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ABSTRACT: Consider a homogeneous time-continuous branching process where individuals
have constant birth rate $\delta$, 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
$E(A_K)=\delta^{K-1}/\left(k(1\vee\delta)^K\right)$, irrespective of the life
length distribution $Q$, subject to the normalizing condition $E(Q)=1$.
04/2013;
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ABSTRACT: 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.
Electronic Journal of Statistics 01/2013; 7:292-322.. · 1.15 Impact Factor
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ABSTRACT: 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.
Mathematical biosciences 06/2012; 240(2):124-31. · 1.30 Impact Factor
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ABSTRACT: We consider a random network evolving in continuous time in which new nodes are born and old may die, and where undirected
edges between nodes are created randomly and may also disappear. The node population is Markovian and so is the creation and
deletion of edges, given the node population. Each node is equipped with a random social index and the intensity at which
a node creates new edges is proportional to the social index, and the neighbour is either chosen uniformly or proportional
to its social index in a modification of the model. We derive properties of the network as time and the node population tends
to infinity. In particular, the degree-distribution is shown to be a mixed Poisson distribution which may exhibit a heavy
tail (e.g. power-law) if the social index distribution has a heavy tail. The limiting results are verified by means of simulations,
and the model is fitted to a network of sexual contacts.
KeywordsRandom networks-Dynamic networks-Birth and death process-Mixed Poisson distribution
Journal of Statistical Physics 04/2012; 139(3):518-535. · 1.40 Impact Factor
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ABSTRACT: 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.
Journal of the Royal Statistical Society Series A (Statistics in Society) 01/2012; 175(1):191–216. · 2.11 Impact Factor
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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. 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.
04/2011;
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ABSTRACT: The birth-death process is widely used in phylogenetics to model speciation and extinction. Recent studies have shown that the inferred rates are sensitive to assumptions about the sampling probability of lineages. Here, we examine the effect of the method used to sample lineages. Whereas previous studies have assumed random sampling (RS), we consider two extreme cases of biased sampling: "diversified sampling" (DS), where tips are selected to maximize diversity and "cluster sampling (CS)," where sample diversity is minimized. DS appears to be standard practice, for example, in analyses of higher taxa, whereas CS may occur under special circumstances, for example, in studies of geographically defined floras or faunas. Using both simulations and analyses of empirical data, we show that inferred rates may be heavily biased if the sampling strategy is not modeled correctly. In particular, when a diversified sample is treated as if it were a random or complete sample, the extinction rate is severely underestimated, often close to 0. Such dramatic errors may lead to serious consequences, for example, if estimated rates are used in assessing the vulnerability of threatened species to extinction. Using Bayesian model testing across 18 empirical data sets, we show that DS is commonly a better fit to the data than complete, random, or cluster sampling (CS). Inappropriate modeling of the sampling method may at least partly explain anomalous results that have previously been attributed to variation over time in birth and death rates.
Molecular Biology and Evolution 04/2011; 28(9):2577-89. · 5.55 Impact Factor
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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 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 $\rho$ (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.
04/2011;
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ABSTRACT: We have evaluated the performance of two classes of probabilistic models for substitution rate variation over phylogenetic trees. In the first class, branch rates are considered to be independent and identically distributed (i.i.d.) stochastic variables. Three versions with respect to the underlying distribution (Gamma, Inverse Gaussian, and LogNormal) are considered. The i.i.d. models are compared with the autocorrelated (AC) model, where rates of adjacent nodes in the tree are AC, so that a node rate is LogNormal distributed around the rate of the parent node. The performance of different models is evaluated using three empirical data sets. For all data sets, it was clear that all tested models extracted substantial knowledge from data when posterior divergence time distributions were compared with the prior distributions and, furthermore, that they clearly outperformed a molecular clock. Moreover, the descriptive power of the i.i.d. models, as evaluated by Bayes factors, was either equal to or clearly better than that of the AC model. The latter effect increased with extended taxon sampling. Likewise, under none of the models could we find compelling evidence, in any of the data sets, for rate correlation between adjacent branches/nodes. These findings challenge previous suggestions of universality of autocorrelation in sequence evolution. We also performed an additional comparison with a divergence time prior including calibration information from fossil evidence. Adding fossil information to the prior had negligible effect on Bayes factors and mainly affected the width of the posterior distribution of the divergence times, whereas the relative position of the mean divergence times were largely unaffected.
Systematic Biology 03/2011; 60(3):329-42. · 10.23 Impact Factor
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ABSTRACT: We analyse a Markovian SIS epidemic amongst a finite population partitioned into households. Since the population is finite, the epidemic will eventually go extinct, i.e., have no more infectives in the population. We study the effects of population size and within household transmission upon the time to extinction. This is done through two approximations. The first approximation is suitable for all levels of within household transmission and is based upon an Ornstein-Uhlenbeck process approximation for the diseases fluctuations about an endemic level relying on a large population. The second approximation is suitable for high levels of within household transmission and approximates the number of infectious households by a simple homogeneously mixing SIS model with the households replaced by individuals. The analysis, supported by a simulation study, shows that the mean time to extinction is minimized by moderate levels of within household transmission.
Journal of Mathematical Biology 12/2010; 61(6):763-79. · 2.96 Impact Factor
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Tom Britton
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ABSTRACT: This paper is a survey paper on stochastic epidemic models. A simple stochastic epidemic model is defined and exact and asymptotic (relying on a large community) properties 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.
Mathematical biosciences 05/2010; 225(1):24-35. · 1.30 Impact Factor
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ABSTRACT: Epidemic models are always simplifications of real world epidemics. Which real world features to include, and which simplifications to make, depend both on the disease of interest and on the purpose of the modelling. In the present paper we discuss some such purposes for which a stochastic model is preferable to a deterministic counterpart. The two main examples illustrate the importance of allowing the infectious and latent periods to be random when focus lies on the probability of a large epidemic outbreak and/or on the initial speed, or growth rate, of the epidemic. A consequence of the latter is that estimation of the basic reproduction number R(0) is sensitive to assumptions about the distributions of the infectious and latent periods when using data from the early stages of an outbreak, which we illustrate with data from the H1N1 influenza A pandemic. Some further examples are also discussed as are some practical consequences related to these stochastic aspects.
Mathematical biosciences 10/2009; 222(2):109-16. · 1.30 Impact Factor
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ABSTRACT: Adipose tissue may contain few large adipocytes (hypertrophy) or many small adipocytes (hyperplasia). We investigated factors of putative importance for adipose tissue morphology.
Subcutaneous adipocyte size and total fat mass were compared in 764 subjects with BMI 18-60 kg/m(2). A morphology value was defined as the difference between the measured adipocyte volume and the expected volume given by a curved-line fit for a given body fat mass and was related to insulin values. In 35 subjects, in vivo adipocyte turnover was measured by exploiting incorporation of atmospheric (14)C into DNA.
Occurrence of hyperplasia (negative morphology value) or hypertrophy (positive morphology value) was independent of sex and body weight but correlated with fasting plasma insulin levels and insulin sensitivity, independent of adipocyte volume (beta-coefficient = 0.3, P < 0.0001). Total adipocyte number and morphology were negatively related (r = -0.66); i.e., the total adipocyte number was greatest in pronounced hyperplasia and smallest in pronounced hypertrophy. The absolute number of new adipocytes generated each year was 70% lower (P < 0.001) in hypertrophy than in hyperplasia, and individual values for adipocyte generation and morphology were strongly related (r = 0.7, P < 0.001). The relative death rate (approximately 10% per year) or mean age of adipocytes (approximately 10 years) was not correlated with morphology.
Adipose tissue morphology correlates with insulin measures and is linked to the total adipocyte number independently of sex and body fat level. Low generation rates of adipocytes associate with adipose tissue hypertrophy, whereas high generation rates associate with adipose hyperplasia.
Diabetes 10/2009; 59(1):105-9. · 8.29 Impact Factor
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ABSTRACT: We analyse a stochastic SIS epidemic amongst a finite population partitioned into households. Since the population is finite, the epidemic will eventually go extinct, i.e., have no more infectives in the population. We study the effects of population size and within household transmission upon the time to extinction. This is done through two approximations. The first approximation is suitable for all levels of within household transmission and is based upon an Ornstein-Uhlenbeck process approximation for the diseases fluctuations about an endemic level relying on a large population. The second approximation is suitable for high levels of within household transmission and approximates the number of infectious households by a simple homogeneously mixing SIS model with the households replaced by individuals. The analysis, supported by a simulation study, shows that the mean time to extinction is minimized by moderate levels of within household transmission.
08/2009;
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ABSTRACT: A Markovian susceptible --> infectious --> removed (SIR) epidemic model is considered in a community partitioned into households. A vaccination strategy, which is implemented during the early stages of the disease following the detection of infected individuals is proposed. In this strategy, the detection occurs while an individual is infectious and other susceptible household members are vaccinated without further delay. Expressions are derived for the influence on the reproduction numbers of this vaccination strategy for equal and unequal household sizes. We fit previously estimated parameters from influenza and use household distributions for Sweden and Tanzania census data. The results show that the reproduction number is much higher in Tanzania (6 compared with 2) due to larger households, and that infected individuals have to be detected (and household members vaccinated) after on average 5 days in Sweden and after 3.3 days in Tanzania, a much smaller difference.
Biometrical Journal 06/2009; 51(3):408-19. · 1.25 Impact Factor
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ABSTRACT: We consider a (social) network whose structure can be represented by a simple random graph having a pre-specified degree distribution. A Markovian susceptible-infectious-removed (SIR) epidemic model is defined on such a social graph. We then consider two real-time vaccination models for contact tracing during the early stages of an epidemic outbreak. The first model considers vaccination of each friend of an infectious individual (once identified) independently with probability p. The second model is related to the first model but also sets a bound on the maximum number an infectious individual can infect before being identified. Expressions are derived for the influence on the reproduction number of these vaccination models. We give some numerical examples and simulation results based on the Poisson and heavy-tail degree distributions where it is shown that the second vaccination model has a bigger advantage compared to the first model for the heavy-tail degree distribution.
Mathematical Biosciences 11/2008; 216(1):1-8. · 1.54 Impact Factor
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ABSTRACT: Cancer cachexia is an important, negative prognostic marker that has been linked to systemic inflammation and cell death through unclear mechanisms. A key feature of cancer cachexia is loss of white adipose tissue (WAT) because of increased adipocyte lipolysis and possibly reduced lipid synthesis (lipogenesis). In this study, the authors investigated whether alterations in fat cell numbers, lipogenesis, or cytokine and/or leukocyte infiltration could account for some of the functional changes observed in WAT in cancer cachexia.
Blood and subcutaneous WAT samples were obtained from a 10 weight-stable patients, from 13 weight losing (cachexia) patients with cancer, and from 5 patients without cancer (noncancer patients) who initially were classified with cancer.
Systemic inflammation (increased circulating levels of interleukin 6 [IL-6]) and enhanced lipolysis were confirmed in the cachectic patients compared with the other patients. However, the messenger RNA expression of IL-6 and other cytokine or leukocyte markers, as well as WAT secretion of IL-6, were not altered in the patients with cachexia. Thus, the elevated serum levels of IL-6 that were observed in cachexia were not derived from WAT. Insulin-induced lipogenesis in adipocytes from patients with cachexia was the same as that in adipocytes from patients with weight-stable cancer and from noncancer patients (2.5-fold maximal stimulation; half-maximum effective concentration, approximately 0.03 nmol/L). Fat cell size was decreased but adipocyte numbers were normal in cancer patients with cachexia, suggesting that there was no major fat cell death.
The current findings indicated that subcutaneous WAT does not contribute to the systemic inflammatory reaction and does not induce adipocyte insulin resistance in cancer cachexia. Moreover, increased fat cell lipolysis, not reduced lipogenesis or adipocyte cell death, appeared to be the primary cause of fat loss in this condition.
Cancer 09/2008; 113(7):1695-704. · 4.77 Impact Factor
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Kirsty L Spalding,
Erik Arner,
Pål O Westermark,
Samuel Bernard,
Bruce A Buchholz,
Olaf Bergmann,
Lennart Blomqvist,
Johan Hoffstedt,
Erik Näslund, Tom Britton,
Hernan Concha,
Moustapha Hassan,
Mikael Rydén,
Jonas Frisén,
Peter Arner
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ABSTRACT: Obesity is increasing in an epidemic manner in most countries and constitutes a public health problem by enhancing the risk for cardiovascular disease and metabolic disorders such as type 2 diabetes. Owing to the increase in obesity, life expectancy may start to decrease in developed countries for the first time in recent history. The factors determining fat mass in adult humans are not fully understood, but increased lipid storage in already developed fat cells (adipocytes) is thought to be most important. Here we show that adipocyte number is a major determinant for the fat mass in adults. However, the number of fat cells stays constant in adulthood in lean and obese individuals, even after marked weight loss, indicating that the number of adipocytes is set during childhood and adolescence. To establish the dynamics within the stable population of adipocytes in adults, we have measured adipocyte turnover by analysing the integration of 14C derived from nuclear bomb tests in genomic DNA. Approximately 10% of fat cells are renewed annually at all adult ages and levels of body mass index. Neither adipocyte death nor generation rate is altered in early onset obesity, suggesting a tight regulation of fat cell number in this condition during adulthood. The high turnover of adipocytes establishes a new therapeutic target for pharmacological intervention in obesity.
Nature 07/2008; 453(7196):783-7. · 36.28 Impact Factor
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ABSTRACT: In phylogenetic inference, the support of an estimated phylogenetic tree topology and its interior branches is usually measured either with non-parametric bootstrap support (BS) values or with Bayesian posterior probabilities (BPPs). Extensive empirical evidence indicates that BPP values are systematically larger than BS when measured on the same data set, but there are no theoretical results supporting such a systematic difference. In the present note, we give a heuristic mathematical argument supporting the empirically observed phenomenon. The argument uses properties of the marginal and profile likelihoods of the normal distribution. The heuristic arguments are supported in a simulation study evaluating different steps in the argument.
Mathematical Medicine and Biology 01/2008; 24(4):401-11. · 1.82 Impact Factor
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ABSTRACT: 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.
Theoretical Population Biology 12/2007; 72(3):389-99. · 1.65 Impact Factor
