Skip to main content
SpringerLink
Log in

Two-group SIR epidemic model with stochastic perturbation

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

A stochastic two-group SIR model is presented in this paper. The existence and uniqueness of its nonnegative solution is obtained, and the solution belongs to a positively invariant set. Furthermore, the globally asymptotical stability of the disease-free equilibrium is deduced by the stochastic Lyapunov functional method if R 0 ≤ 1, which means the disease will die out. While if R 0 > 1, we show that the solution is fluctuating around a point which is the endemic equilibrium of the deterministic model in time average. In addition, the intensity of the fluctuation is proportional to the intensity of the white noise. When the white noise is small, we consider the disease will prevail. At last, we illustrate the dynamic behavior of the model and their approximations via a range of numerical experiments.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Kermack, W. O., Mckendrick, A. G.: Contributions to the mathematical theory of epidemics (Part I). Proc. R. Sm., A115, 700–721 (1927)

    Article  Google Scholar 

  2. Beretta, E., Hara, T., Ma, W., et al.: Global asymptotic stability of an SIR epidemic model with distributed time delay. Nonlinear Anal., 47, 4107–4115 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. Meng, X. Z., Chen, L. S.: The dynamics of a new SIR epidemic model concerning pulse vaccination strategy. Appl. Math. Comput., 197, 528–597 (2008)

    Article  MathSciNet  Google Scholar 

  4. Tchuenche, J. M., Nwagwo, A., Levins, R.: Global behaviour of an SIR epidemic model with time delay. Math. Meth. Appl. Sci., 30, 733–749 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Zhang, F. P., Li, Z. Z., Zhang, F.: Global stability of an SIR epidemic model with constant infectious period. Appl. Math. Comput., 199, 285–291 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Lajmanovich, A., York, J. A.: A deterministic model for gonorrhea in a nonhomogeneous population. Math. Biosci., 28, 221–236 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  7. Beretta, E., Capasso, V.: Global stability results for a multigroup SIR epidemic model. In: (T. G. Hallam, L. J. Gross and S. A. Levin, eds.) Mathematical Ecology, World Scientific, Singapore, 1986, pp. 317–342

    Google Scholar 

  8. Guo, H. B., Li, M. Y., Shuai, Z. S.: Global stability of the endemic equilibrium of multigroup SIR epidemic models. Canad. Appl. Math. Quart., 14, 259–284 (2006)

    MathSciNet  MATH  Google Scholar 

  9. Hethcote, H. W.: An immunization model for a heterogeneous population. Theor. Popu. Biol., 14, 338–349 (1978)

    Article  MathSciNet  Google Scholar 

  10. Arnold, L., Horsthemke, W., Stucki, J. W.: The influence of external real and white noise on the Lotka-Volterra model. Biomedical J., 21, 451–471 (1979)

    MathSciNet  MATH  Google Scholar 

  11. Bandyopadhyay, M., Chattopadhyay, J.: Ratio-dependent predator-prey model: effect of environmental fluctuation and stability. Nonlinearity, 18, 913–936 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  12. Carletti, M., Burrage, K., Burrage, P. M.: Numerical simulation of stochastic ordinary differential equations in biomathematical modelling. Math. Comp. Simul., 64, 271–277 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. Renshaw, E.: Modelling Biological Populations in Space and Time, Cambridge University Press, Cambridge, 1991

    Book  MATH  Google Scholar 

  14. Beretta, E., Kolmanovskii, V., Shaikhet, L.: Stability of epidemic model with time delays influenced by stochastic perturbations. Math. Comput. Simulation, 45, 269–277 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  15. Carletti, M.: On the stability properties of a stochastic model for phage-bacteria interaction in open marine environment. Math. Biosci., 175, 117–131 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  16. Carletti, M.: Mean-square stability of a stochastic model for bacteriophage infection with time delays. Math. Biosci., 210, 395–414 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Yu, J. J., Jiang, D. Q., Shi, N. Z.: Global stability of two-group SIR model with random perturbation. J. Math. Anal. Appl., 360, 235–244 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Dalal, N., Greenhalgh, D., Mao, X. R.: A stochastic model of AIDS and condom use. J. Math. Anal. Appl., 325, 36–53 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. Dalal, N., Greenhalgh, D., Mao, X. R.: A stochastic model for internal HIV dynamics. J. Math. Anal. Appl., 341, 1084–1101 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. Beddington, J. R., May, R. M.: Harvesting natural populations in a randomly fluctuating environment. Science, 197, 463–465 (1977)

    Article  Google Scholar 

  21. Imhof, L., Walcher, S.: Exclusion and persistence in deterministic and stochastic chemostat models. J. Differ. Equations, 217, 26–53 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  22. Mao, X. R.: Stochastic Differential Equations and Applications, Horwood, Chichester, 1997

    MATH  Google Scholar 

  23. Zhang, X. C.: Quasi-sure limit theorem of parabolic stochastic partial differential equations. Acta Mathematica Sinica, English Series, 20, 719–730 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  24. Higham, D. J.: An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Review, 43, 525–546 (2001)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Changshu Institute of Technology, Changshu, 215500, P. R. China

    Chun Yan Ji

  2. School of Mathematics and Statistics, Northeast Normal University, Changchun, 130024, P. R. China

    Chun Yan Ji, Da Qing Jiang & Ning Zhong Shi

Authors
  1. Chun Yan Ji
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Da Qing Jiang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ning Zhong Shi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chun Yan Ji.

Additional information

Supported by National Natural Science Foundation of China (Grant No. 10971021), the Ministry of Education of China (Grant No. 109051), the Ph.D. Programs Foundation of Ministry of China (Grant No. 200918) and the Graduate Innovative Research Project of NENU (Grant No. 09SSXT117)

Electronic supplementary material

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ji, C.Y., Jiang, D.Q. & Shi, N.Z. Two-group SIR epidemic model with stochastic perturbation. Acta. Math. Sin.-English Ser. 28, 2545–2560 (2012). https://doi.org/10.1007/s10114-012-9668-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-012-9668-3

Keywords

MR(2000) Subject Classification

Search

Navigation