Epidemiological Models and Lyapunov Functions

Abstract

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 R 0 ≤ 1, then the disease free equilibrium is globally asymptotically stable. If R 0 > 1, there exists an unique endemic equilibrium which is asymptotically stable on the positive orthant.

Global stability of the endemic equilibrium in infinite dimension: Lyapunov functions and positive operators

“…Recently, even large epidemic systems have been successfully treated by Volterra Lyapunov functions, first systems with an arbitrary number of disease stages, both finite [1,17,25,26,46] and distributed [39,42,[47][48][49][50][51], and then systems with an arbitrary, but finite number of subpopulations [13,18,19,35,36], and finally combinations of both [37]. In most of the latter cases, graph-theoretic methods are used (Metzler matrices are used in [13]).…”

Modelling the Spread of Infectious Diseases in Complex Metapopulations

“…In most of the cases, when a general spatial arrangement of the patches is considered, * E-mail: joan.saldana@udg.edu the graph describing it is assumed to be irreducible, i.e., the set of patches cannot be split in two groups such that there is no migration from one of the groups to the other one (see, for instance, [9,20,13]). This implies that the matrices (a ij ) and (b ij ) containing the non-negative migration rates a ij and b ij from patch j to patch i (i = j) of susceptible and infected individuals, respectively, must be irreducible (which guarantees the existence of a strictly dominant eigenvalue).…”

Construction of control Lyapunov functions for damping stabilization of control affine systems

“…From a stabilization perspective, accessibility properties of this class of systems was studied in [25], stabilization by positive control was presented in [26] and recently by [27]. Lyapunov stabilization for a larger class of population dynamics was presented in [28].…”