framework and asymptotic results have potential application to many disease settings including Ebola dynamics in West Africa, which was the original motivation for this study.     

Bursts of Vertex Activation and Epidemics in Evolving Networks

The dynamic nature of contact patterns creates diverse temporal structures. In particular, empirical studies have shown that contact patterns follow heterogeneous inter-event time intervals, meaning that periods of high activity are followed by long periods of inactivity. To investigate the impact of these heterogeneities in the spread of infection from a theoretical perspective, we propose a stochastic model to generate temporal networks where vertices make instantaneous contacts following heterogeneous inter-event intervals, and may leave and enter the system. We study how these properties affect the prevalence of an infection and estimate , the number of secondary infections of an infectious individual in a completely susceptible population, by modeling simulated infections (SI and SIR) that co-evolve with the network structure. We find that heterogeneous contact patterns cause earlier and larger epidemics in the SIR model in comparison to homogeneous scenarios for a vast range of parameter values, while smaller epidemics may happen in some combinations of parameters. In the case of SI and heterogeneous patterns, the epidemics develop faster in the earlier stages followed by a slowdown in the asymptotic limit. For increasing vertex turnover rates, heterogeneous patterns generally cause higher prevalence in comparison to homogeneous scenarios with the same average inter-event interval. We find that is generally higher for heterogeneous patterns, except for sufficiently large infection duration and transmission probability.     

Modelling sexually transmitted infections: the effect of partnership activity and number of partners on R0

We model a sexually transmitted infection in a network population where individuals have different numbers of partners, separated into steady and casual partnerships, where the risk of transmission is higher in steady partnerships. An individual's number of partners of the two types defines its degree, and the degrees in the community specify the degree distribution. For this structured network population a simple model for disease transmission is defined and the basic reproduction number R0 is derived, R0 being a size-biased (i.e. biasing individuals with many partners) average number of new infections caused by individuals during the early stages of the epidemic. First a homosexual population is considered and then a heterosexual population. The heterosexual model is fitted to data from a census survey on sexual activity from the Swedish island of Gotland. The main empirical finding is that, for relevant transmission rates, the effect that so-called superspreaders have on R0 is over-estimated when not admitting for different types of partnerships.     

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.     

The effect of contact heterogeneity and multiple routes of transmission on final epidemic size

Heterogeneity in the number of potentially infectious contacts amongst members of a population increases the basic reproduction ratio (R(0)) and markedly alters disease dynamics compared to traditional mean-field models. Most models describing transmission on contact networks only account for one specific route of transmission. However, for many infectious diseases multiple routes of transmission exist. The model presented here captures transmission through a well defined network of contacts, complemented by mean-field type transmission amongst the nodes of the network that accounts for alternative routes of transmission. The impact of these combined transmission mechanisms on the final epidemic size is investigated analytically. The analytic predictions for the purely mean-field case and the transmission through the network-only case are confirmed by individual-based network simulations. There is a critical transmission potential above which an increased contribution of the mean-field type transmission increases the final epidemic size while an increased contribution of the transmission through the network decreases it. Below the critical transmission potential the opposite effect is observed.     

Modelling disease spread through random and regular contacts in clustered populations

An epidemic spreading through a network of regular, repeated, contacts behaves differently from one that is spread by random interactions: regular contacts serve to reduce the speed and eventual size of an epidemic. This paper uses a mathematical model to explore the difference between regular and random contacts, considering particularly the effect of clustering within the contact network. In a clustered population random contacts have a much greater impact, allowing infection to reach parts of the network that would otherwise be inaccessible. When all contacts are regular, clustering greatly reduces the spread of infection; this effect is negated by a small number of random contacts.     

Large-sample analysis for a stochastic epidemic model and its parameter estimators

This paper considers the simplest stochastic model for the spread of an epidemic in a closed, homogeneously mixing population. Approximate methods are presented for calculating the probability distribution of the epidemic size (i.e. number of infected individuals). In fact, a functional central limit theorem and a large deviation principle for the epidemic size when the population increases are shown. These results enable us to both obtain a global approximation for the epidemic size and study asymptotic properties of other random variables depending on the complete history of the epidemic. As an application of our results, we derive two sequences of estimators for the contact rate and analyze their asymptotic behaviour.     

Infections on Temporal Networks—A Matrix-Based Approach

We extend the concept of accessibility in temporal networks to model infections with a finite infectious period such as the susceptible-infected-recovered (SIR) model. This approach is entirely based on elementary matrix operations and unifies the disease and network dynamics within one algebraic framework. We demonstrate the potential of this formalism for three examples of networks with high temporal resolution: networks of social contacts, sexual contacts, and livestock-trade. Our investigations provide a new methodological framework that can be used, for instance, to estimate the epidemic threshold, a quantity that determines disease parameters, for which a large-scale outbreak can be expected.     

Incorporating Disease and Population Structure into Models of SIR Disease in Contact Networks

We consider the recently introduced edge-based compartmental models (EBCM) for the spread of susceptible-infected-recovered (SIR) diseases in networks. These models differ from standard infectious disease models by focusing on the status of a random partner in the population, rather than a random individual. This change in focus leads to simple analytic models for the spread of SIR diseases in random networks with heterogeneous degree. In this paper we extend this approach to handle deviations of the disease or population from the simplistic assumptions of earlier work. We allow the population to have structure due to effects such as demographic features or multiple types of risk behavior. We allow the disease to have more complicated natural history. Although we introduce these modifications in the static network context, it is straightforward to incorporate them into dynamic network models. We also consider serosorting, which requires using dynamic network models. The basic methods we use to derive these generalizations are widely applicable, and so it is straightforward to introduce many other generalizations not considered here. Our goal is twofold: to provide a number of examples generalizing the EBCM method for various different population or disease structures and to provide insight into how to derive such a model under new sets of assumptions.     

Dynamics of stochastic epidemics on heterogeneous networks

Epidemic models currently play a central role in our attempts to understand and control infectious diseases. Here, we derive a model for the diffusion limit of stochastic susceptible-infectious-removed (SIR) epidemic dynamics on a heterogeneous network. Using this, we consider analytically the early asymptotic exponential growth phase of such epidemics, showing how the higher order moments of the network degree distribution enter into the stochastic behaviour of the epidemic. We find that the first three moments of the network degree distribution are needed to specify the variance in disease prevalence fully, meaning that the skewness of the degree distribution affects the variance of the prevalence of infection. We compare these asymptotic results to simulation and find a close agreement for city-sized populations.     

Epidemic Spread on Weighted Networks

The contact structure between hosts shapes disease spread. Most network-based models used in epidemiology tend to ignore heterogeneity in the weighting of contacts between two individuals. However, this assumption is known to be at odds with the data for many networks (e.g. sexual contact networks) and to have a critical influence on epidemics'' behavior. One of the reasons why models usually ignore heterogeneity in transmission is that we currently lack tools to analyze weighted networks, such that most studies rely on numerical simulations. Here, we present a novel framework to estimate key epidemiological variables, such as the rate of early epidemic expansion () and the basic reproductive ratio (), from joint probability distributions of number of partners (contacts) and number of interaction events through which contacts are weighted. These distributions are much easier to infer than the exact shape of the network, which makes the approach widely applicable. The framework also allows for a derivation of the full time course of epidemic prevalence and contact behaviour, which we validate with numerical simulations on networks. Overall, incorporating more realistic contact networks into epidemiological models can improve our understanding of the emergence and spread of infectious diseases.     

A stochastic SIR model with contact-tracing: large population limits and statistical inference

This paper is devoted to the presentation and study of a specific stochastic epidemic model accounting for the effect of contact-tracing on the spread of an infectious disease. Precisely, one considers here the situation in which individuals identified as infected by the public health detection system may contribute to detecting other infectious individuals by providing information related to persons with whom they have had possibly infectious contacts. The control strategy, which consists of examining each individual who has been able to be identified on the basis of the information collected within a certain time period, is expected to efficiently reinforce the standard random-screening-based detection and considerably ease the epidemic. In the novel modelling of the spread of a communicable infectious disease considered here, the population of interest evolves through demographic, infection and detection processes, in a way that its temporal evolution is described by a stochastic Markov process, of which the component accounting for the contact-tracing feature is assumed to be valued in a space of point measures. For adequate scalings of the demographic, infection and detection rates, it is shown to converge to the weak deterministic solution of a PDE system, as a parameter n, interpreted as the population size, roughly speaking, becomes larger. From the perspective of the analysis of infectious disease data, this approximation result may serve as a key tool for exploring the asymptotic properties of standard inference methods such as maximum likelihood estimation. We state preliminary statistical results in this context. Eventually, relations of the model with the available data of the HIV epidemic in Cuba, in which country a contact-tracing detection system has been set up since 1986, is investigated and numerical applications are carried out.     

Temporal Percolation of the Susceptible Network in an Epidemic Spreading

In this work, we study the evolution of the susceptible individuals during the spread of an epidemic modeled by the susceptible-infected-recovered (SIR) process spreading on the top of complex networks. Using an edge-based compartmental approach and percolation tools, we find that a time-dependent quantity , namely, the probability that a given neighbor of a node is susceptible at time , is the control parameter of a node void percolation process involving those nodes on the network not-reached by the disease. We show that there exists a critical time above which the giant susceptible component is destroyed. As a consequence, in order to preserve a macroscopic connected fraction of the network composed by healthy individuals which guarantee its functionality, any mitigation strategy should be implemented before this critical time . Our theoretical results are confirmed by extensive simulations of the SIR process.