arXiv subjects

Tom Britton

Publications and source records attributed to Tom Britton.

At least 55 records · Page 3Linked to original sources

arXiv2014

Introduction to statistical inference for infectious diseases

In this paper we first introduce the general stochastic epidemic model for the spread of infectious diseases. Then we give methods for inferring model parameters such as the basic reproduction number $R_0$ and vaccination coverage $v_c$ assuming different types of data from an outbreak such as final outbreak details and temporal data or observations from an ongoing outbreak. Both individual heterogeneities and heterogeneous mixing are discussed. We also provide an overview of statistical methods to perform parameter estimation for stochastic epidemic models. In the last section we describe the problem of early outbreak detection in infectious disease surveillance and statistical models used for this purpose.

stat.ME↗

arXiv2014

Respondent-driven sampling and an unusual epidemic

Respondent-driven sampling (RDS) is frequently used when sampling hard-to-reach and/or stigmatized communities. RDS utilizes a peer-driven recruitment mechanism where sampled individuals pass on participation coupons to at most $c$ of their acquaintances in the community ($c=3$ being a common choice), who then in turn pass on to their acquaintances if they choose to participate, and so on. This process of distributing coupons is shown to behave like a new Reed-Frost type network epidemic model, in which becoming infected corresponds to receiving a coupon. The difference from existing network epidemic models is that an infected individual can not infect (i.e.\ sample) all of its contacts, but only at most $c$ of them. We calculate $R_0$, the probability of a major "outbreak", and the relative size of a major outbreak in the limit of infinite population size and evaluate their adequacy in finite populations. We study the effect of varying $c$ and compare RDS to the corresponding usual epidemic models, i.e.\ the case of $c=\infty$. Our results suggest that the number of coupons has a large effect on RDS recruitment. Additionally, we use our findings to explain previous empirical observations.

physics.soc-ph↗

arXiv2014

On expected durations of birth-death processes, with applications to branching processes and SIS epidemics

We study continuous-time birth-death type processes, where individuals have independent and identically distributed lifetimes, according to a random variable Q, with E[Q]=1, and where the birth rate if the population is currently in state (has size) n is α(n). We focus on two important examples, namely α(n)=λn being a branching process, and α(n)=λn(N-n)/N which corresponds to an SIS epidemic model in a homogeneously mixing community of fixed size N. The processes are assumed to start with a single individual, i.e. in state 1. Let T, A_n, C and S denote the (random) time to extinction, the total time spent in state $n$, the total number of individuals ever alive and the sum of the lifetimes of all individuals in the birth-death process, respectively. The main results of the paper give expressions for the expectation of all these quantities, and shows that these expectations are insensitive to the distribution of Q. We also derive an asymptotic expression for the expected time to extinction of the SIS epidemic, but now starting at the endemic state, which is not independent of the distribution of Q. The results are also applied to the household SIS epidemic, showing that its threshold parameter R_* is insensitive to the distribution of Q, contrary to the household SIR epidemic, for which R_* does depend on Q.

math.PR↗

arXiv2013

Stochastic epidemics in growing populations

Consider a uniformly mixing population which grows as a super-critical linear birth and death process. At some time an infectious disease (of SIR or SEIR type) is introduced by one individual being infected from outside. It is shown that three different scenarios may occur: 1) an epidemic never takes off, 2) an epidemic gets going and grows but at a slower rate than the community thus still being negligible in terms of population fractions, or 3) an epidemic takes off and grows quicker than the community eventually leading to an endemic equilibrium. Depending on the parameter values, either scenario 1 is the only possibility, both scenario 1 and 2 are possible, or scenario 1 and 3 are possible.

math.PR↗

arXiv2013

Random Walks on Directed Networks: Inference and Respondent-driven Sampling

Respondent driven sampling (RDS) is a method often used to estimate population properties (e.g. sexual risk behavior) in hard-to-reach populations. It combines an effective modified snowball sampling methodology with an estimation procedure that yields unbiased population estimates under the assumption that the sampling process behaves like a random walk on the social network of the population. Current RDS estimation methodology assumes that the social network is undirected, i.e. that all edges are reciprocal. However, empirical social networks in general also have non-reciprocated edges. To account for this fact, we develop a new estimation method for RDS in the presence of directed edges on the basis of random walks on directed networks. We distinguish directed and undirected edges and consider the possibility that the random walk returns to its current position in two steps through an undirected edge. We derive estimators of the selection probabilities of individuals as a function of the number of outgoing edges of sampled individuals. We evaluate the performance of the proposed estimators on artificial and empirical networks to show that they generally perform better than existing methods. This is in particular the case when the fraction of directed edges in the network is large.

stat.ME↗

arXiv2013

On the expected time a branching process has K individuals alive

Consider a homogeneous time-continuous branching process where individuals have constant birth rate $δ$, and life length distribution $Q$ having mean $E(Q)=1$. Let $X(u)$ denote the number of individuals alive at time $u$, and assume that $X(0)=1$. Let $K$ be a positive integer and define $A_K:=\int_0^\infty 1_{\{X(u)=K\}}du$, the accumulated time that the branching process has exactly $K$ individuals alive. In this paper we prove that $E(A_K)=δ^{K-1}/\left(k(1\veeδ)^K\right)$, irrespective of the life length distribution $Q$, subject to the normalizing condition $E(Q)=1$.

math.PR↗

arXiv2012

A stochastic model for virus growth in a cell population

A stochastic model for the growth of a virus in a cell population is introduced. The virus has two ways of spreading: either by allowing its host cell to live on and duplicate, or else by multiplying in large numbers within the host cell such that the host cell finally bursts and the viruses then have the chance to enter new uninfected host cells. The model, and in particular the probability of the virus population surviving, is analyzed using the theory of Markov processes together with a coupling argument. Our analysis shows that the optimal strategy of the virus (in terms of survival) is obtained when the virus has no effect on the host cell's life-cycle, in agreement with experimental data about real viruses.

q-bio.CB↗

arXiv2012

A network with tunable clustering, degree correlation and degree distribution, and an epidemic thereon

A random network model which allows for tunable, quite general forms of clustering, degree correlation and degree distribution is defined. The model is an extension of the configuration model, in which stubs (half-edges) are paired to form a network. Clustering is obtained by forming small completely connected subgroups, and positive (negative) degree correlation is obtained by connecting a fraction of the stubs with stubs of similar (dissimilar) degree. An SIR (Susceptible -> Infective -> Recovered) epidemic model is defined on this network. Asymptotic properties of both the network and the epidemic, as the population size tends to infinity, are derived: the degree distribution, degree correlation and clustering coefficient, as well as a reproduction number $R_*$, the probability of a major outbreak and the relative size of such an outbreak. The theory is illustrated by Monte Carlo simulations and numerical examples. The main findings are that clustering tends to decrease the spread of disease, the effect of degree correlation is appreciably greater when the disease is close to threshold than when it is well above threshold and disease spread broadly increases with degree correlation $ρ$ when $R_*$ is just above its threshold value of one and decreases with $ρ$ when $R_*$ is well above one.

math.PR↗

arXiv2012

Respondent-driven Sampling on Directed Networks

Respondent-driven sampling (RDS) is a commonly used substitute for random sampling when studying hidden populations, such as injecting drug users or men who have sex with men, for which no sampling frame is known. The method is an extension of the snowball sample method and can, given that some assumptions are met, generate unbiased population estimates. One key assumption, not likely to be met, is that the acquaintance network in which the recruitment process takes place is undirected, meaning that all recruiters should have the potential to be recruited by the person they recruit. Here we investigate the potential bias of directedness by simulating RDS on real and artificial network structures. We show that directedness is likely to generate bias that cannot be compensated for unless the sampled individuals know how many that potentially may have recruited them (i.e. their indegree), which is unlikely in most situations. Based on one known parameter, we propose an estimator for RDS on directed networks when only outdegrees are observed. By comparison of current RDS estimators' performances on networks with varying structures, we find that our new estimator, together with a recent estimator, which requires the population size as a known quantity, have relatively low level of estimate error and bias. Based on our new estimator, sensitivity analysis can be made by varying values of the known parameter to take uncertainty of network directedness and error in reporting degrees into account. Finally, we have developed a bootstrap procedure for the new estimator to construct confidence intervals.

stat.ME↗

arXiv2012

Inferring global network properties from egocentric data with applications to epidemics

Social networks are rarely observed in full detail. In many situations properties are known for only a sample of the individuals in the network and it is desirable to induce global properties of the full social network from this "egocentric" network data. In the current paper we study a few different types of egocentric data, and show what global network properties are consistent with those egocentric data. Two global network properties are considered: the size of the largest connected component in the network (the giant), and secondly, the possible size of an epidemic outbreak taking place on the network, in which transmission occurs only between network neighbours, and with probability $p$. The main conclusion is that in most cases, egocentric data allow for a large range of possible sizes of the giant and the outbreak. However, there is an upper bound for the latter. For the case that the network is selected uniformly among networks with prescribed egocentric data (satisfying some conditions), the asymptotic size of the giant and the outbreak is characterised.

cs.SI↗

arXiv2011

Inhomogeneous epidemics on weighted networks

A social (sexual) network is modeled by an extension of the configuration model to the situation where edges have weights, e.g. reflecting the number of sex-contacts between the individuals. An epidemic model is defined on the network such that individuals are heterogeneous in terms of how susceptible and infectious they are. The basic reproduction number R_0 is derived and studied for various examples, but also the size and probability of a major outbreak. The qualitative conclusion is that R_0 gets larger as the community becomes more heterogeneous but that different heterogeneities (degree distribution, weight, susceptibility and infectivity) can sometimes have the cumulative effect of homogenizing the community, thus making $R_0$ smaller. The effect on the probability and final size of an outbreak is more complicated.

math.PR↗

arXiv2011

A weighted configuration model and inhomogeneous epidemics

A random graph model with prescribed degree distribution and degree dependent edge weights is introduced. Each vertex is independently equipped with a random number of half-edges and each half-edge is assigned an integer valued weight according to a distribution that is allowed to depend on the degree of its vertex. Half-edges with the same weight are then paired randomly to create edges. An expression for the threshold for the appearance of a giant component in the resulting graph is derived using results on multi-type branching processes. The same technique also gives an expression for the basic reproduction number for an epidemic on the graph where the probability that a certain edge is used for transmission is a function of the edge weight. It is demonstrated that, if vertices with large degree tend to have large (small) weights on their edges and if the transmission probability increases with the edge weight, then it is easier (harder) for the epidemic to take off compared to a randomized epidemic with the same degree and weight distribution. A recipe for calculating the probability of a large outbreak in the epidemic and the size of such an outbreak is also given. Finally, the model is fitted to three empirical weighted networks of importance for the spread of contagious diseases and it is shown that $R_0$ can be substantially over- or underestimated if the correlation between degree and weight is not taken into account.

math.PR↗

arXiv2011

A dynamic network in a dynamic population: asymptotic properties

We derive asymptotic properties for a stochastic dynamic network model in a stochastic dynamic population. In the model, nodes give birth to new nodes until they die, each node being equipped with a social index given at birth. During the life of a node it creates edges to other nodes, nodes with high social index at higher rate, and edges disappear randomly in time. For this model we derive criterion for when a giant connected component exists after the process has evolved for a long period of time, assuming the node population grows to infinity. We also obtain an explicit expression for the degree correlation $ρ$ (of neighbouring nodes) which shows that $ρ$ is always positive irrespective of parameter values in one of the two treated submodels, and may be either positive or negative in the other model, depending on the parameters.

math.PR↗

arXiv2010

Maximizing the size of the giant

We consider two classes of random graphs: $(a)$ Poissonian random graphs in which the $n$ vertices in the graph have i.i.d.\ weights distributed as $X$, where $\mathbb{E}(X) = μ$. Edges are added according to a product measure and the probability that a vertex of weight $x$ shares and edge with a vertex of weight $y$ is given by $1-e^{-xy/(μn)}$. $(b)$ A thinned configuration model in which we create a ground-graph in which the $n$ vertices have i.i.d.\ ground-degrees, distributed as $D$, with $\mathbb{E}(D) = μ$. The graph of interest is obtained by deleting edges independently with probability $1-p$. In both models the fraction of vertices in the largest connected component converges in probability to a constant $1-q$, where $q$ depends on $X$ or $D$ and $p$. We investigate for which distributions $X$ and $D$ with given $μ$ and $p$, $1-q$ is maximized. We show that in the class of Poissonian random graphs, $X$ should have all its mass at 0 and one other real, which can be explicitly determined. For the thinned configuration model $D$ should have all its mass at 0 and two subsequent positive integers.

math.PR↗

arXiv2010

Household epidemic models with varying infection response

This paper is concerned with SIR (susceptible $\to$ infected $\to$ removed) household epidemic models in which the infection response may be either mild or severe, with the type of response also affecting the infectiousness of an individual. Two different models are analysed. In the first model, the infection status of an individual is predetermined, perhaps due to partial immunity, and in the second, the infection status of an individual depends on the infection status of its infector and on whether the individual was infected by a within- or between-household contact. The first scenario may be modelled using a multitype household epidemic model, and the second scenario by a model we denote by the infector-dependent-severity household epidemic model. Large population results of the two models are derived, with the focus being on the distribution of the total numbers of mild and severe cases in a typical household, of any given size, in the event that the epidemic becomes established. The aim of the paper is to investigate whether it is possible to determine which of the two underlying explanations is causing the varying response when given final size household outbreak data containing mild and severe cases. We conduct numerical studies which show that, given data on sufficiently many households, it is generally possible to discriminate between the two models by comparing the Kullback-Leibler divergence for the two fitted models to these data.

math.PR↗

arXiv2010

The Sensitivity of Respondent-driven Sampling Method

Researchers in many scientific fields make inferences from individuals to larger groups. For many groups however, there is no list of members from which to take a random sample. Respondent-driven sampling (RDS) is a relatively new sampling methodology that circumvents this difficulty by using the social networks of the groups under study. The RDS method has been shown to provide unbiased estimates of population proportions given certain conditions. The method is now widely used in the study of HIV-related high-risk populations globally. In this paper, we test the RDS methodology by simulating RDS studies on the social networks of a large LGBT web community. The robustness of the RDS method is tested by violating, one by one, the conditions under which the method provides unbiased estimates. Results reveal that the risk of bias is large if networks are directed, or respondents choose to invite persons based on characteristics that are correlated with the study outcomes. If these two problems are absent, the RDS method shows strong resistance to low response rates and certain errors in the participants' reporting of their network sizes. Other issues that might affect the RDS estimates, such as the method for choosing initial participants, the maximum number of recruitments per participant, sampling with or without replacement and variations in network structures, are also simulated and discussed.

stat.AP↗

arXiv2009

Stochastic epidemic models: a survey

This paper is a survey paper on stochastic epidemic models. A simple stochastic epidemic model is defined and exact and asymptotic model properties (relying on a large community) are presented. The purpose of modelling is illustrated by studying effects of vaccination and also in terms of inference procedures for important parameters, such as the basic reproduction number and the critical vaccination coverage. Several generalizations towards realism, e.g. multitype and household epidemic models, are also presented, as is a model for endemic diseases.

math.PR↗

arXiv2009

Statistical inference for stochastic epidemic models with three levels of mixing

A stochastic epidemic model is defined in which each individual belongs to a household, a secondary grouping (typically school or workplace) and also the community as a whole. Moreover, infectious contacts take place in these three settings according to potentially different rates. For this model we consider how different kinds of data can be used to estimate the infection rate parameters with a view to understanding what can and cannot be inferred, and with what precision. Among other things we find that temporal data can be of considerable inferential benefit compared to final size data, that the degree of heterogeneity in the data can have a considerable effect on inference for non-household transmission, and that inferences can be materially different from those obtained from a model with two levels of mixing. Keywords: Basic reproduction number, Bayesian inference, Epidemic model, Infectious disease data, Markov chain Monte Carlo, Networks.

stat.AP↗