Nonlinear stochastic analysis for a stochastic SIS epidemic model

Volume 10, Issue 9, pp 5116--5124 http://dx.doi.org/10.22436/jnsa.010.09.47

Publication Date: September 30, 2017 Submission Date: May 23, 2017

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 2372 Downloads
• 3991 Views

Authors

Fei Li - College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, P. R. China. Xinzhu Meng - 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, P. R. China. - 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, P. R. China. Yujun Cui - College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, P. R. China.

Abstract

This paper considers a stochastic Susceptible-Infected-Susceptible (SIS) epidemic model with nonlinear saturated incidence. The threshold conditions for disease extinction and stochastic permanence are obtained by using nonlinear stochastic analysis for Feller's test and the canonical probability method. Consequently, this improves and extends some previous results obtained by using Lyapunov method. A series of numerical simulations are carried out to illustrate our theoretical findings.

Share and Cite

Keywords

• Stochastic SIS epidemic model
• Feller's test
• stochastic permanence
• nonlinear saturated incidence

MSC

•  60H10
•  65C30
•  91B70

References

• [1] Y.-L. Cai, Y. Kang, M. Banerjee, W.-M. Wang, A stochastic SIRS epidemic model with infectious force under intervention strategies, J. Differential Equations, 259 (2015), 7463–7502.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [2] T. Feng, X.-Z. Meng, L.-D. Liu, S.-J. Gao, Application of inequalities technique to dynamics analysis of a stochastic eco-epidemiology model , J. Inequal. Appl., 2016 (2016), 29 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [3] S.-J. Gao, L.-S. Chen, J. J. Nieto, A. Torres, Analysis of a delayed epidemic model with pulse vaccination and saturation incidence, Vaccine, 24 (2006), 6037–6045.
  • View Article
  • Google Scholar


• [4] A. Gray, D. Greenhalgh, L. Hu, X. Mao, J. Pan, A stochastic differential equation SIS epidemic model , SIAM J. Appl. Math., 71 (2011), 876–902.
  • View Article
  • MathSciNet
  • Google Scholar


• [5] Z.-X. Hu, W.-B. Ma, S.-G. Ruan, Analysis of SIR epidemic models with nonlinear incidence rate and treatment, Math. Biosci., 238 (2012), 12–20.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [6] C.-Y. Ji, D.-Q. Jiang, Threshold behaviour of a stochastic SIR model, Appl. Math. Model., 38 (2014), 5067–5079.
  • View Article
  • MathSciNet
  • Google Scholar


• [7] Z. Jin, S.-P. Li, X.-G. Zhang, J.-P. Zhang, X.-L. Peng, Epidemiological modeling on complex networks, Complex Systems and Networks, Springer, Berlin (2016)
  • View Article
  • MathSciNet
  • Google Scholar


• [8] I. Karatzas, S. E. Shreve, Brownian motion and stochastic calculus, Springer, New York (1988)
  • Google Scholar


• [9] X.-P. Li, X.-Y. Lin, Y.-Q. Lin, Lyapunov-type conditions and stochastic differential equations driven by G-Brownian motion, J. Math. Anal. Appl., 439 (2016), 235–255.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [10] M. Liu, C.-Z. Bai, Analysis of a stochastic tri-trophic food-chain model with harvesting, J. Math. Biol., 73 (2016), 597–625.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [11] Q. Liu, D.-Q. Jiang, N.-Z. Shi, T. Hayat, B. Ahmad, Stationary distribution and extinction of a stochastic SEIR epidemic model with standard incidence, Phys. A, 476 (2017), 58–69.
  • View Article
  • MathSciNet
  • Google Scholar


• [12] L.-D. Liu, X.-Z. Meng , Optimal harvesting control and dynamics of two species stochastic model with delays, Adv. Difference Equ., 2017 (2017), 17 pages
  • View Article
  • MathSciNet
  • Google Scholar


• [13] X.-Z. Liu, P. Stechlinski , Infectious disease models with time-varying parameters and general nonlinear incidence rate , Appl. Math. Model., 36 (2012), 1974–1994.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [14] M. Liu, K.Wang , Stationary distribution, ergodicity and extinction of a stochastic generalized logistic system, Appl. Math. Lett., 25 (2012), 1980–1985.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [15] J.-L. Lv, K. Wang, Almost sure permanence of stochastic single species models, J. Math. Anal. Appl., 422 (2015), 675–683.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [16] H.-J. Ma, Y.-M. Jia , Stability analysis for stochastic differential equations with infinite Markovian switchings, J. Math. Anal. Appl., 435 (2016), 593–605.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [17] W.-B. Ma, M. Song, Y. Takeuchi , Global stability of an SIR epidemicmodel with time delay, Appl. Math. Lett., 17 (2004), 1141–1145.
  • View Article
  • MathSciNet
  • Google Scholar


• [18] X.-R. Mao , Stochastic Differential Equations and Applications, Woodhead Publishing, Cambridge (2007)
  • Google Scholar


• [19] X.-Z. Meng, Stability of a novel stochastic epidemic model with double epidemic hypothesis, Appl. Math. Comput., 217 (2010), 506–515.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [20] X.-Z. Meng, L.-S. Chen, B. Wu , A delay SIR epidemic model with pulse vaccination and incubation times, Nonlinear Anal., Real World Appl., 11 (2010), 88–98.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [21] X.-Z. Meng, S.-N. Zhao, T. Feng, T.-H. Zhang, Dynamics of a novel nonlinear stochastic SIS epidemic model with double epidemic hypothesis, J. Math. Anal. Appl., 433 (2016), 227–242.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [22] A.-Q. Miao, X.-Y. Wang, T.-Q. Zhang, W. Wang, B. G. S. A. Pradeep, Dynamical analysis of a stochastic SIS epidemic model with nonlinear incidence rate and double epidemic hypothesis, Adv. Difference Equ., 2017 (2017), 27 pages.
  • View Article
  • MathSciNet
  • Google Scholar


• [23] A.-Q. Miao, J. Zhang, T.-Q. Zhang, B. G. S. A. Pradeep, Threshold dynamics of a stochastic SIR model with vertical transmission and vaccination, Comput. Math. Methods Med., 2017 (2017), 10 pages.
  • View Article
  • MathSciNet
  • Google Scholar


• [24] J. J. Nieto, A. Ouahab, R. Rodríguez-López, Random fixed point theorems in partially ordered metric spaces, Fixed Point Theory Appl., 2016 (2016), 19 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [25] X.-P. Sun, Y.-J. Wang, Stability analysis of a stochastic logistic model with nonlinear diffusion term, Appl. Math. Model., 32 (2008), 2067–2075.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [26] X.-Y. Wang, W. Xu, X.-Z. Liu, Stochastic stability of stochastic switched epidemic models with constant and impulsive control schemes, Chaos Solitons Fractals, 78 (2015), 185–193.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [27] C.-Q. Xu, S.-L. Yuan, An analogue of break-even concentration in a simple stochastic chemostat model, Appl. Math. Lett., 48 (2015), 62–68.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [28] X.-H. Zhang, D.-Q. Jiang, A. Alsaedi, T. Hayat, Stationary distribution of stochastic SIS epidemic model with vaccination under regime switching, Appl. Math. Lett., 59 (2016), 87–93.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [29] T.-Q. Zhang, X.-Z. Meng, T.-H. Zhang , Global analysis for a delayed SIV model with direct and environmental transmissions, J. Appl. Anal. Comput., 6 (2016), 479–491.
  • View Article
  • MathSciNet
  • Google Scholar


• [30] T.-Q. Zhang, X.-Z. Meng, T.-H. Zhang, Y. Song, Global dynamics for a new high-dimensional SIR model with distributed delay, Appl. Math. Comput., 218 (2012), 11806–11819.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [31] Z. Zhao, L.-S. Chen, X.-Y. Song, Impulsive vaccination of seir epidemic model with time delay and nonlinear incidence rate, Math. Comput. Simulation, 79 (2008), 500–510.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [32] Y. Zhao, W.-H. Zhang, Observer-based controller design for singular stochastic Markov jump systems with state dependent noise, J. Syst. Sci. Complex., 29 (2016), 946–958.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH