Export file:

Format

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

Content

  • Citation Only
  • Citation and Abstract

Dynamics of an SLIR model with nonmonotone incidence rate and stochastic perturbation

1 School of Mathematics and Statistics, Zhengzhou University, Henan Zhengzhou, 450001, China
2 College of Science, Zhongyuan University of Technology, Henan Zhengzhou, 450007, China
3 School of Mathematics and Statistics, Central China Normal University, Hubei Wuhan 430079, China

Special Issues: Computational models in life sciences

In this paper we study an SLIR epidemic model with nonmonotonic incidence rate, which describes the psychological effect of certain serious diseases on the community when the number of infectives is getting larger. By carrying out a global analysis of the model and studying the stability of the disease-free equilibrium and the endemic equilibrium, we show that either the number of infective individuals tends to zero or the disease persists as time evolves. For the stochastic model, we prove the existence, uniqueness and positivity of the solution of the model. Then, we investigate the stability of the model and we prove that the infective tends asymptotically to zero exponentially almost surely as R0 < 1. We also proved that the SLIR model has the ergodic property as the fluctuation is small, where the positive solution converges weakly to the unique stationary distribution.
  Figure/Table
  Supplementary
  Article Metrics

Keywords epidemic model; nonmonotone incidence rate; psychological effect; global stability; stochastic perturbation; ergodic property

Citation: Jinhui Zhang, Jingli Ren, Xinan Zhang. Dynamics of an SLIR model with nonmonotone incidence rate and stochastic perturbation. Mathematical Biosciences and Engineering, 2019, 16(5): 5504-5530. doi: 10.3934/mbe.2019274

References

  • 1. H. Hethcote, The mathematics of infectious disease, SIAM Rev. 42 (2000), 59–653.
  • 2. V. Capasso and G. Serio, A generalization of the Kermack-Mckendrick deterministic epidemic model, Math. Biosci., 42 (1978), 43–61.
  • 3. W. Liu, S. Levin and Y. Iwasa, Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, J Math. Biol., 23 (1986), 187–204.
  • 4. S. Ruan and W. Wang, Dynamical behavior of an epidemic model with a nonlinear incidence rate, J. Differ. Eq., 188 (2003), 135–163.
  • 5. W. Derrick and P. Van den Driessche, A disease transmission model in a nonconstant population, J. Math. Biol., 31 (1993), 495–512.
  • 6. H. Hethcote and P. Van den Driessche, Some epidemiological models with nonlinear incidence, J. Math. Biol., 29 (1981), 271–287.
  • 7. M. Alexander and S. Moghadas, Periodicity in an epidemic model with a generalized non-linear incidence, Math. Biosci., 189 (2004), 75–96.
  • 8. D. Xiao and S. Ruan, Global analysis of an eqidemic model with nonmonotone incidence rate, Math. Biosci., 208 (2003), 419–429.
  • 9. W. Tan and X. Zhu, A stochastic model for the HIV epidemic in homosexual populations involving age and race, Math. Comput. Model., 24 (1996), 67–105.
  • 10. W. Tan and X. Zhu, A stochastic model of the HIV epidemic for heterosexual transmission involving married couples and prostitutes: I. The probabilities of HIV transmission and pair formation, Math. Comput. Model., 24 (1996), 47–107.
  • 11. W. Tan and X. Zhu, A stochastic model of the HIV epidemic for heterosexual transmission involving married couples and prostitutes: II. The chain multinomial model of the HIV epidemic, Math. Comput. Model., 26 (1997), 17–92.
  • 12. W. Tan and Z. Xiang, A state space model for the HIV epidemic in homosexual populations and some applications, Math. Biosci., 152 (1998), 29–61.
  • 13. J. Beddington and R. May, Harvesting natural populations in a randomly fluctuating environment, Science, 197 (1977), 463–465.
  • 14. N. Dalal, D. Greenhalgh and X. Mao, A stochastic model of AIDS and condom use, J. Math. Anal. Appl., 325 (2007), 36–53.
  • 15. N. Dalal, D. Greenhalgh and X. Mao, A stochastic model for internal HIV dynamics, J. Math. Anal. Appl., 341 (2008), 1084–1101.
  • 16. C. Ji and D. Jiang, Dynamics of a stochastic density dependent predator Cprey system with Beddington CDeAngelis functional response, J. Math. Anal. Appl., 381 (2011), 441–453.
  • 17. D. Jiang, C. Ji, N. Shi, et al., The long time behavior of DI SIR epidemic model with stochastic perturbation, J. Math. Anal. Appl., 372 (2010), 162–180.
  • 18. D. Jiang, J. Yu, C. Ji, et al., Asymptotic behavior of global positive solution to a stochastic SIR model, Math. Comput. Model., 54 (2011), 221–232.
  • 19. Q. Yang, D. Jiang, N. Shi, et al., The ergodicity and extinction of stochastically perturbed sir and seir epidemic models with saturated incidence, J. Math. Anal. Appl., 388 (2012), 248–271.
  • 20. E. Berettaa, V. Kolmanovskiib and L. Shaikhetc, Stability of epidemic model with time delays influenced by stochastic perturbations, Math. Comput. Simulation., 45 (1998), 269–277.
  • 21. M. Carletti, On the stability properties of a stochastic model for phage-bacteria interaction in open marine environment, Math. Biosci., 175 (2002), 117–131.
  • 22. M. Carletti, Numerical simulation of a Campbell-like stochastic delay model for bacteriophage infection, Math. Med. Biol., 23 (2006), 297–310.
  • 23. K. Dietz, Transmission and control of arbovirus diseases, In Proceedings of the Society for Industrial and Applied Mathematics, Epidemiology: Philadelphia, 01 (1975), 104–121.
  • 24. P. Van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 28–29.
  • 25. N. Bhatia and G. Szegö Stability theory of dynamical systems, Springer-Verlag, 1931.
  • 26. J. LaSalle, Stability theory for ordinary differential equations, J. Differ. Eqns., 41 (1968), 57–65.
  • 27. J. LaSalle, The stability of dynamical systems, SIAM Rev., 1976.
  • 28. M. Li and J. Muldowney, A geometric approach to global-stability problems, SIAM J. Math. Anal., 27 (1996), 1070C83.
  • 29. C. Pugh, An improved closing lemma and the general density Theorem, Amer. J. Math., 89 (1976), 1010–1021.
  • 30. C. Pugh and C. Robinson, The C1 closing lemma, including hamiltonians, Ergod. Theor. Dynam. Sys., 3 (1983), 261–313.
  • 31. M. Li and J. Muldowney, On Bendixson's criterion. J Differ. Eq. 106 (1994), 27C39.
  • 32. R. Martin, Logarithmic norms and projections applied to linear differential systems; J. Math. Anal. Appl., 45 (1974), 432–454.
  • 33. W. Coppel, Stability and Asymptotic Behavior of Differential Equations, Am. Math. Monthly, 1965.
  • 34. L. Arnold, Stochastic Differential Equations: Theory and Applications, Wiley, New York, 1972.
  • 35. X. Mao, Stochastic Differential Equations and Applications, Horwood, Chichester, 1997.
  • 36. X. Mao, G. Marion and E. Renshaw, Environmental noise suppresses explosion in population dynamics, Stochastic Process. Appl., 97 (2002), 95–110.
  • 37. T. Caraballo and P. E. Kloeden, The persistence of synchronization under environmental noise, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461 (2005), 2257–2267.
  • 38. R. Z. Hasminskii, Stochastic Stability of Differential Equations, Sijthoff & Noordhoff, Alphen aan den Rijn, The Netherlands, 1980.
  • 39. G. Strang, Linear Algebra and Its Applications, Thomson Learning, Inc., United States, 1988.
  • 40. K. B. Gopal and N. B. Rabi, Stability in distribution for a class of singular diffusions, Ann. Probab., 20 (1992), 312–321.
  • 41. A. Yury and Kutoyants, Statistical Inference for Ergodic Diffusion Processes, Springer, London, 2003.

 

Reader Comments

your name: *   your email: *  

© 2019 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

Copyright © AIMS Press All Rights Reserved