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AbstractThe directed preferential attachment model is revisited. A new exact characterization of the limiting in- and out-degree distribution is given by two independentpure birth processes that are observed at a common exponentially distributed time T(thus creating dependence between in- and out-degree). The characterization gives an explicit form for the joint degree distribution, and this confirms previously derived tail probabilities for the two marginal degree distributions. The new characterization is also used to obtain an explicit expression for tail probabilities in which both degrees are large. A new generalized directed preferential attachment model is then defined and analyzed using similar methods. The two extensions, motivated by empirical evidence, are to allow double-directed (i.e. undirected) edges in the network, and to allow the probability of connecting an ingoing (outgoing) edge to a specified node to also depend on the out-degree (in-degree) of that node.

AbstractWe introduce a weighted configuration model graph, where edge weightscorrespond to the probability of infection in an epidemic on the graph. On these graphs, we study the development of a Susceptible–Infectious–Recovered epidemic using both Reed–Frost and Markovian settings. For the special case of having two different edge types, we determine the basic reproduction numberR0, the probability of a major outbreak, and the relative final size of a major outbreak. Results are compared with those for a calibrated unweighted graph. The degree distributions are based on both theoretical constructs and empirical network data. In addition, bivariate standard normal copulas are used to model the dependence between the degrees of the two edge types, allowing for modeling the correlation between edge types over a wide range. Among the results are that the weighted graph produces much richer results than the unweighted graph. Also, while R0always increases with increasing correlation between the two degrees, this is not necessarily true for the probability of a major outbreak nor for the relative final size of a major outbreak. When using copulas we see that these can produce results that are similar to those of the empirical degree distributions, indicating that in some cases a copula is a viable alternative to using the full empirical data.

AbstractWe couple a multi-type stochastic epidemic process with a directed random graph, where edges have random weights (traversal times). This random graph representation is used to characterise the fractions of individuals infected by the different types of vertices among all infected individuals in the large population limit. For this characterisation, we rely on the theory of multi-type real-time branching processes. We identify a special case of the two-type model in which the fraction of individuals of a certain type infected by individuals of the same type is maximised among all two-type epidemics approximated by branching processes with the same mean offspring matrix.

Susceptible-infectious-recovered-susceptible (SIRS) epidemic models assume that individual immunity wanes in one leap, from complete immunity to complete susceptibility. For many diseases immunity on the contrary wanes gradually, something that has become even more evident during COVID-19 pandemic where also recently infected have a reinfection risk, and booster vaccines are given to increase immunity. Here, a novel mathematical model is presented allowing for the gradual decay of immunity following linear or exponential waning functions. The two new models and the SIRS model are compared assuming all three models have the same cumulative immunity. When no intervention is put in place, we find that the long-term prevalence is higher for the models with gradual waning. If aiming for herd immunity by continuous vaccination, it is shown that larger vaccine quantities are required when immunity wanes gradually compared with results obtained from the SIRS model, and this difference is the biggest for the most realistic assumption of exponentially waning of immunity. For parameter choices fitting to COVID-19, the critical amount of vaccine supply is about 50% higher if immunity wanes linearly, and more than 150% higher when immunity wanes exponentially, when compared with the classic SIRS epidemic model.

Respondent-driven sampling (RDS) is often used to estimate population properties (e.g., sexual risk behavior) in hard-to-reach populations. In RDS, already sampled individuals recruit population members to the sample from their social contacts in an efficient snowball-like sampling procedure. By assuming a Markov model for the recruitment of individuals, asymptotically unbiased estimates of population characteristics can be obtained. Current RDS estimation methodology assumes that the social network is undirected, that is, all edges are reciprocal. However, empirical social networks in general also include a substantial number of nonreciprocal edges. In this article, we develop an estimation method for RDS in populations connected by social networks that include reciprocal and nonreciprocal edges. We derive estimators of the selection probabilities of individuals as a function of the number of outgoing edges of sampled individuals. The proposed estimators are evaluated on artificial and empirical networks and are shown to generally perform better than existing estimators. This is the case in particular when the fraction of directed edges in the network is large.

Respondent-driven sampling (RDS) is often used to estimate population properties (e.g., sexual risk behavior) in hard-to-reach populations. In RDS, already sampled individuals recruit population members to the sample from their social contacts in an efficient snowball-like sampling procedure. By assuming a Markov model for the recruitment of individuals, asymptotically unbiased estimates of population characteristics can be obtained. Current RDS estimation methodology assumes that the social network is undirected, that is, all edges are reciprocal. However, empirical social networks in general also include a substantial number of nonreciprocal edges. In this article, we develop an estimation method for RDS in populations connected by social networks that include reciprocal and nonreciprocal edges. We derive estimators of the selection probabilities of individuals as a function of the number of outgoing edges of sampled individuals. The proposed estimators are evaluated on artificial and empirical networks and are shown to generally perform better than existing estimators. This is the case in particular when the fraction of directed edges in the network is large.

AbstractWe 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) nis a(n). We focus on two important examples, namely a(n) = ? nbeing a branching process, and a(n) = ?n(N- n) / Nwhich 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, An, C, and Sdenote 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 notindependent 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.

Background and AimBrainstem gliomas are rare, accounting for less than 2% of adult gliomas. They are difficult to diagnose and challenging to treat [Hu et al, 2016]. Here, we report a patient with an unusual clinical presentation of brainstem glioma, supported by radiological and histopathological findings.Clinical presentationA 61-year old male retired diver presented with 2-month history of worsening occipital headache, described as a constant pressure worse on lying down with migrainous features of nausea and light-sensitivity, and intermittent episodes of double vision and poor balance.Neurological examination on admission was normal. He progressed to develop double vision, left-sided weakness and worsening swallowing and respiratory function.InvestigationsBlood tests for infections and vasculopathy were negative. Two CSF samples taken showed high protein (1.7g and 3.25g respectively) with the first showing 100% lymphocytes and the latter 56% monocytes and 42% lymphocytes. There were no malignant cells on immunophenotyping.MRIs demonstrated progressive of ring-enhancing lesions in brainstem in keeping with inflammatory con- ditions. Biopsy was not possible due to the centrally-located lesions. Despite receiving wide spectrum antimicrobials, steroids and respiratory support, he died 2 months later. Autopsy of the brain showed a glioblastoma with brainstem infiltration and cystic necrosis in the pons.SummaryGliomatosis pattern and leptomeningeal and nerve root spreading of brainstem glioblastoma can be misleading signs with neuro-infections as a differential. Early neurosurgical intervention may aid diagnosis.kitwu@doctors.org.uk

We present a stochastic epidemic model to study the effect of various preventive measures, such as uniform reduction of contacts and transmission, vaccination, isolation, screening and contact tracing, on a disease outbreak in a homogeneously mixing community. The model is based on an infectivity process, which we define through stochastic contact and infectiousness processes, so that each individual has an independent infectivity profile. In particular, we monitor variations of the reproduction number and of the distribution of generation times. We show that some interventions, i.e. uniform reduction and vaccination, affect the former while leaving the latter unchanged, whereas other interventions, i.e. isolation, screening and contact tracing, affect both quantities. We provide a theoretical analysis of the variation of these quantities, and we show that, in practice, the variation of the generation time distribution can be significant and that it can cause biases in the estimation of reproduction numbers. The framework, because of its general nature, captures the properties of many infectious diseases, but particular emphasis is on COVID-19, for which numerical results are provided.

Consider a random graph where the mean degree is given and fixed. In this paper we derive the maximal size of the largest connected component in the graph. We also study the related question of the largest possible outbreak size of an epidemic occurring ‘on’ the random graph (the graph describing the social structure in the community). More precisely, we look at two different classes of random graphs. First, the Poissonian random graph in which each node iis given an independent and identically distributed (i.i.d.) random weight Xiwith E(Xi)=µ, and where there is an edge between iand jwith probability 1-e-XiXj/(µ n), independently of other edges. The second model is the thinned configuration model in which the nvertices of the ground graph have i.i.d. ground degrees, distributed as D, with 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 qdepends on Xor Dand p. We investigate for which distributions Xand Dwith given µ and p, 1-qis maximized. We show that in the class of Poissonian random graphs, Xshould have all its mass at 0 and one other real, which can be explicitly determined. For the thinned configuration model, Dshould have all its mass at 0 and two subsequent positive integers.

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 ? (of neighbouring nodes) which shows that ? 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.

To identify peripheral lymphovenous communications (LVCs) using labelled erythrocytes and intradermal injection. Intradermal injection delivers macromolecules to loco-regional lymph nodes faster than subcutaneous injection, suggesting easier lymphatic vessel access.

The general stochastic SIR epidemic in a closed population under the influence of a term-time forced environment is considered. An ‘environment’ in this context is any external factor that influences the contact rate between individuals in the population, but is itself unaffected by the population. Here ‘term-time forcing’ refers to discontinuous but cyclic changes in the contact rate. The inclusion of such an environment into the model is done by replacing a single contact rate ? with a cyclically alternating renewal process with k different states denoted {?(t)}t=0. Threshold conditions in terms of R?are obtained, such that R?>1 implies that p, the probability of a large outbreak, is strictly positive. Examples are given where p is evaluated numerically from which the impact of the distribution of the time periods that ?(t) spends in its different states is clearly seen.

To determine how often the sentinel lymph node (SLN) draining the breast is the same node as the SLN draining the upper limb. A common SLN might increase the risk of upper limb breast cancer-related lymphoedema after SLN biopsy.

In this paper a branching process approximation for the spread of a Reed-Frost epidemic on a network with tunable clustering is derived. The approximation gives rise to expressions for the epidemic threshold and the probability of a large outbreak in the epidemic. We investigate how these quantities vary with the clustering in the graph and find that, as the clustering increases, the epidemic threshold decreases. The network is modeled by a random intersection graph, in which individuals are independently members of a number of groups and two individuals are linked to each other if and only if there is at least one group that they are both members of.

Consider a random graph, having a prespecified degree distribution F, but other than that being uniformly distributed, describing the social structure (friendship) in a large community. Suppose that one individual in the community is externally infected by an infectious disease and that the disease has its course by assuming that infected individuals infect their not yet infected friends independently with probability p. For this situation, we determine the values of R0, the basic reproduction number, and t0, the asymptotic final size in the case of a major outbreak. Furthermore, we examine some different local vaccination strategies, where individuals are chosen randomly and vaccinated, or friends of the selected individuals are vaccinated, prior to the introduction of the disease. For the studied vaccination strategies, we determine Rv, the reproduction number, and tv, the asymptotic final proportion infected in the case of a major outbreak, after vaccinating a fraction v.

A stochastic epidemic model is defined in which infected individuals have different severities of disease (e.g. mildly and severely infected) and the severity of an infected individual depends on the severity of the individual he or she was infected by; typically, severe or mild infectives have an increased tendency to infect others severely or, respectively, mildly. Large-population properties of the model are derived, using branching process approximations for the initial stages of an outbreak and density-dependent population processes when a major outbreak occurs. The effects of vaccination are considered, using two distinct models for vaccine action. The consequences of launching a vaccination program are studied in terms of the effect it has on reducing the final size in the event of a major outbreak as a function of the vaccination coverage, and also by determining the critical vaccination coverage above which only small outbreaks can occur.

Patterns of diversification and timing of evolution within Neoaves, which includes almost 95% of all bird species, are virtually unknown. On the other hand, molecular data consistently indicate a Cretaceous origin of many neoavian lineages and the fossil record seems to support an Early Tertiary diversification. Here, we present the first well-resolved molecular phylogeny for Neoaves, together with divergence time estimates calibrated with a large number of stratigraphically and phylogenetically well-documented fossils. Our study defines several well-supported clades within Neoaves. The calibration results suggest that Neoaves, after an initial split from Galloanseres in Mid-Cretaceous, diversified around or soon after the K/T boundary. Our results thus do not contradict palaeontological data and show that there is no solid molecular evidence for an extensive pre-Tertiary radiation of Neoaves.

We consider a stochastic model for the spread of a susceptible–infective–removed (SIR) epidemic among a closed, finite population, in which there are two types of severity of infectious individuals, namely mild and severe. The type of severity depends on the amount of infectious exposure an individual receives, in that infectives are always initially mild but may become severe if additionally exposed. Large-population properties of the model are derived. In particular, a coupling argument is used to provide a rigorous branching process approximation to the early stages of an epidemic, and an embedding argument is used to derive a strong law and an associated central limit theorem for the final outcome of an epidemic in the event of a major outbreak. The basic reproduction number, which determines whether or not a major outbreak can occur given few initial infectives, depends only on parameters of the mild infectious state, whereas the final outcome in the event of a major outbreak depends also on parameters of the severe state. Moreover, the limiting final size proportions need not even be continuous in the model parameters.

cDNA microarrays are used in many contexts to compare mRNA levels between samples of cells. Microarray experiments typically give us expression measurements on 1000-20 000 genes, but with few replicates for each gene. Traditional methods using means and standard deviations to detect differential expression are not satisfactory in this context. A handful of alternative statistics have been developed, including several empirical Bayes methods. In the present paper we present two full hierarchical Bayes models for detecting gene expression, of which one (D) describes our microarray data very well. We also compare the full Bayes and empirical Bayes approaches with respect to model assumptions, false discovery rates and computer running time. The proposed models are compared to existing empirical Bayes models in a simulation study and for a set of data (Yuen et al., 2002), where 27 genes have been categorized by quantitative real-time PCR. It turns out that the existing empirical Bayes methods have at least as good performance as the full Bayes ones.

An increasingly important problem in genome sequencing is the failure of the commonly used shotgun assembly programs to correctly assemble repetitive sequences. The assembly of non-repetitive regions or regions containing repeats considerably shorter than the average read length is in practice easy to solve, while longer repeats have been a difficult problem. We here present a statistical method to separate arbitrarily long, almost identical repeats, which makes it possible to correctly assemble complex repetitive sequence regions. The differences between repeat units may be as low as 1% and the sequencing error may be up to ten times higher. The method is based on the realization that a comparison of only a part of all overlapping sequences at a time in a data set does not generate enough information for a conclusive analysis. Our method uses optimal multi-alignments consisting of all the overlaps of each read. This makes it possible to determine defined nucleotide positions, DNPs, which constitute the differences between the repeat units. Differences between repeats are distinguished from sequencing errors using statistical methods, where the probabilities of obtaining certain combinations of candidate DNPs are calculated using the information from the multi-alignments. The use of DNPs and combinations of DNPs will allow for optimal and rapid assemblies of repeated regions. This method can solve repeats that differ in only two positions in a read length, which is the theoretical limit for repeat separation. We predict that this method will be highly useful in shotgun sequencing in the future.

Abstract.: Empirical evidence shows that childhood diseases persist in large communities whereas in smaller communities the epidemic goes extinct (and is later reintroduced by immigration). The present paper treats a stochastic model describing the spread of an infectious disease giving life-long immunity, in a community where individuals die and new individuals are born. The time to extinction of the disease starting in quasi-stationarity (conditional on non-extinction) is exponentially distributed. As the population size grows the epidemic process converges to a diffusion process. Properties of the limiting diffusion are used to obtain an approximate expression for τ, the mean-parameter in the exponential distribution of the time to extinction for the finite population. The expression is used to study how τ depends on the community size but also on certain properties of the disease/community: the basic reproduction number and the means and variances of the latency period, infectious period and life-length. Effects of introducing a vaccination program are also discussed as is the notion of the critical community size, defined as the size which distinguishes between the two qualitatively different behaviours.

Consider an epidemic outbreak in a large population resulting in an overall proportion infected p~. The proportion needed to be vaccinated in order to prevent the epidemic, the critical vaccination coverage vc, depends on individual and social heterogeneities in the population. In the present note it is shown that vcis larger if, as is likely, individuals differ in terms of susceptibility than if they are equally susceptible.

An epidemic model for the spread of an infectious disease in a population of families is considered. The score test of the hypothesis that there is no higher infectivity between family members is constructed under the assumption that the epidemic process is observed continuously up to some time t. The score process is a martingale as a function of tand by letting the number of families tend to infinity, a central limit theorem for the process can be proved. The central limit theorem not only justifies a normal approximation of the test statistic—it also suggests a smaller variance estimator than expected.

Two stochastic epidemic models, describing the spread of a disease in a population consisting of families of individuals, are considered. Both models allow the probability of infecting individuals to be larger for members of the same family. As the number of families get large while the family sizes remain bounded, weak convergence of the epidemic processes are obtained using theory for Markov counting processes. The results are used to construct score tests of the hypothesis of no extra within family infectivity, when it is assumed that the epidemic is observed only once.

We first study an epidemic amongst a population consisting of individuals with the same infectivity but with varying susceptibilities to the disease. The asymptotic final epidemic size is compared with the corresponding size for a homogeneous population. Then we group a heterogeneous population into households, assuming very high infectivity within households, and investigate how the global infection pressure is affected by rearranging individuals between the households. In both situations considered, it turns out that whether or not homogenizing the individuals or households will result in an increased spread of infection actually depends on the infectiousness of the disease.

When a region tries to prevent an outbreak of an epidemic, two broad strategies are available: limiting the inflow of infected cases by using travel restrictions and quarantines or limiting the risk of local transmission from imported cases by using contact tracing and other community interventions. A number of papers have used epidemiological models to argue that inflow restrictions are unlikely to be effective. We simulate a simple epidemiological model to show that this conclusion changes if containment efforts such as contact tracing have limited capacity. In particular, our results show that moderate travel restrictions can lead to large reductions in the probability of an epidemic when contact tracing is effective but the contact tracing system is close to being overwhelmed.

A 40-year-old male presented as a wake-up stroke with right hemiparesis and aphasia consistent with a left MCA territory infarct. Rapid deterioration within 24 hours resulted in intubation thought to be secondary to malignant MCA syndrome. Within 48 hours, he had acquired new, multi-territory infarcts leading to tetraparesis. Initial ‘routine’ young stroke work-up was unyielding. MRI/MRA suggested vasculitis and CSF analysis revealed raised protein, lymphocytosis, and low glucose ratio with negative gram stain, culture and virology. HIV test was negative. Broad-spectrum antibiotics were commenced with anti-TB cover, pending results. Steroids were administered given diagnostic uncertainty whilst empirical treatment with cyclophosphamide was considered. Interval MRI at one-week post treatment showed significant improvement in vasculitis prompting testing for syphilis. Revisiting the history, the patient was in a same sex relationship and had a prodrome of increasing headaches, anorexia, insomnia, personality change and a palmar rash diagnosed as contact dermatitis, eight weeks prior to admission. Neurosyphilis was confirmed: serum RPR 1:32, TPPA>1:1280, positive EIA; CSF RPR 1:8 and TPPA>1:1280. He was treated with an extended course of antibiotics with slow improvement. We review the challenges in reaching a diagnosis and the latest BASHH guidelines – in this increasingly prevalent disease.

The outbreak of an infectious disease in a human population can lead to individuals responding with preventive measures in an attempt to avoid getting infected. This leads to changes in contact patterns. However, as we show in this paper, rational behaviour at the individual level, such as social distancing from infectious contacts, may not always be beneficial for the population as a whole. We use epidemic network models to demonstrate the potential negative consequences at the population level. We take into account the social structure of the population through several network models. As the epidemic evolves, susceptible individuals may distance themselves from their infectious contacts. Some individuals replace their lost social connections by seeking new ties. If social distancing occurs at a high rate at the beginning of an epidemic, then this can prevent an outbreak from occurring. However, we show that moderate social distancing can worsen the disease outcome, both in the initial phase of an outbreak and the final epidemic size. Moreover, the same negative effect can arise in real-world networks. Our results suggest that one needs to be careful when targeting behavioural changes as they could potentially worsenthe epidemic outcome. Furthermore, network structure crucially influences the way that individual-level measures impact the epidemic at the population level. These findings highlight the importance of careful analysis of preventive measures in epidemic models.

When controlling an emerging outbreak of an infectious disease, it is essential to know the key epidemiological parameters, such as the basic reproduction number R0and the control effort required to prevent a large outbreak. These parameters are estimated from the observed incidence of new cases and information about the infectious contact structures of the population in which the disease spreads. However, the relevant infectious contact structures for new, emerging infections are often unknown or hard to obtain. Here, we show that, for many common true underlying heterogeneous contact structures, the simplification to neglect such structures and instead assume that all contacts are made homogeneously in the whole population results in conservative estimates for R0and the required control effort. This means that robust control policies can be planned during the early stages of an outbreak, using such conservative estimates of the required control effort.

In most countries today, obesity is increasing at an almost epidemic rate and creating a considerable public health problem by elevating the risk of cardiovascular disease and metabolic disorders such as type 2 diabetes. It is possible that, as a result, life expectancy will begin to decline in developed countries; this would be the first such occurrence in recent history. Factors that influence fat mass in adulthood remain incompletely understood, but increased lipid storage in predeveloped fat cells, or adipocytes, is accepted as a major determinant of fat mass in adults. The number of fat cells is constant in both lean and obese adults, even after marked weight loss, suggesting that the number of adipocytes is set in childhood and the adolescent years. It has not been established, however, whether or not the number of adipocytes changes in adulthood.

Advises distributors on the sale opportunities of selling engineered lubes laced with polytetrafluoroethylene (PTFE) resins. Proper maintenance's reduction of a customer's operating costs; Performance features and benefits of PTFE resin-based lubrication; Convincing customers of the effectiveness of one's lube solution; Key role of the supplier.