Research article Special Issues

A mathematical model of tuberculosis (TB) transmission with children and adults groups: A fractional model

  • Received: 01 December 2019 Accepted: 19 February 2020 Published: 17 March 2020
  • MSC : 34A34

  • We consider a novel fractional model to investigates the (tuberculosis) TB model dynamics with two age groups of human, that is, the children and the adults. First, we formulate the model and present the basic results associated to the model. Then, using the fractional operators, Caputo and the Atangana-Baleanu and obtain a generalized model. Further, we give a novel numerical approach for the solution of the fractional model and obtain their approximate solution. We show graphical results with various values of the fractional order. A comparison of the two operators are shown graphically. The results obtained through Atangana-Baleanu operator is flexible than that of Caputo derivative. The infection in tuberculosis (TB) infected people decreases fast when decreasing the fractional order.

    Citation: Fatmawati, Muhammad Altaf Khan, Ebenezer Bonyah, Zakia Hammouch, Endrik Mifta Shaiful. A mathematical model of tuberculosis (TB) transmission with children and adults groups: A fractional model[J]. AIMS Mathematics, 2020, 5(4): 2813-2842. doi: 10.3934/math.2020181

    Related Papers:

  • We consider a novel fractional model to investigates the (tuberculosis) TB model dynamics with two age groups of human, that is, the children and the adults. First, we formulate the model and present the basic results associated to the model. Then, using the fractional operators, Caputo and the Atangana-Baleanu and obtain a generalized model. Further, we give a novel numerical approach for the solution of the fractional model and obtain their approximate solution. We show graphical results with various values of the fractional order. A comparison of the two operators are shown graphically. The results obtained through Atangana-Baleanu operator is flexible than that of Caputo derivative. The infection in tuberculosis (TB) infected people decreases fast when decreasing the fractional order.


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    [1] World Health Organization, Anti-tuberculosis Drug Resistance in the World: Third Global Report, No. WHO / HTM / TB / 2004. 343, World Health Organization, 2004.
    [2] J. Cohen, Extensively drug-resistant TB gets foothold in South Africa, Science 313 (2006), 1554.
    [3] L. B. Reichman, J. H. Tanne, Timebomb: The Global Epidemic of Multi-Drug Resistant Tuberculosis, 2002.
    [4] Y. Zhou, K. Khan, Z. Feng, et al. Projection of tuberculosis incidence with increasing immigration trends, J. Theor. Biol., 254 (2008), 215–228.
    [5] P. Rodrigues, M. G. M. Gomes, C. Rebelo, Drug resistance in tuberculosis: a reinfection model, Theor. Popul. Biol., 71 (2007), 196–212.
    [6] N. Blaser, C. Zahnd, S. Hermans, et al. Tuberculosis in Cape Town: an age-structured transmission model, Epidemics, 14 (2016), 54–61.
    [7] C. P. Bhunu, W. Garira, Z. Mukandavire, et al. Tuberculosis transmission model with chemoprophylaxis and treatment, B. Math. Biol., 70 (2008), 1163–1191.
    [8] Centers for Disease Control and Prevention, TB in Children in the United States, CDC, 2014. Available from: https://www.cdc.gov/tb/topic/populations/tbinchildren/default.htm.
    [9] S. M. Blower, P. M. Small, P. C. Hopewell, Control strategies for tuberculosis epidemics: new models for old problems, Science, 273 (1996), 497–500.
    [10] Fatmawati, H. Tasman, An optimal treatment control of TB-HIV coinfection, International Journal of Mathematics and Mathematical Sciences, 2016 (2016).
    [11] R. I. Hickson, G. N. Mercer, K. M. Lokuge, A metapopulation model of tuberculosis transmission with a case study from high to low burden areas, PLoS One, 7 (2012).
    [12] R. M. G. J. Houben, T. Sumner, A. D. Grant, et al. Ability of preventive therapy to cure latent Mycobacterium tuberculosis infection in HIV-infected individuals in high-burden settings, P. Natl. A. Sci., 111 (2014), 5325–5330.
    [13] R. Kaplan, J. Caldwell, K. Middelkoop, et al. Impact of ART on TB case fatality stratified by CD4 count for HIV-positive TB patients in Cape Town, South Africa (2009-2011), J. Acq. Imm. Def., 66 (2014), 487–494.
    [14] Fatmawati, U. D. Purwati, F. Riyudha, et al. Optimal control of a discrete age-structured model for tuberculosis transmission, Heliyon, 6 (2020).
    [15] S. G, Samko, A. A, Kilbas, O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, 1993.
    [16] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, California, USA, 1999.
    [17] T. Sardar, S. Rana, J. Chattopadhyay, A mathematical model of dengue transmission with memory, Commun. Nonlinear Sci., 22 (2015), 511–525.
    [18] J. Huo, H. Zhao, L. Zhu, The effect on backward bifurcation in a fractional orde HIV model, Nonlinear Analysis: Real World Applications, 26 (2015), 289–305.
    [19] M. Saeedian, M. Khalighi, N. Azimi-Tafreshi, et al. Memory effects on epidemic evolution: the susceptible-infected-recovered epidemic model, Phys. Rev. E., 95 (2017).
    [20] C. M. A. Pinto, A. R. M. Carvalho, The HIV/TB coinfection severity in the presence of TB multidrug resistant strains, Ecol. Complex., 32 (2017), 1–20.
    [21] Fatmawati, E. M. Shaiful, M. I. Utoyo, A fractional order model for HIV dynamics in a two-sex population, International Journal of Mathematics and Mathematical Sciences, 2018 (2018).
    [22] G. C. Wu, Z. G. Deng, D. Baleanu, et al. New variable-order fractional chaotic systems for fast image encryption, Chaos: An Interdisciplinary Journal of Nonlinear Science, 29 (2019).
    [23] G. C. Wu, T. Abdeljawad, J. Liu, et al. Mittag-Leffler stability analysis of fractional discrete-time neural networks via fixed point technique, Nonlinear Analysis: Modelling and Control, 24 (2019), 919–936.
    [24] G. C. Wu, D. Q. Zeng, D. Baleanu, Fractional impulisve differential equations: Exact solutions, integral equations and short memory case, Frac. Calc. Appl. Anal., 22 (2019), 180–192.
    [25] M. Itik, S. P. Banks, Chaos in a three-dimensional cancer model, Int. J. Bifurcat. Chaos, 20 (2010), 71–79.
    [26] M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 1–13.
    [27] M. Caputo, M. Fabrizio, On the notion of fractional derivative and applications to the hysteresis phenomena, Meccanica, 52 (2017), 3043–3052.
    [28] T. Zhang, L. Xiong, Periodic motion for impulsive fractional functional differential equations with piecewise Caputo derivative, Appl. Math. Lett., 101 (2020), 106072.
    [29] A. Atangana, D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model, Therm. Sci., 20 (2016).
    [30] A. Atangana, I. Koca, Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order, Chaos, Solitons & Fractals, 89 (2016), 447–454.
    [31] A. Atangana, K. M. Owolabi, New numerical approach for fractional differential equations, Math. Model. Nat. Pheno., 13 (2018).
    [32] A. Atangana, J. F. Gomez-Aguilar, Decolonisation of fractional calculus rules: Breaking commutativity and associativity to capture more natural phenomena, The European Physical Journal Plus, 133 (2018).
    [33] M. A. Khan, S. Ullah, M. Farooq, A new fractional model for tuberculosis with relapse via Atangana-Baleanu derivative, Chaos, Solitons & Fractals, 116 (2018), 227–238.
    [34] E. Bonyah, Chaos in a 5-D hyperchaotic system with four wings in the light of non-local and non-singular fractional derivatives, Chaos, Solitons & Fractals, 116 (2018), 316–331.
    [35] K. Muhammad Altaf, A. Atangana, Dynamics of Ebola disease in the framework of different fractional derivatives, Entropy, 21 (2019).
    [36] R. Jan, M. A. Khan, P. Kumam, et al, Modeling the transmission of dengue infection through fractional derivatives, Chaos, Solitons & Fractals, 127 (2019), 189–216.
    [37] W. Wang, M. A. Khan, P. Kumam, et al. A comparison study of bank data in fractional calculus, Chaos, Solitons & Fractals, 126 (2019), 369–384.
    [38] M. A. Khan, The dynamics of a new chaotic system through the Caputo-Fabrizio and AtanaganBaleanu fractional operators, Adv. Mech. Eng., 11 (2019).
    [39] Fatmawati, M. A. Khan, M. Azizah, et al. A fractional model for the dynamics of competition between commercial and rural banks in Indonesia, Chaos, Solitons & Fractals, 122 (2019), 32–46.
    [40] S. Ullah, M. A. Khan, M. Farooq, et al. A fractional model for the dynamics of tuberculosis (TB) using Atangana-Baleanu derivative, Discrete Cont. Dyn. S, 13 (2019).
    [41] A. A. Velayati, Tuberculosis in children, International Journal of Mycobacteriology, 5 (2016).
    [42] C. Castillo-Chaves, B. Song, Dynamic models of tuberculosis and their applications, Math. Biosci. Eng., 1 (2004), 361–404.
    [43] Z. M. Odibat, N. T. Shawagfeh, Generalized Taylors formula, Appl. Math. Comput., 186 (2007), 286–293.
    [44] W. Lin, Global existence theory and chaos control of fractional differential equations, J. Math. Anal. Appl., 332 (2007), 709–726.
    [45] P. van den Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29–48.
    [46] K. Diethelm, N. J. Ford, A. D. Freed, A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dynamics, 29 (2002), 3–22.
    [47] K. Diethelm, N. J. Ford, A. D. Freed, Detailed error analysis for a fractional Adams method, Numerical Algorithms, 36 (2004), 31–52.
    [48] C. P. Bhunu, Mathematical analysis of a three-strain tuberculosis transmission model, Appl. Math. Model., 35 (2011), 4647–4660.
    [49] S. Athithan, M. Ghosh, Optimal control of tuberculosis with case detection and treatment, World Journal of Modelling and Simulation, 11 (2015), 111–122.
    [50] Word Health Organization, Factsheet on the World Tuberculosis Report 2017, WHO, 2017. Available from: https://www.who.int/en/news-room/fact-sheets/detail/tuberculosis.
    [51] J. J. Tewa, S. Bowong, B. Mewoli, Mathematical analysis of two-patch model for the dynamical transmission of tuberculosis, Appl. Math. Model., 36 (2012), 2466–2485.
    [52] M. Toufik, A. Atangana, New numerical approximation of fractional derivative with non-local and non-singular kernel: application to chaotic models, The European Physical Journal Plus, 132 (2017), 444.
    [53] L. Xu, H. Hub, F. Qinc, Ultimate boundedness of impulsive fractional differential equations, Appl. Math. Lett., 62 (2016), 110–117.
    [54] L. Xu, J. Li, S. S. Ge, Impuls ivestabilization of fractional differential systems, ISA T., 70 (2017), 125–131.
    [55] L. Xu, X. Chu, H. Hu, Exponential ultimate boundedness of non-autonomous fractional differential systems with time delay and impulses, Appl. Math. Lett., 99 (2020), 106000.
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