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.

Multiple Transmission Pathways and Disease Dynamics in a Waterborne Pathogen Model

“…We do this by finding a Lyapunov function of the general form considered by Korobeinikov and Wake (2002) and Korobeinikov (2004) for the SIR and SEIR models.…”

Global Stability of Equilibria in a Two-Sex HPV Vaccination Model

“…In systems with three or less dimensions, a typical global stability analysis invokes the Poincaré-Bendixson theorem and applies one of the available methods to rule out existence of periodic solutions. Recently, several researchers have successfully applied the Lyapunov direct method to prove the global stability of endemic states in a variety of epidemic and virus population in vivo models with systems of higher dimensions (Korobeinikov, 2004a(Korobeinikov, , 2004b(Korobeinikov, , 2006Korobeinikov and Maini, 2004;Korobeinikov and Wake, 2002;Iwasa et al, 2004;Guo and Li, 2006;McCluskey, 2006). The origin of the Lyapunov functions used in the analysis of these models goes back to Volterra (Harrison, 1979).…”

Modelling multiple relapses in drug epidemics

“…Lyapunov functions of the type used in proving global stability of the drugpersistent equilibrium D * , have been used in the literature especially for ecological models (see [25] and the references cited therein) and these have also been used for epidemic models (see [26,27]). …”