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

Previous Article
Control parameterization for optimal control problems with continuous inequality constraints: New convergence results
NACO Home
This Issue
Next Article
Noether's symmetry Theorem for variational and optimal control problems with time delay

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
Related Papers

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:, , ().
[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:, , ().

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:, , ().

[1]	Yali Yang, Sanyi Tang, Xiaohong Ren, Huiwen Zhao, Chenping Guo. Global stability and optimal control for a tuberculosis model with vaccination and treatment. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 1009-1022. doi: 10.3934/dcdsb.2016.21.1009
[2]	Holly Gaff, Elsa Schaefer. Optimal control applied to vaccination and treatment strategies for various epidemiological models. Mathematical Biosciences & Engineering, 2009, 6 (3) : 469-492. doi: 10.3934/mbe.2009.6.469
[3]	Elena Fimmel, Yury S. Semenov, Alexander S. Bratus. On optimal and suboptimal treatment strategies for a mathematical model of leukemia. Mathematical Biosciences & Engineering, 2013, 10 (1) : 151-165. doi: 10.3934/mbe.2013.10.151
[4]	Cristiana J. Silva, Helmut Maurer, Delfim F. M. Torres. Optimal control of a Tuberculosis model with state and control delays. Mathematical Biosciences & Engineering, 2017, 14 (1) : 321-337. doi: 10.3934/mbe.2017021
[5]	Ellina Grigorieva, Evgenii Khailov, Andrei Korobeinikov. An optimal control problem in HIV treatment. Conference Publications, 2013, 2013 (special) : 311-322. doi: 10.3934/proc.2013.2013.311
[6]	Maria do Rosário de Pinho, Helmut Maurer, Hasnaa Zidani. Optimal control of normalized SIMR models with vaccination and treatment. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 79-99. doi: 10.3934/dcdsb.2018006
[7]	E. Jung, Suzanne Lenhart, Z. Feng. Optimal control of treatments in a two-strain tuberculosis model. Discrete & Continuous Dynamical Systems - B, 2002, 2 (4) : 473-482. doi: 10.3934/dcdsb.2002.2.473
[8]	Sanjukta Hota, Folashade Agusto, Hem Raj Joshi, Suzanne Lenhart. Optimal control and stability analysis of an epidemic model with education campaign and treatment. Conference Publications, 2015, 2015 (special) : 621-634. doi: 10.3934/proc.2015.0621
[9]	Djamila Moulay, M. A. Aziz-Alaoui, Hee-Dae Kwon. Optimal control of chikungunya disease: Larvae reduction, treatment and prevention. Mathematical Biosciences & Engineering, 2012, 9 (2) : 369-392. doi: 10.3934/mbe.2012.9.369
[10]	Kbenesh Blayneh, Yanzhao Cao, Hee-Dae Kwon. Optimal control of vector-borne diseases: Treatment and prevention. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 587-611. doi: 10.3934/dcdsb.2009.11.587
[11]	Cristiana J. Silva, Delfim F. M. Torres. A TB-HIV/AIDS coinfection model and optimal control treatment. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4639-4663. doi: 10.3934/dcds.2015.35.4639
[12]	Urszula Ledzewicz, Mohammad Naghnaeian, Heinz Schättler. Dynamics of tumor-immune interaction under treatment as an optimal control problem. Conference Publications, 2011, 2011 (Special) : 971-980. doi: 10.3934/proc.2011.2011.971
[13]	Joaquim P. Mateus, Paulo Rebelo, Silvério Rosa, César M. Silva, Delfim F. M. Torres. Optimal control of non-autonomous SEIRS models with vaccination and treatment. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1179-1199. doi: 10.3934/dcdss.2018067
[14]	Chao Xu, Yimeng Dong, Zhigang Ren, Huachen Jiang, Xin Yu. Sensor deployment for pipeline leakage detection via optimal boundary control strategies. Journal of Industrial & Management Optimization, 2015, 11 (1) : 199-216. doi: 10.3934/jimo.2015.11.199
[15]	Judy Day, Jonathan Rubin, Gilles Clermont. Using nonlinear model predictive control to find optimal therapeutic strategies to modulate inflammation. Mathematical Biosciences & Engineering, 2010, 7 (4) : 739-763. doi: 10.3934/mbe.2010.7.739
[16]	Honglei Xu, Peng Sui, Guanglu Zhou, Louis Caccetta. Dampening bullwhip effect of order-up-to inventory strategies via an optimal control method. Numerical Algebra, Control & Optimization, 2013, 3 (4) : 655-664. doi: 10.3934/naco.2013.3.655
[17]	Dingjun Yao, Kun Fan. Optimal risk control and dividend strategies in the presence of two reinsurers: Variance premium principle. Journal of Industrial & Management Optimization, 2018, 14 (3) : 1055-1083. doi: 10.3934/jimo.2017090
[18]	Marcelo J. Villena, Mauricio Contreras. Global and local advertising strategies: A dynamic multi-market optimal control model. Journal of Industrial & Management Optimization, 2018, 13 (5) : 1-11. doi: 10.3934/jimo.2018084
[19]	Joanna R. Wares, Joseph J. Crivelli, Chae-Ok Yun, Il-Kyu Choi, Jana L. Gevertz, Peter S. Kim. Treatment strategies for combining immunostimulatory oncolytic virus therapeutics with dendritic cell injections. Mathematical Biosciences & Engineering, 2015, 12 (6) : 1237-1256. doi: 10.3934/mbe.2015.12.1237
[20]	Silvia Martorano Raimundo, Hyun Mo Yang, Ezio Venturino. Theoretical assessment of the relative incidences of sensitive and resistant tuberculosis epidemic in presence of drug treatment. Mathematical Biosciences & Engineering, 2014, 11 (4) : 971-993. doi: 10.3934/mbe.2014.11.971

Impact Factor:

American Institute of Mathematical Sciences