`a`
Mathematical Biosciences and Engineering (MBE)
 

Optimal control of a tuberculosis model with state and control delays
Pages: 321 - 337, Issue 1, February 2017

doi:10.3934/mbe.2017021      Abstract        References        Full text (939.9K)           Related Articles

Cristiana J. Silva - Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal (email)
Helmut Maurer - Institut für Numerische und Angewandte Mathematik, Westfälische Wilhelms Universität Münster, D-48149 Münster, Germany (email)
Delfim F. M. Torres - CIDMA — Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal (email)

1 R. Bellmann and K. L. Cooke, Differential-Difference Equations, Academic Press, New York, 1963.       
2 B. Buonomo and M. Cerasuolo, The effect of time delay in plant-pathogen interactions with host demography, Math. Biosci. Eng., 12 (2015), 473-490.       
3 C. Büskens, Optimierungsmethoden und Sensitivitätsanalyse Für Optimale Steuerprozesse mit Steuer- und Zustands-Beschränkungen, PhD thesis, Institut für Numerische Mathematik, Universität Münster, Germany, 1998.
4 C. Büskens and H. Maurer, SQP methods for solving optimal control problems with control and state constraints: adjoint variables, sensitivity analysis and real-time control, J. Comput. Appl. Math., 120 (2000), 85-108.       
5 C. Castillo-Chavez and Z. Feng, To treat or not to treat: The case of tuberculosis, J. Math. Biol., 35 (1997), 629-656.       
6 T. Cohen and M. Murray, Modeling epidemics of multidrug-resistant M. tuberculosis of heterogeneous fitness, Nat. Med., 10 (2004), 1117-1121.
7 R. V. Culshaw and S. Ruan, A delay-differential equation model of HIV infection of $CD4^+$ T-cells, Math. Biosci., 165 (2000), 27-39.
8 J. Dieudonné, Foundations of Modern Analysis, Academic Press, New York, 1960.       
9 R. Fourer, D. M. Gay and B. W. Kernighan, AMPL: A Modeling Language for Mathematical Programming, Duxbury Press, Brooks-Cole Publishing Company, 1993.
10 L. Göllmann and H. Maurer, Theory and applications of optimal control problems with multiple time-delays, Special Issue on Computational Methods for Optimization and Control, J. Ind. Manag. Optim., 10 (2014), 413-441.       
11 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, J. Theoret. Biol., 248 (2007), 608-617.       
12 J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993.       
13 H. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653.       
14 Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, San Diego, 1993.       
15 M. L. Lambert and P. Van der Stuyft, Delays to tuberculosis treatment: Shall we continue to blame the victim?, Trop. Med. Int. Health, 10 (2005), 945-946.
16 H. Maurer, C. Büskens, J.-H. R. Kim and Y. Kaya, Optimization methods for the verification of second order sufficient conditions for bang-bang controls, Optimal Control Appl. Methods, 26 (2005), 129-156.       
17 N. P. Osmolovskii and H. Maurer, Applications to Regular and Bang-Bang Control: Second-Order Necessary and Sufficient Optimality Conditions in Calculus of Variations and Optimal Control, SIAM Advances in Design and Control, Vol. DC 24, SIAM Publications, Philadelphia, 2012.       
18 P. Rodrigues, C. Rebelo and M. G. M. Gomes, Drug resistance in tuberculosis: A reinfection model, Theor. Popul. Biol., 71 (2007), 196-212.
19 P. Rodrigues, C. J. Silva and D. F. M. Torres, Cost-effectiveness analysis of optimal control measures for tuberculosis, Bull. Math. Biol., 76 (2014), 2627-2645.       
20 H. Schättler, U. Ledzewicz and H. Maurer, Sufficient conditions for strong local optimality in optimal control problems with $L^2$-type objectives and control constraints, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2657-2679.       
21 L. F. Shampine and S. Thompson, Solving DDEs in MATLAB, Appl. Numer. Math., 37 (2001), 441-458.       
22 C. J. Silva and D. F. M. Torres, Optimal control strategies for tuberculosis treatment: A case study in Angola, Numer. Algebra Control Optim., 2 (2012), 601-617.       
23 C. J. Silva and D. F. M. Torres, Optimal Control of Tuberculosis: A Review, Dynamics, Games and Science, CIM Series in Mathematical Sciences 1 (2015), 701-722.
24 C. T. Sreeramareddy, K. V. Panduru, J. Menten and J. Van den Ende, Time delays in diagnosis of pulmonary tuberculosis: A systematic review of literature, BMC Infectious Diseases, 9 (2009), p91.
25 D. G. Storla, S. Yimer and G. A. Bjune, A systematic review of delay in the diagnosis and treatment of tuberculosis, BMC Public Health, 8 (2008), p15.
26 K. Toman, Tuberculosis case-finding and chemotherapy: Questions and answers, WHO Geneva, 1979.
27 P. W. Uys, M. Warren and P. D. van Helden, A threshold value for the time delay to TB diagnosis, PLoS ONE, 2 (2007), e757.
28 P. van den Driessche and J. Watmough, Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission, Math. Biosc., 180 (2002), 29-48.       
29 H. Yang and J. Wei, Global behaviour of a delayed viral kinetic model with general incidence rate, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1573-1582.       
30 A. Wächter and L. T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Math. Program., 106 (2006), 25-57.       
31 Systematic Screening for Active Tuberculosis - Principles and Recommendations, Geneva, World Health Organization, 2013, http://www.who.int/tb/tbscreening/en/.
32 Global Tuberculosis Report 2014, Geneva, World Health Organization, 2014, http://www.who.int/tb/publications/global_report/en/.
33 Centers for Disease and Control Prevention, http://www.cdc.gov/tb/topic/treatment/ltbi.htm

Go to top