G. C. Wake

Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models

Lyapunov functions for classical SIR, SIRS, and SIS epidemiological models are introduced. Global stability of the endemic equilibrium states of the models is thereby established.

The dynamics of a model of a plankton-nutrient interaction

A stability analysis is given for a model of plankton dynamics introduced by Wroblewski et al. (Global Biogeochem. Cycles 2, 199–218, 1988). The detailed dependence of the steady-states and their stability on the various model parameters is explicitly presented and analysed. It is shown that under certain conditions the coexistence of phytoplankton and zooplankton occurs in an orbitally stable oscillatory mode. A distinguished parameter is varied and the steady-states computed. The significance of the lack of stable steady-states leading to periodic population levels is investigated and related to certain oceanographic data.

Critical values for some non-class A geometries in thermal ignition theory

In a previous paper, the authors used a path-following method for the two point boundary value problem governing the ignition of a solid reactant undergoing slow oxidation for symmetric class A geometries and showed the occurrence of multiplicity of steady states. In this paper, the problem is solved in some non-class A geometries (infinite square rod and cube), making use of finite difference discretization of the boundary value problem. It is shown that the multiplicity of steady states changes and that the critical parameters are also different from those found from the shape factor approach.

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.

The distribution of urine deposited on a pasture from grazing animals

Intensive agricultural production practices are known to cause far-reaching effects on water quality. The current paper addresses and quantifies these effects caused by high stocking rates. A set of stochastic difference equations describing the development of the proportion of a grazed field either unaffected by urine deposition, or affected by multiple (1, 2, ...) urine depositions is described. A solution to this set of equations is found for the expected value of multiple (0, 1, 2, ...) urine depositions, and the variances of these depositions. It is assumed that an animal voids urine with a Poisson probability distribution, and that each urine deposition covers a random area with a Gaussian probability density. Given these reasonable assumptions, the probability distributions for each multiplicity of patch distribution can be found numerically. The utility of the results obtained is illustrated for a problem in assessing the nitrogen (N) pollution of ground water from different grazing strategies. It is demonstrated quantitatively that mob stocking (typical of winter management regimes in New Zealand) is often caused by rotational grazing. The latter is often used to optimize grass growth and intake, especially in winter. This increases (more than linearly) the level of N pollution in ground water. This is because of the increased frequency of multiple urine depositions, i.e. more than one urine deposition on the same patch of land in a short time.

A functional differential equation arising in modelling of cell growth

A functional differential equation for the steady size distribution of a population is derived from the usual partial differential equation governing the size distribution, in the particular case where birth occurs by one individual of size x dividing into α new individuals of size x /α. This leads, in the case of constant growth and birth rate functions, to the functional differential equation y ′( x ) = − ay ( x ) + a α y (α x ) together with the integral condition We first look at a number of properties that any solution of this equation and boundary condition must have, and then proceed to find the unique solution by the method of Laplace transforms. Results from number theory on the infinite product found in the solution are presented, and it is shown that y ( x ) tends to a normal distribution as α → 1 + .

Impact of insulin-stimulated glucose removal saturation on dynamic modelling and control of hyperglycaemia

Reported insulin-stimulated glucose removal saturation levels vary widely between individuals and trade off with insulin sensitivity in model-based control methods. A non-linear model and adaptive insulin infusion protocol enabled high-precision blood glucose control in critically ill patients using a constant insulin-stimulated glucose removal saturation parameter. Analysis of clinical trial results with and without saturation modelling indicates the significant impact of this saturation parameter on controller efficacy. Without accounting for saturation, the time-average prediction error during a five-hour trial was up to 17.6%. The average prediction error between the four patients examined in this study was reduced to 5.8% by approximating the saturation parameter. Hence, saturation is an important dynamic that requires good methods of estimation or identification to enable tight glycaemic control.

Global properties of the three-dimensional predator-prey Lotka-Volterra systems

The global properties of the classical three-dimensional Lotka-Volterra two prey-one predator and one prey-two predator systems, under the assumption that competition can be neglected, are analysed with the direct Lyapunov method. It is shown that, except for a pathological case, one species is always driven to extinction, and the system behaves asymptotically as a two-dimensional predator-prey Lotka-Volterra system. The same approach can be easily extended to systems with many prey species and one predator, or many predator species and one prey, and the same conclusion holds. The situation considered is common for New Zealand wild life, where indigenous and introduced species interact with devastating consequences for the indigenous species. According to our results the New Zealand indigenous species are definitely driven to extinction, not only in consequence of unsuccessful competition, but even when competition is absent. This result leads to a better understanding of the mechanism of natural selection, and gives a new insight into pest control practice.

The ignition of hygroscopic combustible materials by water

Abstract The problem of autoignition by the absorption of moisture has been postulated to occur in a number of situations of high practical importance. In this article we examine this phenomenon from a theoretical point of view. The problem is formulated as a combination of classical criticality theory and the critical initial value problem as defined more recently. We shown that the phenomenon of wetting-induced ignition (WII) is possible in certain regions of parameter space, but however close the dry material may have been to criticality before wetting a finite amount of water is required to cause ignition. Equally interesting is the finding, that for a given material, WII is impossible in sufficiently small samples but possible in larger ones. This reveals an important flaw in scaling test procedures.

A mathematical model for pollution in a river and its remediation by aeration

Abstract We present a simple mathematical model for river pollution and investigate the effect of aeration on the degradation of pollutant. The model consists of a pair of coupled reaction–diffusion–advection equations for the pollutant and dissolved oxygen concentrations, respectively. The coupling of these equations occurs because of reactions between oxygen and pollutant to produce harmless compounds. Here we consider the steady-state case in one spatial dimension. For simplified cases the model is solved analytically. We also present a numerical approach to the solution in the general case. The extension to the transient spatial model is relatively straightforward. The study is motivated by the crucial problem of water pollution in many countries and specifically within the Tha Chin River in Thailand. For such real situations, simple models can provide decision support for planning restrictions to be imposed on farming and urban practices.