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Exploring possible causes of a dilution effect using agent-based models
(Massey University, 2024-12-07) Archer, Jason
The relationship between infectious diseases and ecosystems has become a much-discussed topic, with many emerging infectious diseases originating in animal reservoirs. A controversial subject within ecological epidemiology is the dilution effect. It supposes a negative relationship exists between a given ecosystem’s biodiversity and the risk of infectious disease transmission. An appealing prospect in theory, the dilution effect provides a public health incentive for conservation. This idea is particularly relevant for infectious diseases caused by pathogens circulated within natural reservoirs, such as influenza and ebolaviruses. The converse phenomenon, where biodiversity increases the risk, is known as the amplification effect. The debate over the dilution effect has been fervent, particularly surrounding its generalisability and scale dependence. We address these concerns by combining popular Ordinary Differential Equation (ODE) models with two classes of agent-based models, one set on a lattice and the other spatially explicit. Agent-based models are uncommon in ecoepidemiology and have yet to be applied to the topic of dilution. We apply existing methods for quantifying the dilution effect in ODEs to numerical data from simulations and highlight circumstances where observations of dilution and amplification are sensitive to the selected definitions of biodiversity and infection risk. We find that a lattice-based approach is well-suited to capturing spatiotemporal dynamics on longer time scales, while the spatially explicit method effectively describes outbreaks on shorter time scales. We also use a lattice-based model to explore the possible mechanisms of the dilution effect in horticulture. We also highlight two infectious diseases to analyse in more detail: toxoplasmosis and Lyme disease. For toxoplasmosis, we discuss how pathogen influence on prey behaviour can affect infection risk, showing an amplification effect for an example system. With Lyme disease, we discuss the influence of a vector and show how control strategies associated with long-term mitigation of risk can cause a short-term increase in infection. We also show the diluting influence of adding a less competent host, concluding that the dilution effect is not a general phenomenon but a product of scale and the unique properties of individual infections and ecologies.
Bayesian models of age and growth in sharks : a thesis submitted in partial fulfilment of the requirements for the degree of Master of Information Sciences, School of Mathematical and Computational Sciences, Massey University, Auckland, New Zealand
(Massey University, 2023) Bristow, James
Marine ecosystems are under increasing pressure from factors such as overfishing, that have lead to reported declines in chondrichthyan populations across the globe. Approximately 80% of New Zealand chondrichthyian species have no specific management or monitoring, and a lack of species-specific information has led to uncertainty concerning population trends over time. Biological parameters such as growth rates are typically incorporated into fisheries growth models, the most common of which is the von Bertalanffy growth model (VBGM). The common VBGM is monophasic, and is therefore a continuous and monotone curve over all stages of growth. The VBGM has received some criticism concerning the assumption of monophasic growth. One critique is that a single curve fails to account for changes in energy reallocation and growth that occur during the transition between the juvenile and mature life stages of the shark. Conversely, a biphasic growth model may better model the patterns of both juvenile and mature sharks. The biphasic VBGM (BVBGM) has offered promising results when applied to chondrichthyans in previous growth studies. Estimating the ages of chondrichthyans is typically performed by trained readers who count the growth bands deposited within the vertebral centra. However, the reading of vertebrae is subjective, error-prone, and time consuming. While vertebral band pairs are the most commonly used structure for age estimation, there are various sources of bias and uncertainty inherent to the reading process. Deep learning models such as convolutional neural networks (CNNs) have demonstrated promise for the automated interpretation of bony fish otolith growth zones, though the application of CNNs for the automated reading of chondrichthyan vertebrae has yet to be explored. The principal goal of this thesis is to advance methodologies for estimating the growth parameters of sharks via the application of Bayesian models, thereby offering more robust management and conservation of chondrichthyes against various factors such as overfishing. We first explored the potential of biphasic growth models on five species of New Zealand chondrichthyes: Centrophorus squamosus; Isurus oxyrinchus; Lamna nasus; Mustelus lenticulatus; and Prionace glauca. We compared two monophasic and two biphasic growth models using the Pareto-smoothed importance sampling approximation of leave-one-out cross-validation (PSIS-LOO) metric. Biphasic models appeared to provide superior fit for both males and females in the majority of cases, and we were able to improve upon prior examples from the literature where parameter estimates were noted to be biased or poor. Our overall results demonstrated that the popular monophasic VBGM should not be chosen a priori as the only candidate model to describe the growth of chondrichthyans. Instead, we should consider multiple alternative growth models, as informed by statistical evidence and domain expertise. We next explored the feasibility of automating the age estimation of chondrichthyans by training CNNs on an image dataset of Isurus oxyrinchus vertebrae. We evaluated three Bayesian deep learning methods for uncertainty quantification: DeepEnsembling, mixture of Laplace approximations (MoLA), and multi-stochastic weight average Gaussian (MultiSWAG) in terms of predictive power and model calibration. We found that MultiSWAG offered marginally superior predictive performance and model calibration relative to DeepEnsembling and MoLA. Moreover, predictions produced by MultiSWAG typically closely matched the estimates provided by human readers, though the performance of these deep learning models tended to degrade for older age classes. We argue that our results demonstrate promise for emulating trained readers, leading to potential efficiency gains and cost savings. However, we note that there is a lack of evidence that our models are directly counting the bands of the vertebrae, and that further refinement of our CNNs may be required. Our findings have demonstrated the ability of Bayesian methods to perform principled uncertainty quantification and parameter estimation within the context of age and growth modelling. Our Bayesian growth models were able to quantify the epistemic uncertainty of our parameter estimates, offering more robust estimation of growth parameters and superior model fit. Additionally, we were able to incorporate prior information into our growth models, as informed by the available literature and domain expertise. Likewise, the application of Bayesian deep learning facilitated the quantification of epistemic and aleotoric uncertainty for our age estimates, while also offering superior predictive performance and well-calibrated prediction intervals. We showed that Bayesian CNNs could be used to efficiently automate the interpretation of vertebral growth bands for the purposes of age estimation. This study has contributed to the research of New Zealand sharks, marine conservation, and fisheries management by improving methods used to measure and interpret growth parameters. We hope it contributes to the management and persistence of these species for future generations.
Mathematical modelling of heart rate and blood pressure regulation : a thesis presented in partial fulfilment of the requirements for the degree of Master of Science in Mathematics, School of Natural and Computational Sciences, Massey University
(Massey University, 2021) Gibbs, Alexander
A mathematical model of Heart Rate (HR) control has been expanded to include a mechanical model of the heart. The aim is to better understand the significance of Respiratory Sinus Arrhythmia (RSA), a variation of HR with respiration. RSA is found in several species but the physiological benefits and source of it are still under debate. A recently developed model of heart dynamics has been integrated into a model of HR control and gas exchange. HR is assumed to be primarily affected by the parasympathetic signal, with the sympathetic signal taken as a constant in the model. The parasympathetic signal is assumed to be affected by mechanical feedback from the lungs, direct modulation by the respiratory drive, and feedback from the baroreceptors (blood pressure sensors). The inclusion of a mechanical heart model allowed us to better represent the blood pressure in the HR control model and test two hypotheses regarding the function and source of RSA. Our study confirms that the main source of RSA is central modulation of heart rate. However, more work is needed to confirm using this model the hypothesis that RSA minimizes the work done by the heart while maintaining physiological levels of CO2.
Mathematical modelling of the cardiovascular system to study the effects of respiratory sinus arrhythmia and heart failure : this dissertation is submitted for the degree of Doctor of Philosophy, School of Natural and Computational Science, Massey University
(Massey University, 2021) Noreen, Shumaila
This thesis presents the development of lumped parameter models of the cardiovascular system with a specific aim of simulating the system dynamics over a range of heart rates. The models contain several new modelling features that have been introduced progressively throughout the thesis starting with isolated models and continuing with closed loop models of the circulation. Specifically, the contraction of the cardiac chambers is modelled using a time-dependent muscle force with constant elasticity instead of time dependent elasticity. A new hypothesis about the mechanical contraction of the atria generates realistic pressure volume loops. The inter-ventricular interaction is modelled as well. Additionally, hysteresis is incorporated in the aortic valve to produce an end-systolic reverse (negative) ﬂow. Most of the model parameters were taken from the literature and experimental data. Sensitivity analysis was performed on one of the models outputs by changing one parameter at a time; this analysis indicated that the total blood volume is the most influential parameter in the model. The developed models were used to study the effects of Respiratory Sinus Arrhythmia (RSA), variability in heart rate at the frequency of breathing. RSA is an indicator of good health but the mechanism that gives rise to RSA and its function are still debatable. Two potential sources of RSA were incorporated: periodic heart rate that mimics the central regulation of heart rate which originates in the brainstem, and periodic systemic veins resistance that mimics one possible effect of the pleural pressure which drives breathing. The effects of RSA on cardiac output were then studied. The simulations suggest that the mean cardiac output does not change significantly due to RSA at either low or high heart rates. Two types of heart failure were simulated using the new models by changing certain model parameters: systolic and diastolic. Both the systolic and diastolic heart failures caused an accumulation of blood in the lungs. The ejection fraction for diastolic heart failure remained within the normal physiological range while in the case of systolic heart failure the ejection fraction reduced rapidly. These results are consistent with physiological observations.
A multi-compartmental mathematical model of the postprandial human stomach : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Anatomy and Physiology at Massey University, Palmerston North, New Zealand
(Massey University, 2020) Mary Vijay, Nikhila
Computational fluid dynamics of the human stomach helps to understand the gastric processes such as trituration, mixing, and transit of digesta. Their outcomes give greater insight into the design of food and orally dosed drug delivery system. Current models of gastric contractile activity are primarily limited to the gastric antrum and assume global values for the various physiological characteristics. This thesis developed a unified compartmental gastric model with correctly informed anatomical and physiological data. The gastric geometry incorporated the actions of multiple compartments, such as the gastric fundus, body, antrum, pyloric canal, proximal duodenal cap, and the small intestinal brake. Lattice-Boltzmann Method (LBM) is used to simulate the fluid dynamics within the stomach. This thesis quantified the effects of transgastric pressure gradient (TGPG) between the fundus and the duodenum, the effect of antral propagating contraction (APC) amplitude, and the viscosity of the gastric contents on gastric flow, mixing, and gastric emptying. The results of this work suggest that TGPG influences gastric emptying where as APCs do not play major role in gastric emptying. Flow rate without TGPG obtained in this work agrees with previous work (Pal et al., 2004); however, it is higher in the presence of a TGPG. Results show that APCs promote recirculation, and the amplitude of APC is vital in this regard. The 'pendulating' flow of gastric content observed in this work is reported previously in duplex sonography experiments (Hausken et al., 1992). This work quantified the gastric shear rates (0.6 - 2.0 /s). This work also suggests that the viscosity of the content influences gastric fluid dynamics. This work is a simplified first step towards a 3D gastric model. Hence, these simulation studies were performed under two simplifications: dimensionality and rheology, i.e., we have assumed a Newtonian fluid flow in 2D gastric geometry. A 3D gastric model with more rheologically realistic fluid to explore the pseudoplastic fluid dynamics within the stomach in the future is recommended.
Pattern formation in electrically coupled pacemaker cells : a thesis presented in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Manawatū, New Zealand
(Massey University, 2020) Fatoyinbo, Hammed
In this thesis we study electrical activity in smooth muscle cells in the absence of external stimulation. The main goal is to analyse a reaction-diffusion system that models the dynamical behaviour where adjacent cells are coupled through passive electrical coupling. We first analyse the dynamics of an isolated muscle cell for which the model consists of three first-order ordinary differential equations. The cell is either excitable, nonexcitable, or oscillatory depending on the model parameters. To understand this we reduce the model to two equations, nondimensionalise, then perform a detailed numerical bifurcation analysis of the nondimensionalised model. One parameter bifurcation diagrams reveal that even though there is no external stimulus the cell can exhibit two fundamentally distinct types of excitability. By computing two-parameter bifurcation diagrams we are able to explain how the cell transitions between the two types of excitability as parameters are varied. We then study the full reaction-diffusion system first through numerical integration. We show that the system is capable of exhibiting a wide variety of spatiotemporal behaviours such as travelling pulses, travelling fronts, and spatiotemporal chaos. Through a linear stability analysis we are able to show that the spatiotemporal patterns are not due to diffusion-driven instability as is often the case for reaction-diffusion systems. It is as a consequence of the nonlinear dynamics of the reaction terms and coupling effect of diffusion. The precise mechanism is not yet well understood, this will be subject of future work. We then examine travelling wave solutions in detail. In particular we show how they relate to homoclinic and heteroclinic solutions in travelling wave coordinates. Finally we review spectral stability analysis for travelling waves and compute the essential spectrum of travelling waves in our system.
Comparison of oxygen transfer in three types of gas exchangers : this thesis is submitted in partial fulfillment of requirements for the degree of Master of Science in Mathematics, School of Natural and Computational Sciences, Massey University, New Zealand
(Massey University, 2020) Zhao, Run
The avian respiratory system is very different from that of mammals. In birds, air flows through the gas exchange area in one direction while in mammals it flows in two directions - in and out of the gas exchange area. It has been hypothesised that gas exchange occurs more efficiently in avian lungs than in mammalian lungs due to the difference in structure. We test this hypothesis by comparing oxygen exchange in three types of gas exchangers. First, we examine how gas exchange occurs in mammalian lungs, using a well-mixed stationary container of air with blood flowing along one axis. Next, we investigate the case where air flows in the opposite direction to the blood, which is similar to the gas exchange mechanism seen in fish. We then analyse how gas exchange occurs in the avian respiratory system, where the blood flow is perpendicular to the airflow. We compare the results under normal and extreme conditions and conclude that the avian respiratory system exhibits significantly higher gas exchange efficiency compared to mammals, ultimately enabling birds to live in environments where mammals could not survive.
Is our breathing optimal? : this dissertation is submitted for the degree of Doctor of Philosophy, School of Natural and Computational Sciences, Massey University, New Zealand
(Massey University, 2019) Zaidi, Syed Muhammad Faheem
One of the open questions in relation to the control of amplitude and frequency of breathing is why a particular pattern of breathing is observed. This thesis explores the hypothesis that the particular combination of breathing frequency and amplitude realised, is optimal with respect to some objective function. Several objective functions have been suggested in the literature, such as the rate of work during inhalation, the average force exerted by the respiratory muscles, and the weighted sum of volumetric acceleration and work during inhalation; all of these objective functions were studied using 1D models and all provided physiologically acceptable minima under normal conditions. The thesis investigates optimal solutions of mathematical models that range from 2D to 6D and reflect more accurately the coupling between lung mechanics and gas exchange. It shows how published 6D and 5D models can be reduced to new 3D and 2D models. At its simplest, the 2D model consists of two piecewise linear differential equations. The use of higher dimension models require a new definition of the optimization problem as minimizing a given objective function subject to several constraints, such as satisfying the differential equations and maintaining one of the variables at a given average value. The optimal problem can be solved analytically in the case of the simplest 2D model, using concepts from optimal control theory. The analytical solution is used to verify a numerical algorithm that is then used to solve the more complex models. Solutions of the optimization problem for the different objective functions, previously suggested in the literature have been calculated. In all the optimal solutions found in this thesis, the duration of inhalation is equal to the duration of exhalation. However, under normal conditions, the time duration of inhalation is expected to be shorter than that of exhalation. This might be resolved by imposing additional constraints or by proposing a different hypothesis to explain why a particular pattern of breathing is observed.
Dynamical effects of degree correlations in networks of type I model neurons : a dissertation presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Auckland, New Zealand
(Massey University, 2020) Bläsche, Christian
The complex behaviour of human brains arises from the complex interconnection of the well-known building blocks -- neurons. With novel imaging techniques it is possible to monitor firing patterns and link them to brain function or dysfunction. How the network structure affects neuronal activity is, however, poorly understood. In this thesis we study the effects of degree correlations in recurrent neuronal networks on self-sustained activity patterns. Firstly, we focus on correlations between the in- and out-degrees of individual neurons. By using Theta Neurons and Ott/Antonsen theory, we can derive a set of coupled differential equations for the expected dynamics of neurons with equal in-degree. A Gaussian copula is used to introduce correlations between a neuron’s in- and out-degree, and numerical bifurcation analysis is used determine the effects of these correlations on the network's dynamics. We find that positive correlations increase the mean firing rate, while negative correlations have the opposite effect. Secondly, we turn to degree correlations between neurons -- often referred to as degree assortativity -- which describes the increased or decreased probability of connecting two neurons based on their in-or out-degrees, relative to what would be expected by chance. We present an alternative derivation of coarse-grained degree mean field equations utilising Theta Neurons and the Ott/Antonsen ansatz as well, but incorporate actual adjacency matrices. Families of degree connectivity matrices are parametrised by assortativity coefficients and subsequently reduced by singular value decomposition. Thus, we efficiently perform numerical bifurcation analysis on a set of coarse-grained equations. To our best knowledge, this is the first time a study examines the four possible types of degree assortativity separately, showing that two have no effect on the networks' dynamics, while the other two can have a significant effect.
Mathematical modelling of microbial cross-feeding on hydrogen in the human colon: a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Nutritional Science at Massey University, Palmerston North, New Zealand
(Massey University, 2020) Smith, Nick
The microbial population of the human colon contains three subgroups that cross-feed on hydrogen produced during microbial metabolism of carbohydrates: methanogens, sulphate-reducing bacteria (SRB) and reductive acetogens. These microbes and their activities have been linked to various host physiological and health outcomes. This thesis aimed to construct mathematical models for the growth and metabolism of colonic hydrogenotrophs to investigate key factors in hydrogenotroph metabolism and population dynamics that would be difficult to study experimentally. Monoculture models based on Monod kinetics were developed for Methanobrevibacter smithii, Desulfovibrio vulgaris, and Blautia hydrogenotrophica, as representatives of colonic methanogens, SRB and reductive acetogens. The models were parameterised and validated using experimental data. The monoculture models were combined to examine interactions between these microbes, before incorporation into an existing microbial community model, microPop. Adaptations were made to microPop to enable simulation of the colonic environment, investigating the role of hydrogenotrophs in the colon. The D. vulgaris model provided similarly accurate predictions to an existing thermodynamics-based model. The B. hydrogenotrophica model estimated a hydrogen uptake threshold of 86 mM and provided supportive evidence for the confounding effect of growth media on reductive acetogenesis. Growth yield parameters for SRB and methanogenic strains were reduced in co-culture compared to monoculture, while tri-culture modelling identified conditions necessary for the survival of each hydrogenotroph. Substrate competition prevented survival of all three together in continuous culture. The community model predicted colonic pH, short chain fatty acid gradient and dominant microbial groups but could not accurately predict other experimental metabolite and microbial abundance measurements. Investigating the role of colonic sulphate availability showed contrasting predictions: sulphate availability positively correlated with SRB and sulphide concentrations and negatively correlated with methanogen abundance using a continuous representation of the colon, but no effect was predicted using a compartmental representation. This research demonstrates that modelling can extract additional information from existing experimental data. The community model provides a basis for the computational study of the microbiota and hydrogen cross-feeding dynamics in the colon, which can complement or even accelerate experimental research on the influences of the microbiome on the host.

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