Skip to main content

Advertisement

SpringerLink
Log in

Periodic Solution and Ergodic Stationary Distribution of Stochastic SIRI Epidemic Systems with Nonlinear Perturbations

  • Published:
Journal of Systems Science and Complexity Aims and scope Submit manuscript

Abstract

This paper formulates two stochastic nonautonomous SIRI epidemic systems with nonlinear perturbations. The main aim of this study is to investigate stochastic dynamics of the two SIRI epidemic systems and obtain their thresholds. For the nonautonomous stochastic SIRI epidemic system with white noise, the authors provide analytic results regarding the stochastic boundedness, stochastic permanence and persistence in mean. Moreover, the authors prove that the system has at least one nontrivial positive T-periodic solution by using Lyapunov function and Hasminskii’s theory. For the system with Markov conversion, the authors establish sufficient conditions for positive recurrence and existence of ergodic stationary distribution. In addition, sufficient conditions for the extinction of disease are obtained. Finally, numerical simulations are introduced to illustrate the main results.

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. Anderson R, May R, and Medley G, A preliminary study of the transmission dynamics of the human immunodeficiency virus (HIV), the causative agent of AIDS, IMA J. Math. Appl. Med., 1986, 3: 229–263.

    Article  MathSciNet  MATH  Google Scholar 

  2. Herbert H W, The mathematics of infectious diseases, SIAM Rev., 2000, 42: 599–653.

    Article  MathSciNet  MATH  Google Scholar 

  3. Brauer F and Chavez C C, Mathematical Models in Population Biology and Epidemiology, Springer, New York, 2001.

    Book  MATH  Google Scholar 

  4. Gao S J, Chen L S, Nieto J J, et al., Analysis of a delayed epidemic model with pulse vaccination and saturation incidence, Vaccine, 2006, 24: 6037–6045.

    Article  Google Scholar 

  5. Li X Z, Li W S, and Ghosh M, Stability and bifurcation of an SIS epidemic model with treatment, Chaos. Solitons Fractals, 2009, 42: 2822–2832.

    Article  MathSciNet  MATH  Google Scholar 

  6. Ma Z E, Zhou Y C, and Wu J H, Modeling and Dynamics of Infectious Diseases, Higher Education Press, Beijing, 2009.

    Book  MATH  Google Scholar 

  7. Meng X Z, Stability of a novel stochastic epidemic model with double epidemic hypothesis, Appl. Math. Comput., 2010, 217: 506–515.

    MathSciNet  MATH  Google Scholar 

  8. Liu X B and Yang L J, Stability analysis of an SEIQV epidemic model with saturated incidence rate, Nonlinear Anal. Real World Appl., 2012, 13: 2671–2679.

    Article  MathSciNet  MATH  Google Scholar 

  9. Wang W and Ma W B, A diffusive HIV infection model with nonlocal delayed transmission, Appl. Math. Lett., 2018, 75: 96–101.

    Article  MathSciNet  MATH  Google Scholar 

  10. Korobeinikov A, Lyaounov functions and global stability for SIR and SIRS epidemiological models with nonlinear transmission, Bull. Math. Biol., 2006, 30: 615–636.

    Article  MATH  Google Scholar 

  11. Ji C, Jiang D Q, and Shi N Z, Multigroup SIR epidemic model with stochastic perturbation, Phys. A, 2011, 390: 1747–1762.

    Article  Google Scholar 

  12. Gray A, Greenhalgh D, Hu L, et al., A stochastic differential equation SIS epidemic model, SIAM J. Appl. Math., 2011, 71: 876–902.

    Article  MathSciNet  MATH  Google Scholar 

  13. Zhang T Q, Meng X Z, Zhang T H, et al., Global dynamics for a new high-dimensional sir model with distributed delay, Appl. Math. Comput., 2012, 218: 11806–11819.

    MathSciNet  MATH  Google Scholar 

  14. Chen Q L, Teng Z D, Wang L, et al., The existence of codimension-two bifurcation in a discrete SIS epidemic model with standard incidence, Nonlinear Dynam., 2013, 71: 55–73.

    Article  MathSciNet  MATH  Google Scholar 

  15. Wang J L, Muroya Y, and Kuniya T Y, Global stability of a time-delayed multi-group SIS epidemic model with nonlinear incidence rates and patch structure, J. Math. Anal. Appl., 2015, 425: 415–439.

    Article  MathSciNet  MATH  Google Scholar 

  16. Liu Q and Jiang D Q, The threshold of a stochastic delayed SIR epidemic model with vaccination, Phys. A, 2016, 461: 140–147.

    Article  MathSciNet  MATH  Google Scholar 

  17. Meng X Z, Zhao S N, Feng T, et al., Dynamics of a novel nonlinear atochastic SIS epidemic model with double epidemic hypothesis, J. Math. Anal. Appl., 2016, 433: 227–242.

    Article  MathSciNet  MATH  Google Scholar 

  18. Liu G D, Wang X H, Meng X Z, et al., Extinction and persistence in mean of a novel delay impulsive stochastic infected predator-prey system with jumps, Complexity, 2017, 2017(3): 1–15.

    MathSciNet  MATH  Google Scholar 

  19. Miao A Q, Zhang J, Zhang T Q, et al., Threshold dynamics of a stochastic SIR model with vertical transmission and vaccination. Comput. Math. Method. M., 2017, 2017, DOI: https://doi.org/10.1155/2017/4820183.

  20. Miao A Q, Wang X Y, Wang W, et al., Dynamical analysis of a stochastic SIS epidemic model with nonlinear incidence rate and double epidemic hypothesis, Adv. Difference Equ., 2017, 2017: 226.

    Article  MathSciNet  MATH  Google Scholar 

  21. Tudor D, A deterministic model for herpes infections in human and animal polulations, SIAM Rev., 1990, 32: 130–139.

    Article  Google Scholar 

  22. Ding S S and Wang F J, SILI epidemiological model with nonlinear incidence rates, J. Biomath., 1994, 9: 1–59.

    MathSciNet  MATH  Google Scholar 

  23. Blower S, Modeling the genital herpes epidemic, Herpes, 2004, 11(Suppl.3): 138–146.

    Google Scholar 

  24. Wang J L and Shu H Y, Global analysis on a class of multi-group SEIR model with latency and relapse, Math. Biosci. Eng., 2016, 13: 200–225.

    Article  MathSciNet  MATH  Google Scholar 

  25. Fatini M E, Lahrouz A, Pettersson R, et al., Stochastic stability and instability of an epidemic model with relapse, Appl. Math. Comput., 2018, 316: 326–341.

    MathSciNet  MATH  Google Scholar 

  26. Liu Q, Jiang D Q, Hayat T, et al., Stationary distribution and extinction of a stochastic SIRI epidemic model with relapse, Stoch. Anal. Appl., 2018, 36: 138–151.

    Article  MathSciNet  MATH  Google Scholar 

  27. Has’miniskii R, Stochastic Stability of Differential Equations, Sijthoff Noordhoff, Alphen aan den Rijn, 1980.

    Book  Google Scholar 

  28. Mao X R, Stochastic Differential Equations and Their Applications, Horwood Publishing, Chichester, 1997.

    MATH  Google Scholar 

  29. Roberts M G and Saha A K, The asymptotic behaviour of a logistic epidemic model with stochastic disease transmission, Appl. Math. Lett., 1999, 12: 37–41.

    Article  MathSciNet  MATH  Google Scholar 

  30. Du N H and Sam V H, Dynamics of a stochastic Lotka-Volterra model perturbed by white noise, J. Math. Anal. Appl., 2006, 324: 82–97.

    Article  MathSciNet  MATH  Google Scholar 

  31. Zhao Y N, Jiang D Q, and O’Regan D, The extinction and persistence of the stochastic SIS epidemic model with vaccination, Phys. A, 2013, 392: 4916–4927.

    Article  MathSciNet  MATH  Google Scholar 

  32. Ma H J and Jia Y M, Stability analysis for stochastic differential equations with infinite markovian switchings, J. Math. Anal. Appl., 2016, 435: 593–605.

    Article  MathSciNet  MATH  Google Scholar 

  33. Meng X Z and Zhang L, Evolutionary dynamics in a Lotka-Volterra competition model with impulsive periodic disturbance, Math. Methods Appl. Sci., 2016, 39: 177–188.

    Article  MathSciNet  MATH  Google Scholar 

  34. Meng X Z, Wang L, and Zhang T H, Global dynamics analysis of a nonlinear impulsive stochastic chemostat system in a polluted environment, J. Appl. Anal. Comput., 2016, 6: 865–875.

    MathSciNet  Google Scholar 

  35. Liu Q, Jiang D Q, Shi N Z, et al., Stationary distribution and extinction of a stochastic SEIR epidemic model with standard incidence, Phys. A, 2017, 476: 58–69.

    Article  MathSciNet  Google Scholar 

  36. Liu L D and Meng X Z, Optimal harvesting control and dynamics of two-species stochastic model with delays, Adv. Difference Equ., 2017, 2017: 18, https://doi.org/10.1186/s13662-017-1077-6.

    Article  MathSciNet  MATH  Google Scholar 

  37. Zhang S Q, Meng X Z, and Zhang T H, Dynamics analysis and numerical simulations of a stochastic non-autonomous predator-prey system with impulsive effects, Nonlinear Anal. Hybrid Syst., 2017, 26: 19–37.

    Article  MathSciNet  MATH  Google Scholar 

  38. Jódar L, Villanueva R J, and Arenas A, Modeling the spread of seasonal epidemical diseases: Theory and applications, Math. Comput. Model., 2008, 48: 548–557.

    Article  MATH  Google Scholar 

  39. Lin Y G, Jiang D Q, and Liu T H, Nontrivial periodic solution of a stochastic epidemic model with seasonal variation, Appl. Math. Lett., 2015, 45: 103–107.

    Article  MathSciNet  MATH  Google Scholar 

  40. Zhu C and Yin G, Asymptotic properties of hybrid diffusion system, SIAM J. Control. Optim., 2007, 46: 1155–1179.

    Article  MathSciNet  MATH  Google Scholar 

  41. Kutoyants A Y, Statistical Inference for Ergodic Diffusion Processes, Springer, London, 2003.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao, 266590, China

    Weiwei Zhang & Xinzhu Meng

  2. College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, 266590, China

    Weiwei Zhang, Xinzhu Meng & Yulin Dong

  3. State Key Laboratory of Mining Disaster Prevention and Control Co-founded by Shandong Province and the Ministry of Science and Technology, Shandong University of Science and Technology, Qingdao, 266590, China

    Xinzhu Meng

Authors
  1. Weiwei Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xinzhu Meng
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yulin Dong
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xinzhu Meng.

Additional information

This work is supported by the National Natural Science Foundation of China under Grant No. 11371230, the Research Fund for the Taishan Scholar Project of Shandong Province of China, and the SDUST Research Fund under Grant No. 2014TDJH102.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, W., Meng, X. & Dong, Y. Periodic Solution and Ergodic Stationary Distribution of Stochastic SIRI Epidemic Systems with Nonlinear Perturbations. J Syst Sci Complex 32, 1104–1124 (2019). https://doi.org/10.1007/s11424-018-7348-9

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11424-018-7348-9

Keywords

Search

Navigation