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Infectious or contagious diseases can be transmitted from one person to another through social contact networks. In today's interconnected global society, such contagion processes can cause global public health hazards, as exemplified by the ongoing Covid-19 pandemic. It is therefore of great practical relevance to investigate the network transmission of contagious diseases from the perspective of statistical inference. An important and widely studied boundary condition for contagion processes over networks is the so-called epidemic threshold . The epidemic threshold plays a key role in determining whether a pathogen introduced into a social contact network will cause an epidemic or die out. In this paper, we investigate epidemic thresholds from the perspective of statistical network inference. We identify two major challenges that are caused by high computational and sampling complexity of the epidemic threshold. We develop two statistically accurate and computationally efficient approximation techniques to address these issues under the Chung-Lu modeling framework. The second approximation, which is based on random walk sampling, further enjoys the advantage of requiring data on a vanishingly small fraction of nodes. We establish theoretical guarantees for both methods and demonstrate their empirical superiority., (© Indian Statistical Institute 2021.)

Analysing social networks is challenging. Key features of relational data require the use of non-standard statistical methods such as developing system-specific null, or reference, models that randomize one or more components of the observed data. Here we review a variety of randomization procedures that generate reference models for social network analysis. Reference models provide an expectation for hypothesis testing when analysing network data. We outline the key stages in producing an effective reference model and detail four approaches for generating reference distributions: permutation, resampling, sampling from a distribution, and generative models. We highlight when each type of approach would be appropriate and note potential pitfalls for researchers to avoid. Throughout, we illustrate our points with examples from a simulated social system. Our aim is to provide social network researchers with a deeper understanding of analytical approaches to enhance their confidence when tailoring reference models to specific research questions., (© 2021 The Authors. Biological Reviews published by John Wiley & Sons Ltd on behalf of Cambridge Philosophical Society.)

We consider an SIR model with vaccination strategy on a sparse configuration model random graph. We show the convergence of the system when the number of nodes grows and characterize the scaling limits. Then, we prove the existence of optimal controls for the limiting equations formulated in the framework of game theory, both in the centralized and decentralized setting. We show how the characteristics of the graph (degree distribution) influence the vaccination efficiency for optimal strategies, and we compute the limiting final size of the epidemic depending on the degree distribution of the graph and the parameters of infection, recovery and vaccination. We also present several simulations for two types of vaccination, showing how the optimal controls allow to decrease the number of infections and underlining the crucial role of the network characteristics in the propagation of the disease and the vaccination program., (© The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2021.)

In this paper we consider Susceptible [Formula: see text] Infectious [Formula: see text] Recovered (SIR) epidemics on random graphs with clustering. To incorporate group structure of the underlying social network, we use a generalized version of the configuration model in which each node is a member of a specified number of triangles. SIR epidemics on this type of graph have earlier been investigated under the assumption of homogeneous infectivity and also under the assumption of Poisson transmission and recovery rates. We extend known results from literature by relaxing the assumption of homogeneous infectivity both in individual infectivity and between different kinds of neighbours. An important special case of the epidemic model analysed in this paper is epidemics in continuous time with arbitrary infectious period distribution. We use branching process approximations of the spread of the disease to provide expressions for the basic reproduction number [Formula: see text], the probability of a major outbreak and the expected final size. In addition, the impact of random vaccination with a perfect vaccine on the final outcome of the epidemic is investigated. We find that, for this particular model, [Formula: see text] equals the perfect vaccine-associated reproduction number. Generalizations to groups larger than three are discussed briefly.

A Markovian Susceptible [Formula: see text] Infectious [Formula: see text] Recovered (SIR) model is considered for the spread of an epidemic on a configuration model network, in which susceptible individuals may take preventive measures by dropping edges to infectious neighbours. An effective degree formulation of the model is used in conjunction with the theory of density dependent population processes to obtain a law of large numbers and a functional central limit theorem for the epidemic as the population size [Formula: see text], assuming that the degrees of individuals are bounded. A central limit theorem is conjectured for the final size of the epidemic. The results are obtained for both the Molloy-Reed (in which the degrees of individuals are deterministic) and Newman-Strogatz-Watts (in which the degrees of individuals are independent and identically distributed) versions of the configuration model. The two versions yield the same limiting deterministic model but the asymptotic variances in the central limit theorems are greater in the Newman-Strogatz-Watts version. The basic reproduction number [Formula: see text] and the process of susceptible individuals in the limiting deterministic model, for the model with dropping of edges, are the same as for a corresponding SIR model without dropping of edges but an increased recovery rate, though, when [Formula: see text], the probability of a major outbreak is greater in the model with dropping of edges. The results are specialised to the model without dropping of edges to yield conjectured central limit theorems for the final size of Markovian SIR epidemics on configuration-model networks, and for the size of the giant components of those networks. The theory is illustrated by numerical studies, which demonstrate that the asymptotic approximations are good, even for moderate N.

It is well known that many random graphs with infinite variance degrees are ultra-small. More precisely, for configuration models and preferential attachment models where the proportion of vertices of degree at least k is approximately k -(τ - 1) with τ ∈ (2,3), typical distances between pairs of vertices in a graph of size n are asymptotic to 2 log log n | log ( τ - 2 ) | and 4 log log n | log ( τ - 2 ) | , respectively. In this paper, we investigate the behavior of the diameter in such models. We show that the diameter is of order log log n precisely when the minimal forward degree d fwd of vertices is at least 2. We identify the exact constant, which equals that of the typical distances plus 2 / log d fwd . Interestingly, the proof for both models follows identical steps, even though the models are quite different in nature.

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.

This paper is concerned with the analysis of vaccination strategies in a stochastic susceptible [Formula: see text] infected [Formula: see text] 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. Under various vaccine action models, we consider both household-based vaccination schemes, in which the way in which individuals are chosen for vaccination depends on the size of the households in which they reside, and acquaintance vaccination, which targets individuals of high degree in the social network. For both types of vaccination scheme, assuming a large population with few initial infectives, we derive a threshold parameter which determines whether or not a large outbreak can occur and also the probability of a large outbreak and the fraction of the population infected by a large outbreak. The performance of these schemes is studied numerically, focusing on the influence of the household size distribution and the degree distribution of the social network. We find that acquaintance vaccination can significantly outperform the best household-based scheme if the degree distribution of the social network is heavy-tailed. For household-based schemes, when the vaccine coverage is insufficient to prevent a major outbreak and the vaccine is imperfect, we find situations in which both the probability and size of a major outbreak under the scheme which minimises the threshold parameter are larger than in the scheme which maximises the threshold parameter.

The configuration model generates random graphs with any given degree distribution, and thus serves as a null model for scale-free networks with power-law degrees and unbounded degree fluctuations. For this setting, we study the local clustering c ( k ), i.e., the probability that two neighbors of a degree- k node are neighbors themselves. We show that c ( k ) progressively falls off with k and the graph size n and eventually for k = Ω ( n ) settles on a power law c ( k ) ∼ n 5 - 2 τ k - 2 ( 3 - τ ) with τ ∈ ( 2 , 3 ) the power-law exponent of the degree distribution. This fall-off has been observed in the majority of real-world networks and signals the presence of modular or hierarchical structure. Our results agree with recent results for the hidden-variable model and also give the expected number of triangles in the configuration model when counting triangles only once despite the presence of multi-edges. We show that only triangles consisting of triplets with uniquely specified degrees contribute to the triangle counting.

In this paper we study first-passage percolation in the configuration model with empirical degree distribution that follows a power-law with exponent τ ∈ ( 2 , 3 ) . We assign independent and identically distributed (i.i.d.) weights to the edges of the graph. We investigate the weighted distance (the length of the shortest weighted path) between two uniformly chosen vertices, called typical distances. When the underlying age-dependent branching process approximating the local neighborhoods of vertices is found to produce infinitely many individuals in finite time-called explosive branching process-Baroni, Hofstad and the second author showed in Baroni et al. (J Appl Probab 54(1):146-164, 2017) that typical distances converge in distribution to a bounded random variable. The order of magnitude of typical distances remained open for the τ ∈ ( 2 , 3 ) case when the underlying branching process is not explosive. We close this gap by determining the first order of magnitude of typical distances in this regime for arbitrary, not necessary continuous edge-weight distributions that produce a non-explosive age-dependent branching process with infinite mean power-law offspring distributions. This sequence tends to infinity with the amount of vertices, and, by choosing an appropriate weight distribution, can be tuned to be any growing function that is O ( log log n ) , where n is the number of vertices in the graph. We show that the result remains valid for the the erased configuration model as well, where we delete loops and any second and further edges between two vertices.

Recent years have seen a large amount of interest in epidemics on networks as a way of representing the complex structure of contacts capable of spreading infections through the modern human population. The configuration model is a popular choice in theoretical studies since it combines the ability to specify the distribution of the number of contacts (degree) with analytical tractability. Here we consider the early real-time behaviour of the Markovian SIR epidemic model on a configuration model network using a multitype branching process. We find closed-form analytic expressions for the mean and variance of the number of infectious individuals as a function of time and the degree of the initially infected individual(s), and write down a system of differential equations for the probability of extinction by time t that are numerically fast compared to Monte Carlo simulation. We show that these quantities are all sensitive to the degree distribution-in particular we confirm that the mean prevalence of infection depends on the first two moments of the degree distribution and the variance in prevalence depends on the first three moments of the degree distribution. In contrast to most existing analytic approaches, the accuracy of these results does not depend on having a large number of infectious individuals, meaning that in the large population limit they would be asymptotically exact even for one initial infectious individual.

Emergence of new diseases and elimination of existing diseases is a key public health issue. In mathematical models of epidemics, such phenomena involve the process of infections and recoveries passing through a critical threshold where the basic reproductive ratio is 1. In this paper, we study near-critical behaviour in the context of a susceptible-infective-recovered epidemic on a random (multi)graph on n vertices with a given degree sequence. We concentrate on the regime just above the threshold for the emergence of a large epidemic, where the basic reproductive ratio is [Formula: see text], with [Formula: see text] tending to infinity slowly as the population size, n, tends to infinity. We determine the probability that a large epidemic occurs, and the size of a large epidemic. Our results require basic regularity conditions on the degree sequences, and the assumption that the third moment of the degree of a random susceptible vertex stays uniformly bounded as [Formula: see text]. As a corollary, we determine the probability and size of a large near-critical epidemic on a standard binomial random graph in the 'sparse' regime, where the average degree is constant. As a further consequence of our method, we obtain an improved result on the size of the giant component in a random graph with given degrees just above the critical window, proving a conjecture by Janson and Luczak.

We formulate models for the spread of infection on networks that are amenable to analysis in the large population limit. We distinguish three different levels: (1) binding sites, (2) individuals, and (3) the population. In the tradition of physiologically structured population models, the formulation starts on the individual level. Influences from the 'outside world' on an individual are captured by environmental variables. These environmental variables are population level quantities. A key characteristic of the network models is that individuals can be decomposed into a number of conditionally independent components: each individual has a fixed number of 'binding sites' for partners. The Markov chain dynamics of binding sites are described by only a few equations. In particular, individual-level probabilities are obtained from binding-site-level probabilities by combinatorics while population-level quantities are obtained by averaging over individuals in the population. Thus we are able to characterize population-level epidemiological quantities, such as [Formula: see text], r, the final size, and the endemic equilibrium, in terms of the corresponding variables.

A key issue in testing is how many tests are needed for a required level of coverage or fault detection. Estimates are often based on error rates in initial testing, or on code coverage. For example, tests may be run until a desired level of statement or branch coverage is achieved. Combinatorial methods present an opportunity for a different approach to estimating required test set size, using characteristics of the test set. This paper describes methods for estimating the coverage of, and ability to detect, t- way interaction faults of a test set based on a covering array. We also develop a connection between (static) combinatorial coverage and (dynamic) code coverage, such that if a specific condition is satisfied, 100% branch coverage is assured. Using these results, we propose practical recommendations for using combinatorial coverage in specifying test requirements.

Networks have become an important tool for infectious disease epidemiology. Most previous theoretical studies of transmission network models have either considered simple Markovian dynamics at the individual level, or have focused on the invasion threshold and final outcome of the epidemic. Here, we provide a general theory for early real-time behaviour of epidemics on large configuration model networks (i.e. static and locally unclustered), in particular focusing on the computation of the Malthusian parameter that describes the early exponential epidemic growth. Analytical, numerical and Monte-Carlo methods under a wide variety of Markovian and non-Markovian assumptions about the infectivity profile are presented. Numerous examples provide explicit quantification of the impact of the network structure on the temporal dynamics of the spread of infection and provide a benchmark for validating results of large scale simulations., (Copyright © 2015. Published by Elsevier Inc.)

We propose a weighted version of the standard configuration model which allows for a tunable degree-degree correlation. A social network is modeled by a weighted graph generated by this model, where the edge weights indicate the intensity or type of contact between the individuals. An inhomogeneous Reed-Frost epidemic model is then defined on the network, where the inhomogeneity refers to different disease transmission probabilities related to the edge weights. By tuning the model we study the impact of different correlation patterns on the network and epidemics therein. Our results suggest that the basic reproduction number R0 of the epidemic increases (decreases) when the degree-degree correlation coefficient ρ increases (decreases). Furthermore, we show that such effect can be amplified or mitigated depending on the relation between degree and weight distributions as well as the choice of the disease transmission probabilities. In addition, for a more general model allowing additional heterogeneity in the disease transmission probabilities we show that ρ can have the opposite effect on R0., (Copyright © 2014 Elsevier Inc. All rights reserved.)