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.
Similar content being viewed by others
References
Kermack W O, McKendrick A G. Contributions to the mathematical theory of epidemics—Part 1 [J].Proc Roy Soc London Ser A, 1927,115(3):700–721.
Mena-Lorca J, Hethcote H W. Dynamic models of infectious diseases as regulators of population sizes[J].J Math Biol, 1992,30(4):693–716.
Li J, Ma Z. Qualitative analysis of SIS epidemic model with vaccination and varying total population size[J].Math Comput Modelling, 2002,35(11/12):1235–1243.
Heesterbeck J A P, Metz J A J. The saturating contact rate in marriage-and epidemic models[J].J Math Biol, 1993,31(2):529–539.
Brauer F, Van den Driessche P. Models for transmission of disease with immigration of infectives [J].Math Biosci, 2001,171(2):143–154.
Han L, Ma Z, Hethcote H W. Four predator prey models with infectious diseases[J].Math Comput Modelling, 2001,34(7/8):849–858.
LaSalle J P.The Stability of Dynamical System[M]. New York: Academic Press, 1976.
Jeffries C, Klee V, Van den Driessche P. When is a matrix sign stable? [J].Canad J Math, 1977,29(2):315–326.
Author information
Authors and Affiliations
Additional information
Communicated by Li Ji-bin
Foundation item: the National Natural Science Foundation of China (19971066)
Biography: Li Jian-quan (1965-), Associate Professor, Doctor
Rights and permissions
About this article
Cite this article
Jian-quan, L., Juan, Z. & Zhi-en, M. Global analysis of some epidemic models with general contact rate and constant immigration. Appl Math Mech 25, 396–404 (2004). https://doi.org/10.1007/BF02437523
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02437523