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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

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).   Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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).   Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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S. Lenhart and J. T. Workman, "Optimal Control Applied to Biological Models,", Chapman & Hall/CRC, (2007).   Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

W. H. Fleming and R. W. Rishel, "Deterministic and Stochastic Optimal Control,", Applications of Mathematics, (1975).   Google Scholar

[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.  Google Scholar

[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).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

S. Lenhart and J. T. Workman, "Optimal Control Applied to Biological Models,", Chapman & Hall/CRC, (2007).   Google Scholar

[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.  Google Scholar

[13]

L. Pontryagin, V. Boltyanskii, R. Gramkrelidze and E. Mischenko, "The Mathematical Theory of Optimal Processes,", Wiley Interscience, (1962).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[18]

K. Styblo, "Selected Papers, Epidemiology of Tuberculosis,", Royal Netherlands Tuberculosis Association, 24 (1991).   Google Scholar

[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).   Google Scholar

[20]

WHO, Treatment of tuberculosis guidelines,, Fourth edition, (2010).   Google Scholar

[21]

WHO, Global Tuberculosis Control,, WHO Report 2011, (2011).   Google Scholar

[22]

, Available from:, , ().   Google Scholar

[23]

, Available from:, , ().   Google Scholar

[24]

, Available from:, , ().   Google Scholar

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, Available from:, , ().   Google Scholar

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