Habitat fragmentation: simple models for local persistence and the spread of invasive species.

Understanding the persistence and growth of natural populations in environments subject to random localised change is relevant both to the conservation of threatened species and to the control of invasive species. By developing and analysing simple strategic growth models in environments subject to random fragmentation events, we show that simple approximations can be used to predict invasion speeds and extinction probabilities. The rate and size of fragmentation events interact in a nonlinear way, a finding with important consequences for the efficient control of invasive species. Infrequent, large-scale fragmentation events provide more effective means of control than more frequent, smaller scale efforts.

A structured model for COVID-19 spread: modelling age and healthcare inequities.

We use a stochastic branching process model, structured by age and level of healthcare access, to look at the heterogeneous spread of COVID-19 within a population. We examine the effect of control scenarios targeted at particular groups, such as school closures or social distancing by older people. Although we currently lack detailed empirical data about contact and infection rates between age groups and groups with different levels of healthcare access within New Zealand, these scenarios illustrate how such evidence could be used to inform specific interventions. We find that an increase in the transmission rates among children from reopening schools is unlikely to significantly increase the number of cases, unless this is accompanied by a change in adult behaviour. We also find that there is a risk of undetected outbreaks occurring in communities that have low access to healthcare and that are socially isolated from more privileged communities. The greater the degree of inequity and extent of social segregation, the longer it will take before any outbreaks are detected. A well-established evidence for health inequities, particularly in accessing primary healthcare and testing, indicates that Maori and Pacific peoples are at a higher risk of undetected outbreaks in Aotearoa New Zealand. This highlights the importance of ensuring that community needs for access to healthcare, including early proactive testing, rapid contact tracing and the ability to isolate, are being met equitably. Finally, these scenarios illustrate how information concerning contact and infection rates across different demographic groups may be useful in informing specific policy interventions.

Successful contact tracing systems for COVID-19 rely on effective quarantine and isolation.

Models of contact tracing often over-simplify the effects of quarantine and isolation on disease transmission. We develop a model that allows us to investigate the importance of these factors in reducing the effective reproduction number. We show that the reduction in onward transmission during quarantine and isolation has a bigger effect than tracing coverage on the reproduction number. We also show that intuitively reasonable contact tracing performance indicators, such as the proportion of contacts quarantined before symptom onset, are often not well correlated with the reproduction number. We conclude that provision of support systems to enable people to quarantine and isolate effectively is crucial to the success of contact tracing.

Optimal foraging: Lévy pattern or process?

Many different species have been suggested to forage according to a Lévy walk in which the distribution of step lengths is heavy-tailed. Theoretical research has shown that a Lévy exponent of approximately 2 can provide a higher foraging efficiency than other exponents. In this paper, a composite search model is presented for non-destructive foraging behaviour based on Brownian (i.e. non-heavy-tailed) motion. The model consists of an intensive search phase, followed by an extensive phase, if no food is found in the intensive phase. Quantities commonly observed in the field, such as the distance travelled before finding food and the net displacement in a fixed time interval, are examined and compared with the results of a Lévy walk model. It is shown that it may be very difficult, in practice, to distinguish between the Brownian and the Lévy models on the basis of observed data. A mathematical expression for the optimal time to switch from intensive to extensive search mode is derived, and it is shown that the composite search model provides higher foraging efficiency than the Lévy model.

Optimizing the encounter rate in biological interactions: Ballistic versus Lévy versus Brownian strategies.

Bartumeus et al. [Phys. Rev. Lett. 88, 097901 (2002)] compared the efficiency of a Lévy random walk predator strategy with a Brownian random walk strategy in a periodic one-dimensional domain with nonrevisitable moving targets. Their findings from numerical simulations conclude that "a Lévy search strategy is the best option in some, but not all, cases for a random search process." Using the same methodology, we show that the simplest random search strategy of all, ballistic motion in a random direction, outperforms a Lévy strategy in almost every case. We further show that, in the small set of cases where the ballistic strategy is not optimal, the periodic model does not capture the more realistic nonperiodic case. In the nonperiodic case, the ballistic strategy again outperforms the Lévy strategy.

Atherosclerosis and calcium signalling in endothelial cells.

The link between atherosclerosis and regions of disturbed flow and low wall shear stress is now firmly established, but the causal mechanisms underlying the link are not yet understood. It is now recognised that the endothelium is not simply a passive barrier between the blood and the vessel wall, but plays an active role in maintaining vascular homeostasis and participates in the onset of atherosclerosis. Calcium signalling is one of the principal intracellular signalling mechanisms by which endothelial cells (EC) respond to external stimuli, such as fluid shear stress and ligand binding. Previous studies have separately modelled mass transport of chemical species in the bloodstream and calcium dynamics in EC via the inositol trisphosphate (IP(3)) signalling pathway. We review existing models of these two phenomena, before going on to integrate the two components to provide an inclusive new model for the calcium response of the endothelium in an arbitrary vessel geometry. This enables the combined effects of fluid flow and biochemical stimulation on EC to be investigated and is the first time spatially varying, physiological fluid flow-related environmental factors have been combined with intracellular signalling in a mathematical model. Model results show that low endothelial calcium levels in the area of disturbed flow at an arterial widening may be one contributing factor to the onset of vascular disease.

The role of endothelial calcium and nitric oxide in the localisation of atherosclerosis.

A mathematical model of endothelial cell calcium signalling and nitric oxide synthesis under flow conditions is presented. The model is coupled to two important environmental stimuli for endothelial cells: the frictional shear stress exerted on the cell membrane by the blood flow; and the binding of adenosine triphosphate in the bloodstream to cell surface receptors. These stimuli are closely linked to haemodynamic flow conditions and are, in general, spatially varying, allowing the cellular response in different regions of the endothelium to be evaluated. This is used to indicate which areas of the artery wall experience reduced bioavailability of nitric oxide, which is a major factor in the onset of atherosclerosis. The model thus directly addresses the key issue of the causative link, and its underlying biochemical mechanisms, between incidence of atherosclerosis and regions of low wall shear stress (WSS). Model results show that intracellular levels of free calcium and endothelial nitric oxide synthase are lower in endothelial cells adjacent to a region of recirculating flow than in cells adjacent to regions of fully developed arterial flow. This will lead to deficient levels of nitric oxide in the recirculation zone and hence a potentially elevated risk of developing atherosclerotic plaque. This is consistent with the observed spatial correlation between atherosclerosis and regions of disturbed blood flow and low WSS, and provides a mechanism for the localisation of the disease to sites such as arterial bifurcations and bends.

A model of neurovascular coupling and the BOLD response PART II.

A mathematical model is developed which describes a signalling mechanism of neurovascular coupling with a model of a pyramidal neuron and its corresponding fMRI BOLD response. In the first part of two papers (Part I) we described the integration of the neurovascular coupling unit extended to include a complex neuron model, which includes the important Na/K ATPase pump, with a model that provides a BOLD signal taking its input from the cerebral blood flow and the metabolic rate of oxygen consumption. We showed that this produced a viable signal in terms of initial dip, positive and negative BOLD signals. In this paper (PART II) our model predicts the variations of the BOLD response due to variations in neuronal activity and indicates that the BOLD signal could be used as an initial biomarker for neuronal dysfunction or variations in the perfusion of blood to the cerebral tissue. We have compared the simulated hypoxic BOLD response to experimental BOLD signals observed in the hippocampus during hypoxia showing good agreement. This approach of combined quantitative modelling of neurovascular coupling response and its BOLD response will enable more specific assessment of a brain region.

A model of neurovascular coupling and the BOLD response: PART I.

The mechanisms with which neurons communicate with the vasculature to increase blood flow, termed neurovascular coupling is still unclear primarily due to the complex interactions between many parameters and the difficulty in accessing, monitoring and measuring them in the highly heterogeneous brain. Hence a solid theoretical framework based on existing experimental knowledge is necessary to study the relation between neural activity, the associated vasoactive factors released and their effects on the vasculature. Such a framework should also be related to experimental data so that it can be validated against repetitive experiments and generate verifiable hypothesis. We have developed a mathematical model which describes a signaling mechanism of neurovascular coupling with a model of pyramidal neuron and its corresponding fMRI BOLD response. In the first part of two papers we describe the integration of the neurovascular coupling unit extended to include a complex neuron model, which includes the important Na/K ATPase pump, with a model that provides a BOLD signal taking its input from the cerebral blood flow and the metabolic rate of oxygen consumption. We show that this produces a viable signal in terms of initial dip, positive and negative BOLD signals.

A reinforced random walk model of tumour angiogenesis and anti-angiogenic strategies.

It is now well accepted that the growth of a tumour beyond approximately 2 mm in diameter is dependent on its ability to induce the growth of new blood vessels, a process called angiogenesis. This has raised hope that an anti-angiogenic treatment may be effective in the fight against cancer. Here we formulate, using the theory of reinforced random walks, an individual cell-based mathematical model of tumour angiogenesis in response to a diffusible angiogenic factor. The early stages of angiogenesis, in which endothelial cells (EC) escape the parent vessel and invade the extra-cellular matrix, are included in the model, as are the action of a proteolytic enzyme, EC proliferation and capillary branching and anastomosis. The anti-angiogenic potential of angiostatin, a known inhibitor of angiogenesis, is also examined. The capillary networks predicted by the model are in qualitative agreement with experimental observations. Proteolysis and proliferation are shown to be crucial for vascularization, whilst angiostatin is seen to be capable of limiting capillary growth.

Endothelial nitric oxide synthase and calcium production in arterial geometries: an integrated fluid mechanics/cell model.

It is well known that atherosclerosis occurs at very specific locations throughout the human vasculature, such as arterial bifurcations and bends, all of which are subjected to low wall shear stress. A key player in the pathology of atherosclerosis is the endothelium, controlling the passage of material to and from the artery wall. Endothelial dysfunction refers to the condition where the normal regulation of processes by the endothelium is diminished. In this paper, the blood flow and transport of the low diffusion coefficient species adenosine triphosphate (ATP) are investigated in a variety of arterial geometries: a bifurcation with varying inner angle, and an artery bend. A mathematical model of endothelial calcium and endothelial nitric oxide synthase cellular dynamics is used to investigate spatial variations in the physiology of the endothelium. This model allows assessment of regions of the artery wall deficient in nitric oxide (NO). The models here aim to determine whether 3D flow fields are important in determining ATP concentration and endothelial function. For ATP transport, the effects of a coronary and carotid wave form on mass transport is investigated for low Womersley number. For the carotid, the Womersley number is then increased to determine whether this is an important factor. The results show that regions of low wall shear stress correspond with regions of impaired endothetial nitric oxide synthase signaling, therefore reduced availability of NO. However, experimental work is required to determine if this level is significant. The results also suggest that bifurcation angle is an important factor and acute angle bifurcations are more susceptible to disease than large angle bifurcations. It has been evidenced that complex 3D flow fields play an important role in determining signaling within endothelial cells. Furthermore, the distribution of ATP in blood is highly dependent on secondary flow features. The models here use ATP concentration simulated under steady conditions. This has been evidenced to reproduce essential features of time-averaged ATP concentration over a cardiac cycle for small Womersley numbers. However, when the Womersley number is increased, some differences are observed. Transient variations are overall insignificant, suggesting that spatial variation is more important than temporal. It has been determined that acute angle bifurcations are potentially more susceptible to atherogenesis and steady-state ATP transport reproduces essential features of time-averaged pulsatile transport for small Womersley number. Larger Womersley numbers appear to be an important factor in time-dependent mass transfer.

Dynamic myogenic autoregulation in the rat kidney: a whole-organ model.

A transient 1D mathematical model of whole-organ renal autoregulation in the rat is presented, examining the myogenic response on multiple levels of the renal vasculature. Morphological data derived from micro-CT imaging were employed to divide the vasculature via a Strahler ordering scheme. A previously published model of the myogenic response based on wall tension is expanded and adapted to fit the response of each level, corresponding to a distally dominant resistance distribution with the highest contributions localized to the afferent arterioles and interlobular arteries. The mathematical model was further developed to include the effects of in vivo viscosity variation and flow-induced dilation via endothelial nitric oxide production. Computer simulations of the autoregulatory response to pressure perturbations were examined and compared with experimental data. The model supports the hypothesis that change in circumferential wall tension is the catalyst for the myogenic response. The model provides a basis for examining the steady state and transient characteristics of the whole-organ renal myogenic response in the rat, as well as the modulatory influences of metabolic and hemodynamic factors.

Lattice and non-lattice models of tumour angiogenesis.

In order to progress from the relatively harmless avascular state to the potentially lethal vascular state, solid tumours must induce the growth of new blood vessels from existing ones, a process called angiogenesis. The capillary growth centres around endothelial cells: there are several cell-based models of this process in the literature and these have reproduced some of the key microscopic features of capillary growth. The most common approach is to simulate the movement of leading endothelial cells on a regular lattice. Here, we apply a circular random walk model to the process of angiogenesis, and thus allow the cells to move independently of a lattice; the results display good agreement with empirical observations. We also run simulations of two lattice-based models in order to make a critical comparison of the different modelling approaches. Finally, non-lattice simulations are carried out in the context of a realistic model of tumour angiogenesis, and potential anti-angiogenic strategies are evaluated.

A mathematical model of tumour angiogenesis, regulated by vascular endothelial growth factor and the angiopoietins.

Angiogenesis--the growth of new blood vessels from existing ones--is a prerequisite for the growth of solid tumours beyond a diameter of approximately 2 mm. In recent years, the angiopoietins have emerged as important regulators of angiogenesis. They mediate a delicate balance between vascular quiescence, regression and new growth, but their mechanism of action is not fully understood. This work attempts to provide a mathematical description of the role of the angiopoietins in angiogenesis. The model is formulated within the framework of reinforced random walks, which allows easy transition between the continuum (macroscopic) and discrete (microscopic) forms. Model predictions are in qualitative agreement with experimental observations, and may have implications for anti-cancer therapies based on the prevention of angiogenesis.

The role of the angiopoietins in tumour angiogenesis.

Angiogenesis--the growth of new blood vessels from existing ones--is a prerequisite for the growth of solid tumours beyond a diameter of approximately 2 mm. In recent years, the angiopoietins have emerged as important regulators of angiogenesis. They mediate a delicate balance between vascular quiescence, regression and new growth, but their mechanism of action is not fully understood. This work attempts to provide a mathematical description of the role of the angiopoietins in angiogenesis. The model is formulated within the framework of reinforced random walks, which allows easy transition between the continuum (macroscopic) and discrete (microscopic) forms. Model predictions are in qualitative agreement with experimental observations, and may have implications for antiangiogenic cancer therapies.