# Global asymptotic properties of an SEIRS model with multiple infectious stages

Dessalegn Y. Melesse, Abba Gumel

Research output: Contribution to journalArticle

18 Citations (Scopus)

### Abstract

The paper presents a rigorous mathematical analysis of a deterministic model, which uses a standard incidence function, for the transmission dynamics of a communicable disease with an arbitrary number of distinct infectious stages. It is shown, using a linear Lyapunov function, that the model has a globally-asymptotically stable disease-free equilibrium whenever the associated reproduction threshold is less than unity. Further, the model has a unique endemic equilibrium when the threshold exceeds unity. The equilibrium is shown to be locally-asymptotically stable, for a special case, using a Krasnoselskii sub-linearity trick. Finally, a non-linear Lyapunov function is used to show the global asymptotic stability of the endemic equilibrium (for the special case). Numerical simulation results, using parameter values relevant to the transmission dynamics of influenza, are presented to illustrate some of the main theoretical results.

Original language English (US) 202-217 16 Journal of Mathematical Analysis and Applications 366 1 https://doi.org/10.1016/j.jmaa.2009.12.041 Published - Jun 1 2010 Yes

### Fingerprint

Endemic Equilibrium
Lyapunov Function
Asymptotic Properties
Lyapunov functions
Influenza
Globally Asymptotically Stable
Global Asymptotic Stability
Deterministic Model
Asymptotically Stable
Mathematical Analysis
Nonlinear Function
Linearity
Linear Function
Incidence
Exceed
Asymptotic stability
Distinct
Numerical Simulation
Arbitrary
Model

### Keywords

• Equilibria
• Infectious disease
• Lyapunov function
• Reproduction number
• Stability

### ASJC Scopus subject areas

• Analysis
• Applied Mathematics

### Cite this

Global asymptotic properties of an SEIRS model with multiple infectious stages. / Melesse, Dessalegn Y.; Gumel, Abba.

In: Journal of Mathematical Analysis and Applications, Vol. 366, No. 1, 01.06.2010, p. 202-217.

Research output: Contribution to journalArticle

title = "Global asymptotic properties of an SEIRS model with multiple infectious stages",
abstract = "The paper presents a rigorous mathematical analysis of a deterministic model, which uses a standard incidence function, for the transmission dynamics of a communicable disease with an arbitrary number of distinct infectious stages. It is shown, using a linear Lyapunov function, that the model has a globally-asymptotically stable disease-free equilibrium whenever the associated reproduction threshold is less than unity. Further, the model has a unique endemic equilibrium when the threshold exceeds unity. The equilibrium is shown to be locally-asymptotically stable, for a special case, using a Krasnoselskii sub-linearity trick. Finally, a non-linear Lyapunov function is used to show the global asymptotic stability of the endemic equilibrium (for the special case). Numerical simulation results, using parameter values relevant to the transmission dynamics of influenza, are presented to illustrate some of the main theoretical results.",
keywords = "Equilibria, Infectious disease, Lyapunov function, Reproduction number, Stability",
author = "Melesse, {Dessalegn Y.} and Abba Gumel",
year = "2010",
month = "6",
day = "1",
doi = "10.1016/j.jmaa.2009.12.041",
language = "English (US)",
volume = "366",
pages = "202--217",
journal = "Journal of Mathematical Analysis and Applications",
issn = "0022-247X",
number = "1",

}

TY - JOUR

T1 - Global asymptotic properties of an SEIRS model with multiple infectious stages

AU - Melesse, Dessalegn Y.

AU - Gumel, Abba

PY - 2010/6/1

Y1 - 2010/6/1

N2 - The paper presents a rigorous mathematical analysis of a deterministic model, which uses a standard incidence function, for the transmission dynamics of a communicable disease with an arbitrary number of distinct infectious stages. It is shown, using a linear Lyapunov function, that the model has a globally-asymptotically stable disease-free equilibrium whenever the associated reproduction threshold is less than unity. Further, the model has a unique endemic equilibrium when the threshold exceeds unity. The equilibrium is shown to be locally-asymptotically stable, for a special case, using a Krasnoselskii sub-linearity trick. Finally, a non-linear Lyapunov function is used to show the global asymptotic stability of the endemic equilibrium (for the special case). Numerical simulation results, using parameter values relevant to the transmission dynamics of influenza, are presented to illustrate some of the main theoretical results.

AB - The paper presents a rigorous mathematical analysis of a deterministic model, which uses a standard incidence function, for the transmission dynamics of a communicable disease with an arbitrary number of distinct infectious stages. It is shown, using a linear Lyapunov function, that the model has a globally-asymptotically stable disease-free equilibrium whenever the associated reproduction threshold is less than unity. Further, the model has a unique endemic equilibrium when the threshold exceeds unity. The equilibrium is shown to be locally-asymptotically stable, for a special case, using a Krasnoselskii sub-linearity trick. Finally, a non-linear Lyapunov function is used to show the global asymptotic stability of the endemic equilibrium (for the special case). Numerical simulation results, using parameter values relevant to the transmission dynamics of influenza, are presented to illustrate some of the main theoretical results.

KW - Equilibria

KW - Infectious disease

KW - Lyapunov function

KW - Reproduction number

KW - Stability

UR - http://www.scopus.com/inward/record.url?scp=75749117440&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=75749117440&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2009.12.041

DO - 10.1016/j.jmaa.2009.12.041

M3 - Article

AN - SCOPUS:75749117440

VL - 366

SP - 202

EP - 217

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 1

ER -