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    Applications of the Poincare-Hopf Theorem: Epidemic Models and Lotka-Volterra Systems

    83227.pdf (2.436Mb)
    Access Status
    Open access
    Authors
    Ye, Mengbin
    Liu, J.
    Anderson, B.D.O.
    Cao, M.
    Date
    2021
    Type
    Journal Article
    
    Metadata
    Show full item record
    Citation
    Ye, M. and Liu, J. and Anderson, B.D.O. and Cao, M. 2021. Applications of the Poincare-Hopf Theorem: Epidemic Models and Lotka-Volterra Systems. IEEE Transactions on Automatic Control.
    Source Title
    IEEE Transactions on Automatic Control
    DOI
    10.1109/TAC.2021.3064519
    ISSN
    0018-9286
    Faculty
    Faculty of Science and Engineering
    School
    School of Electrical Engineering, Computing and Mathematical Sciences (EECMS)
    Funding and Sponsorship
    http://purl.org/au-research/grants/arc/DP160104500
    http://purl.org/au-research/grants/arc/DP190100887
    Remarks

    Copyright © 2021 IEEE. This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder.

    URI
    http://hdl.handle.net/20.500.11937/83245
    Collection
    • Curtin Research Publications
    Abstract

    This paper focuses on properties of equilibria and their associated regions of attraction for continuous-time nonlinear dynamical systems. The classical Poincar\'e--Hopf Theorem is used to derive a general result providing a sufficient condition for the system to have a unique equilibrium. The condition involves the Jacobian of the system at possible equilibria, and ensures the system is in fact locally exponentially stable. We apply this result to the susceptible-infected-susceptible (SIS) networked model, and a generalised Lotka--Volterra system. We use the result further to extend the SIS model via the introduction of decentralised feedback controllers, which significantly change the system dynamics, rendering existing Lyapunov-based approaches invalid. Using the Poincar\'e--Hopf approach, we identify a necessary and sufficient condition under which the controlled SIS system has a unique nonzero equilibrium (a diseased steady-state), and monotone systems theory is used to show this nonzero equilibrium is attractive for all nonzero initial conditions. A counterpart condition for the existence of a unique equilibrium for a nonlinear discrete-time dynamical system is also presented

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