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

Mengbin Ye*, Ji Liu, Brian D.O. Anderson, Ming Cao

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review

4 Citations (Scopus)
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Abstract

This paper focuses on properties of equilibria and their associated regions of attraction for continuous-time nonlinear dynamical systems. The classical Poincare–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 Poincare–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.
Original languageEnglish
Pages (from-to)1609-1624
Number of pages16
JournalIEEE-Transactions on Automatic Control
Volume67
Issue number4
Early online date8-Mar-2021
DOIs
Publication statusPublished - 30-Mar-2022

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