Table of Contents Author Guidelines Submit a Manuscript
Complexity
Volume 2017, Article ID 4861391, 14 pages
https://doi.org/10.1155/2017/4861391
Research Article

Dynamics of a Nonautonomous Stochastic SIS Epidemic Model with Double Epidemic Hypothesis

1College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China
2State 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

Correspondence should be addressed to Xinzhu Meng; nc.ude.tsuds@601127zxm

Received 25 August 2017; Accepted 11 October 2017; Published 8 November 2017

Academic Editor: Benito M. Chen-Charpentier

Copyright © 2017 Haokun Qi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. W. O. Kermack and A. G. McKendrick, “A contributions to the mathematical theory of epidemics (Part I),” Proceedings of the Royal Society of London A, vol. 115, pp. 700–721, 1927. View at Google Scholar
  2. A. Gray, D. Greenhalgh, L. Hu, X. Mao, and J. Pan, “A stochastic differential equation SIS epidemic model,” SIAM Journal on Applied Mathematics, vol. 71, no. 3, pp. 876–902, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  3. A. Korobeinikov and G. C. Wake, “Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models,” Applied Mathematics Letters, vol. 15, no. 8, pp. 955–960, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  4. H. W. Hethcote and P. van den Driessche, “An SIS epidemic model with variable population size and a delay,” Journal of Mathematical Biology, vol. 34, no. 2, pp. 177–194, 1995. View at Publisher · View at Google Scholar · View at MathSciNet
  5. T. Zhang, X. Meng, Y. Song, and T. Zhang, “A stage-structured predator-prey SI model with disease in the prey and impulsive effects,” Mathematical Modelling and Analysis, vol. 18, no. 4, pp. 505–528, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  6. J. Li and Z. Ma, “Qualitative analyses of SIS epidemic model with vaccination and varying total population size,” Mathematical and Computer Modelling, vol. 35, no. 11-12, pp. 1235–1243, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  7. A. d'Onofrio, “A note on the global behaviour of the network-based SIS epidemic model,” Nonlinear Analysis: Real World Applications, vol. 9, no. 4, pp. 1567–1572, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  8. H. W. Hethcote and P. van den Driessche, “Two SIS epidemiologic models with delays,” Journal of Mathematical Biology, vol. 40, no. 1, pp. 3–26, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  9. T. Zhang, X. Meng, T. Zhang, and Y. Song, “Global dynamics for a new high-dimensional SIR model with distributed delay,” Applied Mathematics and Computation, vol. 218, no. 24, pp. 11806–11819, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  10. S. Gao, L. Chen, J. J. Nieto, and A. Torres, “Analysis of a delayed epidemic model with pulse vaccination and saturation incidence,” Vaccine, vol. 24, no. 35-36, pp. 6037–6045, 2006. View at Publisher · View at Google Scholar · View at Scopus
  11. X. Meng, S. Zhao, T. Feng, and T. Zhang, “Dynamics of a novel nonlinear stochastic SIS epidemic model with double epidemic hypothesis,” Journal of Mathematical Analysis and Applications, vol. 433, no. 1, pp. 227–242, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  12. A. Miao, X. Wang, T. Zhang, W. Wang, and B. Sampath Aruna Pradeep, “Dynamical analysis of a stochastic SIS epidemic model with nonlinear incidence rate and double epidemic hypothesis,” Advances in Difference Equations, 2017:226 pages, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  13. X.-Z. Meng, “Stability of a novel stochastic epidemic model with double epidemic hypothesis,” Applied Mathematics and Computation, vol. 217, no. 2, pp. 506–515, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  14. X. P. Li, X. Y. Lin, and Y. Q. Lin, “Lyapunov-type conditions and stochastic differential equations driven by G-Brownian motion,” Journal of Mathematical Analysis and Applications, vol. 439, no. 1, pp. 235–255, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  15. M. Liu and M. Fan, “Permanence of stochastic Lotka-Volterra systems,” Journal of Nonlinear Science, vol. 27, no. 2, pp. 425–452, 2017. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. L. D. Liu and X. Z. Meng, “Optimal harvesting control and dynamics of two-species stochastic model with delays,” Advances in Difference Equations, vol. 2017, 18 pages, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  17. H. Ma and Y. Jia, “Stability analysis for stochastic differential equations with infinite Markovian switchings,” Journal of Mathematical Analysis and Applications, vol. 435, no. 1, pp. 593–605, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. G. D. Liu, X. H. Wang, X. Z. Meng, and S. J. Gao, “Extinction and persistence in mean of a novel delay impulsive stochastic infected predator-prey system with jumps,” Complexity, vol. 3, pp. 1–15, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  19. C. Tan and W. H. Zhang, “On observability and detectability of continuous-time stochastic Markov jump systems,” Journal of Systems Science and Complexity, vol. 28, no. 4, pp. 830–847, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  20. X. Leng, T. Feng, and X. Meng, “Stochastic inequalities and applications to dynamics analysis of a novel SIVS epidemic model with jumps,” Journal of Inequalities and Applications, Paper No. 138, 25 pages, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  21. G. Li and M. Chen, “Infinite horizon linear quadratic optimal control for stochastic difference time-delay systems,” Advances in Difference Equations, vol. 2015, 14 pages, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  22. Y. Zhao and W. Zhang, “Observer-based controller design for singular stochastic Markov jump systems with state dependent noise,” Journal of Systems Science and Complexity, vol. 29, no. 4, pp. 946–958, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  23. S. Zhang, X. Meng, T. Feng, and T. Zhang, “Dynamics analysis and numerical simulations of a stochastic non-autonomous predator–prey system with impulsive effects,” Nonlinear Analysis: Hybrid Systems, vol. 26, pp. 19–37, 2017. View at Publisher · View at Google Scholar
  24. H.-j. Ma and T. Hou, “A separation theorem for stochastic singular linear quadratic control problem with partial information,” Acta Mathematicae Applicatae Sinica, vol. 29, no. 2, pp. 303–314, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  25. X. Meng, L. Wang, and T. Zhang, “Global dynamics analysis of a nonlinear impulsive stochastic chemostat system in a polluted environment,” Journal of Applied Analysis and Computation, vol. 6, no. 3, pp. 865–875, 2016. View at Google Scholar · View at MathSciNet
  26. X. Liu, Y. Li, and W. Zhang, “Stochastic linear quadratic optimal control with constraint for discrete-time systems,” Applied Mathematics and Computation, vol. 228, pp. 264–270, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  27. X. Z. Meng and X. H. Wang, “Stochastic predator-prey system subject to lévy jump,” Discrete Dynamics in Nature and Society, vol. 2016, Article ID 5749892, 13 pages, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  28. X. Lv, L. Wang, and X. Meng, “Global analysis of a new nonlinear stochastic differential competition system with impulsive effect,” Advances in Difference Equations, vol. 2017, 296 pages, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  29. X. Zhang, D. Jiang, A. Alsaedi, and T. Hayat, “Stationary distribution of stochastic SIS epidemic model with vaccination under regime switching,” Applied Mathematics Letters, vol. 59, pp. 87–93, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  30. F. Li, X. Meng, and Y. Cui, “Nonlinear stochastic analysis for a stochastic SIS epidemic model,” Journal of Nonlinear Sciences and Applications, vol. 10, no. 09, pp. 5116–5124, 2017. View at Publisher · View at Google Scholar
  31. A. Q. Miao, J. Zhang, T. Q. Zhang, and B. G. Pradeep, “Sampath Aruna Pradeep, Threshold dynamics of a stochastic SIR model with vertical transmission and vaccination,” Computational and Mathematical Methods in Medicine, vol. 2017, Article ID 4820183, 10 pages, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  32. Q. Liu, D. Jiang, N. Shi, T. Hayat, and A. Alsaedi, “Nontrivial periodic solution of a stochastic non-autonomous SISV epidemic model,” Physica A: Statistical Mechanics and its Applications, vol. 462, pp. 837–845, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  33. Y. L. Zhou, S. L. Yuan, and D. L. Zhao, “Threshold behavior of a stochastic SIS model with levy jumps,” Applied Mathematics and Computation, vol. 275, pp. 255–267, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  34. T. Feng, X. Meng, L. Liu, and S. Gao, “Application of inequalities technique to dynamics analysis of a stochastic eco-epidemiology model,” Journal of Inequalities and Applications, Paper No. 327, 29 pages, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  35. M. Liu and K. Wang, “Persistence and extinction in stochastic non-autonomous logistic systems,” Journal of Mathematical Analysis and Applications, vol. 375, no. 2, pp. 443–457, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  36. Z. Bai and Y. Zhou, “Existence of two periodic solutions for a non-autonomous SIR epidemic model,” Applied Mathematical Modelling: Simulation and Computation for Engineering and Environmental Systems, vol. 35, no. 1, pp. 382–391, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  37. S. Gao, F. Zhang, and Y. He, “The effects of migratory bird population in a nonautonomous eco-epidemiological model,” Applied Mathematical Modelling: Simulation and Computation for Engineering and Environmental Systems, vol. 37, no. 6, pp. 3903–3916, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  38. T. Kuniya, “Existence of a nontrivial periodic solution in an age-structured SIR epidemic model with time periodic coefficients,” Applied Mathematics Letters, vol. 27, pp. 15–20, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  39. L. Zu, D. Jiang, D. O'Regan, and B. Ge, “Periodic solution for a non-autonomous Lotka-Volterra predator-prey model with random perturbation,” Journal of Mathematical Analysis and Applications, vol. 430, no. 1, pp. 428–437, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  40. Y. Lin, D. Jiang, and T. Liu, “Nontrivial periodic solution of a stochastic epidemic model with seasonal variation,” Applied Mathematics Letters, vol. 45, pp. 103–107, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  41. X. R. Mao, Stochastic Differential Equations and Their Applications, Horwood, Chichester, UK, 1997.
  42. R. Khasminskii, Stochastic Stability of Differential Equations, Springer, Berlin, Germany, 2011.