The structure of typical clusters in large sparse random configurations

The initial purpose of this work is to provide a probabilistic explanation of a recent result on a version of Smoluchowski's coagulation equations in which the number of aggregations is limited. The latter models the deterministic evolution of concentrations of particles in a medium where particles coalesce pairwise as time passes and each particle can only perform a given number of aggregations. Under appropriate assumptions, the concentrations of particles converge as time tends to infinity to some measure which bears a striking resemblance with the distribution of the total population of a Galton-Watson process started from two ancestors. Roughly speaking, the configuration model is a stochastic construction which aims at producing a typical graph on a set of vertices with pre-described degrees. Specifically, one attaches to each vertex a certain number of stubs, and then join pairwise the stubs uniformly at random to create edges between vertices. In this work, we use the configuration model as the stochastic counterpart of Smoluchowski's coagulation equations with limited aggregations. We establish a hydrodynamical type limit theorem for the empirical measure of the shapes of clusters in the configuration model when the number of vertices tends to $\infty$. The limit is given in terms of the distribution of a Galton-Watson process started with two ancestors

Diameters in preferential attachment models

In this paper, we investigate the diameter in preferential attachment (PA-) models, thus quantifying the statement that these models are small worlds. The models studied here are such that edges are attached to older vertices proportional to the degree plus a constant, i.e., we consider affine PA-models. There is a substantial amount of literature proving that, quite generally, PA-graphs possess power-law degree sequences with a power-law exponent \tau>2. We prove that the diameter of the PA-model is bounded above by a constant times \log{t}, where t is the size of the graph. When the power-law exponent \tau exceeds 3, then we prove that \log{t} is the right order, by proving a lower bound of this order, both for the diameter as well as for the typical distance. This shows that, for \tau>3, distances are of the order \log{t}. For \tau\in (2,3), we improve the upper bound to a constant times \log\log{t}, and prove a lower bound of the same order for the diameter. Unfortunately, this proof does not extend to typical distances. These results do show that the diameter is of order \log\log{t}. These bounds partially prove predictions by physicists that the typical distance in PA-graphs are similar to the ones in other scale-free random graphs, such as the configuration model and various inhomogeneous random graph models, where typical distances have been shown to be of order \log\log{t} when \tau\in (2,3), and of order \log{t} when \tau>3

Edge-Based Compartmental Modeling for Infectious Disease Spread Part I: An Overview

The primary tool for predicting infectious disease spread and intervention effectiveness is the mass action Susceptible-Infected-Recovered model of Kermack and McKendrick. Its usefulness derives largely from its conceptual and mathematical simplicity; however, it incorrectly assumes all individuals have the same contact rate and contacts are fleeting. This paper is the first of three investigating edge-based compartmental modeling, a technique eliminating these assumptions. In this paper, we derive simple ordinary differential equation models capturing social heterogeneity (heterogeneous contact rates) while explicitly considering the impact of contact duration. We introduce a graphical interpretation allowing for easy derivation and communication of the model. This paper focuses on the technique and how to apply it in different contexts. The companion papers investigate choosing the appropriate level of complexity for a model and how to apply edge-based compartmental modeling to populations with various sub-structures

Association Between Renal Failure and Foot Ulcer or Lower-Extremity Amputation in Patients With Diabetes

OBJECTIVE—The objective of this study was to evaluate the association between foot ulcers (DFU) and lower-extremity amputation (LEA) and chronic kidney disease (CKD) in patients with diabetes

A preferential attachment model with random initial degrees

In this paper, a random graph process ${G(t)}_{t\geq 1}$ is studied and its degree sequence is analyzed. Let $(W_t)_{t\geq 1}$ be an i.i.d. sequence. The graph process is defined so that, at each integer time $t$, a new vertex, with $W_t$ edges attached to it, is added to the graph. The new edges added at time t are then preferentially connected to older vertices, i.e., conditionally on $G(t-1)$, the probability that a given edge is connected to vertex i is proportional to $d_i(t-1)+\delta$, where $d_i(t-1)$ is the degree of vertex $i$ at time $t-1$, independently of the other edges. The main result is that the asymptotical degree sequence for this process is a power law with exponent $\tau=\min\{\tau_{W}, \tau_{P}\}$, where $\tau_{W}$ is the power-law exponent of the initial degrees $(W_t)_{t\geq 1}$ and $\tau_{P}$ the exponent predicted by pure preferential attachment. This result extends previous work by Cooper and Frieze, which is surveyed.Comment: In the published form of the paper, the proof of Proposition 2.1 is incomplete. This version contains the complete proo

Peripheral tumour targeting using open-source virtual bronchoscopy with electromagnetic tracking: a multi-user pre-clinical study

Objectives: The goal was to demonstrate the utility of open-source tracking and visualisation tools in the targeting of lung cancer. Material and methods: The study demonstrates the first deployment of the Anser electromagnetic (EM) tracking system with the CustusX image-guided interventional research platform to navigate using an endobronchial catheter to injected tumour targets. Live animal investigations validated the deployment and targeting of peripheral tumour models using an innovative tumour marking routine. Results: Novel tumour model deployment was successfully achieved at all eight target sites across two live animal investigations without pneumothorax. Virtual bronchoscopy with tracking successfully guided the tracked catheter to 2–12 mm from the target tumour site. Deployment of a novel marker was achieved at all eight sites providing a reliable measure of targeting accuracy. Targeting accuracy within 10 mm was achieved in 7/8 sites and in all cases, the virtual target distance at marker deployment was within the range subsequently measured with x-ray. Conclusions: Endobronchial targeting of peripheral airway targets is feasible using existing open-source technology. Notwithstanding the shortcomings of current commercial platforms, technological improvements in EM tracking and registration accuracy fostered by open-source technology may provide the impetus for widespread clinical uptake of electromagnetic navigation in bronchoscopy

Intestinal carriage of Staphylococcus aureus: How does its frequency compare with that of nasal carriage and what is its clinical impact?

The bacterial species Staphylococcus aureus, including its methicillin-resistant variant (MRSA), finds its primary ecological niche in the human nose, but is also able to colonize the intestines and the perineal region. Intestinal carriage has not been widely investigated despite its potential clinical impact. This review summarizes literature on the topic and sketches the current state of affairs from a microbiological and infectious diseases' perspective. Major findings are that the average reported detection rate of intestinal carriage in healthy individuals and patients is 20% for S. aureus and 9% for MRSA, which is approximately half of that for nasal carriage. Nasal carriage seems to predispose to intestinal carriage, but sole intestinal carriage occurs relatively frequently and is observed in 1 out of 3 intestinal carriers, which provides a rationale to include intestinal screening for surveillance or in outbreak settings. Colonization of the intestinal tract with S. aureus at a young age occurs at a high frequency and may affect the host's immune system. The frequency of intestinal carriage is generally underestimated and may significantly contribute to bacterial dissemination and subsequent risk of infections. Whether intestinal rather than nasal S. aureus carriage is a primary predictor for infections is still ill-defined