Susceptible-infected-recovered and susceptible-exposed-infected models

Two stochastic epidemic lattice models, the susceptible-infected-recovered and the susceptible-exposed-infected models, are studied on a Cayley tree of coordination number k. The spreading of the disease in the former is found to occur when the infection probability b is larger than b(c) = k/2(k - 1). In the latter, which is equivalent to a dynamic site percolation model, the spreading occurs when the infection probability p is greater than p(c) = 1/(k - 1). We set up and solve the time evolution equations for both models and determine the final and time-dependent properties, including the epidemic curve. We show that the two models are closely related by revealing that their relevant properties are exactly mapped into each other when p = b/[k - (k - 1) b]. These include the cluster size distribution and the density of individuals of each type, quantities that have been determined in closed forms.

The impact of smoke from forest fires on the spectral dispersion of cloud droplet size distributions in the Amazonian region

In this paper, the main microphysical characteristics of clouds developing in polluted and clean conditions in the biomass-burning season of the Amazon region are examined, with special attention to the spectral dispersion of the cloud droplet size distribution and its potential impact on climate modeling applications. The dispersion effect has been shown to alter the climate cooling predicted by the so-called Twomey effect. In biomass-burning polluted conditions, high concentrations of low dispersed cloud droplets are found. Clean conditions revealed an opposite situation. The liquid water content (0.43 +/- 0.19 g m(-3)) is shown to be uncorrelated with the cloud drop number concentration, while the effective radius is found to be very much correlated with the relative dispersion of the size distribution (R(2) = 0.81). The results suggest that an increase in cloud condensation nuclei concentration from biomass-burning aerosols may lead to an additional effect caused by a decrease in relative dispersion. Since the dry season in the Amazonian region is vapor limiting, the dispersion effect of cloud droplet size distributions could be substantially larger than in other polluted regions.

A new scale-invariant ratio and finite-size scaling for the stochastic susceptible-infected-recovered model

The critical behavior of the stochastic susceptible-infected-recovered model on a square lattice is obtained by numerical simulations and finite-size scaling. The order parameter as well as the distribution in the number of recovered individuals is determined as a function of the infection rate for several values of the system size. The analysis around criticality is obtained by exploring the close relationship between the present model and standard percolation theory. The quantity UP, equal to the ratio U between the second moment and the squared first moment of the size distribution multiplied by the order parameter P, is shown to have, for a square system, a universal value 1.0167(1) that is the same for site and bond percolation, confirming further that the SIR model is also in the percolation class.

Bootstrap percolation on homogeneous trees has 2 phase transitions

We study the threshold theta bootstrap percolation model on the homogeneous tree with degree b + 1, 2 <= theta <= b, and initial density p. It is known that there exists a nontrivial critical value for p, which we call p(f), such that a) for p > p(f), the final bootstrapped configuration is fully occupied for almost every initial configuration, and b) if p < p(f) , then for almost every initial configuration, the final bootstrapped configuration has density of occupied vertices less than 1. In this paper, we establish the existence of a distinct critical value for p, p(c), such that 0 < p(c) < p(f), with the following properties: 1) if p <= p(c), then for almost every initial configuration there is no infinite cluster of occupied vertices in the final bootstrapped configuration; 2) if p > p(c), then for almost every initial configuration there are infinite clusters of occupied vertices in the final bootstrapped configuration. Moreover, we show that 3) for p < p(c), the distribution of the occupied cluster size in the final bootstrapped configuration has an exponential tail; 4) at p = p(c), the expected occupied cluster size in the final bootstrapped configuration is infinite; 5) the probability of percolation of occupied vertices in the final bootstrapped configuration is continuous on [0, p(f)] and analytic on (p(c), p(f) ), admitting an analytic continuation from the right at p (c) and, only in the case theta = b, also from the left at p(f).

Ligand-Assisted Preparation of Palladium Supported Nanoparticles: a Step toward Size Control

Supported nanoparticles (SNPs) with narrow size distribution were prepared by H(2) reduction of Pd(2+) previously bound, to ligand-modified silica surfaces. Interestingly, the size of the Pd SNPs was tuned by the ligand grafted on the support surface. Amino- and ethylenediamino-functionalized supports formed Pd(0) SNPs of ca. 6 and 1 nm, respectively. The catalytic properties of both Pd(0) SNPs were investigated.

Effect of sonication on the particle size of montmorillonite clays

This paper reports on the effect of sonication on SAz-1 and SWy-1 montmorillonite suspensions. Changes in the size of the particles of these materials and modifications of their properties have been investigated. The variation of the particle size has been analyzed by DLS (dynamic light scattering). In all cases the clay particles show a bimodal distribution. Sonication resulted in a decrease of the larger modal diameter, as well as a reduction of its volume percentage. Simultaneously, the proportion of the smallest particles increases. After 60 min of sonication, SAz-1 presented a very broad particle size distribution with a modal diameter of 283 nm. On the other hand, the SWy-1 sonicated for 60 min presents a bimodal distribution of particles at 140 and 454 nm. Changes in the properties of the clay suspensions due to sonication were evaluated spectroscopically from dye-clay interactions, using Methylene Blue. The acidic sites present in the interlamellar region, which are responsible for dye protonation, disappeared after sonication of the clay. The changes in the size of the scattering particles and the lack of acidic sites after sonication suggest that sonication induces delamination of the clay particles. (c) 2008 Elsevier Inc. All rights reserved.