Skip to main content
SpringerLink
Log in

Analysis of Stochastic Viral Infection Model with Immune Impairment

  • Original Paper
  • Published:
International Journal of Applied and Computational Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we explore the stochastic viral infection model with immune impairment and show that this model has a unique global solution. Using the Lyapunov method, we investigate the stochastic stability of equilibrium solutions of this model. Finally, sufficient condition for persistence of the disease is established and illustrate our mathematical findings.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Arnold, L.: Stochastic Differential Equations: Theory and Applications. Wiley, New York (1974)

    MATH  Google Scholar 

  2. Chen, L., Chen, J.: Nonlinear Biological Dynamical System. Science Press, Beijing (1993)

    Google Scholar 

  3. Eric, A.V., Noe, C.C., Gerardo, G.A.: Analysis of a viral infection model with immune impairment, intracellular delay and general non-linear incidence rate. Chaos Solitons Fractals 69, 1–9 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  4. Gard, T.C.: Introduction to Stochastic Differential Equations. Marcel Dekker, New York and Basel (1998)

  5. Ji, C., Jiang, D., Shi, N.: Multigroup SIR epidemic model with stochastic perturbation. Phys. A 390, 1747–1762 (2011)

    Article  Google Scholar 

  6. Jia, J., Shi, X.: Analysis of a viral infection model with immune impairment and cure rate. J. Nonlinear Sci. Appl. 9, 3287–3298 (2016)

    MATH  MathSciNet  Google Scholar 

  7. Khasminskii, R.: Stochastic Stability of Differential Equations. Sijthoff & Noordhoff, Alpen (1980)

    Book  MATH  Google Scholar 

  8. Lahrouz, A., Omari, L.: Extinction and stationary distribution of a stochastic SIRS epidemic model with non-linear incidence. Stat. Probab. Lett. 83, 960–968 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  9. Lahrouz, A., Omari, L., Kiouach, D., Belmaâti, A.: Deterministic and stochastic stability of a mathematical model of smoking. Stat. Probab. Lett. 81, 1276–1284 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  10. Liu, M., Wang, K., Wu, Q.: Survival analysis of stochastic competitive models in a polluted environment and stochastic competitive exclusion principle. Bull. Math. Biol. 73, 1969–2012 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  11. Mao, X.: Stochastic Differential Equations and Applications. Horwood Publishing Limited, Chichester (1997)

    MATH  Google Scholar 

  12. Mao, X., Marion, G., Renshaw, E.: Environmental Brownian noise suppresses explossins in population dynamics. Stoch. Process. Appl. 97, 95–110 (2002)

    Article  MATH  Google Scholar 

  13. May, R.M.: Stability and Complexity in Model Ecosystems. Princeton University Press, Princeton (2001)

    MATH  Google Scholar 

  14. Mukhopadhyay, B., Bhattacharyya, R.: Effects of deterministic and random refuge in a prey-predator model with parasite infection. Math. Biosci. 239, 124–130 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  15. Pitchaimani, M., Rajaji, R.: Stochastic asymptotic stability of Nowak–May model with variable diffusion rates. Methodol. Comput. Appl. Probab. 18, 901–910 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  16. Regoes, R.R., Wodarz, D., Nowak, M.A.: Virus dynamics: the effect of target cell limitation and immune responses on virus evolution. J. Theor. Biol. 191, 451–462 (1998)

    Article  Google Scholar 

  17. Wang, K., Wang, W., Pang, H., Liu, X.: Complex dynamic behavior in a viral model with delayed immune response. Phys. D 226, 197–208 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  18. Wang, S., Song, X., Ge, Z.: Dynamics analysis of a delayed viral infection model with immune impairment. Appl. Math. Model 35, 4877–4885 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  19. Wodarz, D., Christensen, J.P., Thomsen, A.R.: The importance of lytic and nonlytic immune responses in viral infections. Trends Immunol. 23, 194–200 (2002)

    Article  Google Scholar 

  20. Xie, Q., Huang, D., Zhang, S., Cao, J.: Analysis of a viral infection model with delayed immune response. Appl. Math. Model. 34, 2388–2395 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  21. Yang, Q., Mao, X.: Stochastic dynamics of SIRS epidemic models with random perturbation. Math. Biosci. Eng. 11(4), 1003–1025 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  22. Zhao, Y., Yuan, S., Ma, J.: Survival and stationary distribution analysis of a stochastic competitive model of three species in a polluted environment. Bull. Math. Biol. 77(7), 1285–1326 (2015)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgements

We would like to thank the anonymous referees for their valuable comments and suggestions. This article was supported by Basic Science Research, University Grant Commission, India.

Author information

Authors and Affiliations

  1. Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai, 600 005, India

    R. Rajaji & M. Pitchaimani

Authors
  1. R. Rajaji
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Pitchaimani
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to R. Rajaji.

Appendix

Appendix

CD4\(^+\)T helper cells are white blood cells that are an essential part of the human immune system. They are often referred to as CD4 cells, \(T-\)helper cells or T4 cells. They are called helper cells because one of their main roles is to send signals to other types of immune cells, including CD8 killer cells, which then destroy the infectious particle.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rajaji, R., Pitchaimani, M. Analysis of Stochastic Viral Infection Model with Immune Impairment. Int. J. Appl. Comput. Math 3, 3561–3574 (2017). https://doi.org/10.1007/s40819-017-0314-8

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40819-017-0314-8

Keywords

Mathematics Subject Classification

Search

Navigation