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Fuzzy Optimization and Decision Making

, Volume 18, Issue 4, pp 475–491 | Cite as

Analysis of uncertain SIS epidemic model with nonlinear incidence and demography

  • Zhiming Li
  • Zhidong TengEmail author
Article
  • 56 Downloads

Abstract

Based on uncertainty theory, this paper studies an uncertain SIS epidemic model with nonlinear incidence and demography. The solution, \(\alpha \)-paths and uncertainty distribution of uncertain model are discussed. Under threshold conditions, extinction and permanence of the disease are studied by \(\alpha \)-paths, which reveal the relationship of deterministic and uncertain models. An example is given to illustrate the above results.

Keywords

Liu process Uncertain SIS epidemic model \(\alpha \)-Path Uncertainty distribution 

Notes

Acknowledgements

This research is funded by the National Natural Science Foundation of China (Grant Nos. 11661076, 61563050) and the Natural Science Foundation of Xinjiang (Grant Nos. 2016D01C043, 2016D03022).

References

  1. Alberto, O. (2002). Stability properties of pulse vaccination strategy in SEIR epidemic model. Mathematical Biosciences, 179, 57–72.MathSciNetCrossRefGoogle Scholar
  2. Bai, Y. Z., & Mu, X. Q. (2018). Global asymptotic stability of a generalized SIRS epidemic model with transfer from infectious to susceptible. Journal of Applied Analysis and Computation, 8, 402–412.MathSciNetGoogle Scholar
  3. Chen, X., & Gao, J. (2013). Uncertain term structure model of interest rate. Soft Computing, 17(4), 597–604.CrossRefGoogle Scholar
  4. Gray, A., Greenhalgh, D., Hu, L., Mao, X., & Pan, J. (2011). A stochastic differential equation SIS epidemic model. SIAM Journal on Applied Mathematics, 71, 876–902.MathSciNetCrossRefGoogle Scholar
  5. Hethcote, H. W. (2000). The mathematics of infectious disease. SIAM Review, 42, 599–653.MathSciNetCrossRefGoogle Scholar
  6. Lahrouz, A., & Omari, L. (2013). Extinction and stationary distribution of a stochastic SIRS epidemic model with non-linear incidence. Statistics & Probability Letters, 83, 960–968.MathSciNetCrossRefGoogle Scholar
  7. Li, B., Yuan, S., & Zhang, W. (2011). Analysis on an epidemic model with a ratio-dependent nonlinear incidence rate. International Journal of Biomathematics, 4, 227–239.MathSciNetCrossRefGoogle Scholar
  8. Li, M., Sheng, Y., Teng, Z., & Miao, H. (2017). An uncertain differential equation for SIS epidemic model. Journal of Intelligent & Fuzzy Systems, 33, 2317–2327.CrossRefGoogle Scholar
  9. Liu, B. (2007). Uncertainty theory (2nd ed.). Berlin: Springer.zbMATHGoogle Scholar
  10. Liu, B. (2008). Fuzzy process, hybird process and uncertain process. Journal of Uncertain Systems, 2, 3–16.Google Scholar
  11. Liu, B. (2009). Some research problems in uncertainty theory. Journal of Uncertain Systems, 3, 3–10.Google Scholar
  12. Liu, B. (2010). Uncertainty theory: A branch of mathematics for modeling human uncertainty. Berlin: Springer.CrossRefGoogle Scholar
  13. Liu, B. (2014). Uncertainty distribution and independence of uncertain processes. Fuzzy Optimization and Decision Making, 13, 259–271.MathSciNetCrossRefGoogle Scholar
  14. Liu, Y. (2012). An analytic method for solving uncertain differential equations. Journal of Uncertain Systems, 6, 244–249.Google Scholar
  15. Tornatore, E., Buccellato, S. M., & Vetro, P. (2005). Stability of a stochastic SIR system. Physica A: Statistical Mechanics and its Applications, 354, 111–126.CrossRefGoogle Scholar
  16. Wang, Y., Liu, L. S., Zhang, X. G., & Wu, Y. H. (2015). Positive solutions of a fractional semipositone differential system arising from the study of HIV infection models. Applied Mathematics and Computation, 258, 312–324.MathSciNetCrossRefGoogle Scholar
  17. Wei, F., & Chen, F. (2016). Stochastic permanence of an SIQS epidmic model with saturated incidence and independent random perturbations. Physica A: Statistical Mechanics and its Applications, 453, 99–107.MathSciNetCrossRefGoogle Scholar
  18. Yao, K. (2015). Uncertain contour process and its application in stock model with floating interest rate. Fuzzy Optimization and Decision Making, 14, 399–424.MathSciNetCrossRefGoogle Scholar
  19. Yao, K., & Chen, X. (2013). A numerical method for solving uncertain differential equations. Journal of Intelligent & Fuzzy Systems, 25, 825–832.MathSciNetzbMATHGoogle Scholar
  20. Yuan, S., & Li, B. (2009). Global dynamics of an epidemic model with a ratio-dependent nonlinear incidence rate. Discrete Dynamics in Nature and Society.  https://doi.org/10.1155/2009/609306.CrossRefGoogle Scholar
  21. Zhao, Y., Jiang, D., & O’Regan, D. (2013). The extinctioin and persistence of stochastic SIS epidemic model with vaccination. Physica A: Statistical Mechanics and its Applications, 392, 4916–4927.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.College of Mathematics and System ScienceXinjiang UniversityUrumqiChina

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