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Mathematical models applied to epidemiology are useful tools that help understand how infectious diseases spread in populations, and hence support public-health decisions. Over the last 250 years, these modelling tools have have developed at an increasing rate, both on the theoretical and computational sides. This thesis explores various modelling techniques to address debated or unanswered questions about the transmission dynamics of infectious diseases, in particular sexually transmitted ones. The role of sero-discordant couples (when only one partner is infected) in the HIV epidemic in Sub-Saharan Africa is controversial. Their importance compared to other sexual transmission routes is critical when designing intervention policies. In chapter 2, I used a compartmental model with an original partnership process to show that infection of uncoupled individuals is usually the predominant route, while transmission within discordant couples is also important, but to a lesser extent. Despite the availability of inexpensive antimicrobial treatment, syphilis remains prevalent worldwide, affecting millions of individuals. Development of a syphilis vaccine would be a potentially promising step towards control, but the value of dedicating resources to vaccine development should be evaluated in the context of the anticipated benefits. In chapter 3, I explored the potential impact of a hypothetical syphilis vaccine on morbidity from both syphilis and HIV using an agent-based model. My results suggest that an efficacious vaccine has the potential to sharply reduce syphilis under a wide range of scenarios, while expanded treatment interventions are likely to be substantially less effective. General concepts in epidemic modelling, that could be applied to any disease, are still debated. In particular, a rigorous definition and analysis of the generation interval – the interval between the time that an individual is infected by an infector and the time this infector was infected – needed clarification. Indeed, the generation interval is a fundamental quantity when modelling and forecasting epidemics. Chapter 4 clarifies its theoretical framework, explains how its distribution changes as an epidemic progresses and discuss how empirical generation-interval data can be used to correctly inform mathematical models. / Thesis / Doctor of Philosophy (PhD)

In this thesis, we consider a stochastic SIRS model of EEG data. The model is built over three different network structures: a random network, a scale-free network, and a small-world network. These models are then fit to an EEG signal from a control individual and an EEG signal from an individual experiencing an epileptic seizure. We are interested in determining whether these models can distinguish between the two data sets, and whether any of the network structures offer a significantly better fit to the data than others; there is also a broader interest in the effects of different network structures on the time series characteristics of an SIRS system. / Thesis / Master of Science (MSc)

The aim of this thesis was to identify ways in which the velocity of a track cyclist could be increased, primarily through the reduction of aerodynamic drag, and to determine which factors had the most significant impact on athlete performance. An appropriate test method was set up in the wind tunnel at the University of Canterbury to measure the aerodynamic drag of different cycling positions and equipment, including helmets, skinsuits, frames and wheels, in order to measure the impact of specific changes on athlete performance. A mathematical model of the Individual Pursuit (IP) event was also created to calculate the velocity profile and finishing time for athletes competing under different race conditions. The model was created in Microsoft Excel and used first principles to analyse the forces acting on a cyclist, which lead to the development of equations for power supply and demand. The mathematical model was validated using SRM data for eleven, elite track cyclists, and was found to be accurate to 0.31s (0.16%). An analysis of changes made to the bike, athlete, and environmental conditions using the mathematical model showed that the drag area and air density had the greatest impact on the finishing time. The model was then used to predict the finishing times for different pacing strategies by generating different power profiles for a given athlete with a fixed stock of energy (the work done remained the same for all generated power profiles) in order to identify the optimal pacing strategy for the IP. The length of time spent in the initial acceleration phase was found to have a significant impact on the results, although all strategies simulated with an initial acceleration phase resulted in a faster finishing time than all other strategies simulated. Results from the wind tunnel tests showed that, in general, changes made to the position of the cyclist had the greatest impact on the aerodynamic drag compared to changes made to the equipment. Multiple changes in position had a greater impact on drag than individual changes in position, but the changes were not additive; the total gain or loss in drag for multiple changes in position was not the sum of individual gains or losses in drag. Actual gains and losses also varied significantly between athletes, primarily due to differences in body size and shape, riding experience, and reference position from which changes were made from. Changes in position that resulted in a reduction of the frontal area, such as lowering the handlebars and head, were the most successful at reducing the aerodynamic drag, and a change in skinsuit was found to have the greatest impact on drag out of all equipment changes, primarily due to the choice of material and seam placement. The mathematical model was used to quantify the impact of changes in position and equipment made in the wind tunnel on the overall finishing time for a given athlete competing in an IP event. Time savings of up to 8 seconds were seen for multiple changes in position, and up to 5 seconds for changes to the equipment. Overall this thesis highlights the significance of aerodynamics on athlete performance in track cycling, suggesting that it is worthwhile spending time and money on research and technology to find new ways to reduce the aerodynamic drag and maximise the speed of cyclists. Although this thesis primarily concentrates on the Individual Pursuit event in track cycling, the same principles can be applied to other cycling disciplines, as well as to other sports.

Finding a common language in describing and interpreting multivariate data associated with rehabilitation and disturbance ecology, has became a major challenge. The main objective of this study is to find and evaluate mathematical models to describe ecosystem change based on selected indicators of change. Existing data from a previous rehabilitation project on Hendrina Power Station (Mpumalanga, South Africa) was used as a database for this study and this study aims to report on the development of models concentrating on radar graphs and a model based on matrix mathematics. The main groups of organisms selected for the construction of models, were vegetation, soil mesofauna and ant species. The datasets were limited to some indicative species and their mean abundances were determined. The grids that were used were randomly chosen and the models were constructed. Radar graphs were constructed to model the suite of species identified, through a sensitivity analysis, to indicate possible rehabilitation success over time and was applied to the different rehabilitation ages. The surface areas under the radar graphs were determined and compared for the different rehabilitation ages in the same year of survey. Correlation graphs were drawn between the surface area and the rehabilitation ages. These graphs did not indicate much relevance in indicating rehabilitation success, but the radar graphs proved to be good indicators of change in abundance of the selected species over time. iv The vegetation species, Eragrostis curvula, was the only species that showed a strong significant positive relationship with rehabilitation age and could be considered a good rehabilitation species and indicator of rehabilitation success. After the evaluation of this model, Eragrostis curvula, and two additional ant species, Tetramorium setigerum and Lepisiota laevis, were added. These species that were added, showed an increase in abundance over time, as found in a previous study. These radar graphs also did not indicate much relevance and it can be concluded that the radar graphs can only be used for a visual representation of the changes in abundance of the relevant species over time. This study also refers to a matrix model. This model focused on the interactions between the different variables selected. The percentage carbon in the soil were also added to the list of species. Model fitting graphs were constructed and correlations were drawn between the species that had significant values in the interaction table. This model could be useful for future studies, but more data and replication is necessary, over a longer period of time. This will serve to eliminate possible shortcomings of the model. / Thesis (M. Environmental Science (Biodiversity and Conservation Biology))--North-West University, Potchefstroom Campus, 2007.

Digitized by Kansas Correctional Industries

In this thesis mathematical models describing the growth of a solid tumour in the presence of an immune response are presented. Specifically, attention is focused on the interactions between cytotoxic T-lymphocytes (CTLs) and tumour cells in a small, avascular multicellular tumour. At this stage of the disease the CTLs and the tumour cells are considered to be in a state of dynamic equilibrium or cancer dormancy. The precise biochemical and cellular mechanisms by which CTLs can control a cancer and keep it in a dormant state are still not completely understood from a biological and immunological point of view. The mathematical models focus on the spatio-temporal dynamics of tumour cells, immune cells, chemokines and “chemo-repellors” in an immunogenic tumour. The CTLs and tumour cells are assumed to migrate and interact with each other in such a way that lymphocyte-tumour cell complexes are formed. These complexes result in either the death of the tumour cells (the normal situation) or the inactivation of the lymphocytes and consequently the survival of the tumour cells. In the latter case, we assume that each tumour cell which survives its “brief encounter” with the CTLs undergoes certain beneficial phenotypic changes. We explore the dynamics of the model under these assumptions and show that the process of the immuno-evasion can arise as a consequence of these encounters.Our computational simulations suggest that the proposed mechanism is able to mimic various dynamics of immunoevasion during the lifespan of a mouse. We also highlight the differential spatiotemporal contributions to evasion due, respectively, to: i) a decrease in the probability pi of being lethally hit; ii) a decrease in the probability, embedded in k+ i , that a tumour cell is recognized by a CTL. In particular, our model suggests that a decrease in the parameters pi is needed to produce evasion, which does not occur in the case where pi remains constant at its baseline level inferred from the experimental data. However, the role of the parameters k+ i is important since it can greatly accelerate the simulated process. Moreover, our computational simulations also show that the proposed mechanism can also deeply affect the spatial patterning of the tumour. In particular, our model suggests that to have a uniform invasion profile for the tumour cells necessitates also having a decrease in the recognition rate, embedded in the parameters k+ i . These parameters also differentially shape the spatial distribution of the various classes of tumour cells. Also in this thesis, we discuss mathematical models of the interactions between a tumour and both the innate and the cellular part of the adaptive immune system. We have developed and formulated spatiotemporal models of the interactions between macrophages, natural killer cells, cytotoxic T lymphocytes and tumour cells. In addition to presenting computational simulations of our ODE and PDE models, we investigate the linear stability analysis of steady states of the model and the effect of the initial conditions on the behaviour of the ODE solution. We show that limit cycle behaviour could be obtained by making some changes in the parameter values, which gave us oscillations in the solution of the ODE and PDE systems. We observe that there is a slowly damped oscillation in the behaviour of the tumour, natural killer and CTL cells. Also we note that the solution converges to the second steady state where the tumour size is small (dormant state).A model of cancer invasion and metastasis is also discussed in this thesis. This model attempts to describe the interactions between cancer cells, urokinase plasminogen activator (uPA), plasminogen activator inhibitor-1 (PAI-1), plasmin, extracellular matrix (ECM) and the immune response. The mathematical model focuses on the effect of the immune response on cancer invasion by assuming that there is some form of limit cycle behaviour between the cancer cells and the effector cells. The work we present in this chapter develops a mathematical model for tumour invasion with an immune response using a continuum model in 1 and 2 space dimensions. This model consists of a system of nonlinear partial differential equations and examines the effector cell response the tumour invasion. This model consists of effector cells, tumour cells, ECM, uPA, PAI-1, and plasmin. First, we set all spatial components of the model to zero and consider only the reaction kinetics in order to compare between the behaviour of our model and the original Chaplain and Lolas model (Chaplain and Lolas, 2005). The spatially homogeneous simulation shows the behaviour of solutions have regular oscillations because there is a closed orbit. Second, we present the computational results of the spatio-temporal model, and we note from these simulations that the tumour size of our model is smaller than the tumour size of the Chaplain and Lolas model because the immune cells are interacting with the tumour cells, and also the degradation of ECM is less than that in the Chaplain and Lolas model. In addition, the number of tumour cell clusters in our model is less than those in the Chaplain and Lolas model. Also we found the tumour clusters of the mathematical model which was discussed in this chapter to have the same range than the tumour clusters of the Chaplain and Lolas model. The final model presented in this thesis is a mathematical model of cancer cells and effector cells which exhibit standing-wave behaviour between them. We show tha the wave of invading cancer cells can be stopped by the wave of effector cells or ECM.This model also focuses on the effect of the mutation of cancer cells to another subpopulation which is more malignant and which has the ability to invade the ECM or the effector cells to occupy space. The numerical simulations discussed in this chapter are essentially associated with an initial model of two equations representing the effector cells and tumour cells, such that there is a standing wave between these species. We note that the solution of the mathematical model is a travelling wave and also has a standing wave solution (i.e. the wave of effector cells stops the wave of tumour cells when they meet). This phenomenon occurs when the two diffusion coefficients are the same. We calculate the wave speed to illustrate that the speed tends to zero when the two waves meet - a positive speed of tumour cells refers to an invading tumour, and a negative speed refers to the decreasing of effector cells. After this we modify the model by adding an equation for a second cancer cell population T2, which is a sub-population 2 of tumour cells. This is to reflect the fact that cancer is a progressive disease, and as such it becomes more malignant as the cancer cells undergo successive mutations. We show in this case how the new type of cancer cells start to invade the effector cells after the failure of the first type. The third model discussed in this chapter is arrived at by adding an ECM equation to the second model, and it explains how the standing wave arise from two types of equations - the first one contains diffusion, and the second one has no diffusion.All the mathematical models in this thesis use numerical analysis of nonlinear partial differential equations and computational simulations to obtain insight into the underlying biological systems. The systems of nonlinear partial differential equations were numerically solved by a PDE solver in MATLAB for 1D and COMSOL for 2D. We used the MATLAB PDE solver pdepe which uses the method described in Skeel and Berzins (1990) for the spatial discretisation and the MATLAB routine ode15s for the time integration.The numerical simulations demonstrate the existence of cell distributions that are quasi-stationary in time and heterogeneous in space.

A mathematical model of the interaction between chronic lymphocytic leukemia (CLL) and CD4+ (helper) T cells was developed to study the role of T cells in cancer survival. In particular, a system of four nonlinear advection diffusion reaction partial differential equations were used to simulate spatial effects such as chemical diffusion and chemotaxis on CLL survival and proliferation.

69 leaves : tables ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--University of Adelaide, Dept. of Applied Mathematics, 1979