Lyapunov functions and global properties for SEIR and SEIS epidemic models

Abstract

Abstract. Explicit Lyapunov functions for SIR and SEIR compartmental epidemic models with nonlinear incidence of the form βI p S q for the case p ≤ 1 are constructed. Global stability of the models is thereby established.1. Introduction. It is traditionally postulated that the spread of an infection occurs according to the principle of mass action and associated with it the bilinear incidence rate. However, there are a variety of reasons why this standard bilinear incidence rate may require modification. The incidence rate of the form βI p S q , were S and I are respectively the number of susceptible and infective individuals in the population (or the fractions of susceptible and infective), and β, p and q are positive constants, is the most common nonlinear incidence rate. In recent years models with this incidence rate were considered by several authors, for example Liu et al. [11,12] In this note applying the direct Lyapunov method, we consider global properties of SIR and SEIR models with the incidence rate of the form βI p S q for the particular case p ≤ 1. We construct a Lyapunov function which is a generalization of the Lyapunov functions constructed earlier for models with bilinear incidence [10,8]. We show that the condition p ≤ 1 is a sufficient condition for global stability, and that the global properties of the systems do not depend on the value of q.

Global dynamics of a staged-progression model with amelioration for infectious diseases

“…Our global stability result for P * generalizes earlier results in [7,20]. The proof utilizes a global Lyapunov function motivated by the work in [13][14][15]. We have shown that introducing amelioration at stage k of the disease progression indeed may increase the basic reproduction number, and hence may have a negative effect on the disease control in the population.…”

Lyapunov functions and global properties for SEIR and SEIS epidemic models

“…Consequently, the function (4) is indeed a Lyapunov function [14]. This function is a generalization of the Lyapunov functions constructed in [10,8] for the case of the nonlinear incidence rate βI p S q .…”

Global properties of basic virus dynamics models

“…Hethcote (2000); Korobeinikov (2004)). Indeed, if the equation for the recovered population R is omitted (the constant population size assumption allows us to do so), the system (1.1) is equivalent to the SEIR model: x corresponds to susceptible population S, y to exposed population E, and ν to infective population I.…”

“…For the SEIR model there is a global Lyapunov function (Korobeinikov, 2004) which allows a straightforward investigating of global properties of the system (1.1) as well. The following theorem holds for the system.…”