## A new fractional model for tuberculosis with relapse via Atangana-Baleanu derivative.(English)Zbl 1442.92150

Summary: In this work, a new fractional order epidemic model for the tuberculosis (TB) disease with relapse using Atangana-Baleanu derivative is formulated. The basic reproduction number of the model is investigated using next generation technique. The fixed point theorem is applied to show the existence and uniqueness of solution for the model. A newly proposed numerical scheme in literature is implemented for the iterative solution of the proposed fractional model. The total new and relapse notified TB cases in Khyber Pakhtunkhwa Pakistan from 2002 to 2017 are used to parameterized the model parameters and provided a good fit to the real data. Finally, numerical results are obtained for different values of the fractional order $$\tau$$ and the model parameters, in order to validate the importance of the arbitrary order derivative. It is noticed that the non-integer order derivative provides more realistic and deeper information about the complexity of the dynamics of TB model with relapse.

### MSC:

 92D30 Epidemiology 34A08 Fractional ordinary differential equations 34C60 Qualitative investigation and simulation of ordinary differential equation models 34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
Full Text:

### References:

 [2] Kim, S.; Aurelio, A.; Jung, E., Mathematical model and intervention strategies for mitigating tuberculosis in the philippines, J Theor Biol, 443, 100-112 (2018) · Zbl 1397.92643 [4] Yang, Y.; Li, J.; Zhou, Y., Global stability of two tuberculosis models with treatment and self-cure, Rocky Mountain J Math, 1367-1386 (2012) · Zbl 1251.92024 [5] Van Den Driessche, P.; Wang, L.; Zou, X., Modeling diseases with latency and relapse, Math Biosci Eng, 4, 2, 205 (2007) · Zbl 1123.92018 [6] Wallis, R. S., Mathematical models of tuberculosis reactivation and relapse, Front Microbiol, 7, 669 (2016) [7] Yang, Y.; Wu, J.; Li, J.; Xu, X., Tuberculosis with relapse: a model, Math Popul Stud, 24, 1, 3-20 (2017) [8] Podlubny, I., Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications (1999), Elsevier · Zbl 0924.34008 [9] Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional integrals and derivatives, Theory and Applications, Gordon and Breach, Yverdon, 1993, 44 (1993) [10] Caputo, M.; Fabrizio, M., A new definition of fractional derivative without singular kernel, Prog Fract Differ Appl, 1, 2, 1-13 (2015) [11] Goufo, E. F.D., A biomathematical view on the fractional dynamics of cellulose degradation, Fract Calc Appl Anal, 18, 3, 554-564 (2015) · Zbl 1316.26004 [12] Atangana, A.; Goufo, E. F.D., Computational analysis of the model describing HIV infection of cd4, Biomed Res Int, 2014, 1-7 (2014) [13] Goufo, E. F.D.; Maritz, R.; Munganga, J., Some properties of the Kermack-Mckendrick epidemic model with fractional derivative and nonlinear incidence, Adv Differ Equ, 2014, 1, 278 (2014) · Zbl 1344.92160 [14] Goufo, D.; Franc, E.; Oukouomi Noutchie, S. C.; Mugisha, S., A fractional SEIR epidemic model for spatial and temporal spread of measles in metapopulations, Abstr Appl Anal, 2014 (2014) [15] Sweilam, N. H.; AL-Mekhlafi, S. M., Numerical study for multi-strain tuberculosis (TB) model of variable-order fractional derivatives, J Adv Res, 7, 2, 271-283 (2016) [16] Baleanu, D.; Magin, R. L.; Bhalekar, S.; Daftardar-Gejji, V., Chaos in the fractional order nonlinear Bloch equation with delay, Commun Nonlinear Sci Numer Simul, 25, 1-3, 41-49 (2015) [17] Ullah, S.; Khan, M. A.; Farooq, M., A new fractional model for the dynamics of the hepatitis b virus using the Caputo-Fabrizio derivative, Eur Phys J Plus, 133, 6, 237 (2018) [18] Abdulhameed, M.; Vieru, D.; Roslan, R., Magnetohydrodynamic electroosmotic flow of maxwell fluids with Caputo-Fabrizio derivatives through circular tubes, Comput Math Appl, 74, 10, 2503-2519 (2017) · Zbl 1394.76143 [19] Firoozjaee, M.; Jafari, H.; Lia, A.; Baleanu, D., Numerical approach of Fokker-Planck equation with Caputo-Fabrizio fractional derivative using Ritz approximation, J Comput Appl Math, 339, 367-373 (2018) · Zbl 1393.65029 [20] Atangana, A.; Baleanu, D., New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model, Therm Sci, 20, 2, 763-769 (2016) [21] Baleanu, D.; Jajarmi, A.; Bonyah, E.; Hajipour, M., New aspects of poor nutrition in the life cycle within the fractional calculus, Adv Differ Equ, 2018, 1, 230 (2018) [22] Atangana, A.; Koca, I., Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order, Chaos Solitons Fractals, 89, 447-454 (2016) · Zbl 1360.34150 [23] Alkahtani, B. S.T., Chua’s circuit model with Atangana-Baleanu derivative with fractional order, Chaos Solitons Fractals, 89, 547-551 (2016) · Zbl 1360.34160 [24] Atangana, A., Non validity of index law in fractional calculus: a fractional differential operator with Markovian and non-Markovian properties, Phys A, 505, 688-706 (2018) [25] Atangana, A.; Gómez-Aguilar, J., Decolonisation of fractional calculus rules: breaking commutativity and associativity to capture more natural phenomena, Eur Phys J Plus, 133, 1-22 (2018) [26] Alkahtani, B. S.T.; Atangana, A.; Koca, I., Novel analysis of the fractional Zika model using the adams type predictor-corrector rule for non-singular and non-local fractional operators, J Nonlinear Sci Appl, 10, 6, 3191-3200 (2017) [27] Ullah, S.; Khan, M. A.; Farooq, M., Modeling and analysis of the fractional HBV model with Atangana-Baleanu derivative, Eur Phys J Plus, 133, 8, 313 (2018) [29] Van den Driessche, P.; Watmough, J., Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math Biosci, 180, 1-2, 29-48 (2002) · Zbl 1015.92036 [30] Toufik, M.; Atangana, A., New numerical approximation of fractional derivative with non-local and non-singular kernel: application to chaotic models, Eur Phys J Plus, 132, 10, 444 (2017)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.