Wound healing is a complex process, in which a sequence of interrelated events at both the cell and tissue levels interact and contribute to the reduction in wound size. For diabetic patients, many of these processes are compromised, so that wound healing slows down and in some cases halts. In this thesis we develop a series of increasingly detailed mathematical models to describe and investigate healing of full thickness skin wounds. We begin by developing a time-dependent ordinary differential equation model. This phenomenological model focusses on the main processes contributing to closure of a full thickness wound: proliferation in the epidermis and growth and contraction in the dermis. Model simulations suggest that the relative contributions of growth and contraction to healing of the dermis are altered in diabetic wounds. We investigate further the balance between growth and contraction by developing a more detailed, spatially-resolved model using continuum mechanics. Due to the initial large retraction of the wound edge upon injury, we adopt a non-linear elastic framework. Morphoelasticity theory is applied, with the total deformation of the material decomposed into an addition of mass and an elastic response. We use the model to investigate how interactions between growth and stress influence dermal wound healing. The model reveals that contraction alone generates unrealistically high tension in the dermal tissue and, hence, volumetric growth must contribute to healing. We show that, in the simplified case of homogeneous growth, the tissue must grow anisotropically in order to reduce the size of the wound and we postulate mechanosensitive growth laws consistent with this result. After closure the surrounding tissue remodels, returning to its residually stressed state. We identify the steady state growth profile associated with this remodelled state. The model is used to predict the outcome of rewounding experiments as a method of quantifying the amount of stress in the tissue and the application of pressure treatments to control tissue synthesis. The thesis concludes with an extension to the spatially-resolved mechanical model to account for the effects of the biochemical environment. Partial differential equations describing the dynamics of fibroblasts and a regulating growth factor are coupled to equations for the tissue mechanics, described in the morphoelastic framework. By accounting for biomechanical and biochemical stimuli the model allows us to formulate mechanistic laws for growth and contraction. We explore how disruption of mechanical and chemical feedback can lead to abnormal wound healing and use the model to identify specific treatments for normalising healing in these cases.

Cell migration, the active translocation of cells is involved in various biological processes, e.g. development of tissues and organs, tumor invasion and wound healing. Cell migration behavior can be divided into two distinct classes: single cell migration and collective cell migration. Single cell migration describes the migration of cells without interaction with other cells in their environment. Collective cell migration is the joint, active movement of multiple cells, e.g. in the form of strands, cohorts or sheets which emerge as the result of individual cell-cell interactions. Collective cell migration can be observed during branching morphogenesis, vascular sprouting and embryogenesis. Experimental studies of single cell migration have been extensive. Collective cell migration is less well investigated due to more difficult experimental conditions than for single cell migration. Especially, experimentally identifying the impact of individual differences in cell phenotypes on individual cell migration behavior inside cell populations is challenging because the tracking of individual cell trajectories is required. In this thesis, a novel mathematical modeling approach, individual-based lattice-gas cellular automata (IB-LGCA), that allows to investigate the migratory behavior of individual cells inside migrating cell populations by enabling the tracking of individual cells is introduced. Additionally, stochastic differential equation (SDE) approximations of individual cell trajectories for IB-LGCA models are constructed. Such SDE approximations allow the analytical description of the trajectories of individual cells during single cell migration. For a complete analytical description of the trajectories of individual cell during collective cell migration the aforementioned SDE approximations alone are not sufficient. Analytical approximations of the time development of selected observables for the cell population have to be added. What observables have to be considered depends on the specific cell migration mechanisms that is to be modeled. Here, partial integro-differential equations (PIDE) that approximate the time evolution of the expected cell density distribution in IB-LGCA are constructed and coupled to SDE approximations of individual cell trajectories. Such coupled PIDE and SDE approximations provide an analytical description of the trajectories of individual cells in IB-LGCA with density-dependent cell-cell interactions. Finally, an IB-LGCA model and corresponding analytical approximations were applied to investigate the impact of changes in cell-cell and cell-ECM forces on the migration behavior of an individual, labeled cell inside a population of epithelial cells. Specifically, individual cell migration during the epithelial-mesenchymal transition (EMT) was considered. EMT is a change from epithelial to mesenchymal cell phenotype which is characterized by cells breaking adhesive bonds with surrounding epithelial cells and initiating individual migration along the extracellular matrix (ECM). During the EMT, a transition from collective to single cell migration occurs. EMT plays an important role during cancer progression, where it is believed to be linked to metastasis development. In the IB-LGCA model epithelial cells are characterized by balanced cell-cell and cell-ECM forces. The IB-LGCA model predicts that the balance between cell-cell and cell-ECM forces can be disturbed to some degree without being accompanied by a change in individual cell migration behavior. Only after the cell force balance has been strongly interrupted mesenchymal migration behavior is possible. The force threshold which separates epithelial and mesenchymal migration behavior in the IB-LGCA has been identified from the corresponding analytical approximation. The IB-LGCA model allows to obtain quantitative predictions about the role of cell forces during EMT which in the context of mathematical modeling of EMT is a novel approach.

Philosophiae Doctor - PhD / Fitted Numerical Methods for Delay Di erential Equations Arising in Biology E.B.M. Bashier PhD thesis, Department of Mathematics and Applied Mathematics,Faculty of Natural Sciences, University of the Western Cape. This thesis deals with the design and analysis of tted numerical methods for some delay di erential models that arise in biology. Very often such di erential equations are very complex in nature and hence the well-known standard numerical methods seldom produce reliable numerical solutions to these problems. Ine ciencies of these methods are mostly accumulated due to their dependence on crude step sizes and unrealistic stability conditions.This usually happens because standard numerical methods are initially designed to solve a class of general problems without considering the structure of any individual problems. In this thesis, issues like these are resolved for a set of delay di erential equations. Though the developed approaches are very simplistic in nature, they could solve very complex problems as is shown in di erent chapters.The underlying idea behind the construction of most of the numerical methods in this thesis is to incorporate some of the qualitative features of the solution of the problems into the discrete models. Resulting methods are termed as tted numerical methods. These methods have high stability properties, acceptable (better in many cases) orders of convergence, less computational complexities and they provide reliable solutions with less CPU times as compared to most of the other conventional solvers. The results obtained by these methods are comparable to those found in the literature. The other salient feature of the proposed tted methods is that they are unconditionally stable for most of the problems under consideration.We have compared the performances of our tted numerical methods with well-known software packages, for example, the classical fourth-order Runge-Kutta method, standard nite di erence methods, dde23 (a MATLAB routine) and found that our methods perform much better. Finally, wherever appropriate, we have indicated possible extensions of our approaches to cater for other classes of problems. May 2009.

The paper is focused on the Predator-Prey model modified in the case of harvesting one or both populations. Firstly there is given a short description of the basic model and the sensitivity analysis. The first essential modification is percentage harvesting. This model could be easily converted to the basic one using a substitution. The next modification is constant harvesting. Solving this system requires linearization, which was properly done and brought valuable results applicable even for the basic or the percentage harvesting model. The next chapter describes regulation models, which could be used especially in applying environmental policies. All reasonable regulation models are shown after distinguishing between discrete and continuous harvesting. The last chapter contains an algorithm for maximizing the profit of a harvester using econometrical modelling tools.

We first discuss some fundamental results such as equilibria, linearization, and stability of nonlinear dynamical systems arising in mathematical modeling. Next we study the dynamics in planar systems such as limit cycles, the Poincaré-Bendixson theorem, and some of its useful consequences. We then study the interaction between two and three different cell populations, and perform stability and bifurcation analysis on the systems. We also analyze the impact of immunotherapy on the tumor cell population numerically.

Cell migration, the active translocation of cells is involved in various biological processes, e.g. development of tissues and organs, tumor invasion and wound healing. Cell migration behavior can be divided into two distinct classes: single cell migration and collective cell migration. Single cell migration describes the migration of cells without interaction with other cells in their environment. Collective cell migration is the joint, active movement of multiple cells, e.g. in the form of strands, cohorts or sheets which emerge as the result of individual cell-cell interactions. Collective cell migration can be observed during branching morphogenesis, vascular sprouting and embryogenesis. Experimental studies of single cell migration have been extensive. Collective cell migration is less well investigated due to more difficult experimental conditions than for single cell migration. Especially, experimentally identifying the impact of individual differences in cell phenotypes on individual cell migration behavior inside cell populations is challenging because the tracking of individual cell trajectories is required. In this thesis, a novel mathematical modeling approach, individual-based lattice-gas cellular automata (IB-LGCA), that allows to investigate the migratory behavior of individual cells inside migrating cell populations by enabling the tracking of individual cells is introduced. Additionally, stochastic differential equation (SDE) approximations of individual cell trajectories for IB-LGCA models are constructed. Such SDE approximations allow the analytical description of the trajectories of individual cells during single cell migration. For a complete analytical description of the trajectories of individual cell during collective cell migration the aforementioned SDE approximations alone are not sufficient. Analytical approximations of the time development of selected observables for the cell population have to be added. What observables have to be considered depends on the specific cell migration mechanisms that is to be modeled. Here, partial integro-differential equations (PIDE) that approximate the time evolution of the expected cell density distribution in IB-LGCA are constructed and coupled to SDE approximations of individual cell trajectories. Such coupled PIDE and SDE approximations provide an analytical description of the trajectories of individual cells in IB-LGCA with density-dependent cell-cell interactions. Finally, an IB-LGCA model and corresponding analytical approximations were applied to investigate the impact of changes in cell-cell and cell-ECM forces on the migration behavior of an individual, labeled cell inside a population of epithelial cells. Specifically, individual cell migration during the epithelial-mesenchymal transition (EMT) was considered. EMT is a change from epithelial to mesenchymal cell phenotype which is characterized by cells breaking adhesive bonds with surrounding epithelial cells and initiating individual migration along the extracellular matrix (ECM). During the EMT, a transition from collective to single cell migration occurs. EMT plays an important role during cancer progression, where it is believed to be linked to metastasis development. In the IB-LGCA model epithelial cells are characterized by balanced cell-cell and cell-ECM forces. The IB-LGCA model predicts that the balance between cell-cell and cell-ECM forces can be disturbed to some degree without being accompanied by a change in individual cell migration behavior. Only after the cell force balance has been strongly interrupted mesenchymal migration behavior is possible. The force threshold which separates epithelial and mesenchymal migration behavior in the IB-LGCA has been identified from the corresponding analytical approximation. The IB-LGCA model allows to obtain quantitative predictions about the role of cell forces during EMT which in the context of mathematical modeling of EMT is a novel approach.

Conservation in the EU and all over the world aims at reducing biodiversity loss which has become a great issue in the last decades. However, despite existing efforts, Earth is assumed to face a sixth mass extinction. One major challenge for conservation is to reconcile the targets with conflicting interests, e.g. for food production in intensively used agricultural landscapes. Agriculture is an example of a coupled human-environment system that is approached in this thesis with the help of mathematical models from two directions. Firstly, the ecological subsystem is considered to find processes relevant for the effect of habitat connectivity on population abundances. Modelling theory predicts that the species-specific growth parameters (intrinsic growth rate and carrying capacity) indicate whether dispersal has a positive or negative effect on the total population size at equilibrium (r-K relationship). We use laboratory experiments in combination with a system of ordinary differential equations and deliver the first empirical evidence for a negative effect of dispersal on the population size in line with this theory. The result is of particular relevance for the design of dispersal corridors or stepping stones which are meant to increase connectivity between habitats. These measures might not be effective for biological conservation. A second population model, consisting of two coupled Ricker maps with a mate-finding Allee effect, is analyzed in order to examine the effect of bistability due to the Allee effect in combination with overcompensation in a spatial system. The interplay can cause complex population dynamics including multiple coexisting attractors, long transients and sudden population collapses. Essential extinction teaches us that not only small populations are prone to extinction but chaotic dynamics can drive a population extinct in a short period of time as well. By a comprehensive model analysis, we find that dispersal can prevent essential extinction of a population. In the context of conservation that is: habitat connectivity can promote rescue effects to save a population that exhibits an Allee effect. The two findings of the first part of this thesis have contrasting implications for conservation which shows that universal recommendations regarding habitat connectivity are impossible without knowledge of the specific system. Secondly, a model for the socio-economic subsystem is presented. Agri-environment schemes (AES) are payments that compensate farmers for forgone profits on the condition that they improve the ecological state of the agricultural system. However, classical economic models that describe the cost-effectiveness of AES often do not take the social network of farmers into account. Numerical simulations of the socio-economic model presented in this thesis suggest that social norms can hinder farmers from scheme participation. Moreover, social norms lead to multistability in farmers’ land-use decision behaviour. Informational campaigns potentially decrease the threshold towards more long-term scheme participation and might be a good tool to complement compensation payments if social norms affect land-use decisions. Finally, a coupled human-environment system is analyzed. An integrated economicecological model is studied to investigate the cost-effectiveness of AES if the species of concern exhibits an Allee effect. A numerical model analysis indicates large trade-offs between agricultural production and persistence probability. Moreover, conservation success strongly depends on the initial population size, meaning that conservation is well advised to start before the species is threatened. Spatial aggregation of habitat can promote rescue effects, suggesting land-sparing solutions for conservation. In that case,agglomeration bonuses may serve to increase the effectiveness of AES. Possible causes for population declines are diverse and can be a combination of human influences, e.g. due to habitat degradation and inherent ecosystem properties. That complicates the task of conservation. The models presented in this thesis simplify complex systems in order to extract processes relevant for biological conservation. The analysis of spatial effects and dynamical model complexity, e.g. due to Allee effects or a nonlinear utility function, allows us improve the understanding of coupled human-environment systems.

No description available.

Despite a decade of publications investigating the effect of germline polymorphisms on both toxicity related to treatment with 5-fluorouracil-based (5-FU) chemotherapy and prognosis following diagnosis with colorectal cancer (CRC), few genetic biomarkers have been identified convincingly. For 5-FU toxicity and CRC prognosis, in four results chapters, this thesis aims to validate previously-reported genetic biomarkers, identify new markers, determine the mechanistic basis of associated polymorphisms, and expand upon methods in the field. The first three results chapters investigate genetic biomarkers for the prediction of toxicity caused by 5-FU-based treatment, particularly for the 5-FU prodrug capecitabine (Xeloda®, Roche). In the first, a systematic review and meta-analysis is performed for all variants that have been previously studied for an association with toxicity caused by any 5-FU-based drug regimen. 16 studies are analysed, including 36 previously-studied variants. Four variants show strong evidence of affecting a patient’s risk of global (any) 5-FU-related toxicity upon analysis of both the existing data and over 900 patients from the QUASAR2 trial of capecitabine +/- bevacizumab (Avastin®, Roche/Genentech): DPYD 2846, DPYD *2A, TYMS 5’VNTR and TYMS 3’UTR. Next, 1,456 polymorphisms in 25 genes involved in the activation, action or degradation of 5-FU are investigated in 1,046 patients from QUASAR2. At a Bonferroni-corrected p-value threshold of 3.43e-05, three novel associations with capecitabine-related toxicity are identified in DPYD (rs12132152, rs7548189, A551T) and the previously-identified TYMS 5’VNTR and 3’UTR toxicity polymorphisms are refined to a tagging SNP (rs2612091) downstream of TYMS and intronic to the adjacent ENOSF1, the latter of which appears to be functional. Finally, a genome-wide investigation of 4.77 million directly genotyped or imputed SNPs identifies one variant (rs2093152 on chr20) as significantly associated with capecitabine-related diarrhoea (p<5e-08), though no associations meet this threshold for global toxicity. In the study of CRC prognosis, a severe left truncation to the VICTOR trial is defined and shown to probably reduce statistical power but not bias effect estimates. Applying standard and novel genome-wide analysis approaches, a set of 43 SNPs are prioritised for future work. With over one million new CRC cases annually, this work helps define biomarkers that could become broadly applicable in the clinical setting.

In under 6 decades dengue has emerged from South East Asia to become the most widespread arbovirus affecting human populations. Recent dramatic increases in epidemic dengue fever have mainly been attributed to factors such as vector expansion and ongoing ecological, climate and socio-demographic changes. The failure to control the virus in endemic regions and prevent global spread of its mosquito vectors and genetic variants, underlines the urgency to reassess previous research methods, hypotheses and empirical observations. This thesis comprises a set of studies that integrate currently neglected and emerging epidemiological, ecological and evolutionary factors into unified mathematical frameworks, in order to better understand the contemporary population biology of the dengue virus. The observed epidemiological dynamics of dengue are believed to be driven by selective forces emerging from within-host cross-immune reactions during sequential, heterologous infections. However, this hypothesis is mainly supported by modelling approaches that presume all hosts to contribute equally and significantly to the selective effects of cross-immunity both in time and space. In the research presented in this thesis it is shown that the previously proposed effects of cross-immunological reactions are weakened in agent-based modelling approaches, which relax the common deterministic and homogeneous mixing assumptions in host-host and host-pathogen interactions. Crucially, it is shown that within these more detailed models, previously reported universal signatures of dengue's epidemiology and population genetics can be reproduced by demographic and natural stochastic processes alone. While this contrasts with the proposed role of cross-immunity, it presents demographic stochasticity as a parsimonious mechanism that integrates, for the first time, multi-scale features of dengue's population biology. The implications of this research are applicable to many other pathogens, involving challenging new ways of determining the underlying causes of the complex phylodynamics of antigenically diverse pathogens.