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An SIRS epidemic model

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Abstract

This paper considers an SIRS epidemic model that incorporates constant immigration rate, a general population-size dependent contact rate and proportional transfer rate from the infective class to susceptible class. A threshold parameter σ is identified. If σ⩽1, the diseasefree equilibrium is globally stable. If σ>1, a unique endemic equilibrium is locally asymptotically stable. For two important special cases of mass action incidence and standard incidence, global stability of the endemic equilibrium is proved provided the threshold is larger than unity. Some previous results are extended and improved.

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References

  1. Brauer, F., Van Den Driessche, P., Models for transmission of disease with immigration of infectives, Math. Biosci., 2001,171:143–154.

    Article  MATH  MathSciNet  Google Scholar 

  2. Chen Junjie, Zhang Nansong, An HIV/AIDS transmission model with infectious stages and constant recruitment, Acta Math. Appl. Sinica—Chinese series, 2002, 25(3):538–546 (in Chinese).

    MATH  Google Scholar 

  3. Zhang Juan, Ma Zhien, Dynamic behavior of the SEIS model with saturating contact rate, J. Xian Jiaotong Univ., 2002,36(2):204–207.

    MATH  MathSciNet  Google Scholar 

  4. Zhou Jinshi, An epidemiological model with population size dependent incidence, Rocky Mountain J. Math., 1994,24:429–445.

    Article  MATH  MathSciNet  Google Scholar 

  5. Roberts, M. G., Kao, R. R., The dynamics of an infectious disease in a population with birth pulses, Math. Biosci., 1998,149:23–36.

    Article  MATH  Google Scholar 

  6. Greenhalgh, D., Hopf bifurcation in epidemic models with a latent period and nonpermanent immunity, Math. Comput. Modelling, 1997,25(2):85–107.

    Article  MATH  MathSciNet  Google Scholar 

  7. Thieme, H. R., Persistence under relaxed point-dissipativity (with application to an endemic model), SIAM J. Math. Anal., 1993,24(2):407–435.

    Article  MATH  MathSciNet  Google Scholar 

  8. Wang Fujun, Two epidemic models with emigration of population, Appl. Math. J. Chinese Univ. Ser. A, 1991,6(2):297–303.

    MATH  Google Scholar 

  9. Hethcote, H. W., Qualitative analyses of communicable disease models, Math. Biosci., 1976,28:335–356.

    Article  MATH  MathSciNet  Google Scholar 

  10. Mena-Lorca, J., Hethcote, H. W., Dynamic models of infectious diseases as regulators of population sizes, J. Math. Biol., 1992,30:693–716.

    Article  MATH  MathSciNet  Google Scholar 

  11. Liao Xiaoxin, Theory Methods and Application of Stability, Wuhan: Huazhong Science and Engineering University Press, 1999.

    Google Scholar 

  12. Gao Weibing, An Introduction to Dynamical Stability, Beijing:Higher Education Press, 1987.

    Google Scholar 

  13. Jeffries, C., Klee, V., Van Den Driessche, P., When is a matrix sign stable? Canad. J. Math., 1977,29:315–326.

    MATH  MathSciNet  Google Scholar 

  14. Busenberg, S., Van Den Driessche, P., Analysis of a disease transmission model in a population with varying size, J. Math. Biol., 1990,28:257–270.

    Article  MATH  MathSciNet  Google Scholar 

  15. Genik, L., Van Den Driessche, P., A model for diseases without immunity in a variable size population, Canadian Appl. Math. Quart., 1998,6(1):5–16.

    MATH  Google Scholar 

  16. Fan Meng, Li Michael Y, Wang Ke, Global stability of an SEIS epidemic model with recruitment and a varying total population size, Math. Biosci., 2001,170:199–208.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Dept. of Math., Zhejiang Univ., 310027, Hangzhou, China

    Chen Junjie

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  1. Chen Junjie
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Supported by the Science and Technology Foundation of Zhejiang University (107000-544301).

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Chen, J. An SIRS epidemic model. Appl. Math. Chin. Univ. 19, 101–108 (2004). https://doi.org/10.1007/s11766-004-0027-8

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  • DOI: https://doi.org/10.1007/s11766-004-0027-8

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