### A comparison of some control strategies for a non-integer order tuberculosis model

#### Abstract

The aim of this paper is to investigate some optimal control strategies for a generalized tuberculosis model consisting of four compartments. We construct the model with the use of Caputo time fractional derivative. Contribution of distancing control, latent case finding control, case holding control and their combinations are discussed and the optimality system is obtained based on the Hamiltonian principle. Additionally, the non-negativity and boundedness of the solution are shown. We present some illustrative examples to determine the most effective strategy to minimize the number of infected people and maximize the number of susceptible individuals. Moreover, we discuss the contribution of the Caputo derivative and the order of the fractional derivative to efficiency of the control strategies.

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World Health Organization. (2017). Tuberculosis fact sheet. Geneva, Switzerland. Available from https://www.who.int/tb/publications/global report/gtbr2017 main text.pdf. [Accessed 6 July 2018].

Feng, Z., Castillo-Chavez, C., & Capurro, A.F. (2000). A model for tuberculosis with exogenous reinfection. Theoretical Population Biology, 57 (3), 235–247.

Small, P.M., & Fujiwara, P.I. (2001). Management of tuberculosis in the United States,

Bailey, N.T., et al., (1975). The mathematical theory of infectious diseases and its applications. Charles Griffin & Company Ltd, 5a Crendon Street, High Wycombe, Bucks HP13 6LE.

Waaler, H., Geser, A., & Andersen, S. (1962). The use of mathematical models in the study of the epidemiology of tuberculosis. American Journal of Public Health and the Nations Health, 52 (6), 1002–1013.

Hethcote, H.W. (2000). The mathematics of infectious diseases. SIAM Review, 42 (4), 599–653.

Choisy, M., Gu´egan, J.F., & Rohani, P. (2007). Mathematical modeling of infectious diseases dynamics. Encyclopedia of infectious diseases: Modern methodologies, 379–404.

Castillo-Chavez, C., & Feng, Z. (1997) To treat or not to treat: the case of tuberculosis. Journal of Mathematical Biology, 35 (6), 629–656.

Castillo-Ch´avez, C., Feng, Z., et al. (1998). Mathematical models for the disease dynamics of tuberculosis. in: G. S. Mary Ann Horn, G. Webb (Eds.), Advances in Mathematical Population Dynamics-Molecules, Cells and Man, Vanderbilt University Press, 117–128.

Liu, L., Zhao, X.Q., & Zhou, Y., (2010). A tuberculosis model with seasonality. Bulletin of Mathematical Biology, 72 (4), 931–952. A comparison of some control strategies for a non-integer order tuberculosis model 29

Podlubny, I. (1999). Fractional differential equations, Vol. 198 of Mathematics in Science and Engineering, Academic Press, Inc., San Diego, CA, An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications.

Yang, X.J., & Machado, J.A.T. (2017). A new fractional operator of variable order: application in the description of anomalous diffusion. Physica A: Statistical Mechanics and its Applications, 481, 276–283.

Moghaddam, B.P., & Machado, J.A.T. (2017). Extended algorithms for approximating variable order fractional derivatives with applications. Journal of Scientific Computing, 71 (3), 1351–1374.

Moghaddam,B.P., & Machado, J.A.T. (2017). SM-algorithms for approximating the variable-order fractional derivative of high order. Fundamenta Informaticae, 151 (1-4), 293–311.

Koca, I. (2018). Analysis of rubella disease model with non-local and non-singular fractional derivatives. An International Journal of Optimization and Control: Theories and Applications, 8 (1), 17–25.

Sweilam, N.H., Al-Mekhlafi, S.M., & Assiri, T.A.R. (2017). Numerical study for time delay multistrain tuberculosis model of fractional order. Complexity, 2017, Article ID 1047381, 14 pages.

Wojtak, W., Silva, C.J., & Torres, D.F. (2018). Uniform asymptotic stability of a fractional tuberculosis model. Mathematical Modelling of Natural Phenomena, 13 (1), 9.

Pinto, C.M.A., & Carvalho, A.R.M. (2017). The HIV/TB coinfection severity in the presence of TB multi-drug resistant strains. Ecological complexity, 32, 1–20.

Carvalho, A.R.M., & Pinto, C.M.A. (2018). Non-integer order analysis of the impact of diabetes and resistant strains in a model for TB infection. Communications in Nonlinear Science and Numerical Simulation, 61, 104–126.

De Pillis, L.G., & Radunskaya, A. (2001). A mathematical tumor model with immune resistance and drug therapy: An optimal control approach. Computational and Mathematical Methods in Medicine, 3 (2), 79–100.

Okosun, K.O., Ouifki, R., & Marcus, N. (2011). Optimal control analysis of a malaria disease transmission model that includes treatment and vaccination with waning immunity. Biosystems, 106 (2-3), 136–145.

Area, I, Nda¨Irou, F., Nieto, J.J., Silva, C.J., & Torres, D.F.M. (2018). Ebola model and optimal control with vaccination constraints. Journal of Industrial & Management Optimization, 14 (2), 427–446.

Hamdache, A., Saadi, S., & Elmouki, I. (2016). Free terminal time optimal control problem for the treatment of HIV infection. An International Journal of Optimization and Control: Theories & Applications, 6 (1), 33–51.

Sweilam, N.H., & Al-Mekhlafi, S.M. (2016). On the optimal control for fractional multistrain TB model. Optimal Control Applications and Methods, 37 (6), 1355–1374.

Silva, C.J., & Torres, D.F.M. (2013). Optimal control for a tuberculosis model with reinfection and post-exposure interventions. Mathematical Biosciences, 244 (2), (2013) 154–164.

Bowong, S. (2010). Optimal control of the transmission dynamics of tuberculosis. Non-linear Dynamics, 61 (4), 729–748.

Moualeu, D.P., Weiser, M., Ehrig, R., & Deuflhard, P. (2015). Optimal control for a tuberculosis model with undetected cases in Cameroon. Communications in Nonlinear Science and Numerical Simulation, 20 (3), 986–1003.

Jung, E., Lenhart, S., & Feng, Z. (2002). Optimal control of treatments in a two-strain tuberculosis model. Discrete and Continuous Dynamical Systems Series B, 2 (4), 473–482.

Silva, C.J., & Torres, D.F.M. (2015). Optimal control of tuberculosis: A review, in: Dynamics, Games and Science, Springer, 701–722.

Kim, S., Aurelio, A. de Los Reyes V., & Jung, E. (2018). Mathematical model and intervention strategies for mitigating tuberculosis in the Philippines. Journal of Theoretical Biology, 443, 100–112.

Odibat, Z.M., & Shawagfeh, N.T. (2007). Generalized Taylor’s formula. Applied Mathematics and Computation, 186 (1), 286–293.

Diethelm, K. (2013). A fractional calculus based model for the simulation of an outbreak of dengue fever. Nonlinear Dynamics, 71 (4), 613–619.

Pontryagin, L., Boltyanskii, V., Gamkrelidze, R., & Mishchenko, E. (1962). The Mathematical Theory of Optimal Processes. John Wiley & Sons, New York.

Lin, Y., & Xu, C. (2007). Finite difference/spectral approximations for the timefractional diffusion equation. Journal of Computational Physics, 225 (2), 1533–1552.

Lenhart, S., Workman, J.T. (2007). Optimal Control Applied to Biological Models. Chapman & Hall, CRC Press.[1] World Health Organization. (2017). Tuberculosis fact sheet. Geneva, Switzerland. Available from https://www.who.int/tb/publications/global report/gtbr2017 main text.pdf. [Accessed 6 July 2018].

Feng, Z., Castillo-Chavez, C., & Capurro, A.F. (2000). A model for tuberculosis with exogenous reinfection. Theoretical Population Biology, 57 (3), 235–247.

Small, P.M., & Fujiwara, P.I. (2001). Management of tuberculosis in the United States,

Bailey, N.T., et al., (1975). The mathematical theory of infectious diseases and its applications. Charles Griffin & Company Ltd, 5a Crendon Street, High Wycombe, Bucks HP13 6LE.

Waaler, H., Geser, A., & Andersen, S. (1962). The use of mathematical models in the study of the epidemiology of tuberculosis. American Journal of Public Health and the Nations Health, 52 (6), 1002–1013.

Hethcote, H.W. (2000). The mathematics of infectious diseases. SIAM Review, 42 (4), 599–653.

Choisy, M., Gu´egan, J.F., & Rohani, P. (2007). Mathematical modeling of infectious diseases dynamics. Encyclopedia of infectious diseases: Modern methodologies, 379–404.

Castillo-Chavez, C., & Feng, Z. (1997) To treat or not to treat: the case of tuberculosis. Journal of Mathematical Biology, 35 (6), 629–656.

Castillo-Ch´avez, C., Feng, Z., et al. (1998). Mathematical models for the disease dynamics of tuberculosis. in: G. S. Mary Ann Horn, G. Webb (Eds.), Advances in Mathematical Population Dynamics-Molecules, Cells and Man, Vanderbilt University Press, 117–128.

Liu, L., Zhao, X.Q., & Zhou, Y., (2010). A tuberculosis model with seasonality. Bulletin of Mathematical Biology, 72 (4), 931–952. A comparison of some control strategies for a non-integer order tuberculosis model 29

Podlubny, I. (1999). Fractional differential equations, Vol. 198 of Mathematics in Science and Engineering, Academic Press, Inc., San Diego, CA, An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications.

Yang, X.J., & Machado, J.A.T. (2017). A new fractional operator of variable order: application in the description of anomalous diffusion. Physica A: Statistical Mechanics and its Applications, 481, 276–283.

Moghaddam, B.P., & Machado, J.A.T. (2017). Extended algorithms for approximating variable order fractional derivatives with applications. Journal of Scientific Computing, 71 (3), 1351–1374.

Moghaddam,B.P., & Machado, J.A.T. (2017). SM-algorithms for approximating the variable-order fractional derivative of high order. Fundamenta Informaticae, 151 (1-4), 293–311.

Koca, I. (2018). Analysis of rubella disease model with non-local and non-singular fractional derivatives. An International Journal of Optimization and Control: Theories and Applications, 8 (1), 17–25.

Sweilam, N.H., Al-Mekhlafi, S.M., & Assiri, T.A.R. (2017). Numerical study for time delay multistrain tuberculosis model of fractional order. Complexity, 2017, Article ID 1047381, 14 pages.

Wojtak, W., Silva, C.J., & Torres, D.F. (2018). Uniform asymptotic stability of a fractional tuberculosis model. Mathematical Modelling of Natural Phenomena, 13 (1), 9.

Pinto, C.M.A., & Carvalho, A.R.M. (2017). The HIV/TB coinfection severity in the presence of TB multi-drug resistant strains. Ecological complexity, 32, 1–20.

Carvalho, A.R.M., & Pinto, C.M.A. (2018). Non-integer order analysis of the impact of diabetes and resistant strains in a model for TB infection. Communications in Nonlinear Science and Numerical Simulation, 61, 104–126.

De Pillis, L.G., & Radunskaya, A. (2001). A mathematical tumor model with immune resistance and drug therapy: An optimal control approach. Computational and Mathematical Methods in Medicine, 3 (2), 79–100.

Okosun, K.O., Ouifki, R., & Marcus, N. (2011). Optimal control analysis of a malaria disease transmission model that includes treatment and vaccination with waning immunity. Biosystems, 106 (2-3), 136–145.

Area, I, Nda¨Irou, F., Nieto, J.J., Silva, C.J., & Torres, D.F.M. (2018). Ebola model and optimal control with vaccination constraints. Journal of Industrial & Management Optimization, 14 (2), 427–446.

Hamdache, A., Saadi, S., & Elmouki, I. (2016). Free terminal time optimal control problem for the treatment of HIV infection. An International Journal of Optimization and Control: Theories & Applications, 6 (1), 33–51.

Sweilam, N.H., & Al-Mekhlafi, S.M. (2016). On the optimal control for fractional multistrain TB model. Optimal Control Applications and Methods, 37 (6), 1355–1374.

Silva, C.J., & Torres, D.F.M. (2013). Optimal control for a tuberculosis model with reinfection and post-exposure interventions. Mathematical Biosciences, 244 (2), (2013) 154–164.

Bowong, S. (2010). Optimal control of the transmission dynamics of tuberculosis. Non-linear Dynamics, 61 (4), 729–748.

Moualeu, D.P., Weiser, M., Ehrig, R., & Deuflhard, P. (2015). Optimal control for a tuberculosis model with undetected cases in Cameroon. Communications in Nonlinear Science and Numerical Simulation, 20 (3), 986–1003.

Jung, E., Lenhart, S., & Feng, Z. (2002). Optimal control of treatments in a two-strain tuberculosis model. Discrete and Continuous Dynamical Systems Series B, 2 (4), 473–482.

Silva, C.J., & Torres, D.F.M. (2015). Optimal control of tuberculosis: A review, in: Dynamics, Games and Science, Springer, 701–722.

Kim, S., Aurelio, A. de Los Reyes V., & Jung, E. (2018). Mathematical model and intervention strategies for mitigating tuberculosis in the Philippines. Journal of Theoretical Biology, 443, 100–112.

Odibat, Z.M., & Shawagfeh, N.T. (2007). Generalized Taylor’s formula. Applied Mathematics and Computation, 186 (1), 286–293.

Diethelm, K. (2013). A fractional calculus based model for the simulation of an outbreak of dengue fever. Nonlinear Dynamics, 71 (4), 613–619.

Pontryagin, L., Boltyanskii, V., Gamkrelidze, R., & Mishchenko, E. (1962). The Mathematical Theory of Optimal Processes. John Wiley & Sons, New York.

Lin, Y., & Xu, C. (2007). Finite difference/spectral approximations for the time fractional diffusion equation. Journal of Computational Physics, 225 (2), 1533–1552.

Lenhart, S., Workman, J.T. (2007). Optimal Control Applied to Biological Models. Chapman & Hall, CRC Press.

DOI: http://dx.doi.org/10.11121/ijocta.01.2019.00657

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