Ranking Problems Arising from ODE Models on Networks

  • Author / Creator
    Biglands, Adrian J.
  • 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
  • Graduation date
    Fall 2015
  • Type of Item
  • Degree
    Doctor of Philosophy
  • DOI
  • License
    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
  • Institution
    University of Alberta
  • Degree level
  • 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