[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Springer Nature Link
Log in

A mathematical study of an imprecise SIR epidemic model with treatment control

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

Abstract

In this paper, a Susceptible–Infected–Recovered (SIR) model with imprecise biological parameters is studied. Due to the lack of precise numerical data of the biological parameters here the model with imprecise data is considered. Sometimes it is not possible to collect the numerical data as a fixed value, but the interval in which it belongs to can easily be determined. For this reason an SIR model is introducing with the interval numbers as parametric functional form. Then the existence of equilibrium points with their feasibility criteria is checked and discussed about the stability of the system. Optimal control problem is formulated and solved. Numerical simulation works are given in support of the analytical results.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /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 includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

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

Similar content being viewed by others

References

  1. Arqub, O.A., Al-Smadi, M., Momani, S., Hayat, T.: Application of reproducing kernel algorithm for solving second-order, two-point fuzzy boundary value problems. Soft Comput. (2016). doi:10.1007/s00500-016-2262-3

  2. Arqub, O.A., Al-Smadi, M., Momani, S., Hayat, T.: Numerical solutions of fuzzy differential equations using reproducing kernel Hilbert space method. Soft Comput. (2015). doi:10.1007/s00500-015-1707-4

  3. Arqub, O.A.: Adaptation of reproducing kernel algorithm for solving fuzzy Fredholm–Volterra integro differential equations. Neural Comput. Appl. (2015). doi:10.1007/s00521-015-2110-x

  4. Arqub, O.A., El-Ajou, A.: Solution of the fractional epidemic model by homotopy analysis method. J. King Saud Univ. Sci. 25, 73–81 (2013)

    Article  Google Scholar 

  5. Beretta, E., Hara, T., Ma, W., Takeuchi, Y.: Global asymptotic stability of an SIR epidemic model with distributed time delay. Nonlinear Anal. 47, 4107–4115 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  6. Beretta, E., Breda, D.: An SEIR epidemic model with constant latency time and infectious period. Math. Biosci. Eng. 8, 931–952 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bernoulli, D.: Reflexions sur les avantages de linoculation, Mercure de France (1760) 173–190. English translation by R. Pulskamp, Department of Mathematics and Computer Science, Xavier University, Cincinnati (2009); http://www.cs.xu.edu/math/Sources/DanBernoulli/1760-reflections-inoculation (1760)

  8. Birkhoff, G., Rota, G.C.: Ordinary Differential Equations. Ginn, Boston (1982)

    MATH  Google Scholar 

  9. Buonomo, B., D’Onofrio, A., Lacitignola, D.: Global stability of an SIR epidemic model with information dependent vaccination. Math. Biosci. 216, 9–16 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Buonomo, B., Lacitignola, D.: On the backward bifurcation of a vaccination model with nonlinear incidence. Nonlinear Anal.: Model. Control 16, 30–46 (2011)

    MathSciNet  MATH  Google Scholar 

  11. Eckalbar, J.C., Eckalbar, W.L.: Dynamics of an epidemic model with quadratic treatment. Nonlinear Anal.: Real World Appl. 12, 320–332 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hamer, W.H.: Epidemic disease in England. Lancet 1, 733–739 (1906)

    Google Scholar 

  13. Hu, Z., Ma, W., Ruan, S.: Analysis of SIR epidemic models with nonlinear incidencerate and treatment. Math. Biosci. 238, 12–20 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Jana, S., Haldar, P., Kar, T.K.: Complex dynamics of an epidemic model with vaccination and treatment controls. J. Dyn. Control Int. (2015). doi:10.1007/s40435-015-0189-7

  15. Jana, S., Nandi, S.K., Kar, T.K.: Complex dynamics of an SIR epidemic model with saturated incidence rate and treatment. Acta Biotheor. 64, 6584 (2016). doi:10.1007/s10441-015-9273-9

    Article  Google Scholar 

  16. Jana, S., Kar, T.K.: A mathematical study of a prey–predator model in relevance to pest control. Nonlinear Dyn. 74, 667–683 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kar, T.K., Batabyal, A.: Stability analysis and optimal control of an SIR epidemic model with vaccination. BioSystems 104, 127–135 (2011)

    Article  Google Scholar 

  18. Kar, T.K., Jana, S.: Application of three controls optimally in a vector-borne disease-a mathematical study. Commun. Nonlinear Sci. Numer. Simul. 18, 2868–2884 (2013a)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kar, T.K., Jana, S.: A theoretical study on mathematical modelling of an infectious disease with application of optimal control. BioSystems 111, 37–50 (2013b)

    Article  Google Scholar 

  20. Kar, T.K., Mondal, P.K.: Global dynamics and bifurcation in delayed SIR epidemic model. Nonlinear Anal. Real World Appl. 12, 2058–2068 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. Keeling, M.J., Rohani, P.: Modeling Infectious Diseases in Humans and Animals. Princeton University Press, NJ (2008)

    MATH  Google Scholar 

  22. Kermack, W.O., McKendric, A.G.: Contribution to the mathematical theory of epidemics. Proc. R. Soc. Lond. Ser. A 115, 700–721 (1927)

    Article  MATH  Google Scholar 

  23. Laarabi, H., Abta, A., Hattaf, K.: Optimal control of a delayed SIRS epidemic model with vaccination and treatment. Acta Biotheor. (2015). doi:10.1007/s10441-015-9244-1

  24. Lenhart, S., Workman, J.T.: Optimal Control Applied to Biological Model. Mathematical and Computational Biology Series Chapman & Hall/CRC (2007)

  25. Makinde, O.D.: Adomian decomposition approach to a SIR epidemic model with constant vaccination strategy. Appl. Math. Comput. 184, 842–848 (2007)

    MathSciNet  MATH  Google Scholar 

  26. Mondal, P.K., Jana, S., Haldar, p, Kar, T.K.: Dynamical behavior of an epidemic model in a fuzzy transmission. Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 23(5), 651–665 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  27. Okosun, K.O., Ouifki, R., Marcus, N.: Optimal control analysis of a malaria disease transmission model that includes treatment and vaccination with waning immunity. Biosystems 106, 136–145 (2011)

    Article  Google Scholar 

  28. Okosun, K.O., Makinde, O.D.: Impact of chemo-therapy on optimal control of malaria disease with infected immigrants. BioSystems 104(1), 32–41 (2011)

    Article  Google Scholar 

  29. Pal, D., Mahapatra, G., Samanta, G.P.: Optimal harvesting of prey–predator system with interval biological parameters: a bioeconomic model. Math. Biosci. (2012). doi:10.1016/j.mbs.2012.11.007

  30. Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mishchenko, E.F.: The Mathematical Theory of Optimal Processes. Wiley, New York (1962)

    Google Scholar 

  31. Qiu, Z., Feng, Z.: Transmission dynamics of an influenza model with vaccination and antiviral treatment. Bull. Math. Biol. 72, 1–33 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  32. Sahu, G.P., Dhar, J.: Analysis of an SVEIS epidemic model with partial temporary immunity and saturation incidence rate. Appl. Math. Model. 36, 908–923 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  33. Sharma, S., Samanta G.P.: Optimal harvesting of a two species competition model with imprecise biological parameters. Nonlinear Dyn. (2014). doi:10.1007/s11071-014-1354-9

  34. Tchuenche, J.M., Khamis, S.A., Agusto, F.B., Mpeshe, S.C.: Optimal control and sensitivity analysis of an influenza model with treatment and vaccination. Acta. Biotheor. 59, 1–28 (2011)

    Article  Google Scholar 

  35. van den Driessche, P., Watmough, J.: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of diseases transmission. Math. Biosci. 180, 29–48 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  36. Wang, W.: Backward bifurcation of an epidemic model with treatment. Math. Biosci. 201, 58–71 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  37. Wu, L., Feng, Z.: Homoclinic bifurcation in an SIQR model for childhood diseases. J. Differ. Equ. 168, 150–167 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  38. Yi, N., Zhang, Q., Mao, K., Yang, D., Li, Q.: Analysis and control of an SEIR epidemic system with nonlinear transmission rate. Math. Comput. Model. 50, 1498–1513 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  39. Zadeh, L. A.: Fuzzy sets. Inf. Control 8, 338–353 (1965)

  40. Zhang, Y., Agarwal, P., Bhatnagar, V., Balochian, S., Yan, J.: Swarm intelligence and its applications. Sci. World J., Volume 2013, Article ID 528069, 3 p (2013). doi:10.1155/2013/528069

  41. Zhang, Y., Balochian, S., Agarwal, P., Bhatnagar, V., Housheya, O.J.: Artificial Intelligence and Its Applications 2014, Mathematical Problems in Engineering Volume 2016, Article ID 3871575, 6 p (2016). doi:10.1155/2016/3871575

  42. Zhang, F.-F., Jin, Z., Sun, G.-Q.: Bifurcation analysis of a delayed epidemic model. Appl. Math. Comput. 216, 753–767 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  43. Zhang, X., Liu, X.: Backward Bifurcation of an epidemic model with saturated treatment. J. Math. Anal. Appl. 348, 433–443 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  44. Zhou, Y., Yang, K., Zhou, K., Liang, Y.: Optimal vaccination policies for an SIR model with limited resources. Acta Biotheor. 62, 171–181 (2014)

    Article  Google Scholar 

  45. Zhou, L., Fan, M.: Dynamics of an SIR epidemic model with limited resources visited. Nonlinear Anal.: Real world Appl. 13, 312–324 (2012)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore, 721101, India

    Anjana Das & M. Pal

Authors
  1. Anjana Das
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Pal
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anjana Das.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Das, A., Pal, M. A mathematical study of an imprecise SIR epidemic model with treatment control. J. Appl. Math. Comput. 56, 477–500 (2018). https://doi.org/10.1007/s12190-017-1083-6

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12190-017-1083-6

Keywords

Mathematics Subject Classification

Search

Navigation