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Optimal control strategy for adherence to different treatment regimen in various stages of tuberculosis infection

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Abstract

In this article, we propose a new mathematical model for tuberculosis considering the infectivity of both smear-positive and smear-negative individuals, searching for an efficient control strategy that may be followed to curtail the disease. We have employed different treatment regimens in various stages of tuberculosis infection. The fundamental epidemic threshold quantity \(R_0\) is inspected by the next-generation matrix method. The forward normalized sensitivity indices of the model parameters connected with \(R_0\) are computed to scale their impacts on the basic reproduction number. An optimal control problem is constructed considering three different treatment regimens in different possible stages of TB, and the control problem is solved analytically. The simulation results suggest that the combined implementation of all the controls optimally is the best policy to minimize the tuberculosis prevalence with the least interventions implementations costs.

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References

  1. S. Khajanchi, D.K. Das, T.K. Kar, Physica A 497, 52 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  2. Z. Feng, C. Castillo-Chavez, A.F. Capurro, Theor. Popul. Biol. 57, 235 (2000)

    Article  Google Scholar 

  3. S. Mushayabasa, C.P. Bhunu, J. Biol. Phys. 39, 723 (2013)

    Article  Google Scholar 

  4. D.K. Das, S. Khajanchi, T.K. Kar, Appl. Math. Comput. 366, 124732 (2020)

    MathSciNet  Google Scholar 

  5. D.K. Das, T.K. Kar, J. Math. Anal. Appl. 492, 124407 (2020)

    Article  MathSciNet  Google Scholar 

  6. S. Pandey, V.K. Chadha, R. Laxminarayan, N. Arinaminpathy, Int. J. Tuberc. Lung Dis. 21, 366 (2017)

    Article  Google Scholar 

  7. D.K. Das, S. Khajanchi, T.K. Kar, Chaos Solitons Fractals 130, 109450 (2020)

    Article  MathSciNet  Google Scholar 

  8. S. Athithan, M. Ghosh, Int. J. Dyn. Cont. 1, 223 (2013)

    Article  Google Scholar 

  9. D. Okuonghae, A. Korobeinikov, Math. Modell. Nat. Phenom. 2, 113 (2007)

    Article  Google Scholar 

  10. R.M. Des Prez, C. Muschenheim, J. Chronic Dis. 15, 599 (1962)

    Article  Google Scholar 

  11. H.W. Kim, J.S. Kim, Tuberc. Respiratory Dis. 81, 6 (2018)

    Article  Google Scholar 

  12. C.P. Bhunu, W. Garira, Z. Mukandavire, M. Zimba, Bull. Math. Biol. 70, 1163 (2008)

    Article  MathSciNet  Google Scholar 

  13. P. Desikan, Ind. J. Med. Res. 137, 442 (2013)

    Google Scholar 

  14. L.S.G. Linguissi, C.J. Vouvoungui, P. Poulain, G.B. Essassa, S. Kwedi, F. Ntoumi, B.M.C. Res, Notes 8, 804 (2015)

    Google Scholar 

  15. E. Hernandez-Garduno, V. Cook, D. Kunimoto, R.K. Elwood, W.A. Black, J.M. FitzGerald, Thorax 59, 286 (2004)

    Article  Google Scholar 

  16. D.K. Das, T.K. Kar, Chaos Solitons Fractals 146, 110879 (2021)

    Article  Google Scholar 

  17. A.K. Dutt, W.W. Stead, Semin. Respir. Infect. 9, 113 (1994)

    Google Scholar 

  18. R. Colebunders, I. Bastian, Int. J. Tuberc. Lung Dis. 4, 97 (2000)

    Google Scholar 

  19. L.C. Campos, M.V.V. Rocha, D.M.C. Willers, D.R. Silva, PloS One 11, e0147933 (2016)

    Article  Google Scholar 

  20. T.K. Kar, S. Jana, Commun. Nonlinear Sci. Numer. Simul. 18, 2868 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  21. T.K. Kar, S. Jana, BioSystems 111, 37 (2013)

    Article  ADS  Google Scholar 

  22. G. Birkoff, G.C. Rota, Ordinary Differential Equations Ginn. Wiley, Boston (1982)

  23. P. Van den Driessche, J. Watmough, Math. Biosci. 180, 29 (2002)

    Article  MathSciNet  Google Scholar 

  24. G. Zaman, Y.H. Kang, I.H. Jung, BioSystems 93, 240 (2008)

    Article  Google Scholar 

  25. H.R. Joshi, Optim. Con. Appl. Methods 23, 199 (2002)

    Article  Google Scholar 

  26. A. Khatua, T.K. Kar, S.K. Nandi, S. Jana, Y. Kang, Energ. Ecol. Environ. 5, 389 (2020)

    Article  Google Scholar 

  27. A. Khatua, T.K. Kar, Eur. Phys. J. Plus 135, 1 (2020)

    Article  Google Scholar 

  28. D.K. Das, A. Khatua, T.K. Kar, S. Jana, Appl. Math. Comput. 404, 126207 (2021)

    Google Scholar 

  29. W.H. Fleming, R.W. Rishel, Deterministic and Stochastic Optimal Control (Sringer-Verlag, New York, 1975)

    Book  Google Scholar 

  30. D.L. Lukes, Differential Equations: Classical to Controlled (Academic Press, New York, 1982)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  32. N. Chitnis, J.M. Hyman, J.M. Cushing, Bull. Math. Biol. 70, 1272 (2008)

    Article  MathSciNet  Google Scholar 

  33. S. Lenhart, J.T. Workman, Optimal Control Applied to Biological Models, Mathematical and Computational Biology Series (Chapman & Hall/CRC, Boca Raton, 2007)

    Book  Google Scholar 

  34. T.A. Yıldız, E. Karaoğlu, Nonlinear Dyn. 97, 2643 (2019)

    Article  Google Scholar 

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Acknowledgements

The research of A. Khatua is financially supported by Department of Science and Technology-INSPIRE, Government of India (No. DST/INSPIRE Fellowship/2016/IF160667, dated: September 21, 2016). We are also grateful to the anonymous reviewers and the Editor-in-Chief for their valuable comments which have helped us a lot to improve the quality and the presentation of the manuscript significantly.

Funding

The research of A. Khatua is financially supported by Department of Science and Technology-INSPIRE, Government of India (No. DST/INSPIRE Fellowship/2016/IF160667, dated: 21st September, 2016).

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Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah, West Bengal, 711103, India

    Anupam Khatua, Dhiraj Kumar Das & Tapan Kumar Kar

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  1. Anupam Khatua
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  2. Dhiraj Kumar Das
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  3. Tapan Kumar Kar
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Correspondence to Anupam Khatua.

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Khatua, A., Das, D.K. & Kar, T.K. Optimal control strategy for adherence to different treatment regimen in various stages of tuberculosis infection. Eur. Phys. J. Plus 136, 801 (2021). https://doi.org/10.1140/epjp/s13360-021-01811-3

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  • DOI: https://doi.org/10.1140/epjp/s13360-021-01811-3

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