Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models

Korobeinikov, Andrei; Wake, G. C.

doi:10.1016/s0893-9659(02)00069-1

“…Lyapunov function for the Lotka-Volterra prey-predator system (Goh, 1980;Takeuchi, 1996), and it was also successfully applied for epidemiological models with the bilinear incidence rate (Korobeinikov and Wake, 2002;Korobeinikov, 2004,a). For the incidence rate of the form βI p S q the function V (S, I) takes the form…”

Section: Discussionmentioning

confidence: 99%

Lyapunov Functions and Global Stability for SIR and SIRS Epidemiological Models with Non-Linear Transmission

Korobeinikov

¹

2006

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“…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 .…”

Section: Proofmentioning

confidence: 95%

Lyapunov functions and global properties for SEIR and SEIS epidemic models

Korobeinikov

¹

2004

Mathematical Medicine and Biology

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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.

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“…This Lyapunov function has been constructed for three-and four-compartmental models [10,8] with the standard bilinear incidence.…”

Section: Proofmentioning

confidence: 99%

Lyapunov functions and global properties for SEIR and SEIS epidemic models

Korobeinikov¹

2004

Mathematical Medicine and Biology

Self Cite

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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.

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Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models

Abstract: Lyapunov functions for classical SIR, SIRS, and SIS epidemiological models are introduced. Global stability of the endemic equilibrium states of the models is thereby established

Cited by 308 publications

References 2 publications

Lyapunov Functions and Global Stability for SIR and SIRS Epidemiological Models with Non-Linear Transmission

Lyapunov Functions and Global Stability for SIR and SIRS Epidemiological Models with Non-Linear Transmission

Lyapunov functions and global properties for SEIR and SEIS epidemic models

Lyapunov functions and global properties for SEIR and SEIS epidemic models

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