Numerical analysis for the time distributed-order and Riesz space fractional diffusions on bounded domains

Subdiffusion equations with distributed-order fractional derivatives describe some important physical phenomena. In this paper, we consider the time distributed-order and Riesz space fractional diffusions on bounded domains with Dirichlet boundary conditions. Here, the time derivative is defined as the distributed-order fractional derivative in the Caputo sense, and the space derivative is defined as the Riesz fractional derivative. First, we discretize the integral term in the time distributed-order and Riesz space fractional diffusions using numerical approximation. Then the given equation can be written as a multi-term time–space fractional diffusion. Secondly, we propose an implicit difference method for the multi-term time–space fractional diffusion. Thirdly, using mathematical induction, we prove the implicit difference method is unconditionally stable and convergent. Also, the solvability for our method is discussed. Finally, two numerical examples are given to show that the numerical results are in good agreement with our theoretical analysis.

Stability and convergence of a new finite volume method for a two-sided space-fractional diffusion equation

In this paper, we consider a two-sided space-fractional diffusion equation with variable coefficients on a finite domain. Firstly, based on the nodal basis functions, we present a new fractional finite volume method for the two-sided space-fractional diffusion equation and derive the implicit scheme and solve it in matrix form. Secondly, we prove the stability and convergence of the implicit fractional finite volume method and conclude that the method is unconditionally stable and convergent. Finally, some numerical examples are given to show the effectiveness of the new numerical method, and the results are in excellent agreement with theoretical analysis.

A Novel High Order Space-Time Spectral Method for the Time Fractional Fokker--Planck Equation

The fractional Fokker-Planck equation is an important physical model for simulating anomalous diffusions with external forces. Because of the non-local property of the fractional derivative an interesting problem is to explore high accuracy numerical methods for fractional differential equations. In this paper, a space-time spectral method is presented for the numerical solution of the time fractional Fokker-Planck initial-boundary value problem. The proposed method employs the Jacobi polynomials for the temporal discretization and Fourier-like basis functions for the spatial discretization. Due to the diagonalizable trait of the Fourier-like basis functions, this leads to a reduced representation of the inner product in the Galerkin analysis. We prove that the time fractional Fokker-Planck equation attains the same approximation order as the time fractional diffusion equation developed in [23] by using the present method. That indicates an exponential decay may be achieved if the exact solution is sufficiently smooth. Finally, some numerical results are given to demonstrate the high order accuracy and efficiency of the new numerical scheme. The results show that the errors of the numerical solutions obtained by the space-time spectral method decay exponentially.

Weather variability and influenza A (H7N9) transmission in Shanghai, China: A Bayesian spatial analysis

Abstract Background A novel avian influenza A (H7N9) virus was first found in humans in Shanghai, and infected over 433 patients in China. To date, very little is known about the spatiotemporal variability or environmental drivers of the risk of H7N9 infection. This study explored the spatial and temporal variation of H7N9 infection and assessed the effects of temperature and rainfall on H7N9 incidence. Methods A Bayesian spatial conditional autoregressive (CAR) model was used to assess the spatiotemporal distribution of the risk of H7N9 infection in Shanghai, by district and fortnight for the period 19th February–14th April 2013. Data on daily laboratory-confirmed H7N9 cases, and weather variability including temperature (°C) and rainfall (mm) were obtained from the Chinese Information System for Diseases Control and Prevention and Chinese Meteorological Data Sharing Service System, respectively, and aggregated by fortnight. Results High spatial variations in the H7N9 risk were mainly observed in the east and centre of Shanghai municipality. H7N9 incidence rate was significantly associated with fortnightly mean temperature (Relative Risk (RR): 1.54; 95% credible interval (CI): 1.22–1.94) and fortnightly mean rainfall (RR: 2.86; 95% CI: 1.47–5.56). Conclusion There was a substantial variation in the spatiotemporal distribution of H7N9 infection across different districts in Shanghai. Optimal temperature and rainfall may be one of the driving forces for H7N9.

A framework for understanding and generating integrated solutions for residential peak energy demand

Supplying peak energy demand in a cost effective, reliable manner is a critical focus for utilities internationally. Successfully addressing peak energy concerns requires understanding of all the factors that affect electricity demand especially at peak times. This paper is based on past attempts of proposing models designed to aid our understanding of the influences on residential peak energy demand in a systematic and comprehensive way. Our model has been developed through a group model building process as a systems framework of the problem situation to model the complexity within and between systems and indicate how changes in one element might flow on to others. It is comprised of themes (social, technical and change management options) networked together in a way that captures their influence and association with each other and also their influence, association and impact on appliance usage and residential peak energy demand. The real value of the model is in creating awareness, understanding and insight into the complexity of residential peak energy demand and in working with this complexity to identify and integrate the social, technical and change management option themes and their impact on appliance usage and residential energy demand at peak times.

Inferring lung cancer risk factor patterns through joint Bayesian spatio-temporal analysis

Background: Preventing risk factor exposure is vital to reduce the high burden from lung cancer. The leading risk factor for developing lung cancer is tobacco smoking. In Australia, despite apparent success in reducing smoking prevalence, there is limited information on small area patterns and small area temporal trends. We sought to estimate spatio-temporal patterns for lung cancer risk factors using routinely collected population-based cancer data. Methods: The analysis used a Bayesian shared component spatio-temporal model, with male and female lung cancer included separately. The shared component reflected exposure to lung cancer risk factors, and was modelled over 477 statistical local areas (SLAs) and 15 years in Queensland, Australia. Analyses were also run adjusting for area-level socioeconomic disadvantage, Indigenous population composition, or remoteness. Results: Strong spatial patterns were observed in the underlying risk factor exposure for both males (median Relative Risk (RR) across SLAs compared to the Queensland average ranged from 0.48-2.00) and females (median RR range across SLAs 0.53-1.80), with high exposure observed in many remote areas. Strong temporal trends were also observed. Males showed a decrease in the underlying risk across time, while females showed an increase followed by a decrease in the final two years. These patterns were largely consistent across each SLA. The high underlying risk estimates observed among disadvantaged, remote and indigenous areas decreased after adjustment, particularly among females. Conclusion: The modelled underlying exposure appeared to reflect previous smoking prevalence, with a lag period of around 30 years, consistent with the time taken to develop lung cancer. The consistent temporal trends in lung cancer risk factors across small areas support the hypothesis that past interventions have been equally effective across the state. However, this also means that spatial inequalities have remained unaddressed, highlighting the potential for future interventions, particularly among remote areas.

The Lymphedema Evaluation in Gynecological cancer Study (LEGS): Design of a prospective, longitudinal, cohort study

Background The Lymphoedema Evaluation in Gynecological cancer Study (LEGS) was a longitudinal, observational, cohort study prospectively evaluating the incidence and risk factors of lower-limb lymphedema after treatment for gynecological cancer. Here we describe the study protocol and characteristics of the sample. Methods Women with a newly diagnosed gynecological cancer between June 1, 2008 and February 28, 2011, aged 18 years or older, and treated at one of six hospitals in Queensland, Australia, were eligible. Lymphedema was assessed by circumference measurements, bioimpedance spectroscopy, and self-reported swelling. LEGS incorporated a cohort of patients requiring surgery for benign gynecological conditions for comparison purposes. Data were collected prior to surgery and at regular intervals thereafter up to 2-years post-diagnosis. Results 546 women participated (408 cancer, 138 benign), with a 24-month retention rate of 78%. Clinical and treatment characteristics of participants were similar to the Queensland gynecological cancer population, except for a higher proportion of early-stage cervical cancers recruited to LEGS compared with Queensland proportions (89% versus 55%, respectively). Discussion Few imbalances were observed between participants with complete and incomplete follow-up data. The prospective design and collection of objective and patient-reported outcome data will allow comprehensive assessment of incidence and risk factors of lower-limb lymphedema.

Specificity and context in post-exercise recovery: It is not a one-size-fits-all approach

The concept of specificity of exercise prescription and training is a longstanding and widely accepted foundation of the exercise sciences. Simply, the principle holds that training adaptations are achieved relative to the stimulus applied. That is, the manipulation of training variables (e.g. intensity or loading, mode, volume and frequency) directly influences the acute training stimulus, and so the long-term adaptive response (Young et al., 2001; Bird et al., 2005). Translating this concept to practice then recommends that exercise be prescribed specific to the desired outcomes, and the more closely this is achieved, the greater the performance gain is likely to be. However, the cardiovascular and metabolic adaptations traditionally associated with long, slow distance training types, similarly achieved using high-intensity training methods (for a review see Gibala et al., 2012), highlights understanding of underlying physiology as paramount for effective training program design. Various other factors including illness, sleep and psychology also impact on the training stimulus (Halson, 2014) and must be managed collectively with appropriate post-exercise recovery to continue performance improvements and reduce overtraining and injury risks (Kenttä and Hassmén, 1998).

Normative requirements for regulatory compliance: An abstract formal framework

By definition, regulatory rules (in legal context called norms) intend to achieve specific behaviour from business processes, and might be relevant to the whole or part of a business process. They can impose conditions on different aspects of process models, e.g., control-flow, data and resources etc. Based on the rules sets, norms can be classified into various classes and sub-classes according to their effects. This paper presents an abstract framework consisting of a list of norms and a generic compliance checking approach on the idea of (possible) execution of processes. The proposed framework is independent of any existing formalism, and provides a conceptually rich and exhaustive ontology and semantics of norms needed for business process compliance checking. The possible uses of the proposed framework include to compare different compliance management frameworks (CMFs).

Melanoma cell colony expansion parameters revealed by approximate Bayesian computation

In vitro studies and mathematical models are now being widely used to study the underlying mechanisms driving the expansion of cell colonies. This can improve our understanding of cancer formation and progression. Although much progress has been made in terms of developing and analysing mathematical models, far less progress has been made in terms of understanding how to estimate model parameters using experimental in vitro image-based data. To address this issue, a new approximate Bayesian computation (ABC) algorithm is proposed to estimate key parameters governing the expansion of melanoma cell (MM127) colonies, including cell diffusivity, D, cell proliferation rate, λ, and cell-to-cell adhesion, q, in two experimental scenarios, namely with and without a chemical treatment to suppress cell proliferation. Even when little prior biological knowledge about the parameters is assumed, all parameters are precisely inferred with a small posterior coefficient of variation, approximately 2–12%. The ABC analyses reveal that the posterior distributions of D and q depend on the experimental elapsed time, whereas the posterior distribution of λ does not. The posterior mean values of D and q are in the ranges 226–268 µm2h−1, 311–351 µm2h−1 and 0.23–0.39, 0.32–0.61 for the experimental periods of 0–24 h and 24–48 h, respectively. Furthermore, we found that the posterior distribution of q also depends on the initial cell density, whereas the posterior distributions of D and λ do not. The ABC approach also enables information from the two experiments to be combined, resulting in greater precision for all estimates of D and λ.

Prior and posterior linear pooling for combining expert opinions: Uses and impact on Bayesian Networks

The use of expert knowledge to quantify a Bayesian Network (BN) is necessary when data is not available. This however raises questions regarding how opinions from multiple experts can be used in a BN. Linear pooling is a popular method for combining probability assessments from multiple experts. In particular, Prior Linear Pooling (PrLP), which pools opinions then places them into the BN is a common method. This paper firstly proposes an alternative pooling method, Posterior Linear Pooling (PoLP). This method constructs a BN for each expert, then pools the resulting probabilities at the nodes of interest. Secondly, it investigates the advantages and disadvantages of using these pooling methods to combine the opinions of multiple experts. Finally, the methods are applied to an existing BN, the Wayfinding Bayesian Network Model, to investigate the behaviour of different groups of people and how these different methods may be able to capture such differences. The paper focusses on 6 nodes Human Factors, Environmental Factors, Wayfinding, Communication, Visual Elements of Communication and Navigation Pathway, and three subgroups Gender (female, male),Travel Experience (experienced, inexperienced), and Travel Purpose (business, personal) and finds that different behaviors can indeed be captured by the different methods.

The effect of BCMO1 gene variants on macular pigment optical density in young healthy Caucasians

Background Serum lutein (L) and zeaxanthin (Z) positively correlate with macular pigment optical density (MPOD), hence the latter is a valuable indirect tool for measuring L and Z content in the macula. L and Z have been attributed antioxidant capacity and protection from certain retinal diseases but their uptake within the eye is thought to depend on genetic, age and environmental factors. In particular gene variants within beta-carotene monooxygenase (BCMO1) are thought to modulate MPOD in the macula. Objectives: To determine the effect of BCMO1 single nucleotide polymorphisms (SNPs) rs11645428, rs6420424 and rs6464851 on macular pigment optical density (MPOD) in a cohort of young healthy participants of Caucasian origin with normal ocular health. Design In this cohort study, MPOD was assessed in 46 healthy participants (22 male and 24 female) with a mean age of 24 ± 4.0 years (range 19-33). The three SNPs, rs11645428, rs6420424, rs6564851 that have established associations with MPOD were determined using MassEXTEND (hME) Sequenom assay. One-way analysis of variance (ANOVA) was performed on groups segregated into homozygous and heterozygous BCMO1 genotypes. Correlations between body mass index (BMI), iris colour, gender, central retinal thickness (CRT), diet and MPOD were investigated. Results MPOD did not significantly vary with BCMO1 rs11645428 (F2,41 = 0.700, p = 0.503), rs6420424 (F2,41 = 0.210, p = 0.801) nor rs6464851 homozygous or heterozygous genotypes (F2,41 = 0,13, p = 0.88), in this young healthy cohort. The combination of these three SNPs into triple genotypes based on plasma conversion efficiency did not affect MPOD (F2,41 = 0.07, p = 0.9). There was a significant negative correlation with MPOD and central retinal thickness (r = - 0.39, p = 0.01) but no significant correlation between BMI, iris colour, gender and MPOD. Conclusion Our results indicate that macular pigment deposition within the central retina is not dependent on BCMO1 gene variants in young healthy people. We propose that MPOD is saturated in younger persons and/or other gene variant combinations determine its deposition.

Simulating droplet motion on virtual leaf surfaces

A curvilinear thin film model is used to simulate the motion of droplets on a virtual leaf surface, with a view to better understand the retention of agricultural sprays on plants. The governing model, adapted from Roy et al. (2002 J. Fluid Mech. 454, 235–261) with the addition of a disjoining pressure term, describes the gravity- and curvature driven flow of a small droplet on a complex substrate: a cotton leaf reconstructed from digitized scan data. Coalescence is the key mechanism behind spray coating of foliage, and our simulations demonstrate that various experimentally observed coalescence behaviours can be reproduced qualitatively. By varying the contact angle over the domain, we also demonstrate that the presence of a chemical defect can act as an obstacle to the droplet’s path, causing break-up. In simulations on the virtual leaf, it is found that the movement of a typical spray size droplet is driven almost exclusively by substrate curvature gradients. It is not until droplet mass is sufficiently increased via coalescence that gravity becomes the dominating force.

Toward making inroads in reducing the disparity of lung health in Australian Indigenous and New Zealand Maori children

Globally, Indigenous populations, which include Aboriginal and Torres Strait islanders in Australia and Māori people in New Zealand (NZ), have poorer health than their non-Indigenous counterparts (1). Indigenous peoples worldwide face substantial challenges in poverty, education, employment, housing, and disconnection from ancestral lands (1). While addressing social determinants of health is a priority, solving clinical issues is equally important. Indeed, ignoring the latter until social issues improve risks further disparity as this may take generations. A systematic overview of interventions addressing social determinants of health found a striking lack of reliable evaluations (2). Where evidence was available, health improvement associated with interventions was modest or uncertain (2). Thus, advances in healthcare remain essential and these require the best evidence available in preventing and managing common illnesses, including respiratory illnesses

GPU accelerated algorithms for computing matrix function vector products with applications to exponential integrators and fractional diffusion

The efficient computation of matrix function vector products has become an important area of research in recent times, driven in particular by two important applications: the numerical solution of fractional partial differential equations and the integration of large systems of ordinary differential equations. In this work we consider a problem that combines these two applications, in the form of a numerical solution algorithm for fractional reaction diffusion equations that after spatial discretisation, is advanced in time using the exponential Euler method. We focus on the efficient implementation of the algorithm on Graphics Processing Units (GPU), as we wish to make use of the increased computational power available with this hardware. We compute the matrix function vector products using the contour integration method in [N. Hale, N. Higham, and L. Trefethen. Computing Aα, log(A), and related matrix functions by contour integrals. SIAM J. Numer. Anal., 46(5):2505–2523, 2008]. Multiple levels of preconditioning are applied to reduce the GPU memory footprint and to further accelerate convergence. We also derive an error bound for the convergence of the contour integral method that allows us to pre-determine the appropriate number of quadrature points. Results are presented that demonstrate the effectiveness of the method for large two-dimensional problems, showing a speedup of more than an order of magnitude compared to a CPU-only implementation.