A survey on Lyapunov functions for epidemic compartmental models

Abstract

In this survey, we propose an overview on Lyapunov functions for a variety of compartmental models in epidemiology. We exhibit the most widely employed functions, and provide a commentary on their use. Our aim is to provide a comprehensive starting point to readers who are attempting to prove global stability of systems of ODEs. The focus is on mathematical epidemiology, however some of the functions and strategies presented in this paper can be adapted to a wider variety of models, such as prey-predator or rumor spreading.

Keywords: Compartmental models; Disease free and endemic equilibria; Epidemic models; Global stability; Lyapunov functions; Ordinary differential equations.

Fig. 1

Power levels of Lyapunov functions (3) (a), (4) (b), (5) (c), and (6) (d). Values of the parameters are $\mathrm{\Lambda }=0.8$, $\mu =1$, $\delta =1$, $\gamma =1$ in all the figures, $\beta =1$ in a, so that ${\mathcal{R}}_0=4/15<1$, and $\beta =4$ in b, c and d, so that ${\mathcal{R}}_0=16/15>1$. We represent $V(S,I)=k$, with $k\in \{0.1,0.25,0.5,1,1.5,2,2.5\}$ in a, $k\in \{0.001,0.01,0.025,0.05,0.1,0.2\}$ in b and c, and $k\in \{0.01,0.025,0.05,0.1,0.2,0.5\}$ in d. Black dots represent the globally stable equilibrium the system converges to, and correspond to $V(S,I)=0$