Discretization of Fractional Differential Equations by a Piecewise Constant Approximation

There has recently been considerable interest in using a nonstandard piecewise approximation to formulate fractional order differential equations as difference equations that describe the same dynamical behaviour and are more amenable to a dynamical systems analysis. Unfortunately, due to mistakes in the fundamental papers, the difference equations formulated through this process do not capture the dynamics of the fractional order equations. We show that the correct application of this nonstandard piecewise approximation leads to a one parameter family of fractional order differential equations that converges to the original equation as the parameter tends to zero. A closed formed solution exists for each member of this family and leads to the formulation of a difference equation that is of increasing order as time steps are taken. Whilst this does not lead to a simplified dynamical analysis it does lead to a numerical method for solving the fractional order differential equation. The method is shown to be equivalent to a quadrature based method, despite the fact that it has not been derived from a quadrature. The method can be implemented with non-uniform time steps. An example is provided showing that the difference equation can correctly capture the dynamics of the underlying fractional differential equation

Distinct forms of the ß subunit of GTP-binding regulatory proteins identified by molecular cloning

Two distinct β subunits of guanine nucleotide-binding regulatory proteins have been identified by cDNA cloning and are referred to as β 1 and β 2 subunits. The bovine transducin β subunit (β 1) has been cloned previously. We have now isolated and analyzed cDNA clones that encode the β 2 subunit from bovine adrenal, bovine brain, and a human myeloid leukemia cell line, HL-60. The 340-residue Mr 37,329 β 2 protein is 90% identical with β 1 in predicted amino acid sequence, and it is also organized as a series of repetitive homologous segments. The major mRNA that encodes the bovine β 2 subunit is 1.7 kilobases in length. It is expreβed at lower levels than β 1 subunit mRNA in all tiβues examined. The β 1 and β 2 meβages are expreβed in cloned human cell lines. Hybridization of cDNA probes to bovine DNA showed that β 1 and β 2 are encoded by separate genes. The amino acid sequences for the bovine and human β 2 subunit are identical, as are the amino acid sequences for the bovine and human β 1 subunit. This evolutionary conservation suggests that the two β subunits have different roles in the signal transduction process

Estimation and Validation of a Multiattribute Model of Alzheimer Disease Progression

OBJECTIVES: To estimate and validate a multiattribute model of the clinical course of Alzheimer disease (AD) from mild AD to death in a high-quality prospective cohort study, and to estimate the impact of hypothetical modifications to AD progression rates on costs associated with Medicare and Medicaid services. DATA AND METHODS: The authors estimated sex-specific longitudinal Grade of Membership (GoM) models for AD patients (103 men, 149 women) in the initial cohort of the Predictors Study (1989-2001) based on 80 individual measures obtained every 6 mo for 10 y. These models were replicated for AD patients (106 men, 148 women) in the 2nd Predictors Study cohort (1997-2007). Model validation required that the disease-specific transition parameters be identical for both Predictors Study cohorts. Medicare costs were estimated from the National Long Term Care Survey. RESULTS: Sex-specific models were validated using the 2nd Predictors Study cohort with the GoM transition parameters constrained to the values estimated for the 1st Predictors Study cohort; 57 to 61 of the 80 individual measures contributed significantly to the GoM models. Simulated, cost-free interventions in the rate of progression of AD indicated that large potential cost offsets could occur for patients at the earliest stages of AD. CONCLUSIONS: AD progression is characterized by a small number of parameters governing changes in large numbers of correlated indicators of AD severity. The analysis confirmed that the progression of AD represents a complex multidimensional physiological process that is similar across different study cohorts. The estimates suggested that there could be large cost offsets to Medicare and Medicaid from the slowing of AD progression among patients with mild AD. The methodology appears generally applicable in AD modeling

Predicting First Traversal Times for Virions and Nanoparticles in Mucus with Slowed Diffusion

AbstractParticle-tracking experiments focusing on virions or nanoparticles in mucus have measured mean-square displacements and reported diffusion coefficients that are orders of magnitude smaller than the diffusion coefficients of such particles in water. Accurate description of this subdiffusion is important to properly estimate the likelihood of virions traversing the mucus boundary layer and infecting cells in the epithelium. However, there are several candidate models for diffusion that can fit experimental measurements of mean-square displacements. We show that these models yield very different estimates for the time taken for subdiffusive virions to traverse through a mucus layer. We explain why fits of subdiffusive mean-square displacements to standard diffusion models may be misleading. Relevant to human immunodeficiency virus infection, using computational methods for fractional subdiffusion, we show that subdiffusion in normal acidic mucus provides a more effective barrier against infection than previously thought. By contrast, the neutralization of the mucus by alkaline semen, after sexual intercourse, allows virions to cross the mucus layer and reach the epithelium in a short timeframe. The computed barrier protection from fractional subdiffusion is some orders of magnitude greater than that derived by fitting standard models of diffusion to subdiffusive data

Triplet energy differences and the low lying structure of Ga 62

Background: Triplet energy differences (TED) can be studied to yield information on isospin-non-conserving interactions in nuclei. Purpose: The systematic behavior of triplet energy differences (TED) of T=1, J\u3c0=2+ states is examined. The A=62 isobar is identified as having a TED value that deviates significantly from an otherwise very consistent trend. This deviation can be attributed to the tentative assignments of the pertinent states in Ga62 and Ge62. Methods: An in-beam \u3b3-ray spectroscopy experiment was performed to identify excited states in Ga62 using Gamma-Ray Energy Tracking In-Beam Nuclear Array with the S800 spectrometer at NSCL using a two-nucleon knockout approach. Cross-section calculations for the knockout process and shell-model calculations have been performed to interpret the population and decay properties observed. Results: Using the systematics as a guide, a candidate for the transition from the T=1, 2+ state is identified. However, previous work has identified similar states with different J\u3c0 assignments. Cross-section calculations indicate that the relevant T=1, 2+ state should be one of the states directly populated in this reaction. Conclusions: As spins and parities were not measurable, it is concluded that an unambiguous identification of the first T=1, 2+ state is required to reconcile our understanding of TED systematics

Genetic risk prediction of atrial fibrillation

Background—Atrial fibrillation (AF) has a substantial genetic basis. Identification of individuals at greatest AF risk could minimize the incidence of cardioembolic stroke. Methods—To determine whether genetic data can stratify risk for development of AF, we examined associations between AF genetic risk scores and incident AF in five prospective studies comprising 18,919 individuals of European ancestry. We examined associations between AF genetic risk scores and ischemic stroke in a separate study of 509 ischemic stroke cases (202 cardioembolic [40%]) and 3,028 referents. Scores were based on 11 to 719 common variants (≥5%) associated with AF at P-values ranging from <1x10-3 to <1x10-8 in a prior independent genetic association study. Results—Incident AF occurred in 1,032 (5.5%) individuals. AF genetic risk scores were associated with new-onset AF after adjusting for clinical risk factors. The pooled hazard ratio for incident AF for the highest versus lowest quartile of genetic risk scores ranged from 1.28 (719 variants; 95%CI, 1.13-1.46; P=1.5x10-4) to 1.67 (25 variants; 95%CI, 1.47-1.90; P=9.3x10-15). Discrimination of combined clinical and genetic risk scores varied across studies and scores (maximum C statistic, 0.629-0.811; maximum ΔC statistic from clinical score alone, 0.009-0.017). AF genetic risk was associated with stroke in age- and sex-adjusted models. For example, individuals in the highest versus lowest quartile of a 127-variant score had a 2.49-fold increased odds of cardioembolic stroke (95%CI, 1.39-4.58; P=2.7x10-3). The effect persisted after excluding individuals (n=70) with known AF (odds ratio, 2.25; 95%CI, 1.20-4.40; P=0.01). Conclusions—Comprehensive AF genetic risk scores were associated with incident AF beyond associations for clinical AF risk factors, though offered small improvements in discrimination. AF genetic risk was also associated with cardioembolic stroke in age- and sex-adjusted analyses. Efforts are warranted to determine whether AF genetic risk may improve identification of subclinical AF or help distinguish between stroke mechanisms

Household air pollution in low- and middle-income countries: health risks and research priorities

Household air pollution (HAP), which results from incomplete combustion of the solid fuels traditionally used for cooking and heating, affects the homes of nearly 3 billion people. It is the leading environmental cause of death and disability worldwide, with highest risks for women and children due to their domestic roles. The high levels of pollutants found in HAP cause a range of diseases, in addition to burns and scalds and injuries or violence experienced during fuel collection. Additionally, household solid fuel use can pose substantive environmental risks, including degradation from fuel gathering as well as climate change from release of both CO2 and short-lived climate forcers, such as black carbon, during combustion. Despite the broad support to find solutions, only a few solid fuel interventions have shown that they might improve health over the long term, especially when implemented at the scale required (Box 1)

Private Sector Union Density and the Wage Premium: Past, Present, and Future

The rise and decline of private sector unionization were among the more important features of the U.S. labor market during the twentieth century. Following a dramatic spurt in unionization after passage of the depression-era National Labor Relations Act (NLRA) of 1935, union density peaked in the mid-1950s, and then began a continuous decline. At the end of the century, the percentage of private wage and salary workers who were union members was less than 10 percent, not greatly different from union density prior to the NLRA