Massey Documents by Type

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How immune dynamics shape multi-season epidemics: a continuous-discrete model in one dimensional antigenic space.
(Springer, 2024-03-27) Roberts MG; Hickson RI; McCaw JM
We extend a previously published model for the dynamics of a single strain of an influenza-like infection. The model incorporates a waning acquired immunity to infection and punctuated antigenic drift of the virus, employing a set of coupled integral equations within a season and a discrete map between seasons. The long term behaviour of the model is demonstrated by examples where immunity to infection depends on the time since a host was last infected, and where immunity depends on the number of times that a host has been infected. The first scenario leads to complicated dynamics in some regions of parameter space, and to regions of parameter space with more than one attractor. The second scenario leads to a stable fixed point, corresponding to an identical epidemic each season. We also examine the model with both paradigms in combination, almost always but not exclusively observing a stable fixed point or periodic solution. Adding stochastic perturbations to the between season map fails to destroy the model's qualitative dynamics. Our results suggest that if the level of host immunity depends on the elapsed time since the last infection then the epidemiological dynamics may be unpredictable.
Optimal nutritional intake for fetal growth
(American Institute of Mathematical Sciences, 2011) Kiataramkul C; Wake G; Ben Tal A; Lenbury Y
The regular nutritional intake of an expectant mother clearly affects the weight development of the fetus. Assuming the growth of the fetus follows a deterministic growth law, like a logistic equation, albeit dependent on the nutritional intake, the ideal solution is usually determined by the birthweight being pre-assigned, for example, as a percentage of the mother⿿s average weight. This problem can then be specified as an optimal control problem with the daily intake as the control, which appears in a Michaelis-Menten relationship, for which there are well-developed procedures to follow. The best solution is determined by requiring minimum total intake under which the preassigned birth weight is reached. The algorithm has been generalized to the case where the fetal weight depends in a detailed way on the cumulative intake, suitably discounted according to the history. The optimality system is derived and then solved numerically using an iterative method for the specific values of parameter. The procedure is generic and can be adapted to any growth law and any parameterisation obtained by the detailed physiology
Analysis of a dynamical system of animal growth and composition : a thesis submitted in partial fulfilment of the requirements for the degree of Master of Science in Mathematics at Massey University, Albany, New Zealand
(Massey University, 2010) Abdul Latif, Nurul Syaza
This thesis investigates the analysis of the extended model of animal growth proposed by Oliviera et al (personal communication, July 2009). This mechanistic model of animal growth based on a detailed representation of energy dynamics focussing on the interaction between four compartment of body composition; nutrient level, fat content, visceral protein and non-visceral protein. The model is mathematically analysed and the behaviour of the model for different feeding level is examined. The animal growth model exhibits thresholds typical of nonlinear systems and multiple stable steady states which have distinct basins of stability which depend on the value of the large number of physiologically-determined parameters. These have not been previously explored theoretically and these are done in this thesis. The model demonstrates richer behaviour where path-following techniques are used to explore the distribution in parameter space of the varying phenomenology.
Pattern formation in a neural field model : a thesis presented in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Auckland, New Zealand
(Massey University, 2008) Elvin, Amanda Jane
In this thesis I study the effects of gap junctions on pattern formation in a neural field model for working memory. I review known results for the base model (the “Amari model”), then see how the results change for the “gap junction model”. I find steady states of both models analytically and numerically, using lateral inhibition with a step firing rate function, and a decaying oscillatory coupling function with a smooth firing rate function. Steady states are homoclinic orbits to the fixed point at the origin. I also use a method of piecewise construction of solutions by deriving an ordinary differential equation from the partial integro-differential formulation of the model. Solutions are found numerically using AUTO and my own continuation code in MATLAB. Given an appropriate level of threshold, as the firing rate function steepens, the solution curve becomes discontinuous and stable homoclinic orbits no longer exist in a region of parameter space. These results have not been described previously in the literature. Taking a phase space approach, the Amari model is written as a four-dimensional, reversible Hamiltonian system. I develop a numerical technique for finding both symmetric and asymmetric homoclinic orbits. I discover a small separate solution curve that causes the main curve to break as the firing rate function steepens and show there is a global bifurcation. The small curve and the global bifurcation have not been reported previously in the literature. Through the use of travelling fronts and construction of an Evans function, I show the existence of stable heteroclinic orbits. I also find asymmetric steady state solutions using other numerical techniques. Various methods of determining the stability of solutions are presented, including a method of eigenvalue analysis that I develop. I then find both stable and transient Turing structures in one and two spatial dimensions, as well as a Type-I intermittency. To my knowledge, this is the first time transient Turing structures have been found in a neural field model. In the Appendix, I outline numerical integration schemes, the pseudo-arclength continuation method, and introduce the software package AUTO used throughout the thesis.

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