Skip to main content

Advertisement

SpringerLink
Log in

Optimal control strategies for tuberculosis dynamics with exogenous reinfections in case of treatment at home and treatment in hospital

  • Original paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

We propose an extended mathematical model for tuberculosis dynamics including five groups as susceptible individuals, S(t); latent, infected with TB but not infectious individuals, E(t); individuals who have active TB and treated at home, \(I_1(t)\); individuals who have active TB and treated in hospital \(I_2(t)\) and recovered individuals R(t). We incorporate exogenous reinfection and infection of recovered individuals to extend the model. Since healthcare facilities are limited and treatment in hospital might hard to afford for the people living in resource-poor countries, being treated at home or hospital are two important cases affecting the transmission and eradication of the disease. Therefore, an optimal control problem (OCP) is constructed governed by the extended TB model where the importance of being treated at home and in hospital can be revealed and the success of the treatment can be measured mathematically in case of limited number of healthcare facilities and treatment expenses. The aim of applying optimal control strategy to this new SEIR model is to minimize the number of latent and infectious individuals as well as the cost of applying three controls, namely distancing control, \(u_1(t)\); effort that prevents the failure of treatment at home for active TB infectious individuals, \(u_2(t)\); and effort that prevents the failure of treatment in hospital for active TB infectious individuals, \(u_3(t)\). We set two different controls \(u_2(t)\) and \(u_3(t)\) special to the subgroups \(I_1(t)\) and \(I_2(t)\) so that we can catch the contribution of being treated in hospital or being treated at home. While constructing the OCP, the form of the reproduction number is used so that we can construct the OCP in a way to decrease the reproduction number. In addition, local asymptotic stability of the disease-free equilibrium (DFE) point is proven with the use of the reproduction number; but, global asymptotic stability of the DFE point may not be achieved due to exogenous reinfection. It means that the disease might become persistent even if there is a small infectious group of people in the population. Therefore, the need for optimal control strategy is revealed even for a small infectious subgroup. At the end, we present some numerical results to obtain the optimal intervention strategies in case the transition rate from/to home to/from hospital is large.

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.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  1. Agusto, F.: Mathematical model of Ebola transmission dynamics with relapse and reinfection. Math. Biosci. 283, 48–59 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  2. Allen, L.J.: Introduction to Mathematical Biology. Pearson/Prentice Hall, Upper Saddle River (2007)

    Google Scholar 

  3. Allen, L.J., Brauer, F., Van den Driessche, P., Wu, J.: Mathematical Epidemiology, vol. 1945. Springer, Berlin (2008)

    Book  Google Scholar 

  4. Bhunu, C., Mushayabasa, S., Tchuenche, J.: A theoretical assessment of the effects of smoking on the transmission dynamics of tuberculosis. Bull. Math. Biol. 73(6), 1333–1357 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Birkhoff, G., Rota, G.: Ordinary Differential Equations. Needham Heights, Ginn, Needham (1982)

    MATH  Google Scholar 

  6. Bowong, S.: Optimal control of the transmission dynamics of tuberculosis. Nonlinear Dyn. 61(4), 729–748 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bowong, S., Kurths, J.: Modeling and analysis of the transmission dynamics of tuberculosis without and with seasonality. Nonlinear Dyn. 67(3), 2027–2051 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bowong, S., Tewa, J.J.: Global analysis of a dynamical model for transmission of tuberculosis with a general contact rate. Commun. Nonlinear Sci. Numer. Simul. 15(11), 3621–3631 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Canetti, G., Sutherland, I., Svandova, E.: Endogenous reactivation and exogenous reinfection: their relative importance with regard to the development of non-primary tuberculosis. Bull. Int. Union Against Tuberc. 47, 116 (1972)

    Google Scholar 

  10. Carvalho, A.R., Pinto, C.M.: Non-integer order analysis of the impact of diabetes and resistant strains in a model for TB infection. Commun. Nonlinear Sci. Numer. Simul. 61, 104–126 (2018)

    Article  MathSciNet  Google Scholar 

  11. Castillo-Chavez, C., Feng, Z.: To treat or not to treat: the case of tuberculosis. J. Math. Biol. 35(6), 629–656 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  12. Castillo-Chavez, C., Feng, Z., Huang, W.: On the computation of R\(_0\) and its role on global stability. Math. Approaches Emerg. Reemerg. Infect. Dis.: Introd. 1, 229 (2002)

    Article  MATH  Google Scholar 

  13. Choi, S., Jung, E., Castillo-Chavez, C.: Optimal treatment strategies for tuberculosis with exogenous reinfection. Technical Report, MTBI-07-04M. Arizona State University (2010)

  14. Choi, S., Jung, E., Lee, S.M.: Optimal intervention strategy for prevention tuberculosis using a smoking-tuberculosis model. J. Theor. Biol. 380, 256–270 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. Davenne, T., McShane, H.: Why don’t we have an effective tuberculosis vaccine yet? Expert Rev. Vaccin. 15(8), 1009–1013 (2016)

    Article  Google Scholar 

  16. Van den Driessche, P., Watmough, J.: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180(1–2), 29–48 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  17. Feng, Z., Castillo-Chavez, C., Capurro, A.F.: A model for tuberculosis with exogenous reinfection. Theor. Popul. Biol. 57(3), 235–247 (2000)

    Article  MATH  Google Scholar 

  18. Fleming, W.H., Rishel, R.W.: Deterministic and Stochastic Optimal Control. Applications of Mathematics, vol. 1. Springer, Berlin (1975)

    Book  MATH  Google Scholar 

  19. Gao, Dp, Huang, Nj: Optimal control analysis of a tuberculosis model. Appl. Math. Model. 58, 47–64 (2018)

    Article  MathSciNet  Google Scholar 

  20. Gumel, A.B.: Global dynamics of a two-strain avian influenza model. Int. J. Comput. Math. 86(1), 85–108 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  22. Huo, H.F., Zou, M.X.: Modelling effects of treatment at home on tuberculosis transmission dynamics. Appl. Math. Model. 40(21–22), 9474–9484 (2016)

    Article  MathSciNet  Google Scholar 

  23. Jung, E., Lenhart, S., Feng, Z.: Optimal control of treatments in a two-strain tuberculosis model. Discret. Contin. Dyn. Syst. Ser. B 2(4), 473–482 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  24. Khajanchi, S., Banerjee, S.: Influence of multiple delays in brain tumor and immune system interaction with t11 target structure as a potent stimulator. Math. Biosci. 302, 116–130 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  25. Khajanchi, S., Banerjee, S.: A strategy of optimal efficacy of T11 target structure in the treatment of brain tumor. J. Biol. Syst. 27, 1–31 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  26. Khajanchi, S., Das, D.K., Kar, T.K.: Dynamics of tuberculosis transmission with exogenous reinfections and endogenous reactivation. Phys. A: Stat. Mech. Appl. 497, 52–71 (2018)

    Article  MathSciNet  Google Scholar 

  27. Khajanchi, S., Ghosh, D.: The combined effects of optimal control in cancer remission. Appl. Math. Comput. 271, 375–388 (2015)

    MathSciNet  MATH  Google Scholar 

  28. Kumar, A., Srivastava, P.K.: Vaccination and treatment as control interventions in an infectious disease model with their cost optimization. Commun. Nonlinear Sci. Numer. Simul. 44, 334–343 (2017)

    Article  MathSciNet  Google Scholar 

  29. Lenhart, S., Workman, J.T.: Optimal Control Applied to Biological Models. Chapman & Hall, CRC Press, Boca Raton (2007)

    MATH  Google Scholar 

  30. Liu, Wm, Hethcote, H.W., Levin, S.A.: Dynamical behavior of epidemiological models with nonlinear incidence rates. J. Math. Biol. 25(4), 359–380 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  31. Ma, Z., Zhou, Y., Wang, W., Jin, Z.: Mathematical Modeling and Studying of Dynamic Models of Infectious Diseases. Science Press, London (2004)

    Google Scholar 

  32. Mallela, A., Lenhart, S., Vaidya, N.K.: HIV-TB co-infection treatment: modeling and optimal control theory perspectives. J. Comput. Appl. Math. 307, 143–161 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  33. McAsey, M., Mou, L., Han, W.: Convergence of the forward–backward sweep method in optimal control. Comput. Optim. Appl. 53(1), 207–226 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  34. Moualeu, D.P., Weiser, M., Ehrig, R., Deuflhard, P.: Optimal control for a tuberculosis model with undetected cases in Cameroon. Commun. Nonlinear Sci. Numer. Simul. 20(3), 986–1003 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  35. Omran, A.R.: The epidemiologic transition: a theory of the epidemiology of population change. Milbank Q. 83(4), 731–757 (2005)

    Article  Google Scholar 

  36. Perelson, A.S., Kirschner, D.E., De Boer, R.: Dynamics of HIV infection of CD4+ T cells. Math. Biosci. 114(1), 81–125 (1993)

    Article  MATH  Google Scholar 

  37. Pinto, C.M., Carvalho, A.R.: The HIV/TB coinfection severity in the presence of TB multi-drug resistant strains. Ecol. Complex. 32, 1–20 (2017)

    Article  MathSciNet  Google Scholar 

  38. Pinto, C.M., Carvalho, A.R.: The impact of pre-exposure prophylaxis (PrEP) and screening on the dynamics of HIV. J. Comput. Appl. Math. 339, 231–244 (2018)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  40. Silva, C.J., Torres, D.F.: Optimal control for a tuberculosis model with reinfection and post-exposure interventions. Math. Biosci. 244(2), 154–164 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  41. Silva, C.J., Torres, D.F.: Optimal control of tuberculosis: a review. In: Bourguignon, J.P., Jeltsch, R., Pinto, A.A., Viana, A. (eds.) Dynamics, Games and Science, pp. 701–722. Springer, Berlin (2015)

    Chapter  Google Scholar 

  42. Uplekar, M., Raviglione, M., et al.: Hospital or home? Scripting a high point in the history of TB care and control. WHO South-East Asia J. Public Health 1(2), 220 (2012)

    Article  Google Scholar 

  43. Waaler, H., Geser, A., Andersen, S.: The use of mathematical models in the study of the epidemiology of tuberculosis. Am. J. Public Health N. Health 52(6), 1002–1013 (1962)

    Article  Google Scholar 

  44. World Health Organization: Global Tuberculosis Report 2017. Technical Report (2018). WHO/HTM/TB/2017.23

Download references

Acknowledgements

The authors would like to thank the reviewers for their comments and suggestions that helped to improve the manuscript.

Funding

There is no funding for this research.

Author information

Authors and Affiliations

  1. Department of Logistics Management, University of Turkish Aeronautical Association, 06790, Ankara, Turkey

    Tuğba Akman Yıldız

  2. Department of Electrical and Electronical Engineering, University of Turkish Aeronautical Association, 06790, Ankara, Turkey

    Esra Karaoğlu

Authors
  1. Tuğba Akman Yıldız
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Esra Karaoğlu
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors read and approved the final manuscript.

Corresponding author

Correspondence to Tuğba Akman Yıldız.

Ethics declarations

Conflict of interest

The authors have no conflict of interest concerning the publication of this manuscript.

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

Akman Yıldız, T., Karaoğlu, E. Optimal control strategies for tuberculosis dynamics with exogenous reinfections in case of treatment at home and treatment in hospital. Nonlinear Dyn 97, 2643–2659 (2019). https://doi.org/10.1007/s11071-019-05153-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-019-05153-9

Keywords

Mathematics Subject Classification

Search

Navigation