Optimal Control and Sensitivity Analysis of a Fractional Order TB Model
AbstractA Caputo fractional-order mathematical model for the transmission dynamics of tuberculosis (TB) was recently proposed in [Math. Model. Nat. Phenom. 13 (2018), no. 1, Art. 9]. Here, a sensitivity analysis of that model is done, showing the importance of accuracy of parameter values. A fractional optimal control (FOC) problem is then formulated and solved,with the rate of treatment as the control variable. Finally, a cost-effectiveness analysis is performed to assess the cost and the effectiveness of the control measures during the intervention, showing in which conditions FOC is useful with respect to classical (integer-order) optimal control.
H. M. Ali, F. Lobo Pereira and S. M. A. Gama, A new approach to the Pontryagin maximum principle for nonlinear fractional optimal control problems, Math. Methods Appl. Sci. 39 (2016), no. 13, 3640–3649.
R. Almeida, S. Pooseh and D. F. M. Torres, Computational methods in the fractional calculus of variations, Imperial College Press,London, 2015.
N. Chitnis, J. M. Hyman and J. M. Cushing, Determining important parameters in the spread of malaria throughthe sensitivityanalysis of a mathematical model, Bull. Math. Biol. 70 (2008), no. 5, 1272–1296.
A. Debbouche, J. J. Nieto and D. F. M. Torres, Optimal solutions to relaxation in multiple control problems of Sobolev type with nonlocal nonlinear fractional differential equations, J. Optim. Theory Appl. 174 (2017), no. 1, 7–31. arXiv:1504.05153
R. Denysiuk, C. J. Silva and D. F. M. Torres, Multiobjective approach to optimal control for a tuberculosis model, Optim. Methods Softw. 30 (2015), no. 5, 893–910. arXiv:1412.0528
K. Diethelm, N. J. Ford, A. D. Freed and Yu. Luchko, Algorithms for the fractional calculus: a selection of numerical methods,Comput. Methods Appl. Mech. Engrg. 194 (2005), no. 6-8, 743–773.
S. Jahanshahi and D. F. M. Torres, A simple accurate method for solving fractional variational and optimal control problems, J. Optim. Theory Appl. 174 (2017), no. 1, 156–175. arXiv:1601.06416
R. Kamocki, Pontryagin maximum principle for fractional ordinary optimal control problems, Math. Methods Appl. Sci. 37 (2014),no. 11, 1668–1686.
A. B. Malinowska and D. F. M. Torres, Introduction to the fractional calculus of variations, Imperial College Press, London, 2012.
M. A. Mikucki, Sensitivity analysis of the basic reproduction number and other quantities for infectious disease models, PhD thesis, Colorado State University, 2012.
K. O. Okosun, O. Rachid and N. Marcus, Optimal control strategies and cost-effectiveness analysis of a malaria model, Biosystems 111 (2013), no. 2, 83–101.
I. Podlubny, Fractional differential equations, Mathematics in Science and Engineering, 198, Academic Press, Inc., San Diego, CA, 1999.
D. R. Powell, J. Fair, R. J. LeClaire, L. M. Moore and D. Thompson, Sensitivity analysis of an infectious disease model, Proc. Int.System Dynamics Conference, Boston, MA, 2005.
H. S. Rodrigues, M. T. T. Monteiro and D. F. M. Torres, Sensitivity analysis in a dengue epidemiological model, Conference Papers in Mathematics 2013 (2013), Art. ID 721406, 7 pp. arXiv:1307.0202
P. Rodrigues, C. J. Silva and D. F. M. Torres, Cost-effectiveness analysis of optimal control measures for tuberculosis, Bull. Math.Biol. 76 (2014), no. 10, 2627–2645. arXiv:1409.3496
M. R. Sidi Ammi, I. Jamiai and D. F. M. Torres, Global existence of solutions for a fractional Caputo nonlocal thermistor problem,Adv. Difference Equ. 2017 (2017), Paper no. 363, 14 pp. arXiv:1711.00143
C. J. Silva, H. Maurer and D. F. M. Torres, Optimal control of a tuberculosis model with state and control delays, Math. Biosci. Eng.14 (2017), no. 1, 321–337. arXiv:1606.08721
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), no. 3, 601–617. arXiv:1203.3255
C. J. Silva and D. F. M. Torres, Optimal control for a tuberculosis model with reinfection and post-exposure interventions, Math.Biosci. 244 (2013), no. 2, 154–164. arXiv:1305.2145
C. J. Silva and D. F. M. Torres, Optimal control of tuberculosis: a review, in Dynamics, games and science, 701–722, CIM Ser. Math.Sci., 1, Springer, Cham, 2015. arXiv:1406.3456
N. H. Sweilam and S. M. Al-Mekhlafi, Numerical study for multi strain tuberculosis (TB) model of variable-order fractional derivatives, J. Adv. Res. 7 (2016), no. 2, 271–283.
P. van den Driessche, Reproduction numbers of infectious disease models, Infect. Disease Model. 2 (2017), no. 3, 288–303.
WHO, Global Tuberculosis Report 2016, World Health Organization, 2016.
W. Wojtak, C. J. Silva and D. F. M. Torres, Uniform asymptotic stability of a fractional tuberculosis model, Math. Model. Nat.Phenom. 13 (2018), no. 1, Art. 9, 10 pp. arXiv:1801.07059
Y. Yang, J. Li, Z. Ma and L. Liu, Global stability of two models with incomplete treatment for tuberculosis, Chaos Solitons Fractals 43 (2010), no. 1-12, 79–85.
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).