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 & 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, (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. 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. doi: 10.1001/jama.282.7.677.
[4]	W. H. Fleming and R. W. Rishel, "Deterministic and Stochastic Optimal Control,", Applications of Mathematics, (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. 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, (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. 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. 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. 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. doi: 10.3934/mbe.2011.8.307.
[11]	S. Lenhart and J. T. Workman, "Optimal Control Applied to Biological Models,", Chapman & Hall/CRC, (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. 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. 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. 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. doi: 10.1056/NEJM200107193450307.
[17]	K. Styblo, State of art: epidemiology of tuberculosis,, Bull. Int. Union Tuberc., 53 (1978), 141.
[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, (1991).
[20]	WHO, Treatment of tuberculosis guidelines,, Fourth edition, (2010).
[21]	WHO, Global Tuberculosis Control,, WHO Report 2011, (2011).
[22]	, Available from:, , ().
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show all references

References:

[1]	L. Cesari, "Optimization-Theory and Applications. Problems with Ordinary Differential Equations,", Applications of Mathematics 17, (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. 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. doi: 10.1001/jama.282.7.677.
[4]	W. H. Fleming and R. W. Rishel, "Deterministic and Stochastic Optimal Control,", Applications of Mathematics, (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. 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, (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. 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. 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. 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. doi: 10.3934/mbe.2011.8.307.
[11]	S. Lenhart and J. T. Workman, "Optimal Control Applied to Biological Models,", Chapman & Hall/CRC, (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. 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. 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. 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. doi: 10.1056/NEJM200107193450307.
[17]	K. Styblo, State of art: epidemiology of tuberculosis,, Bull. Int. Union Tuberc., 53 (1978), 141.
[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, (1991).
[20]	WHO, Treatment of tuberculosis guidelines,, Fourth edition, (2010).
[21]	WHO, Global Tuberculosis Control,, WHO Report 2011, (2011).
[22]	, Available from:, , ().
[23]	, Available from:, , ().
[24]	, Available from:, , ().
[25]	, Available from:, , ().
[26]	, Available from:, , ().
[27]	, Available from:, , ().
[28]	, Available from:, , ().
[29]	, Available from:, , ().
[30]	, Available from:, , ().
[31]	, Available from:, , ().
[32]	, Available from:, , ().

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