Tightness for the interface of the one-dimensional contact process

We consider a symmetric, finite-range contact process with two types of infection; both have the same (supercritical) infection rate and heal at rate 1, but sites infected by Infection 1 are immune to Infection 2. We take the initial configuration where sites in $(-\infty,0]$ have Infection 1 and sites in $[1,\infty)$ have Infection 2, then consider the process $\rho_t$ defined as the size of the interface area between the two infections at time $t$. We show that the distribution of $\rho_t$ is tight, thus proving a conjecture posed by Cox and Durrett in [Bernoulli 1 (1995) 343--370].Comment: Published in at http://dx.doi.org/10.3150/09-BEJ236 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Limiting shape for first-passage percolation models on random geometric graphs

Let a random geometric graph be defined in the supercritical regime for the existence of a unique infinite connected component in Euclidean space. Consider the first-passage percolation model with independent and identically distributed random variables on the random infinite connected component. We provide sufficient conditions for the existence of the asymptotic shape and we show that the shape is an Euclidean ball. We give some examples exhibiting the result for Bernoulli percolation and the Richardson model. For the Richardson model we further show that it converges weakly to a branching process in the joint limit of large intensities and slow passing times

Limiting shape for first-passage percolation models on random geometric graphs

Let a random geometric graph be defined in the supercritical regime for the existence of a unique infinite connected component in Euclidean space. Consider the first-passage percolation model with independent and identically distributed random variables on the random infinite connected component. We provide sufficient conditions for the existence of the asymptotic shape and we show that the shape is an Euclidean ball. We give some examples exhibiting the result for Bernoulli percolation and the Richardson model. For the Richardson model we further show that it converges weakly to a branching process in the joint limit of large intensities and slow passing times

Limiting shape for first-passage percolation models on random geometric graphs

Let a random geometric graph be defined in the supercritical regime for the existence of a unique infinite connected component in Euclidean space. Consider the first-passage percolation model with independent and identically distributed random variables on the random infinite connected component. We provide sufficient conditions for the existence of the asymptotic shape and we show that the shape is an Euclidean ball. We give some examples exhibiting the result for Bernoulli percolation and the Richardson model. For the Richardson model we further show that it converges weakly to a branching process in the joint limit of large intensities and slow passing times

A Note on Truncated Long-Range Percolation with Heavy Tails on Oriented Graphs

We consider oriented long-range percolation on a graph with vertex set $\mathbb{Z}^d \times \mathbb{Z}_+$ and directed edges of the form $\langle (x,t), (x+y,t+1)\rangle$, for $x,y$ in $\mathbb{Z}^d$ and $t \in \mathbb{Z}_+$. Any edge of this form is open with probability $p_y$, independently for all edges. Under the assumption that the values $p_y$ do not vanish at infinity, we show that there is percolation even if all edges of length more than $k$ are deleted, for $k$ large enough. We also state the analogous result for a long-range contact process on $\mathbb{Z}^d$.Comment: 12 pages, 2 figure

Pain and quality of life in patients undergoing radiotherapy for spinal metastatic disease treatment

Abstract Background Radiotherapy is an important tool in the control of pain in patients with spinal metastatic disease. We aimed to evaluate pain and of quality of life of patients with spinal metastatic disease undergoing radiotherapy with supportive treatment. Methods The study enrolled 30 patients. From January 2008 to January 2010, patients selection included those treated with a 20 Gy tumour dose in five fractions. Patients completed the visual analogue scale for pain assessment and the SF-36 questionnaire for quality of life assessment. Results The most frequent primary sites were breast, multiple myeloma, prostate and lymphoma. It was found that 14 spinal metastatic disease patients (46.66%) had restricted involvement of three or fewer vertebrae, while 16 patients (53.33%) had cases involving more than three vertebrae. The data from the visual analogue scale evaluation of pain showed that the average initial score was 5.7 points, the value 30 days after the end of radiotherapy was 4.60 points and the average value 6 months after treatment was 4.25 points. Notably, this final value was 25.43% lower than the value from the initial analysis. With regard to the quality of life evaluation, only the values for the functional capability and social aspects categories of the questionnaire showed significant improvement. Conclusion Radiotherapy with supportive treatment appears to be an important tool for the treatment of pain in patients with spinal metastatic disease