Modelling leptospirosis in livestock.

New Zealand has one of the highest (per capita) incidences of human leptospirosis in the world. It is the highest occurring occupational disease in New Zealand, often transmitted from livestock such as deer, sheep and cattle to humans. A cyclical model, showing the dynamics of infection of leptospirosis in farmed livestock in New Zealand, is presented. The limit cycle, bifurcation diagram and quasi-R0 value of the system are determined. Leptospire death rate is used as a control parameter. Previously published parameter values are used in a case study to produce figures demonstrating analytical results. The model is used to predict conditions under which the infection will persist in the population.

Solutions to an advanced functional partial differential equation of the pantograph type.

A model for cells structured by size undergoing growth and division leads to an initial boundary value problem that involves a first-order linear partial differential equation with a functional term. Here, size can be interpreted as DNA content or mass. It has been observed experimentally and shown analytically that solutions for arbitrary initial cell distributions are asymptotic as time goes to infinity to a certain solution called the steady size distribution. The full solution to the problem for arbitrary initial distributions, however, is elusive owing to the presence of the functional term and the paucity of solution techniques for such problems. In this paper, we derive a solution to the problem for arbitrary initial cell distributions. The method employed exploits the hyperbolic character of the underlying differential operator, and the advanced nature of the functional argument to reduce the problem to a sequence of simple Cauchy problems. The existence of solutions for arbitrary initial distributions is established along with uniqueness. The asymptotic relationship with the steady size distribution is established, and because the solution is known explicitly, higher-order terms in the asymptotics can be readily obtained.

A new, improved and generalizable approach for the analysis of biological data generated by -omic platforms.

The principles embodied by the Developmental Origins of Health and Disease (DOHaD) view of 'life history' trajectory are increasingly underpinned by biological data arising from molecular-based epigenomic and transcriptomic studies. Although a number of 'omic' platforms are now routinely and widely used in biology and medicine, data generation is frequently confounded by a frequency distribution in the measurement error (an inherent feature of the chemistry and physics of the measurement process), which adversely affect the accuracy of estimation and thus, the inference of relationships to other biological measures such as phenotype. Based on empirical derived data, we have previously derived a probability density function to capture such errors and thus improve the confidence of estimation and inference based on such data. Here we use published open source data sets to calculate parameter values relevant to the most widely used epigenomic and transcriptomic technologies Then by using our own data sets, we illustrate the benefits of this approach by specific application, to measurement of DNA methylation in this instance, in cases where levels of methylation at specific genomic sites represents either (1) a response variable or (2) an independent variable. Further, we extend this formulation to consideration of the 'bivariate' case, in which the co-dependency of methylation levels at two distinct genomic sites is tested for biological significance. These tools not only allow greater accuracy of measurement and improved confidence of functional inference, but in the case of epigenomic data at least, also reveal otherwise cryptic information.

The application of a model of glucose and insulin dynamics to explain an observed effect of leptin administration in reversal of developmental programming.

A dynamical model describing the glucose-insulin physiological system was applied to an experiment on the administration of the adipokine leptin between neonatal days 3 and 13 to rats whose dams were subject to different levels of nutrition during gestation. The effect of leptin treatment on the glucose-insulin equilibrium point and on the levels of other associated metabolites showed a significant change in direction that depended on the level of prenatal nutrition. Leptin has been shown to affect two factors that affect the equilibrium levels of glucose and insulin, gluconeogenesis and insulin sensitivity. Each effect is described by a parameter in the dynamical model. Mathematical analysis shows that changes in these parameters in the manner promoted by leptin would indeed increase or decrease the glucose-insulin equilibria depending on their initial equilibrium levels which might be induced by the level of prenatal nutrition. This analysis explains the results of the leptin experiment in the context of the dynamics of the glucocorticoid system. It also proposes a physiological mechanism for the expression of plasticity in the organism based on the status of the glucose and insulin equilibria.

On a cell-growth model for plankton.

The frequency distribution of diatoms (microscopic unicellular alga with silicified cell-walls, found as plankton) is shown to evolve in time as a steady-size distribution with constant shape, scaled by time. This distribution is preserved when the division occurs at a fixed size into two daughter cells of half-size. In cases where the parameters for growth, division frequency, dispersion and mortality are constants, the frequency distributions can be found explicitly and thus provide a benchmark for computations in more complex cases.

Analysis of a predator-prey system with predator switching.

In this paper, we consider an interaction of prey and predator species where prey species have the ability of group defence. Thresholds, equilibria and stabilities are determined for the system of ordinary differential equations. Taking carrying capacity as a bifurcation parameter, it is shown that a Hopf bifurcation can occur implying that if the carrying capacity is made sufficiently large by enrichment of the environment, the model predicts the eventual extinction of the predator providing strong support for the so-called 'paradox of enrichment'.

Adaptive bolus-based set-point regulation of hyperglycemia in critical care.

Critically ill patients are often hyperglycemic and extremely diverse in their dynamics. Consequently, fixed protocols and sliding scales can result in error and poor control. A two-compartment glucose-insulin system model that accounts for time-varying insulin sensitivity and endogenous glucose removal, along with two different saturation kinetics is developed and verified in proof-of-concept clinical trials for adaptive control of hyperglycemia. The adaptive control algorithm monitors the physiological status of a critically ill patient, allowing real-time tight glycemic regulation. The bolus-based insulin administration approach is shown to result in safe, targeted stepwise glycemic reduction for three critically ill patients.

Modeling the interaction of milking frequency and nutrition on mammary gland growth and lactation.

A mathematical model of biological mechanisms regulating lactation is constructed. In particular, the model allows prediction of the effect of milking frequency on milk yield and mammary regression, and the interaction of nutrition and milking frequency in determining yield. Possible interactions of nutrition with milking frequency on alveolar dynamics are highlighted. The model is based upon the association of prolonged engorgement (as a consequence of milk accumulation) of active secretory alveoli with changes in gene expression that result in impairment and, ultimately, cessation of milk secretion. The emptying of alveoli at milking, following alveolar contraction induced by oxytocin, prevents this process and also allows quiescent alveoli to reactivate. Prolonged engorgement results in apoptosis of the secretory cells and, hence, regression of the mammary gland. Milk yield is linked to alveolar populations, with secretion rates being modulated by nutrition and udder fill effects. The model was used to investigate different management scenarios, and is in agreement with experimental results. The model shows that while milking frequency drives alveolar population, and therefore potential milk production, actual production varies considerably with nutrition. A significant portion of the loss associated with once-daily milking was due to udder fill rather than loss of secretory tissue. The model showed qualitative agreement with experimental data, on the acute and chronic effects of temporary once-daily milking.

Rethinking sedation and agitation management in critical illness.

OBJECTIVE: To examine difficulties in sedation management in the critically ill patient and explore how a semi automated sedation controller can improve agitation control. To present recent work on measurements of agitation, dynamic systems modelling and control of patient agitation response. DATA SOURCES: Articles and peer-reviewed studies identified through a PUBMED search and selected original works from the biomedical engineering literature of relevance to agitation control and management. SUMMARY OF REVIEW: Over-sedation has an adverse impact on intensive care resources. Interventions to constrain sedation delivery through development of protocols or regular cessation of infusions result in reduction in resource utilisation, but have not significantly addressed existing difficulties in agitation control. We develop a paradigm in which control of agitation in critically ill patients becomes the primary objective of sedation management. This principle is central to the function of a nurse-managed semi-automated sedation delivery device. The clinical application of this device using subjective assessments of agitation is presented. A framework for the development of improved automated sedation delivery systems using objective measurements of agitation and control, based on agitation feedback, is described. Using dynamic systems modelling and a simulated nurse, a bolus-driven approach significantly reduced agitation and minimised drug utilisation. This result challenges the current practice of sedating patients using continuous infusions. CONCLUSIONS: A simple computerised interface with an algorithm that continually reduces the infusion rate in the absence of agitation has successfully been introduced into clinical practice. Nursing staff reported high levels of satisfaction with this device and it has enabled detailed data on patterns of sedation administration to be extracted for analysis. This data has been used to validate a model of the fundamental agitation-sedation dynamics.

Active insulin infusion using optimal and derivative-weighted control.

Close control of blood glucose levels significantly reduces vascular complications in Type I diabetes. A control method for the automation of insulin infusion that utilizes emerging technologies in blood glucose biosensors is presented. The controller developed provides tighter, more optimal control of blood glucose levels, while accounting for variation in patient response, insulin employed and sensor bandwidth. Particular emphasis is placed on controller simplicity and robustness necessary for medical devices and implants.A PD controller with heavy emphasis on the derivative term is found to outperform the typically used proportional-weighted controllers in glucose tolerance and multi-meal tests. Simulation results show reductions of over 50% in the magnitude and duration of blood glucose excursions from basal levels. A closed-form steady state optimal solution is also developed as a benchmark, and results in a flat glucose response. The impact and trade-offs associated with sensor bandwidth, sensor lag and proportional versus derivative-based control methods are illustrated. Overall, emerging blood glucose sensor technologies that enable frequent measurement are shown to enable more effective, automated control of blood glucose levels within a tight, acceptable range for Type I and II diabetic individuals.

Influence of milking frequency on mammary gland dynamics.

The effects of milking frequency on milk production is a key question for the dairy industry. Milk production is related to the number of active alveoli in the mammary gland and movement between active and quiescent alveolar pools is influenced by the milking frequency. In this paper, we analyse a mechanistic model based on known biological inputs that describes the effect of milking frequency on the alveolar composition of the mammary gland. It is shown that the model can qualitatively reproduce the correct alveolar dynamics. We also investigate the model robustness and parameter sensitivity. Additionally, by making the plausible assumption that the senescence rate of alveoli is proportional to the number of quiescent alveoli present, we obtain an analytical solution requiring periodic resetting.

A differential-delay model of pasture accumulation and loss in controlled grazing systems.

A grazing population dynamics model is proposed where organisms in a grazed population have a fixed life span. The motivating context is that of ruminants grazing grass-dominant pasture. The model takes the form of a differential-delay equation in which the rate of loss of pasture due to senescence at some time depends on the rate at which leaves are reaching maturity at that time. Comparisons are made with data from a continuous grazing experiment due to Bircham and Hodgson (Grass and Forage Science, 38:323-331, 1985), leading to a prediction of 21.9 days for herbage life span. Predictions of herbage utilization are consistent with measured data. The model predicts lower senescence in swards in regrowth than in grazed swards at the same herbage mass. Solutions and equilibria are obtained for the linear form of the model with continuous grazing pressure. Solutions and bounds are obtained for the linear model with intermittent grazing pressure, and its usefulness in modeling grazed pastures is discussed. A delay model is a simple but powerful means of including the concept of fixed herbage life span in grazing modeling. Questions of herbage life span and percentage utilization are naturally contained in the mechanism of a differential-delay model. There are not so well handled by models that treat senescence of herbage empirically.

The regulation of an age-structured population by a fatal disease.

A model describing the effect of a fatal disease on an age-structured population which would otherwise grow is presented and analysed. If the disease is capable of regulating host numbers, there is an endemic steady age distribution (SAD), for which an analytic expression is obtained under some simplifying assumptions. The ability of the disease to regulate the population depends on a parameter R(alpha), which is defined in terms of the given age-dependent birth and death rates, and where alpha is the age-dependent disease-induced death rate. If R(alpha) < 1 the endemic SAD is attained, while R(alpha) > 1 means the disease cannot control the population's size. The number R(0) is the expected number of offspring produced by each individual in the absence of the disease; for a growing population we require R(0) > 1. A stability analysis is also performed and it is conjectured that the endemic SAD is locally asymptotically stable whenever it is attained. This is demonstrated explicitly for a very simple example where all rates are taken as constant.

Thresholds and stability analysis of models for the spatial spread of a fatal disease.

A simple two-class (susceptibles and infectives) model describing the dynamics of a fatal disease in a variable-size population is presented and analysed. Spatial dependence is introduced into the model by considering two different mechanisms for the geographic spread of the disease: nonlocal interaction between susceptibles and infectives, and migratory spread of the animals. The steady states and their stability for these spatially dependent models are deduced; no spatially heterogeneous steady states were possible. For nonlocal interaction, there were two spatially uniform steady states: the trivial state (no infectives or susceptibles), which was unstable, and the endemic state (constant proportion of the population infected), which was locally asymptotically stable. With migratory spread, the number of spatially uniform steady states was dependent on the boundary conditions imposed. With hostile (Dirichlet) boundary conditions, only the trivial steady state was possible and its local stability found to depend on the rate of diffusion of the total population. With no-flux (Neumann) boundary conditions, the steady states are the trivial and endemic states; these were unstable and locally asymptotically stable, respectively.

Mathematical analysis of perifusion data: models predicting elution concentration.

System models are constructed and analyzed for combined convective flow and for dispersion in distorting concentrations of a chemical signal as it passes through a packed column. We derive general analytical solutions for these models. The results have applications to analyses such as in biological experiments involving hormonal stimulation of perifused cells, elution chromatography, adsorption columns, and studies of groundwater flow. The models reveal that the column distorts an incoming signal (such as a change in solute concentration in the flowing liquid) at the inlet. This distortion is greatest at low values of the Peclet number of the flow and is small at larger values. We explore the effects of the approximations inherent in the mathematical models of the system. Specification of the boundary conditions of the problem are shown to be particularly important. With the use of incorrect models, it is possible to obtain accurate interpolations to data obtained from perfusion experiments. However, the parameters derived (in particular the dispersion constant and the peak concentration of a solute concentration pulse) may be considerably in error. This may lead to errors when these parameter estimates are used to predict results in other experimental situations.