1. |
- Lu, Xin, et al.
(författare)
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Respondent-driven sampling on directed networks
- 2013
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Ingår i: Electronic Journal of Statistics. - 1935-7524. ; 7, s. 292-322
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Tidskriftsartikel (refereegranskat)abstract
- Respondent-driven sampling (RDS) is a widely used method for generating chain-referral samples from hidden populations. It is an extension of the snowball sampling 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. Using a mean-field approach, we develop an estimator which is based on prior information about the average indegrees of estimated variables. When the indegree is known, such as for RDS studies over internet social networks, the estimator can greatly reduce estimate error and bias as compared with current methods; when the indegree is not known, which is most common for interview-based RDS studies, the estimator can through sensitivity analysis be used as a tool to account for uncertainties of network directedness and error in self-reported degree data. The performance of the new estimator, together with previous RDS estimators, is investigated thoroughly by simulations on networks with varying structures. We have applied the new estimator on an empirical RDS study for injecting drug users in New York City.
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2. |
- Britton, Tom, et al.
(författare)
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A dynamic network in a dynamic population: asymptotic properties
- 2011
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Ingår i: Journal of Applied Probability. - : Cambridge University Press (CUP). - 1475-6072 .- 0021-9002. ; 48:4, s. 1163-1178
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Tidskriftsartikel (refereegranskat)abstract
- 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 a criterion for when a giant connected component exists after the process has evolved for a long period of time, assuming that the node population grows to infinity. We also obtain an explicit expression for the degree correlation rho (of neighbouring nodes) which shows that rho 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.
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3. |
- Spricer, Kristoffer, 1966-
(författare)
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Random networks with weights and directions, and epidemics thereon
- 2018
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Networks, consisting of nodes and of edges, can be used to model numerous phenomena, e.g, web pages linking to each other or interactions between people in a population. Edges can be directed, such as a one way link from one web page to another, or undirected (bi-directional), such as physical contacts between pairs of people, which potentially could spread an infection either way between them. Edges can also have weights associated with them, in this thesis corresponding to the probability that an infection is transmitted on the edge.Empirical networks are often only partially known, in the form of ego-centric network data where only a subset of the nodes and the number of adjacent edges of each node have been observed. This situation lends itself well to analysis through the undirected or partially directed configuration model - a random network model where the number of edges of each node (the degree) is given but where the way these edges are connected is random.The four papers in this thesis are concerned with the properties of the configuration model and with the usefulness of it with respect to its ability to model the spread of epidemics on empirical networks. Paper I proves the asymptotic convergence to a given degree distribution for the partially directed configuration model. In Paper II it is shown that epidemics on some empirical and theoretically constructed networks grow exponentially, similarly to what can be seen on the corresponding configuration models. Finally, in Papers III and IV, large population analytical results for the reproduction number, the probability of a large epidemic outbreak and the final size of such an outbreak are derived assuming a configuration model network with weighted and/or partially directed edges. These results are then evaluated on several large empirical networks upon which epidemics are simulated. We find that on some of these networks the analytical expressions are compatible with the results of the simulations. This makes the model useful as a tool for analyzing such networks.
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4. |
- Britton, Tom, et al.
(författare)
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A weighted configuration model and inhomogeneous epidemics
- 2011
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Ingår i: Journal of statistical physics. - : Springer Science and Business Media LLC. - 0022-4715 .- 1572-9613. ; 145, s. 1368-1384
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Tidskriftsartikel (refereegranskat)abstract
- 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 (reflecting how closely ‘connected’ the corresponding vertices are). 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 R0 can be substantially over- or underestimated if the correlation between degree and weight is not taken into account.
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5. |
- Ball, Frank, et al.
(författare)
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A network with tunable clustering, degree correlation and degree distribution, and an epidemic thereon
- 2013
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Ingår i: Journal of Mathematical Biology. - : Springer Science and Business Media LLC. - 0303-6812 .- 1432-1416. ; 66:4-5, s. 979-1019
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Tidskriftsartikel (refereegranskat)abstract
- 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 , 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 (1) clustering tends to decrease the spread of disease, (2) the effect of degree correlation is appreciably greater when the disease is close to threshold than when it is well above threshold and (3) disease spread broadly increases with degree correlation when is just above its threshold value of one and decreases with when is well above one.
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6. |
- Ball, Frank, et al.
(författare)
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A stochastic SIR network epidemic model with preventive dropping of edges
- 2019
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Ingår i: Journal of Mathematical Biology. - : Springer Science and Business Media LLC. - 0303-6812 .- 1432-1416. ; 78:6, s. 1875-1951
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Tidskriftsartikel (refereegranskat)abstract
- A Markovian Susceptible Infectious 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 N, 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 R0 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 R0>1, 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.
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7. |
- Ball, Frank, et al.
(författare)
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AN EPIDEMIC IN A DYNAMIC POPULATION WITH IMPORTATION OF INFECTIVES
- 2017
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Ingår i: The Annals of Applied Probability. - 1050-5164 .- 2168-8737. ; 27:1, s. 242-274
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Tidskriftsartikel (refereegranskat)abstract
- Consider a large uniformly mixing dynamic population, which has constant birth rate and exponentially distributed lifetimes, with mean population size n. A Markovian SIR (susceptible -> infective -> recovered) infectious disease, having importation of infectives, taking place in this population is analysed. The main situation treated is where n -> infinity, keeping the basic reproduction number R-0 as well as the importation rate of infectives fixed, but assuming that the quotient of the average infectious period and the average lifetime tends to 0 faster than 1/log n. It is shown that, as n -> infinity, the behaviour of the 3-dimensional process describing the evolution of the fraction of the population that are susceptible, infective and recovered, is encapsulated in a 1-dimensional regenerative process S = {S(t); t >= 0} describing the limiting fraction of the population that are susceptible. The process S grows deterministically, except at one random time point per regenerative cycle, where it jumps down by a size that is completely determined by the waiting time since the start of the regenerative cycle. Properties of the process S, including the jump size and stationary distributions, are determined.
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8. |
- Ball, Frank, et al.
(författare)
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Epidemics on networks with preventive rewiring
- 2022
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Ingår i: Random structures & algorithms (Print). - : Wiley. - 1042-9832 .- 1098-2418. ; 61:2, s. 250-297
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Tidskriftsartikel (refereegranskat)abstract
- A stochastic SIR (susceptible infective recovered) model is considered for the spread of an epidemic on a network, described initially by an Erdős–Rényi random graph, in which susceptible individuals connected to infectious neighbors may drop or rewire such connections. A novel construction of the model is used to derive a deterministic model for epidemics started with a positive fraction initially infected and prove convergence of the scaled stochastic model to that deterministic model as the population size . For epidemics initiated by a single infective that take off, we prove that for part of the parameter space, in the limit as , the final fraction infected is discontinuous in the infection rate at its threshold , thus not converging to 0 as . The discontinuity is particularly striking when rewiring is necessarily to susceptible individuals in that jumps from 0 to 1 as passes through .
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9. |
- Ball, Frank, et al.
(författare)
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Household epidemic models with varying infection response
- 2011
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Ingår i: Journal of Mathematical Biology. - : Springer Science and Business Media LLC. - 0303-6812 .- 1432-1416. ; 63:2, s. 309-337
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Tidskriftsartikel (refereegranskat)abstract
- This paper is concerned with SIR (susceptible -> infected -> 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.
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10. |
- Ball, Frank, et al.
(författare)
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ON EXPECTED DURATIONS OF BIRTH-DEATH PROCESSES, WITH APPLICATIONS TO BRANCHING PROCESSES AND SIS EPIDEMICS
- 2016
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Ingår i: Journal of Applied Probability. - : Cambridge University Press (CUP). - 0021-9002 .- 1475-6072. ; 53:1, s. 203-215
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Tidskriftsartikel (refereegranskat)abstract
- 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 alpha(n). We focus on two important examples, namely alpha(n) = lambda n being a branching process, and alpha(n) = lambda n(N-n)/N which corresponds to an SIS (susceptible -> infective -> susceptible) 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. We give expressions for the expectation of all these quantities and show 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, in contrast to the household SIR (susceptible -> infective -> recovered) epidemic, its threshold parameter R-* is insensitive to the distribution of Q.
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