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# Ranking Problems Arising from ODE Models on Networks

• Author / Creator
• 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^* = (x1^* , . . . , xn^)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 xi^* 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^* = (I1^* , . . . , In^)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^
= (x1^* , . . . , xn^*)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.

• Subjects / Keywords
Fall 2015
• Type of Item
Thesis
• Degree
Doctor of Philosophy
• DOI
https://doi.org/10.7939/R34X54T73
This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for non-commercial purposes. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.
• Language
English
• Institution
University of Alberta
• Degree level
Doctoral
• Department
• Specialization
• Applied Mathematics
• Supervisor / co-supervisor and their department(s)
• Examining committee members and their departments
• Julien Arino, Department of Mathematical and Statistical Sciences
• James Muldowney, Department of Mathematical and Statistical Sciences
• Anthony Lau, Department of Mathematical and Statistical Sciences
• Davin McLaughlin, Department of Mathematical and Statistical Sciences
• Michael Yi Li, Department of Mathematical and Statistical Sciences
• Hao Wang, Department of Mathematical and Statistical Sciences