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

  • Weiwei Zhang
  • Xinzhu MengEmail author
  • Yulin Dong


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.


Extinction and stochastic permanence Markov chain periodic solution stationary distribution and ergodicity stochastic SIRI epidemic model 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    Herbert H W, The mathematics of infectious diseases, SIAM Rev., 2000, 42: 599–653.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Brauer F and Chavez C C, Mathematical Models in Population Biology and Epidemiology, Springer, New York, 2001.CrossRefzbMATHGoogle Scholar
  4. [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.CrossRefGoogle Scholar
  5. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Ma Z E, Zhou Y C, and Wu J H, Modeling and Dynamics of Infectious Diseases, Higher Education Press, Beijing, 2009.CrossRefzbMATHGoogle Scholar
  7. [7]
    Meng X Z, Stability of a novel stochastic epidemic model with double epidemic hypothesis, Appl. Math. Comput., 2010, 217: 506–515.MathSciNetzbMATHGoogle Scholar
  8. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Wang W and Ma W B, A diffusive HIV infection model with nonlocal delayed transmission, Appl. Math. Lett., 2018, 75: 96–101.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Korobeinikov A, Lyaounov functions and global stability for SIR and SIRS epidemiological models with nonlinear transmission, Bull. Math. Biol., 2006, 30: 615–636.CrossRefzbMATHGoogle Scholar
  11. [11]
    Ji C, Jiang D Q, and Shi N Z, Multigroup SIR epidemic model with stochastic perturbation, Phys. A, 2011, 390: 1747–1762.CrossRefGoogle Scholar
  12. [12]
    Gray A, Greenhalgh D, Hu L, et al., A stochastic differential equation SIS epidemic model, SIAM J. Appl. Math., 2011, 71: 876–902.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [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.MathSciNetzbMATHGoogle Scholar
  14. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    Liu Q and Jiang D Q, The threshold of a stochastic delayed SIR epidemic model with vaccination, Phys. A, 2016, 461: 140–147.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [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.MathSciNetGoogle Scholar
  19. [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: 10.1155/2017/4820183.Google Scholar
  20. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    Tudor D, A deterministic model for herpes infections in human and animal polulations, SIAM Rev., 1990, 32: 130–139.CrossRefGoogle Scholar
  22. [22]
    Ding S S and Wang F J, SILI epidemiological model with nonlinear incidence rates, J. Biomath., 1994, 9: 1–59.MathSciNetzbMATHGoogle Scholar
  23. [23]
    Blower S, Modeling the genital herpes epidemic, Herpes, 2004, 11(Suppl.3): 138–146.Google Scholar
  24. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [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.MathSciNetGoogle Scholar
  26. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    Has’miniskii R, Stochastic Stability of Differential Equations, Sijthoff Noordhoff, Alphen aan den Rijn, 1980.CrossRefGoogle Scholar
  28. [28]
    Mao X R, Stochastic Differential Equations and Their Applications, Horwood Publishing, Chichester, 1997.zbMATHGoogle Scholar
  29. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  30. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  31. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  32. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  33. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  34. [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.MathSciNetGoogle Scholar
  35. [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.MathSciNetCrossRefGoogle Scholar
  36. [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, Scholar
  37. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  38. [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.CrossRefzbMATHGoogle Scholar
  39. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    Zhu C and Yin G, Asymptotic properties of hybrid diffusion system, SIAM J. Control. Optim., 2007, 46: 1155–1179.MathSciNetCrossRefzbMATHGoogle Scholar
  41. [41]
    Kutoyants A Y, Statistical Inference for Ergodic Diffusion Processes, Springer, London, 2003.zbMATHGoogle Scholar

Copyright information

© Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of Electrical Engineering and AutomationShandong University of Science and TechnologyQingdaoChina
  2. 2.College of Mathematics and Systems ScienceShandong University of Science and TechnologyQingdaoChina
  3. 3.State Key Laboratory of Mining Disaster Prevention and Control Co-founded by Shandong Province and the Ministry of Science and TechnologyShandong University of Science and TechnologyQingdaoChina

Personalised recommendations