Applications of the Poincaré–Hopf Theorem: Epidemic Models and Lotka–Volterra Systems

This article focuses on properties of equilibria and their associated regions of attraction for continuous-time nonlinear dynamical systems. The classical Poincaré–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 that the system is in fact locally exponentially stable. We apply this result to the susceptible–infected–susceptible (SIS) networked epidemic model, and a generalized Lotka–Volterra system. We use the result further to extend the SIS model via the introduction of decentralized feedback controllers, which significantly change the system dynamics, rendering existing Lyapunov-based approaches invalid. Using the Poincaré–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 that 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.