Applied Mathematics and Mechanics

, Volume 25, Issue 4, pp 396–404

# Global analysis of some epidemic models with general contact rate and constant immigration

• Li Jian-quan
• Zhang Juan
• Ma Zhi-en
Article

## Abstract

An epidemic models of SIR type and SIRS type with general contact rate and constant immigration of each class were discussed by means of theory of limit system and suitable Liapunov functions. In the absence of input of infectious individuals, the threshold of existence of endemic equilibrium is found. For the disease-free equilibrium and the endemic equilibrium of corresponding SIR model, the sufficient and necessary conditions of global asymptotical stabilities are all obtained. For corresponding SIRS model, the sufficient conditions of global asymptotical stabilities of the disease-free equilibrium and the endemic equilibrium are obtained. In the existence of input of infectious individuals, the models have no disease-free equilibrium. For corresponding SIR model, the endemic equilibrium is globally asymptotically stable; for corresponding SIRS model, the sufficient conditions of global asymptotical stability of the endemic equilibrium are obtained.

## Key words

epidemic model equilibrium global asymptotical stability limit system

O175.12

92D30 92C60

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© Editorial Committee of Applied Mathematics and Mechanics 2004

## Authors and Affiliations

• Li Jian-quan
• 1
• 2
• Zhang Juan
• 1
• Ma Zhi-en
• 1
1. 1.Department of Applied MathematicsXi’an Jiaotong UniversityXi’anP. R. China
2. 2.Telecommunication Engineering InstituteAir Force Engineering UniversityXi’anP. R. China