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

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