Novel analytical and numerical methods for solving fractional dynamical systems

During the past three decades, the subject of fractional calculus (that is, calculus of integrals and derivatives of arbitrary order) has gained considerable popularity and importance, mainly due to its demonstrated applications in numerous diverse and widespread fields in science and engineering. For example, fractional calculus has been successfully applied to problems in system biology, physics, chemistry and biochemistry, hydrology, medicine, and finance. In many cases these new fractional-order models are more adequate than the previously used integer-order models, because fractional derivatives and integrals enable the description of the memory and hereditary properties inherent in various materials and processes that are governed by anomalous diffusion. Hence, there is a growing need to find the solution behaviour of these fractional differential equations. However, the analytic solutions of most fractional differential equations generally cannot be obtained. As a consequence, approximate and numerical techniques are playing an important role in identifying the solution behaviour of such fractional equations and exploring their applications. The main objective of this thesis is to develop new effective numerical methods and supporting analysis, based on the finite difference and finite element methods, for solving time, space and time-space fractional dynamical systems involving fractional derivatives in one and two spatial dimensions. A series of five published papers and one manuscript in preparation will be presented on the solution of the space fractional diffusion equation, space fractional advectiondispersion equation, time and space fractional diffusion equation, time and space fractional Fokker-Planck equation with a linear or non-linear source term, and fractional cable equation involving two time fractional derivatives, respectively. One important contribution of this thesis is the demonstration of how to choose different approximation techniques for different fractional derivatives. Special attention has been paid to the Riesz space fractional derivative, due to its important application in the field of groundwater flow, system biology and finance. We present three numerical methods to approximate the Riesz space fractional derivative, namely the L1/ L2-approximation method, the standard/shifted Gr¨unwald method, and the matrix transform method (MTM). The first two methods are based on the finite difference method, while the MTM allows discretisation in space using either the finite difference or finite element methods. Furthermore, we prove the equivalence of the Riesz fractional derivative and the fractional Laplacian operator under homogeneous Dirichlet boundary conditions – a result that had not previously been established. This result justifies the aforementioned use of the MTM to approximate the Riesz fractional derivative. After spatial discretisation, the time-space fractional partial differential equation is transformed into a system of fractional-in-time differential equations. We then investigate numerical methods to handle time fractional derivatives, be they Caputo type or Riemann-Liouville type. This leads to new methods utilising either finite difference strategies or the Laplace transform method for advancing the solution in time. The stability and convergence of our proposed numerical methods are also investigated. Numerical experiments are carried out in support of our theoretical analysis. We also emphasise that the numerical methods we develop are applicable for many other types of fractional partial differential equations.

Important omitted spatial variables in safety models: Understanding contributing crash causes at intersections

Advances in safety research—trying to improve the collective understanding of motor vehicle crash causation—rests upon the pursuit of numerous lines of inquiry. The research community has focused on analytical methods development (negative binomial specifications, simultaneous equations, etc.), on better experimental designs (before-after studies, comparison sites, etc.), on improving exposure measures, and on model specification improvements (additive terms, non-linear relations, etc.). One might think of different lines of inquiry in terms of ‘low lying fruit’—areas of inquiry that might provide significant improvements in understanding crash causation. It is the contention of this research that omitted variable bias caused by the exclusion of important variables is an important line of inquiry in safety research. In particular, spatially related variables are often difficult to collect and omitted from crash models—but offer significant ability to better understand contributing factors to crashes. This study—believed to represent a unique contribution to the safety literature—develops and examines the role of a sizeable set of spatial variables in intersection crash occurrence. In addition to commonly considered traffic and geometric variables, examined spatial factors include local influences of weather, sun glare, proximity to drinking establishments, and proximity to schools. The results indicate that inclusion of these factors results in significant improvement in model explanatory power, and the results also generally agree with expectation. The research illuminates the importance of spatial variables in safety research and also the negative consequences of their omissions.

The use of ZIP and CART to model cryptosporidiosis in relation to climatic variables

This research assesses the potential impact of weekly weather variability on the incidence of cryptosporidiosis disease using time series zero-inflated Poisson (ZIP) and classification and regression tree (CART) models. Data on weather variables, notified cryptosporidiosis cases and population size in Brisbane were supplied by the Australian Bureau of Meteorology, Queensland Department of Health, and Australian Bureau of Statistics, respectively. Both time series ZIP and CART models show a clear association between weather variables (maximum temperature, relative humidity, rainfall and wind speed) and cryptosporidiosis disease. The time series CART models indicated that, when weekly maximum temperature exceeded 31°C and relative humidity was less than 63%, the relative risk of cryptosporidiosis rose by 13.64 (expected morbidity: 39.4; 95% confidence interval: 30.9–47.9). These findings may have applications as a decision support tool in planning disease control and risk management programs for cryptosporidiosis disease.

Evaluation of financial incentives as a quality improvement strategy in the public hospital context : clinician attitudes, design variables, and economic costs

In 2008, a three-year pilot ‘pay for performance’ (P4P) program, known as ‘Clinical Practice Improvement Payment’ (CPIP) was introduced into Queensland Health (QHealth). QHealth is a large public health sector provider of acute, community, and public health services in Queensland, Australia. The organisation has recently embarked on a significant reform agenda including a review of existing funding arrangements (Duckett et al., 2008). Partly in response to this reform agenda, a casemix funding model has been implemented to reconnect health care funding with outcomes. CPIP was conceptualised as a performance-based scheme that rewarded quality with financial incentives. This is the first time such a scheme has been implemented into the public health sector in Australia with a focus on rewarding quality, and it is unique in that it has a large state-wide focus and includes 15 Districts. CPIP initially targeted five acute and community clinical areas including Mental Health, Discharge Medication, Emergency Department, Chronic Obstructive Pulmonary Disease, and Stroke. The CPIP scheme was designed around key concepts including the identification of clinical indicators that met the set criteria of: high disease burden, a well defined single diagnostic group or intervention, significant variations in clinical outcomes and/or practices, a good evidence, and clinician control and support (Ward, Daniels, Walker & Duckett, 2007). This evaluative research targeted Phase One of implementation of the CPIP scheme from January 2008 to March 2009. A formative evaluation utilising a mixed methodology and complementarity analysis was undertaken. The research involved three research questions and aimed to determine the knowledge, understanding, and attitudes of clinicians; identify improvements to the design, administration, and monitoring of CPIP; and determine the financial and economic costs of the scheme. Three key studies were undertaken to ascertain responses to the key research questions. Firstly, a survey of clinicians was undertaken to examine levels of knowledge and understanding and their attitudes to the scheme. Secondly, the study sought to apply Statistical Process Control (SPC) to the process indicators to assess if this enhanced the scheme and a third study examined a simple economic cost analysis. The CPIP Survey of clinicians elicited 192 clinician respondents. Over 70% of these respondents were supportive of the continuation of the CPIP scheme. This finding was also supported by the results of a quantitative altitude survey that identified positive attitudes in 6 of the 7 domains-including impact, awareness and understanding and clinical relevance, all being scored positive across the combined respondent group. SPC as a trending tool may play an important role in the early identification of indicator weakness for the CPIP scheme. This evaluative research study supports a previously identified need in the literature for a phased introduction of Pay for Performance (P4P) type programs. It further highlights the value of undertaking a formal risk assessment of clinician, management, and systemic levels of literacy and competency with measurement and monitoring of quality prior to a phased implementation. This phasing can then be guided by a P4P Design Variable Matrix which provides a selection of program design options such as indicator target and payment mechanisms. It became evident that a clear process is required to standardise how clinical indicators evolve over time and direct movement towards more rigorous ‘pay for performance’ targets and the development of an optimal funding model. Use of this matrix will enable the scheme to mature and build the literacy and competency of clinicians and the organisation as implementation progresses. Furthermore, the research identified that CPIP created a spotlight on clinical indicators and incentive payments of over five million from a potential ten million was secured across the five clinical areas in the first 15 months of the scheme. This indicates that quality was rewarded in the new QHealth funding model, and despite issues being identified with the payment mechanism, funding was distributed. The economic model used identified a relative low cost of reporting (under $8,000) as opposed to funds secured of over $300,000 for mental health as an example. Movement to a full cost effectiveness study of CPIP is supported. Overall the introduction of the CPIP scheme into QHealth has been a positive and effective strategy for engaging clinicians in quality and has been the catalyst for the identification and monitoring of valuable clinical process indicators. This research has highlighted that clinicians are supportive of the scheme in general; however, there are some significant risks that include the functioning of the CPIP payment mechanism. Given clinician support for the use of a pay–for-performance methodology in QHealth, the CPIP scheme has the potential to be a powerful addition to a multi-faceted suite of quality improvement initiatives within QHealth.

Whole-proteome phylogeny of large dsDNA viruses and parvoviruses through a composition vector method related to dynamical language model

Background The vast sequence divergence among different virus groups has presented a great challenge to alignment-based analysis of virus phylogeny. Due to the problems caused by the uncertainty in alignment, existing tools for phylogenetic analysis based on multiple alignment could not be directly applied to the whole-genome comparison and phylogenomic studies of viruses. There has been a growing interest in alignment-free methods for phylogenetic analysis using complete genome data. Among the alignment-free methods, a dynamical language (DL) method proposed by our group has successfully been applied to the phylogenetic analysis of bacteria and chloroplast genomes. Results In this paper, the DL method is used to analyze the whole-proteome phylogeny of 124 large dsDNA viruses and 30 parvoviruses, two data sets with large difference in genome size. The trees from our analyses are in good agreement to the latest classification of large dsDNA viruses and parvoviruses by the International Committee on Taxonomy of Viruses (ICTV). Conclusions The present method provides a new way for recovering the phylogeny of large dsDNA viruses and parvoviruses, and also some insights on the affiliation of a number of unclassified viruses. In comparison, some alignment-free methods such as the CV Tree method can be used for recovering the phylogeny of large dsDNA viruses, but they are not suitable for resolving the phylogeny of parvoviruses with a much smaller genome size.

Variables governing the dynamics of attacker-defender systems in team sport

This project was a step forward in the examination and identification of key variables on the perception, decision making and action of team sport athletes through theoretical insights provided by the ecological dynamics perspective. The methodology drew on experiential knowledge of elite coaches to drive further empirical investigation into the specific task, environmental and personal constraints that shape the behaviour of athletes in specific performance contexts. The thesis has provided an effective rationale for further investigation into the emergent perception, decision making and action demanded of athletes in these unpredictable, fluent, fast-paced environments.

Some novel techniques of parameter estimation for the dynamical models in biological systems

Inverse problems based on using experimental data to estimate unknown parameters of a system often arise in biological and chaotic systems. In this paper, we consider parameter estimation in systems biology involving linear and non-linear complex dynamical models, including the Michaelis–Menten enzyme kinetic system, a dynamical model of competence induction in Bacillus subtilis bacteria and a model of feedback bypass in B. subtilis bacteria. We propose some novel techniques for inverse problems. Firstly, we establish an approximation of a non-linear differential algebraic equation that corresponds to the given biological systems. Secondly, we use the Picard contraction mapping, collage methods and numerical integration techniques to convert the parameter estimation into a minimization problem of the parameters. We propose two optimization techniques: a grid approximation method and a modified hybrid Nelder–Mead simplex search and particle swarm optimization (MH-NMSS-PSO) for non-linear parameter estimation. The two techniques are used for parameter estimation in a model of competence induction in B. subtilis bacteria with noisy data. The MH-NMSS-PSO scheme is applied to a dynamical model of competence induction in B. subtilis bacteria based on experimental data and the model for feedback bypass. Numerical results demonstrate the effectiveness of our approach.

Profiling and characterisation of key weather variables and their implication on building design

Local climate is a critical element in the design of buildings. In this paper, ten years of historical weather data in Australia's all eight capital cities are analyzed to characterize the variation profiles of climatic variables. The method of descriptive statistics is employed. Either the pattern of cumulative distribution and/or the profile of percentage distribution are used to graphically illustrate the similarity and difference between different study locations. It is found that although the weather variables vary with different locations, except for the extreme parts, there is often a good, nearly linear relation between weather variable and its cumulative percentage for the majority of middle part. The implication of these extreme parts and the slopes of the middle parts on building design is also discussed.

On the significance of omitted variables in intersection crash modeling

Advances in safety research—trying to improve the collective understanding of motor vehicle crash causes and contributing factors—rest upon the pursuit of numerous lines of research inquiry. The research community has focused considerable attention on analytical methods development (negative binomial models, simultaneous equations, etc.), on better experimental designs (before-after studies, comparison sites, etc.), on improving exposure measures, and on model specification improvements (additive terms, non-linear relations, etc.). One might logically seek to know which lines of inquiry might provide the most significant improvements in understanding crash causation and/or prediction. It is the contention of this paper that the exclusion of important variables (causal or surrogate measures of causal variables) cause omitted variable bias in model estimation and is an important and neglected line of inquiry in safety research. In particular, spatially related variables are often difficult to collect and omitted from crash models—but offer significant opportunities to better understand contributing factors and/or causes of crashes. This study examines the role of important variables (other than Average Annual Daily Traffic (AADT)) that are generally omitted from intersection crash prediction models. In addition to the geometric and traffic regulatory information of intersection, the proposed model includes many spatial factors such as local influences of weather, sun glare, proximity to drinking establishments, and proximity to schools—representing a mix of potential environmental and human factors that are theoretically important, but rarely used. Results suggest that these variables in addition to AADT have significant explanatory power, and their exclusion leads to omitted variable bias. Provided is evidence that variable exclusion overstates the effect of minor road AADT by as much as 40% and major road AADT by 14%.

Analytical solutions for the two- and three-dimensional time-fractional telegraph equations by the method of separating variables

In this paper, a method of separating variables is effectively implemented for solving a time-fractional telegraph equation (TFTE) in two and three dimensions. We discuss and derive the analytical solution of the TFTE in two and three dimensions with nonhomogeneous Dirichlet boundary condition. This method can be extended to other kinds of the boundary conditions.

A history of heart interventions moderates the relationship between psychological variables and the presence of chest pain in older women with self-reported coronary heart disease

Objectives: This study examines the hypothesis that a past history of heart interventions will moderate the relationship between psychosocial factors (stressful life events, social support, perceived stress, having a current partner, having a past diagnosis of depression or anxiety over the past 3 years, time pressure, education level, and the mental health index) and the presence of chest pain in a sample of older women. Design: Longitudinal survey over a 3-year period. Methods: The sample was taken from a prospective cohort study of 10,432 women initially aged between 70 and 75 years, who were surveyed in 1996 and then again in 1999. Two groups of women were identified: those reporting to have heart disease but no past history of heart interventions (i.e., coronary artery bypass graft/angioplasty) and those reporting to have heart disease with a past history of heart interventions. Results: Binary logistic regression analysis was used to show that for the women with self-reported coronary heart disease but without a past history of heart intervention, feelings of time pressure as well as the number of stressful life events experienced in the 12 months prior to 1996 were independent risk factors for the presence of chest pain, even after accounting for a range of traditional risk factors. In comparison, for the women with self-reported coronary heart disease who did report a past history of heart interventions, a diagnosis of depression in the previous 3 years was the significant independent risk factor for chest pain even after accounting for traditional risk factors. Conclusion: The results indicate that it is important to consider a history of heart interventions as a moderator of the associations between psychosocial variables and the frequency of chest pain in older women. Statement of Contribution: What is already known on this subject? Psychological factors have been shown to be independent predictors of a range of health outcomes in individuals with coronary heart disease, including the presence of chest pain. Most research has been conducted with men or with small samples of women; however, the evidence does suggest that these relationships exist in women as well as in men. What does this study add? Most studies have looked at overall relationships between psychological variables and health outcomes. The few studies that have looked at moderators have mainly examined gender as a moderator. To our knowledge, this is the first published study to examine a history of heart interventions as a moderator of the relationship between psychological variables and the presence of chest pain.

A critical evaluation and comparison of two formative measures of system quality using criterion variables

Evaluating the validity of formative variables has presented ongoing challenges for researchers. In this paper we use global criterion measures to compare and critically evaluate two alternative formative measures of System Quality. One model is based on the ISO-9126 software quality standard, and the other is based on a leading information systems research model. We find that despite both models having a strong provenance, many of the items appear to be non-significant in our study. We examine the implications of this by evaluating the quality of the criterion variables we used, and the performance of PLS when evaluating formative models with a large number of items. We find that our respondents had difficulty distinguishing between global criterion variables measuring different aspects of overall System Quality. Also, because formative indicators “compete with one another” in PLS, it may be difficult to develop a set of measures which are all significant for a complex formative construct with a broad scope and a large number of items. Overall, we suggest that there is cautious evidence that both sets of measures are valid and largely equivalent, although questions still remain about the measures, the use of criterion variables, and the use of PLS for this type of model evaluation.