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Ranking Problems Arising from ODE Models on Networks Open Access


Other title
Eigenvector Centrality
ODE Network
Equilibrium Ranking
Type of item
Degree grantor
University of Alberta
Author or creator
Biglands, Adrian J.
Supervisor and department
Michael Yi Li, Department of Mathematical and Statistical Sciences
Examining committee member and department
Davin McLaughlin, Department of Mathematical and Statistical Sciences
Hao Wang, Department of Mathematical and Statistical Sciences
Julien Arino, Department of Mathematical and Statistical Sciences
James Muldowney, Department of Mathematical and Statistical Sciences
Michael Yi Li, Department of Mathematical and Statistical Sciences
Anthony Lau, Department of Mathematical and Statistical Sciences
Department of Mathematical and Statistical Sciences
Applied Mathematics
Date accepted
Graduation date
Doctor of Philosophy
Degree level
The use of ordinary differential equations modelled on networks has become an increasingly important technique in many areas of research. The local behaviour of the system is modelled with differential equations and interactions between members or nodes are described using weighted digraphs. For instance, in public health nodes can represent different groups of people affected by an infectious disease, while edges in the network represent the cross-infection between the groups. The local behaviours of the disease in each group are described with ODE dynamics. In ecology, the spatial dispersal of one or more species considers the habitation patches as nodes and the edges between nodes describe the movement of the species between patches. This thesis develops a method of ranking the nodes of an ODE network at a positive equilibrium x^* = (x_1^* , . . . , x_n^*)T . Such a ranking is called an equilibrium ranking. More specifically, assuming an ODE system modelled on a network (G,B) has a positive equilibrium x^*, we associate x_i^* to node i of the network. These positive equilibrium values x_i^* are used to rate, and hence rank, the individual nodes of the network. Such an equilibrium ranking reflects both the graph structure and the local ODE parameters of the model. In my dissertation I investigate equilibrium ranking for several ODE networks including SIR epidemiology models with n different groups or spatial regions, single or multiple species ecological models, and coupled oscillator models from engineering. For an SIR model the equilibrium ranking can be obtained using the equilibrium values of the disease prevalence vector I^* = (I_1^* , . . . , I_n^*)T . This will indicate which of the n groups or patches has the highest number of infected individuals per capita. In the single species ecology model an equilibrium ranking vector comes from the species density vector x^* = (x_1^* , . . . , x_n^*)T , and reflects overpopulation or extinction on different patches or in different groups. The dependence of equilibrium ranking on both graph structure and local parameters is also investigated. In particular, the dependence of the equilibrium ranking is considered for several digraph structures including rooted trees, loop digraphs, unicyclic and multi-cyclic digraphs. This will allow researchers to fix the network structure (G,B) of the system and focus on how the dynamics play a role in the importance of nodes in a network.
Permission is hereby granted to the University of Alberta Libraries to reproduce single copies of this thesis and to lend or sell such copies for private, scholarly or scientific research purposes only. The author reserves all other publication and other rights in association with the copyright in the thesis and, except as herein before provided, neither the thesis nor any substantial portion thereof may be printed or otherwise reproduced in any material form whatsoever without the author's prior written permission.
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