Epidemiological Models and Lyapunov Functions

TLDR

The models studied include differential infectivity and staged progression frameworks. The paper surveys global stability results for deterministic compartmental epidemiological models. The authors employ Lyapunov methods to provide simple proofs of classical stability results and to establish new global stability theorems for differential susceptibility and infectivity models with mass action and multiple compartments. They show that when R0 ≤ 1 the disease‑free equilibrium is globally asymptotically stable, whereas for R0 > 1 a unique endemic equilibrium exists and is asymptotically stable on the positive orthant.

Abstract

We give a survey of results on global stability for deterministic compartmental epidemiological models. Using Lyapunov techniques we revisit a classical result, and give a simple proof. By the same methods we also give a new result on differential susceptibility and infectivity models with mass action and an arbitrary number of compartments. These models encompass the so-called differential infectivity and staged progression models. In the two cases we prove that if the basic reproduction ratio ≤ 1, then the disease free equilibrium is globally asymptotically stable. If > 1, there exists an unique endemic equilibrium which is asymptotically stable on the positive orthant.