PySSM : a Python module for Bayesian inference of linear Gaussian state space models

PySSM is a Python package that has been developed for the analysis of time series using linear Gaussian state space models (SSM). PySSM is easy to use; models can be set up quickly and efficiently and a variety of different settings are available to the user. It also takes advantage of scientific libraries Numpy and Scipy and other high level features of the Python language. PySSM is also used as a platform for interfacing between optimised and parallelised Fortran routines. These Fortran routines heavily utilise Basic Linear Algebra (BLAS) and Linear Algebra Package (LAPACK) functions for maximum performance. PySSM contains classes for filtering, classical smoothing as well as simulation smoothing.

Interaction of myxomatosis and Rabbit Haemorrhagic Disease in wild rabbit

Increasing resistance of rabbits to myxomatosis in Australia has led to the exploration of Rabbit Haemorrhagic Disease, also called Rabbit Calicivirus Disease (RCD) as a possible control agent. While the initial spread of RCD in Australia resulted in widespread rabbit mortality in affected areas, the possible population dynamic effects of RCD and myxomatosis operating within the same system have not been properly explored. Here we present early mathematical modelling examining the interaction between the two diseases. In this study we use a deterministic compartment model, based on the classical SIR model in infectious disease modelling. We consider, here, only a single strain of myxomatosis and RCD and neglect latent periods. We also include logistic population growth, with the inclusion of seasonal birth rates. We assume there is no cross-immunity due to either disease. The mathematical model allows for the possibility of both diseases to be simultaneously present in an individual, although results are also presented for the case where co infection is not possible, since co-infection is thought to be rare and questions exist as to whether it can occur. The simulation results of this investigation show that it is a crucial issue and should be part of future field studies. A single simultaneous outbreak of RCD and myxomatosis was simulated, while ignoring natural births and deaths, appropriate for a short timescale of 20 days. Simultaneous outbreaks may be more common in Queensland. For the case where co-infection is not possible we find that the simultaneous presence of myxomatosis in the population suppresses the prevalence of RCD, compared to an outbreak of RCD with no outbreak of myxomatosis, and thus leads to a less effective control of the population. The reason for this is that infection with myxomatosis removes potentially susceptible rabbits from the possibility of infection with RCD (like a vaccination effect). We found that the reduction in the maximum prevalence of RCD was approximately 30% for an initial prevalence of 20% of myxomatosis, for the case where there was no simultaneous outbreak of myxomatosis, but the peak prevalence was only 15% when there was a simultaneous outbreak of myxomatosis. However, this maximum reduction will depend on other parameter values chosen. When co-infection is allowed then this suppression effect does occur but to a lesser degree. This is because the rabbits infected with both diseases reduces the prevalence of myxomatosis. We also simulated multiple outbreaks over a longer timescale of 10 years, including natural population growth rates, with seasonal birth rates and density dependent(logistic) death rates. This shows how both diseases interact with each other and with population growth. Here we obtain sustained outbreaks occurring approximately every two years for the case of a simultaneous outbreak of both diseases but without simultaneous co-infection, with the prevalence varying from 0.1 to 0.5. Without myxomatosis present then the simulation predicts RCD dies out quickly without further introduction from elsewhere. With the possibility of simultaneous co-infection of rabbits, sustained outbreaks are possible but then the outbreaks are less severe and more frequent (approximately yearly). While further model development is needed, our work to date suggests that: 1) the diseases are likely to interact via their impacts on rabbit abundance levels, and 2) introduction of RCD can suppress myxomatosis prevalence. We recommend that further modelling in conjunction with field studies be carried out to further investigate how these two diseases interact in the population.

A sequential Monte Carlo algorithm to incorporate model uncertainty in Bayesian sequential design

Here we present a sequential Monte Carlo (SMC) algorithm that can be used for any one-at-a-time Bayesian sequential design problem in the presence of model uncertainty where discrete data are encountered. Our focus is on adaptive design for model discrimination but the methodology is applicable if one has a different design objective such as parameter estimation or prediction. An SMC algorithm is run in parallel for each model and the algorithm relies on a convenient estimator of the evidence of each model which is essentially a function of importance sampling weights. Other methods for this task such as quadrature, often used in design, suffer from the curse of dimensionality. Approximating posterior model probabilities in this way allows us to use model discrimination utility functions derived from information theory that were previously difficult to compute except for conjugate models. A major benefit of the algorithm is that it requires very little problem specific tuning. We demonstrate the methodology on three applications, including discriminating between models for decline in motor neuron numbers in patients suffering from neurological diseases such as Motor Neuron disease.

Change Point Estimation in Monitoring Survival Time

Precise identification of the time when a change in a hospital outcome has occurred enables clinical experts to search for a potential special cause more effectively. In this paper, we develop change point estimation methods for survival time of a clinical procedure in the presence of patient mix in a Bayesian framework. We apply Bayesian hierarchical models to formulate the change point where there exists a step change in the mean survival time of patients who underwent cardiac surgery. The data are right censored since the monitoring is conducted over a limited follow-up period. We capture the effect of risk factors prior to the surgery using a Weibull accelerated failure time regression model. Markov Chain Monte Carlo is used to obtain posterior distributions of the change point parameters including location and magnitude of changes and also corresponding probabilistic intervals and inferences. The performance of the Bayesian estimator is investigated through simulations and the result shows that precise estimates can be obtained when they are used in conjunction with the risk-adjusted survival time CUSUM control charts for different magnitude scenarios. The proposed estimator shows a better performance where a longer follow-up period, censoring time, is applied. In comparison with the alternative built-in CUSUM estimator, more accurate and precise estimates are obtained by the Bayesian estimator. These superiorities are enhanced when probability quantification, flexibility and generalizability of the Bayesian change point detection model are also considered.

New exact solutions for Hele-Shaw flow in doubly connected regions

Radial Hele-Shaw flows are treated analytically using conformal mapping techniques. The geometry of interest has a doubly-connected annular region of viscous fluid surrounding an inviscid bubble that is either expanding or contracting due to a pressure difference caused by injection or suction of the inviscid fluid. The zero-surface-tension problem is ill-posed for both bubble expansion and contraction, as both scenarios involve viscous fluid displacing inviscid fluid. Exact solutions are derived by tracking the location of singularities and critical points in the analytic continuation of the mapping function. We show that by treating the critical points, it is easy to observe finite-time blow-up, and the evolution equations may be written in exact form using complex residues. We present solutions that start with cusps on one interface and end with cusps on the other, as well as solutions that have the bubble contracting to a point. For the latter solutions, the bubble approaches an ellipse in shape at extinction.

Modelling the interaction of keratinocytes and fibroblasts during normal and abnormal wound healing processes

The crosstalk between fibroblasts and keratinocytes is a vital component of the wound healing process, and involves the activity of a number of growth factors and cytokines. In this work, we develop a mathematical model of this crosstalk in order to elucidate the effects of these interactions on the regeneration of collagen in a wound that heals by second intention. We consider the role of four components that strongly affect this process: transforming growth factor-beta, platelet-derived growth factor, interleukin-1 and keratinocyte growth factor. The impact of this network of interactions on the degradation of an initial fibrin clot, as well as its subsequent replacement by a matrix that is mainly comprised of collagen, is described through an eight-component system of nonlinear partial differential equations. Numerical results, obtained in a two-dimensional domain, highlight key aspects of this multifarious process such as reepithelialisation. The model is shown to reproduce many of the important features of normal wound healing. In addition, we use the model to simulate the treatment of two pathological cases: chronic hypoxia, which can lead to chronic wounds; and prolonged inflammation, which has been shown to lead to hypertrophic scarring. We find that our model predictions are qualitatively in agreement with previously reported observations, and provide an alternative pathway for gaining insight into this complex biological process.

Optimising container processes at multimodal seaport terminals : an integrated approach and application

A Multimodal Seaport Container Terminal (MSCT) is a complex system which requires careful planning and control in order to operate efficiently. It consists of a number of subsystems that require optimisation of the operations within them, as well as synchronisation of machines and containers between the various subsystems. Inefficiency in the terminal can delay ships from their scheduled timetables, as well as cause delays in delivering containers to their inland destinations, both of which can be very costly to their operators. The purpose of this PhD thesis is to use Operations Research methodologies to optimise and synchronise these subsystems as an integrated application. An initial model is developed for the overall MSCT; however, due to a large number of assumptions that had to be made, as well as other issues, it is found to be too inaccurate and infeasible for practical use. Instead, a method of developing models for each subsystem is proposed that then be integrated with each other. Mathematical models are developed for the Storage Area System (SAS) and Intra-terminal Transportation System (ITTS). The SAS deals with the movement and assignment of containers to stacks within the storage area, both when they arrive and when they are rehandled to retrieve containers below them. The ITTS deals with scheduling the movement of containers and machines between the storage areas and other sections of the terminal, such as the berth and road/rail terminals. Various constructive heuristics are explored and compared for these models to produce good initial solutions for large-sized problems, which are otherwise impractical to compute by exact methods. These initial solutions are further improved through the use of an innovative hyper-heuristic algorithm that integrates the SAS and ITTS solutions together and optimises them through meta-heuristic techniques. The method by which the two models can interact with each other as an integrated system will be discussed, as well as how this method can be extended to the other subsystems of the MSCT.

Critical time scales for advection–diffusion–reaction processes

The concept of local accumulation time (LAT) was introduced by Berezhkovskii and coworkers in 2010–2011 to give a finite measure of the time required for the transient solution of a reaction–diffusion equation to approach the steady–state solution (Biophys J. 99, L59 (2010); Phys Rev E. 83, 051906 (2011)). Such a measure is referred to as a critical time. Here, we show that LAT is, in fact, identical to the concept of mean action time (MAT) that was first introduced by McNabb in 1991 (IMA J Appl Math. 47, 193 (1991)). Although McNabb’s initial argument was motivated by considering the mean particle lifetime (MPLT) for a linear death process, he applied the ideas to study diffusion. We extend the work of these authors by deriving expressions for the MAT for a general one–dimensional linear advection–diffusion–reaction problem. Using a combination of continuum and discrete approaches, we show that MAT and MPLT are equivalent for certain uniform–to-uniform transitions; these results provide a practical interpretation for MAT, by directly linking the stochastic microscopic processes to a meaningful macroscopic timescale. We find that for more general transitions, the equivalence between MAT and MPLT does not hold. Unlike other critical time definitions, we show that it is possible to evaluate the MAT without solving the underlying partial differential equation (pde). This makes MAT a simple and attractive quantity for practical situations. Finally, our work explores the accuracy of certain approximations derived using the MAT, showing that useful approximations for nonlinear kinetic processes can be obtained, again without treating the governing pde directly.

A continuum model of the growth of engineered epidermal skin substitutes

We present a porous medium model of the growth and deterioration of the viable sublayers of an epidermal skin substitute. It consists of five species: cells, intracellular and extracellular calcium, tight junctions, and a hypothesised signal chemical emanating from the stratum corneum. The model is solved numerically in Matlab using a finite difference scheme. Steady state calcium distributions are predicted that agree well with the experimental data. Our model also demonstrates epidermal skin substitute deterioration if the calcium diffusion coefficient is reduced compared to reported values in the literature.

Using approximate Bayesian computation to estimate transmission rates of nosocomial pathogens

In this paper, we apply a simulation based approach for estimating transmission rates of nosocomial pathogens. In particular, the objective is to infer the transmission rate between colonised health-care practitioners and uncolonised patients (and vice versa) solely from routinely collected incidence data. The method, using approximate Bayesian computation, is substantially less computer intensive and easier to implement than likelihood-based approaches we refer to here. We find through replacing the likelihood with a comparison of an efficient summary statistic between observed and simulated data that little is lost in the precision of estimated transmission rates. Furthermore, we investigate the impact of incorporating uncertainty in previously fixed parameters on the precision of the estimated transmission rates.

Models of collective cell behaviour with crowding effects : comparing lattice-based and lattice-free approaches

Individual-based models describing the migration and proliferation of a population of cells frequently restrict the cells to a predefined lattice. An implicit assumption of this type of lattice based model is that a proliferative population will always eventually fill the lattice. Here we develop a new lattice-free individual-based model that incorporates cell-to-cell crowding effects. We also derive approximate mean-field descriptions for the lattice-free model in two special cases motivated by commonly used experimental setups. Lattice-free simulation results are compared to these mean-field descriptions and to a corresponding lattice-based model. Data from a proliferation experiment is used to estimate the parameters for the new model, including the cell proliferation rate, showing that the model fits the data well. An important aspect of the lattice-free model is that the confluent cell density is not predefined, as with lattice-based models, but an emergent model property. As a consequence of the more realistic, irregular configuration of cells in the lattice-free model, the population growth rate is much slower at high cell densities and the population cannot reach the same confluent density as an equivalent lattice-based model.

An example of cognitive obstacles in advanced integration: the case of scalar line integrals

Cognitive obstacles that arise in the teaching and learning of scalar line integrals, derived from cognitive aids provided to students when first learning about integration of single variable functions are described. A discussion of how and why the obstacles cause students problems is presented and possible strategies to overcome the obstacles are outlined.

Mean-field descriptions of collective migration with strong adhesion

Random walk models based on an exclusion process with contact effects are often used to represent collective migration where individual agents are affected by agent-to-agent adhesion. Traditional mean field representations of these processes take the form of a nonlinear diffusion equation which, for strong adhesion, does not predict the averaged discrete behavior. We propose an alternative suite of mean-field representations, showing that collective migration with strong adhesion can be accurately represented using a moment closure approach.

A practical approach for identifying expected solution performance and robustness in operations research applications

A practical approach for identifying solution robustness is proposed for situations where parameters are uncertain. The approach is based upon the interpretation of a probability density function (pdf) and the definition of three parameters that describe how significant changes in the performance of a solution are deemed to be. The pdf is constructed by interpreting the results of simulations. A minimum number of simulations are achieved by updating the mean, variance, skewness and kurtosis of the sample using computationally efficient recursive equations. When these criterions have converged then no further simulations are needed. A case study involving several no-intermediate storage flow shop scheduling problems demonstrates the effectiveness of the approach.

Computationally efficient methods for solving time-variable-order time-space fractional reaction-diffusion equation

Fractional differential equations are becoming more widely accepted as a powerful tool in modelling anomalous diffusion, which is exhibited by various materials and processes. Recently, researchers have suggested that rather than using constant order fractional operators, some processes are more accurately modelled using fractional orders that vary with time and/or space. In this paper we develop computationally efficient techniques for solving time-variable-order time-space fractional reaction-diffusion equations (tsfrde) using the finite difference scheme. We adopt the Coimbra variable order time fractional operator and variable order fractional Laplacian operator in space where both orders are functions of time. Because the fractional operator is nonlocal, it is challenging to efficiently deal with its long range dependence when using classical numerical techniques to solve such equations. The novelty of our method is that the numerical solution of the time-variable-order tsfrde is written in terms of a matrix function vector product at each time step. This product is approximated efficiently by the Lanczos method, which is a powerful iterative technique for approximating the action of a matrix function by projecting onto a Krylov subspace. Furthermore an adaptive preconditioner is constructed that dramatically reduces the size of the required Krylov subspaces and hence the overall computational cost. Numerical examples, including the variable-order fractional Fisher equation, are presented to demonstrate the accuracy and efficiency of the approach.