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Optimal Control of Tuberculosis: A Review

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Dynamics, Games and Science

Part of the book series: CIM Series in Mathematical Sciences ((CIMSMS,volume 1))

Abstract

We review the optimal control of systems modeling the dynamics of tuberculosis. Time dependent control functions are introduced in the mathematical models, representing strategies for the improvement of the treatment and cure of active infectious and/or latent individuals. Optimal control theory allows then to find the optimal way to implement the strategies, minimizing the number of infectious and/or latent individuals and keeping the cost of implementation as low as possible. An optimal control problem is proposed and solved, illustrating the procedure. Simulations show an effective reduction in the number of infectious individuals.

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References

  1. Anita, S., Arnautu, V., Capasso, V.: An Introduction to Optimal Control Problems in Life Sciences and Economics: From Mathematical Models to Numerical Simulation with MATLAB. Modeling and Simulation in Science, Engineering and Technology, XII. Birkhäuser, Basel (2011)

    Google Scholar 

  2. Aparitio, J.P., Capurro, A.F., Castillo-Chavez, C.: Markers of disease evolution: the case of tuberculosis. J. Theor. Biol. 212(2), 227–237 (2002)

    Article  Google Scholar 

  3. Behncke, H.: Optimal control of deterministic epidemics. Optim. Control Appl. Methods 21, 269–285 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  4. Blower, S., Small, P., Hopewell, P.: Control strategies for tuberculosis epidemics: new models for old problems. Science 273, 497–500 (1996)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Bowong, S., Alaoui, A.M.A.: Optimal interventions strategies for tuberculosis. Commun. Nonlinear Sci. Numer. Simul. 18, 1441–1453 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. 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 

  8. Castillo-Chavez, C., Feng, Z.: Mathematical models for the disease dynamics of tuberculosis. In: Horn, M.A., Simonett, G., Webb, G. (eds.) Advances in Mathematical Population Dynamics-Molecules, Cells and Man, pp. 117–128. Vanderbilt University Press, Nashville (1998)

    Google Scholar 

  9. Castillo-Chavez, C., Feng, Z.: Global stability of an age-structure model for TB and its applications to optimal vaccination strategies. Math. Biosci. 151(2), 135–154 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cesari, L.: Optimization — Theory and Applications. Problems with Ordinary Differential Equations. Applications of Mathematics, vol. 17. Springer, New York (1983)

    Google Scholar 

  11. Chaulet, P.: Treatment of Tuberculosis: Case Holding Until Cure. WHO/TB/83, 141. World Health Organization, Geneva (1983)

    Google Scholar 

  12. Chiang, C.Y., Riley, L.W.: Exogenous reinfection in tuberculosis. Lancet Infect. Dis. 5, 629–636 (2005)

    Article  Google Scholar 

  13. Cohen, T., Murray, M.: Modeling epidemics of multidrug-resistant M. tuberculosis of heterogeneous fitness. Nat. Med. 10(10), 1117–1121 (2004)

    Google Scholar 

  14. Dye, C., Garnett, G.P., Sleeman, K., Williams, B.G.: Prospects for worldwide tuberculosis control under the who dots strategy. Directly observed short-course therapy. Lancet 352(9144), 1886–1891 (1998)

    Google Scholar 

  15. Eisen, M.: Mathematical Models in Cell Biology and Cancer Chemotherapy. Lectures Notes in Biomathematics, vol. 30. Springer, Berlin (1979)

    Google Scholar 

  16. Emvudu, Y., Demasse, R., Djeudeu, D.: Optimal control of the lost to follow up in a tuberculosis model. Comput. Math. Methods Med. 2011, 12 (2011). Art. ID 398476

    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. Springer, New York (1975)

    Book  MATH  Google Scholar 

  19. Frieden, T., Driver, R.C.: Tuberculosis control: pas 10 years and future progress. Tuberculosis 83, 82–85 (2003)

    Article  Google Scholar 

  20. Gaff, H., Schaefer, E.: Optimal control applied to vaccination and treatment strategies for various epidemiologic models. Math. Biosci. Eng. 6, 469–492 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  21. Gomes, M., Franco, A., Gomes, M., Medley, G.: The reinfection threshold promotes variability in tuberculosis epidemiology and vaccine efficacy. Proc. R. Soc. B 271(1539), 617–623 (2004)

    Article  Google Scholar 

  22. Gomes, M.G.M., Rodrigues, P., Hilker, F.M., Mantilla-Beniers, N.B., Muehlen, M., Paulo, A.C., Medley, G.F.: Implications of partial immunity on the prospects for tuberculosis control by post-exposure interventions. J. Theor. Biol. 248(4), 608–617 (2007)

    Article  MathSciNet  Google Scholar 

  23. Gumel, A.B., Shivakumar, P.N., Sahai, B.M.: A mathematical model for the dynamics of HIV-1 during the typical course of infection. In: Proceedings of the Third World Congress of Nonlinear Analysts, vol. 47, pp. 2073–2083 (2001)

    MathSciNet  Google Scholar 

  24. Hattaf, K., Rachik, M., Saadi, S., Tabit, Y., Yousfi, N.: Optimal control of tuberculosis with exogenous reinfection. Appl. Math. Sci. (Ruse) 3(5–8), 231–240 (2009)

    Google Scholar 

  25. Hethcote, H.: A thousand and one epidemic models. In: Levin, S.A. (ed.) Frontiers in Theoretical Biology, pp. 504–515. Springer, Berlin (1994)

    Chapter  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  28. Karrakchou, J., Rachik, M., Gourari, S.: Optimal control and infectiology: application to an HIV/AIDS model. Appl. Math. Comput. 177, 807–818 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  29. Ledzewicz, U., Schättler, H.: Optimal bang-bang controls for a 2-compartment model in cancer chemotherapy. J. Optim. Theory Appl. 114, 609–637 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  30. Ledzewicz, U., Schättler, H.: Anti-Angiogenic therapy in cancer treatment as an optimal control problem. SIAM J. Control. Optim. 46, 1052–1079 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  31. Ledzewicz, U., Schättler, H.: Optimal and suboptimal protocols for a class of mathematical models of tumor anti-angiogenesis. J. Theor. Biol. 252, 295–312 (2008)

    Article  Google Scholar 

  32. Ledzewicz, U., Schättler, H.: On optimal singular controls for a general SIR-model with vaccination and treatment. Discrete Contin. Dyn. Syst. Supplement, 981–990 (2011)

    Google Scholar 

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

    MATH  Google Scholar 

  34. Martin, R., Teo, K.L.: Optimal Control of Drug Administration in Cancer Chemotherapy. World Scientific, Singapore (1994)

    MATH  Google Scholar 

  35. Okuonghae, D., Aihie, V.U.: Optimal control measures for tuberculosis mathematical models including immigration and isolation of infective. J. Bio. Syst. 18(1), 17–54 (2010)

    Article  MathSciNet  Google Scholar 

  36. Pontryagin, L., Boltyanskii, V., Gramkrelidze, R., Mischenko, E.: The Mathematical Theory of Optimal Processes. Wiley Interscience, New York (1962)

    MATH  Google Scholar 

  37. Raviglione, M.C.: Evolution of WHO, 1948–2001 policies for tuberculosis control. Lancet 359, 775–780 (2002)

    Article  Google Scholar 

  38. Raviglione, M.C., Dye, C., Schmizt, S., Kochi, A.: For the global surveillance and monitoring project: assessment of worldwide tuberculosis control. Lancet 350, 624–629 (1997)

    Article  Google Scholar 

  39. Reichman, L.B., Hershfield, E.S.: Tuberculosis: A Comprehensive International Approach. Dekker, New York (2000)

    Google Scholar 

  40. Rodrigues, P., Rebelo, C., Gomes, M.G.M.: Drug resistance in tuberculosis: a reinfection model. Theor. Popul. Biol. 71, 196–212 (2007)

    Article  MATH  Google Scholar 

  41. Rodrigues, H.S., Monteiro, M.T.T., Torres, D.F.M.: Dynamics of dengue epidemics when using optimal control. Math. Comput. Model. 52(9–10), 1667–1673 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  42. Rodrigues, H.S., Monteiro, M.T.T., Torres, D.F.M.: Bioeconomic perspectives to an optimal control dengue model. Int. J. Comput. Math. 90(10), 2126–2136 (2013)

    Article  MATH  Google Scholar 

  43. Rodrigues, H.S., Monteiro, M.T.T., Torres, D.F.M.: Dengue in Cape Verde: vector control and vaccination. Math. Popul. Stud. 20(4), 208–223 (2013)

    Article  MathSciNet  Google Scholar 

  44. Rodrigues, P., Silva, C.J., Torres, D.F.M.: Optimal control strategies for reducing the number of active infected individuals with tuberculosis. In: Proceedings of the SIAM Conference on Control and Its Applications (CT13) 8–10 July, pp. 44–50. SIAM, San Diego (2013)

    Google Scholar 

  45. Rodrigues, H.S., Monteiro, M.T.T., Torres, D.F.M.: Optimal control and numerical software: an overview. In: Systems Theory: Perspectives, Applications and Developments, pp. 93–110. Nova Science Publishers, New York (2014)

    Google Scholar 

  46. Rodrigues, H.S., Monteiro, M.T.T., Torres, D.F.M.: Vaccination models and optimal control strategies to dengue. Math. Biosci. 247(1), 1–12 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  47. Silva, C.J., Torres, D.F.M.: Optimal control strategies for tuberculosis treatment: a case study in Angola. Numer. Algebra Control Optim. 2(3), 601–617 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  48. Silva, C.J., Torres, D.F.M.: Optimal control applied to tuberculosis models. The IEA-EEF European Congress of Epidemiology 2012: epidemiology for a fair and healthy society. Eur. J. Epidemiol. 27, S140–S141 (2012)

    Google Scholar 

  49. Silva, C.J., Torres, D.F.M.: An optimal control approach to malaria prevention via insecticide-treated nets. In: Conference Papers in Mathematics, 8 pp. (2013). Art. ID 658468

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  51. Silva, C.J., Torres, D.F.M.: Modeling TB-HIV syndemic and treatment. J. Appl. Math. (2014). Art. ID 248407. http://dx.doi.org/10.1155/2014/248407

  52. Small, P.M., Fujiwara, P.I.: Management of tuberculosis in the United States. N. Engl. J. Med. 345(3), 189–200 (2001)

    Article  Google Scholar 

  53. Styblo, K.: State of art: epidemiology of tuberculosis. Bull. Int. Union Tuberc. 53, 141–152 (1978)

    Google Scholar 

  54. Swan, G.W.: Role of optimal control in cancer chemotherapy. Math. Biosci. 101, 237–284 (1990)

    Article  MATH  Google Scholar 

  55. Swierniak, A.: Optimal treatment protocols in leukemia – modelling the proliferation cycle. In: Proceedings of the 12th IMACS World Congress, vol. 4, pp. 170–172. Baltzer, Basel, Paris (1988)

    Google Scholar 

  56. Swierniak, A.: Cell cycle as an object of control. J. Biol. Syst. 3, 41–54 (1995)

    Article  Google Scholar 

  57. Verver, S., Warren, R.M., Beyers, N., Richardson, M., van der Spuy, G.D., Borgdorff, M.W., Enarson, D.A., Behr, M.A., van Helden, P.D.: Rate of reinfection tuberculosis after successful treatment is higher than rate of new tuberculosis. Am. J. Respir. Crit. Care Med. 171, 1430–1435 (2005)

    Article  Google Scholar 

  58. Vynnycky, E., Fine, P.E.: The natural history of tuberculosis: the implications of age-dependent risks of disease and the role of reinfection. Epidemiol. Infect. 119(2), 183–201 (1997)

    Article  Google Scholar 

  59. Whang, S., Choi, S., Jung, E.: A dynamic model for tuberculosis transmission and optimal treatment strategies in South Korea. J. Theor. Biol. 279, 120–131 (2011)

    Article  MathSciNet  Google Scholar 

  60. WHO.: Global Tuberculosis Control. WHO Report, Geneva (2012)

    Google Scholar 

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Acknowledgements

This work was partially presented at the Thematic session Control of diseases and epidemics, MECC 2013—International Conference Planet Earth, Mathematics of Energy and Climate Change, 25–27 March 2013, Calouste Gulbenkian Foundation (FCL), Lisbon, Portugal. It was supported by Portuguese funds through the Center for Research and Development in Mathematics and Applications (CIDMA), and The Portuguese Foundation for Science and Technology (FCT), within project UID/MAT/04106/2013. Silva was also supported by FCT through the post-doc fellowship SFRH/BPD/72061/2010, Torres by project PTDC/EEI-AUT/1450/2012. The authors are very grateful to two anonymous referees, for valuable remarks and comments, which significantly contributed to the quality of the paper.

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  1. Department of Mathematics, Center for Research and Development in Mathematics and Applications (CIDMA), University of Aveiro, 3810–193, Aveiro, Portugal

    Cristiana J. Silva & Delfim F. M. Torres

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  1. Cristiana J. Silva
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Correspondence to Delfim F. M. Torres .

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  1. IHES Le Bois-Marie, Bures-sur-Yvette, France

    Jean-Pierre Bourguignon

  2. Department of Mathematics, ETH Zürich; Seminar für Angewandte Mathematik, Zürich, Switzerland

    Rolf Jeltsch

  3. Department of Mathematics, University of Porto; Faculty of Sciences, Porto, Portugal

    Alberto Adrego Pinto

  4. Instituto de Matemática Pura e Aplicada, IMPA, Rio de Janeiro, Rio de Janeiro, Brazil

    Marcelo Viana

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Silva, C.J., Torres, D.F.M. (2015). Optimal Control of Tuberculosis: A Review. In: Bourguignon, JP., Jeltsch, R., Pinto, A., Viana, M. (eds) Dynamics, Games and Science. CIM Series in Mathematical Sciences, vol 1. Springer, Cham. https://doi.org/10.1007/978-3-319-16118-1_37

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