Skip to main content
Log in

Mathematical analysis of a time delay visceral leishmaniasis model

  • Original Research
  • Published:
Journal of Applied Mathematics and Computing Aims and scope Submit manuscript

Abstract

In this paper, we discuss some of the dynamical characteristics of a visceral leishmaniasis (VL) model with time delay. We have derived sufficient conditions to ensure the stability of the considered delayed VL model at the steady states. Taking the time delay as a bifurcation parameter, we have established a criteria for the existence of Hopf bifurcation of the considered model. Moreover, conditions for global stability of the steady states are also presented. Finally, some numerical simulations are given to show the effectiveness of our theoretical results.

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

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Hethcote, H.W.: The mathematics of infectious diseases. SIAM Rev. 42(4), 599–653 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Anderson, R.M., May, R.M.: Infectious Diseases of Humans: Dynamics and Control. Oxford University Press, Oxford (1992)

    Google Scholar 

  3. Hethcote, H.W.: Qualitative analyses of communicable disease models. Math. Biosci. 28(3–4), 335–356 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  4. Hu, Z., Ma, W., Ruan, S.: Analysis of sir epidemic models with nonlinear incidence rate and treatment. Math. Biosci. 238(1), 12–20 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Tian, J.P., Wang, J.: Global stability for cholera epidemic models. Math. Biosci. 232(1), 31–41 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Liu, X., Yang, L.: Stability analysis of an seiqv epidemic model with saturated incidence rate. Nonlinear Anal. Real World Appl. 13(6), 2671–679 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chitnis, N., Cushing, J.M., Hyman, J.: Bifurcation analysis of a mathematical model for malaria transmission. SIAM J. Appl. Math. 67(1), 24–45 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. Lv, C., Huang, L., Yuan, Z.: Global stability for an HIV-1 infection model with beddington-deangelis incidence rate and ctl immune response. Commun. Nonlinear Sci. Numer. Simul. 19(1), 121–127 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ponte-Sucre, A.: Introduction: leishmaniasis—the biology of a parasite. In: Ponte-Sucre, A., Diaz, E., Padrón-Nieves, M. (eds.) Drug Resistance in Leishmania Parasites, pp. 1–12. Springer, Vienna (2013)

    Chapter  Google Scholar 

  10. Oryan, A., Akbari, M.: Worldwide risk factors in leishmaniasis. Asian Pac. J. Trop. Med. 9(10), 925–932 (2016)

    Article  Google Scholar 

  11. CDC division of parasitic diseases, leishmaniasis. http://www.dpd.cdc.gov/dpdx/HTML/Leishmaniasis.htm. Accessed 21 Jan 2019

  12. Desjeux, P.: Leishmaniasis: current situation and new perspectives. Comp. Immunol. Microbiol. Infect. Dis. 27(5), 305–318 (2004)

    Article  Google Scholar 

  13. Chappuis, F., Sundar, S., Hailu, A., Ghalib, H., Rijal, S., Peeling, R.W., Alvar, J., Boelaert, M.: Visceral leishmaniasis: what are the needs for diagnosis, treatment and control? Nat. Rev. Microbiol. 5(11supp), S7 (2007)

    Article  Google Scholar 

  14. World Health Organization: Investing to Overcome the Global Impact of Neglected Tropical Diseases: Third WHO Report on Neglected Tropical Diseases 2015, vol. 3. World Health Organization (2015)

  15. Alvar, J., Yactayo, S., Bern, C.: Leishmaniasis and poverty. Trends Parasitol. 22(12), 552–557 (2006)

    Article  Google Scholar 

  16. Agyingi, E.O., Ross, D.S., Bathena, K.: A model of the transmission dynamics of leishmaniasis. J. Biol. Syst. 19(2), 237–250 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Bi, K., Chen, Y., Zhao, S., Kuang, Y., John Wu, C.-H.,: Current visceral leishmaniasis research: a research review to inspire future study. BioMed Res. Int. 2018, Article ID 9872095 (2018)

  18. Boukhalfa, F., Helal, M., Lakmeche, A.: Mathematical analysis of visceral leishmaniasis model. Res. Appl. Math. 1, Article ID 101263 (2017)

  19. ELmojtaba, I.M., Mugisha, J., Hashim, M.H.: Mathematical analysis of the dynamics of visceral leishmaniasis in the Sudan. Appl. Math. Comput. 217(6), 2567–2578 (2010)

    MathSciNet  MATH  Google Scholar 

  20. ELmojtaba, I.M., Mugisha, J., Hashim, M.H.: Vaccination model for visceral leishmaniasis with infective immigrants. Math. Methods Appl. Sci. 36(2), 216–226 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  21. Ready, P.D.: Epidemiology of visceral leishmaniasis. Clin. Epidemiol. 6, 147 (2014)

    Article  Google Scholar 

  22. Zhao, S., Kuang, Y., Wu, C.-H., Ben-Arieh, D., Ramalho-Ortigao, M., Bi, K.: Zoonotic visceral leishmaniasis transmission: modeling, backward bifurcation, and optimal control. J. Math. Biol. 73(6–7), 1525–1560 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  23. Biswas, S.: Mathematical modeling of visceral leishmaniasis and control strategies. Chaos Solitons Fractals 104, 546–556 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kuang, Y.: Delay Differential Equations: With Applications in Population Dynamics, vol. 191. Academic Press, Cambridge (1993)

    MATH  Google Scholar 

  25. Wanjun, X., Soumen, K., Maitra, S.: Dynamics of a delayed seiq epidemic model. Adv. Differ. Equ. 2018(336), 1–21 (2018)

    MathSciNet  MATH  Google Scholar 

  26. Basir, F.A.: Dynamics of infectious diseases with media coverage and two time delay. Math. Models Comput. Simul. 10(6), 770–783 (2018)

    Article  MathSciNet  Google Scholar 

  27. Shu, L., Weiming, Y.: Cholera model incorporating media coverage with multiple delays. Math. Methods Appl. Sci. 42(2), 419–439 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  28. Ephraim, A., Tamas, W.: Analysis of a model of leishmaniasis with multiple time lags in all populations. Math. Comput. Appl. 24(63), 1–16 (2019)

    MathSciNet  Google Scholar 

  29. Das, P., Mukherjee, D., Sarkar, A.: Effect of delay on the model of American cutaneous leishmaniasis. J. Biol. Syst. 15(02), 139–147 (2007)

    Article  MATH  Google Scholar 

  30. Roy, P.K., Biswas, D., Basir, F.: Transmission dynamics of cutaneous leishmaniasis: a delay-induced mathematical study. J. Med. Res. Dev. 4(2), 11–23 (2015)

    Google Scholar 

  31. Shimozako, H.J., Wu, J., Massad, E.: Mathematical modelling for zoonotic visceral leishmaniasis dynamics: a new analysis considering updated parameters and notified human Brazilian data. Infect. Dis. Model. 2(2), 143–160 (2017)

    Google Scholar 

  32. Rihan, F.A., Rahman, D.A., Lakshmanan, S., Alkhajeh, A.: A time delay model of tumour-immune system interactions: global dynamics, parameter estimation, sensitivity analysis. Appl. Math. Comput. 232, 606–623 (2014)

    MathSciNet  MATH  Google Scholar 

  33. Diekmann, O., Heesterbeek, J.A.P., Metz, J.A.: On the definition and the computation of the basic reproduction ratio \(R_0\) in models for infectious diseases in heterogeneous populations. J. Math. Biol. 28(4), 365–382 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  34. Gu, K., Chen, J., Kharitonov, V.L.: Stability of Time-Delay Systems. Springer, Berlin (2003)

    Book  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to acknowledge financial support from Sultan Qaboos University, Oman and United Arab Emirates University, UAE through the joint research Grant No. CL/SQU-UAEU/17/01. The authors acknowledge, with thanks, the comments of an anonymous reviewer, which enhanced the clarity and readability of the paper.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Ibrahim M. Elmojtaba.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gandhi, V., Al-Salti, N.S. & Elmojtaba, I.M. Mathematical analysis of a time delay visceral leishmaniasis model. J. Appl. Math. Comput. 63, 217–237 (2020). https://doi.org/10.1007/s12190-019-01315-5

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12190-019-01315-5

Keywords

Mathematics Subject Classification

Navigation