Export file:

Format

  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text

Content

  • Citation Only
  • Citation and Abstract

Dynamical analysis of a stochastic SIRS epidemic model with saturating contact rate

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

In this paper, a stochastic SIRS epidemic model with saturating contact rate is constructed. First, for the deterministic system, the stability of the equilibria is discussed by using eigenvalue theory. Second, for the stochastic system, the threshold conditions of disease extinction and persistence are established. Our results indicate that a large environmental noise intensity can suppress the spread of disease. Conversely, if the intensity of environmental noise is small, the system has a stationary solution which indicates the disease is persistent. Eventually, we introduce some computer simulations to validate the theoretical results.
  Figure/Table
  Supplementary
  Article Metrics

References

1. S. Ullah, M. A. Khan, M. Farooq, T. Gul, Modeling and analysis of Tuberculosis (TB) in Khyber Pakhtunkhwa, Pakistan, Math. Comput. Simulation, 165 (2019), 181-199.

2. G. Sun, J. Xie, S. Huang, Z. Jin, M. Li, L. Liu, Transmission dynamics of cholera: Mathematical modeling and control strategies, Commun. Nonlinear Sci. Numer. Simul., 45 (2017), 235-244.

3. Y. Bai, X. Mu, Global asymptotic stability of a generalized SIRS epidemic model with transfer from infectious to susceptible, J. Appl. Anal. Comput., 8 (2018), 402-412.

4. J. Lai, S. Gao, Y. Liu, X. Meng, Impulsive switching epidemic model with benign worm defense and quarantine strategy, Complexity, 540 (2020), 1-12.

5. Q. Liu, D. Jiang, Threshold behavior in a stochastic SIR epidemic model with Logistic birth, Phys. A, 2020 (2020), 123488.

6. D. Zhao, S. Yuan, Threshold dynamics of the stochastic epidemic model with jump-diffusion infection force, J. Appl. Anal. Comput., 9 (2019), 440-451.

7. B. Zhang, Y. Cai, B. Wang, W. Wang, Dynamics and asymptotic profiles of steady states of an SIRS epidemic model in spatially heterogenous environment, Math. Biosci. Eng., 17 (2019), 893-909.

8. Y. Tu, S. Gao, Y. Liu, D. Chen, Y. Xu, Transmission dynamics and optimal control of stagestructured HLB model, Math. Biosci. Eng., 16 (2019), 5180.

9. H. W. Hethcote, Qualitative analyses of communicable disease models, Math. Biosci., 28 (1976), 335-356.

10. J. Chen, An SIRS epidemic model, Appl. Math. J. Chinese Univ. Ser. B, 19 (2004), 101-108.

11. T. Li, F. Zhang, H. Liu, Y. Chen, Threshold dynamics of an SIRS model with nonlinear incidence rate and transfer from infectious to susceptible, Appl. Math. Lett., 70 (2017), 52-57.

12. Y. Wang, G. Liu, Dynamics analysis of a stochastic SIRS epidemic model with nonlinear incidence rate and transfer from infectious to susceptible, Math. Biosci. Eng., 16 (2019), 6047-6070.

13. H. R. Thieme, C. Castillo-Chavez, On the role of variable infectivity in the dynamics of the human immunodeficiency virus epidemic, Mathematical and Statistical Approaches to AIDS Epidemiology, Springer, Berlin, 1989.

14. J. Heesterbeek, J. A. Metz, The saturating contact rate in marriage and epidemic models, J. Math. Biol., 31 (1993), 529-539.

15. H. Zhang, Y. Li, W. Xu, Global stability of an SEIS epidemic model with general saturation incidence, Appl. Math., 2013 (2013), 1-11.

16. G. Lan, Y. Huang, C. Wei, S. Zhang, A stochastic SIS epidemic model with saturating contact rate, Phys. A, 529 (2019), 121504.

17. Y. Cai, J. Jiao, Z. Gui, Y. Liu, W. Wang, Environmental variability in a stochastic epidemic model, Appl. Math. Comput., 329 (2018), 210-226.

18. D. Kiouach, L. Boulaasair, Stationary distribution and dynamic behaviour of a stochastic SIVR epidemic model with imperfect vaccine, J. Appl. Math., 2018 (2018), 1-11.

19. X. Zhang, H. Peng,, Stationary distribution of a stochastic cholera epidemic model with vaccination under regime switching, Appl. Math. Lett., 102 (2020), 106095.

20. W. Wang, C. Ji, Y. Bi, S.Liu, Stability and asymptoticity of stochastic epidemic model with interim immune class and independent perturbations, Appl. Math. Lett., 104 (2020), 106245.

21. R. Lu, F. Wei, Persistence and extinction for an age-structured stochastic SVIR epidemic model with generalized nonlinear incidence rate, Phys. A, 513 (2019), 572-587.

22. T. Feng, Z. Qiu, X. Meng, Dynamics of a stochastic hepatitis-c virus system with host immunity, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 6367.

23. 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.

24. W. Guo, Q. Zhang, X. Li, W. Wang, Dynamic behavior of a stochastic SIRS epidemic model with media coverage, Math. Methods Appl. Sci., 41 (2018), 5506-5525.

25. Y. Zhang, K. Fan, S. Gao, Y. Liu, S. Chen., Ergodic stationary distribution of a stochastic SIRS epidemic model incorporating media coverage and saturated incidence rate, Phys. A, 514 (2019), 671-685.

26. Y. Cai, Y. Kang, M. Banerjee, W. Wang, A stochastic SIRS epidemic model with infectious force under intervention strategies, J. Differ. Equations, 259 (2015), 7463-7502.

27. D. Jiang, Q. Liu, N. Shi, T. Hayat, A. Alsaedi, P. Xia, Dynamics of a stochastic HIV-1 infection model with logistic growth, Phys. A, 469 (2017), 706-717.

28. T. Feng, Z. Qiu, Global analysis of a stochastic TB model with vaccination and treatment, Discrete Contin. Dyn. Syst. Ser. B, 24 (2018), 2923.

29. S. Zhao, S. Yuan, H. Wang, Threshold behavior in a stochastic algal growth model with stoichiometric constraints and seasonal variation, J. Differ. Equations, 268 (2020), 5113-5139.

30. X. Ji, S. Yuan, T. Zhang, H. Zhu, Stochastic modeling of algal bloom dynamics with delayed nutrient recycling, Math. Biosci. Eng., 16 (2019), 1-24.

31. Q. Liu, D. Jiang, T. Hayat, A. Alsaedi, B. Ahmad, A stochastic SIRS epidemic model with logistic growth and general nonlinear incidence rate, Phys. A, 551 (2020), 124152.

32. S. Cai, Y. Cai, X. Mao, A stochastic differential equation SIS epidemic model with two correlated Brownian motions, Nonlinear Dynam., 97 (2019), 2175-2187.

33. Q. Liu, D. Jiang, N. Shi, T. Hayat, A. Alsaedi, The threshold of a stochastic SIS epidemic model with imperfect vaccination, Math. Comput. Simulation, 144 (2018), 78-90.

34. R. Khasminskii, Stochastic Stability of Differential Equations, Springer, Heidelberg, 2011.

35. D. Xu, Y. Huang, Z. Yang, Existence theorems for periodic Markov process and stochastic functional differential equations, Discrete Contin. Dyn. Syst., 24 (2009), 1005.

© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Article outline

Show full outline
Copyright © AIMS Press All Rights Reserved