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
[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. |