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

TL;DR

Using the Poincaré–Hopf approach, a necessary and sufficient condition is identified, 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 non zero equilibrium is attractive for all nonzero initial conditions.

Abstract

This paper focuses on the equilibria and their regions of attraction for continuous-time nonlinear dynamical systems. The classical Poincaré--Hopf Theorem from differential topology is used to derive a result on a sufficient condition on the Jacobian for the existence of a unique equilibrium for the system, which is in fact locally exponentially stable. We show how to apply this result to the deterministic SIS networked model, and a nonlinear Lotka--Volterra system. We apply the result further to establish an impossibility conclusion for a class of distributed feedback controllers whose goal is to drive the SIS network to the zero equilibrium. Specifically, we show that if the uncontrolled system has a unique nonzero equilibrium (a diseased steady-state), then the controlled system also has a unique nonzero equilibrium. Applying results from monotone dynamical systems theory, we further show that both the uncontrolled and controlled system will converge to their nonzero equilibrium from all nonzero initial conditions exponentially fast. A counterpart sufficient condition for the existence of a unique equilibrium for a nonlinear discrete-time dynamical system is also presented.

Figures (3)

• Figure 1: An illustration of the compact manifolds $\mathcal{M}_\epsilon$ and $\tilde{\mathcal{M}}_\epsilon$ for Eq. (\ref{['eq:SIS_contact_network']}), with $n = 2$. The cube $\Xi_n$ is in light grey, with dotted black borders, and corners indicated. The dashed red line identifies the boundary of $\mathcal{M}_\epsilon$ (defined in Eq. (\ref{['eq:M_definition']})), and notice the lower corner point of $(\epsilon y_1, \epsilon y_2)$ with exaggerated size (in reality, $\epsilon > 0$ is small). The solid red line identifies $\partial\tilde{\mathcal{M}}_\epsilon$, with the shaded red area being $\mathop{\mathrm{\mathrm{Int}}}\nolimits(\tilde{\mathcal{M}}_\epsilon)$. One can see that $\tilde{\mathcal{M}}_\epsilon$ is simply $\mathcal{M}_\epsilon$ but with the corners rounded so that $\tilde{\mathcal{M}}_\epsilon$ is smooth. The $(1,1)$ corner is magnified to give a clear view. The rounding of corners is exaggerated for clarity; in reality, one only requires an arbitrarily small smoothing of each corner or edge. With reference to Eq. (\ref{['eq:boundary_point']}), black arrows denote canonical unit vectors $\mathbf{e}_i, i = 1, 2$ (with direction), and blue arrows show the vector field $f$ pointing inward at example points on $\partial\tilde{\mathcal{M}}_\epsilon$.
• Figure 2: Vector field of an uncontrolled SIS network model with $2$ nodes. The red dot identifies the unique endemic equilibrium $x^* =[0.4413, 0.2973]^\top$.
• Figure 3: Vector field of a controlled SIS network model with $2$ nodes. The red dot identifies the unique endemic equilibrium $\bar{x}^* = [0.15, 0.1142]^\top$. Although the feedback control $h_i(x_i(t))$ shifts $\bar{x}^*$ closer to the origin (the healthy equilibrium) compared to $x^* = [0.4413, 0.2973]^\top$ of the uncontrolled network (see Fig. \ref{['fig:SIS_nocontrol']}), all trajectories of the controlled SIS network converge to $\bar{x}^*$ except $x(0) = \boldsymbol{0}_2$.

Theorems & Definitions (39)

• Proposition 1: lajmanovich1976SISnetworkfall2007SIS_model
• Proposition 2: Nagumo's Theorem blanchini1999set_invariance
• Definition 1: Pointing inward
• Proposition 3: The Poincaré-Hopf Theoremmilnor1997topology
• Theorem 1: Unique Equilibrium
• proof
• Remark 1
• Remark 2
• Lemma 1
• Lemma 2: berman1979nonnegative_matrices