Optimal control strategies for tuberculosis treatment: A case study in Angola

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2012, 2(3): 601-617. doi: 10.3934/naco.2012.2.601

Optimal control strategies for tuberculosis treatment: A case study in Angola

Cristiana J. Silva ^1, and Delfim F. M. Torres ^2,

1.	Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal
2.	CIDMA — Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal

Received March 2012 Revised March 2012 Published August 2012

Abstract
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We apply optimal control theory to a tuberculosis model given by a system of ordinary differential equations. Optimal control strategies are proposed to minimize the cost of interventions. Numerical simulations are given using data from Angola.

Keywords: Optimal control, epidemiology, treatment strategies, tuberculosis, Angola..

Mathematics Subject Classification: Primary: 92D30; Secondary: 49M0.

Citation: Cristiana J. Silva, Delfim F. M. Torres. Optimal control strategies for tuberculosis treatment: A case study in Angola. Numerical Algebra, Control and Optimization, 2012, 2 (3) : 601-617. doi: 10.3934/naco.2012.2.601

References:

[1]	L. Cesari, "Optimization-Theory and Applications. Problems with Ordinary Differential Equations," Applications of Mathematics 17, Springer-Verlag, New York, 1983.
[2]	A. d'Onofrio, U. Ledzewicz, H. Maurer and H. Schättler, On optimal delivery of combination therapy for tumors, Math. Biosci., 222 (2009), 13-26. doi: 10.1016/j.mbs.2009.08.004.
[3]	C. Dye, S. Scheele, P. Dolin, V. Pathania and M. C. Raviglione, Global burden of tuberculosis. Estimated incidence, prevalence, and mortality by country, Journal of the American Medical Association, 282 (1999), 677-686. doi: 10.1001/jama.282.7.677.
[4]	W. H. Fleming and R. W. Rishel, "Deterministic and Stochastic Optimal Control," Applications of Mathematics, No. 1, Springer-Verlag, Berlin-New York, 1975.
[5]	M. G. M. Gomes, P. Rodrigues, F. M. Hilker, N. B. Mantilla-Beniers, M. Muehlen, A. C. Paulo and G. F. Medley, Implications of partial immunity on the prospects for tuberculosis control by post-exposure interventions, Journal of Theoretical Biology, 248 (2007), 608-617. doi: 10.1016/j.jtbi.2007.06.005.
[6]	L. S. Jennings, M. E. Fisher, K. L. Teo and C. J. Goh, "MISER3 Optimal Control Software: Theory and User Manual," Version 3.3, Department of Mathematics, The University of Western Australia, Nedlands, Australia, 2004.
[7]	E. Jung, S. Lenhart and Z. Feng, Optimal control of treatments in a two-strain tuberculosis model, Discrete and Continuous Dynamical Systems - Series B, 2 (2002), 473-482. doi: 10.3934/dcdsb.2002.2.473.
[8]	M. E. Kruk, N. R. Schwalbe and C. A. Aguiar, Timing of default from tuberculosis treatment: a systematic review, Tropical Medicine and International Health, 13 (2008), 703-712. doi: 10.1111/j.1365-3156.2008.02042.x.
[9]	U. Ledzewicz, J. Marriott, H. Maurer and H. Schättler, Realizable protocols for optimal administration of drugs in mathematical models for anti-angiogenic treatment, Math. Med. Biol., 27 (2010), 157-179. doi: 10.1093/imammb/dqp012.
[10]	U. Ledzewicz, H. Maurer and H. Schättler, Optimal and suboptimal protocols for a mathematical model for tumor anti-angiogenesis in combination with chemotherapy, Math. Biosci. Eng., 8 (2011), 307-323. doi: 10.3934/mbe.2011.8.307.
[11]	S. Lenhart and J. T. Workman, "Optimal Control Applied to Biological Models," Chapman & Hall/CRC, Boca Raton, FL, 2007.
[12]	R. C. Loxton, K. L. Teo and V. Rehbock, Optimal control problems with multiple characteristic time points in the objective and constraints, Automatica J. IFAC, 44 (2008), 2923-2929. doi: 10.1016/j.automatica.2008.04.011.
[13]	L. Pontryagin, V. Boltyanskii, R. Gramkrelidze and E. Mischenko, "The Mathematical Theory of Optimal Processes," Wiley Interscience, 1962.
[14]	H. S. Rodrigues, M. T. T. Monteiro and D. F. M. Torres, Dynamics of dengue epidemics when using optimal control, Math. Comput. Modelling, 52 (2010), 1667-1673. doi: 10.1016/j.mcm.2010.06.034.
[15]	H. S. Rodrigues, M. T. T. Monteiro, D. F. M. Torres and A. Zinober, Dengue disease, basic reproduction number and control, Int. J. Comput. Math., 89 (2012), 334-346. doi: 10.1080/00207160.2011.554540.
[16]	P. M. Small and P. I. Fujiwara, Management of tuberculosis in the United States, N. Engl. J. Med., 345 (2001), 189-200. doi: 10.1056/NEJM200107193450307.
[17]	K. Styblo, State of art: epidemiology of tuberculosis, Bull. Int. Union Tuberc., 53 (1978), 141-152.
[18]	K. Styblo, "Selected Papers, Epidemiology of Tuberculosis," Royal Netherlands Tuberculosis Association, 24, 1991.
[19]	K. L. Teo, C. J. Goh and K. H. Wong, "A Unified Computational Approach to Optimal Control Problems," Pitman Monographs and Surveys in Pure and Applied Mathematics, 55, Longman Sci. Tech., Harlow, 1991.
[20]	WHO, Treatment of tuberculosis guidelines, Fourth edition, WHO, Geneva, 2010. Available from: http://www.who.int/tb/publications/tb_treatmentguidelines/en/index.html.
[21]	WHO, Global Tuberculosis Control, WHO Report 2011, Geneva, 2011. Available from: http://www.who.int/tb/publications/global_report/en/index.html.
[22]	, Available from: http://allafrica.com/stories/201103240476.html.
[23]	, Available from: http://apps.who.int/ghodata/?theme=country.
[24]	, Available from: http://www.avert.org/tuberculosis.htm.
[25]	, Available from: http://www.minsa.gov.ao/VerNoticia.aspx?id=10499.
[26]	, Available from: https://projects.coin-or.org/Ipopt.
[27]	, Available from: http://www.ampl.com.
[28]	, Available from: http://tomdyn.com.
[29]	, Available from: http://www.minsa.gov.ao/VerNoticia.aspx?id=8797.
[30]	, Available from: http://www.tradingeconomics.com/angola/tuberculosis-case-detection-rate-all-forms-wb-data.html.
[31]	, Available from: http://www.usaid.gov/our_work/global_health/id/tuberculosis/countries/africa/angola_profile.html.
[32]	, Available from: http://www.who.int/en/.

show all references

References:

[1]	L. Cesari, "Optimization-Theory and Applications. Problems with Ordinary Differential Equations," Applications of Mathematics 17, Springer-Verlag, New York, 1983.
[2]	A. d'Onofrio, U. Ledzewicz, H. Maurer and H. Schättler, On optimal delivery of combination therapy for tumors, Math. Biosci., 222 (2009), 13-26. doi: 10.1016/j.mbs.2009.08.004.
[3]	C. Dye, S. Scheele, P. Dolin, V. Pathania and M. C. Raviglione, Global burden of tuberculosis. Estimated incidence, prevalence, and mortality by country, Journal of the American Medical Association, 282 (1999), 677-686. doi: 10.1001/jama.282.7.677.
[4]	W. H. Fleming and R. W. Rishel, "Deterministic and Stochastic Optimal Control," Applications of Mathematics, No. 1, Springer-Verlag, Berlin-New York, 1975.
[5]	M. G. M. Gomes, P. Rodrigues, F. M. Hilker, N. B. Mantilla-Beniers, M. Muehlen, A. C. Paulo and G. F. Medley, Implications of partial immunity on the prospects for tuberculosis control by post-exposure interventions, Journal of Theoretical Biology, 248 (2007), 608-617. doi: 10.1016/j.jtbi.2007.06.005.
[6]	L. S. Jennings, M. E. Fisher, K. L. Teo and C. J. Goh, "MISER3 Optimal Control Software: Theory and User Manual," Version 3.3, Department of Mathematics, The University of Western Australia, Nedlands, Australia, 2004.
[7]	E. Jung, S. Lenhart and Z. Feng, Optimal control of treatments in a two-strain tuberculosis model, Discrete and Continuous Dynamical Systems - Series B, 2 (2002), 473-482. doi: 10.3934/dcdsb.2002.2.473.
[8]	M. E. Kruk, N. R. Schwalbe and C. A. Aguiar, Timing of default from tuberculosis treatment: a systematic review, Tropical Medicine and International Health, 13 (2008), 703-712. doi: 10.1111/j.1365-3156.2008.02042.x.
[9]	U. Ledzewicz, J. Marriott, H. Maurer and H. Schättler, Realizable protocols for optimal administration of drugs in mathematical models for anti-angiogenic treatment, Math. Med. Biol., 27 (2010), 157-179. doi: 10.1093/imammb/dqp012.
[10]	U. Ledzewicz, H. Maurer and H. Schättler, Optimal and suboptimal protocols for a mathematical model for tumor anti-angiogenesis in combination with chemotherapy, Math. Biosci. Eng., 8 (2011), 307-323. doi: 10.3934/mbe.2011.8.307.
[11]	S. Lenhart and J. T. Workman, "Optimal Control Applied to Biological Models," Chapman & Hall/CRC, Boca Raton, FL, 2007.
[12]	R. C. Loxton, K. L. Teo and V. Rehbock, Optimal control problems with multiple characteristic time points in the objective and constraints, Automatica J. IFAC, 44 (2008), 2923-2929. doi: 10.1016/j.automatica.2008.04.011.
[13]	L. Pontryagin, V. Boltyanskii, R. Gramkrelidze and E. Mischenko, "The Mathematical Theory of Optimal Processes," Wiley Interscience, 1962.
[14]	H. S. Rodrigues, M. T. T. Monteiro and D. F. M. Torres, Dynamics of dengue epidemics when using optimal control, Math. Comput. Modelling, 52 (2010), 1667-1673. doi: 10.1016/j.mcm.2010.06.034.
[15]	H. S. Rodrigues, M. T. T. Monteiro, D. F. M. Torres and A. Zinober, Dengue disease, basic reproduction number and control, Int. J. Comput. Math., 89 (2012), 334-346. doi: 10.1080/00207160.2011.554540.
[16]	P. M. Small and P. I. Fujiwara, Management of tuberculosis in the United States, N. Engl. J. Med., 345 (2001), 189-200. doi: 10.1056/NEJM200107193450307.
[17]	K. Styblo, State of art: epidemiology of tuberculosis, Bull. Int. Union Tuberc., 53 (1978), 141-152.
[18]	K. Styblo, "Selected Papers, Epidemiology of Tuberculosis," Royal Netherlands Tuberculosis Association, 24, 1991.
[19]	K. L. Teo, C. J. Goh and K. H. Wong, "A Unified Computational Approach to Optimal Control Problems," Pitman Monographs and Surveys in Pure and Applied Mathematics, 55, Longman Sci. Tech., Harlow, 1991.
[20]	WHO, Treatment of tuberculosis guidelines, Fourth edition, WHO, Geneva, 2010. Available from: http://www.who.int/tb/publications/tb_treatmentguidelines/en/index.html.
[21]	WHO, Global Tuberculosis Control, WHO Report 2011, Geneva, 2011. Available from: http://www.who.int/tb/publications/global_report/en/index.html.
[22]	, Available from: http://allafrica.com/stories/201103240476.html.
[23]	, Available from: http://apps.who.int/ghodata/?theme=country.
[24]	, Available from: http://www.avert.org/tuberculosis.htm.
[25]	, Available from: http://www.minsa.gov.ao/VerNoticia.aspx?id=10499.
[26]	, Available from: https://projects.coin-or.org/Ipopt.
[27]	, Available from: http://www.ampl.com.
[28]	, Available from: http://tomdyn.com.
[29]	, Available from: http://www.minsa.gov.ao/VerNoticia.aspx?id=8797.
[30]	, Available from: http://www.tradingeconomics.com/angola/tuberculosis-case-detection-rate-all-forms-wb-data.html.
[31]	, Available from: http://www.usaid.gov/our_work/global_health/id/tuberculosis/countries/africa/angola_profile.html.
[32]	, Available from: http://www.who.int/en/.

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