-
Previous Article
Noether's symmetry Theorem for variational and optimal control problems with time delay
- NACO Home
- This Issue
-
Next Article
Control parameterization for optimal control problems with continuous inequality constraints: New convergence results
Optimal control strategies for tuberculosis treatment: A case study in Angola
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 |
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/. |
[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 and 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 and Continuous Dynamical Systems - B, 2018, 23 (1) : 79-99. doi: 10.3934/dcdsb.2018006 |
[7] |
M. Teresa T. Monteiro, Isabel Espírito Santo, Helena Sofia Rodrigues. An optimal control problem applied to a wastewater treatment plant. Discrete and Continuous Dynamical Systems - S, 2022, 15 (3) : 587-601. doi: 10.3934/dcdss.2021153 |
[8] |
E. Jung, Suzanne Lenhart, Z. Feng. Optimal control of treatments in a two-strain tuberculosis model. Discrete and Continuous Dynamical Systems - B, 2002, 2 (4) : 473-482. doi: 10.3934/dcdsb.2002.2.473 |
[9] |
Cristiana J. Silva, Delfim F. M. Torres. A TB-HIV/AIDS coinfection model and optimal control treatment. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 4639-4663. doi: 10.3934/dcds.2015.35.4639 |
[10] |
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 and Continuous Dynamical Systems - S, 2018, 11 (6) : 1179-1199. doi: 10.3934/dcdss.2018067 |
[11] |
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 |
[12] |
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 |
[13] |
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 |
[14] |
Kbenesh Blayneh, Yanzhao Cao, Hee-Dae Kwon. Optimal control of vector-borne diseases: Treatment and prevention. Discrete and Continuous Dynamical Systems - B, 2009, 11 (3) : 587-611. doi: 10.3934/dcdsb.2009.11.587 |
[15] |
Chao Xu, Yimeng Dong, Zhigang Ren, Huachen Jiang, Xin Yu. Sensor deployment for pipeline leakage detection via optimal boundary control strategies. Journal of Industrial and Management Optimization, 2015, 11 (1) : 199-216. doi: 10.3934/jimo.2015.11.199 |
[16] |
Maciej Leszczyński, Urszula Ledzewicz, Heinz Schättler. Optimal control for a mathematical model for anti-angiogenic treatment with Michaelis-Menten pharmacodynamics. Discrete and Continuous Dynamical Systems - B, 2019, 24 (5) : 2315-2334. doi: 10.3934/dcdsb.2019097 |
[17] |
Andrea Signori. Penalisation of long treatment time and optimal control of a tumour growth model of Cahn–Hilliard type with singular potential. Discrete and Continuous Dynamical Systems, 2021, 41 (6) : 2519-2542. doi: 10.3934/dcds.2020373 |
[18] |
Nasser H. Sweilam, Taghreed A. Assiri, Muner M. Abou Hasan. Optimal control problem of variable-order delay system of advertising procedure: Numerical treatment. Discrete and Continuous Dynamical Systems - S, 2022, 15 (5) : 1247-1268. doi: 10.3934/dcdss.2021085 |
[19] |
Jesús-Javier Chi-Domínguez, Francisco Rodríguez-Henríquez. Optimal strategies for CSIDH. Advances in Mathematics of Communications, 2022, 16 (2) : 383-411. doi: 10.3934/amc.2020116 |
[20] |
Marcelo J. Villena, Mauricio Contreras. Global and local advertising strategies: A dynamic multi-market optimal control model. Journal of Industrial and Management Optimization, 2019, 15 (3) : 1017-1048. doi: 10.3934/jimo.2018084 |
Impact Factor:
Tools
Metrics
Other articles
by authors
[Back to Top]