    ## A fractional model for the dynamics of tuberculosis infection using Caputo-Fabrizio derivative

 1 Department of Mathematics, University of Peshawar, Peshawar, Khyber Pakhtunkhwa, Pakistan 2 Department of Mathematics, City University of Science and Information Technology, Khyber Pakhtunkhwa, Pakistan 3 Departement de Mathematiques, FSTE Université Moulay Ismail, BP.509 Boutalamine 52000 Errachidia, Morocco 4 Department of Mathematics and Computer Science, Faculty of Arts and Sciences, Cankaya University Ankara, Turkey

* Corresponding author: hammouch.zakia@gmail.com

Received  September 2018 Revised  October 2018 Published  March 2019

In the present paper, we study the dynamics of tuberculosis model using fractional order derivative in Caputo-Fabrizio sense. The number of confirmed notified cases reported by national TB program Khyber Pakhtunkhwa, Pakistan, from the year 2002 to 2017 are used for our analysis and estimation of the model biological parameters. The threshold quantity $\mathcal{R}_0$ and equilibria of the model are determined. We prove the existence of the solution via fixed-point theory and further examine the uniqueness of the model variables. An iterative solution of the model is computed using fractional Adams-Bashforth technique. Finally, the numerical results are presented by using the estimated values of model parameters to justify the significance of the arbitrary fractional order derivative. The graphical results show that the fractional model of TB in Caputo-Fabrizio sense gives useful information about the complexity of the model and one can get reliable information about the model at any integer or non-integer case.

Citation: Saif Ullah, Muhammad Altaf Khan, Muhammad Farooq, Zakia Hammouch, Dumitru Baleanu. A fractional model for the dynamics of tuberculosis infection using Caputo-Fabrizio derivative. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020057
  , World Health Organization Media Centre. Available: , Available from: http://apps.who.int/iris/bitstream/10665/136607/1/ccsbrief_pak_en.pdf.Accessed2016. Google Scholar  T. Abdeljawad and D. Baleanu, On fractional derivatives with exponential kernel and their discrete versions, J. Report. Math. Phy., 80 (2017), 11-27.  doi: 10.1016/S0034-4877(17)30059-9.  Google Scholar  T. Abdeljawad, Fractional operators with exponential kernels and a Lyapunov type inequality, Phy. A: Stat. Mech. Appl., 313 (2017), 1-12.  doi: 10.1186/s13662-017-1285-0.  Google Scholar  T. Abdeljawad, Q. M. Al-Mdallal and M. A. Hajji, Arbitrary order fractional difference operators with discrete exponential kernels and applications, Dis. Dyn. Nat. Soci., 2017 (2017), Art. ID 4149320, 8 pp. doi: 10.1155/2017/4149320.  Google Scholar  T. Abdeljawad and D. Baleanu, Monotonicity results for fractional difference operators with discrete exponential kernels, Adv. Differ. Equ., 78 (2017), 1-9.  doi: 10.1186/s13662-017-1126-1.  Google Scholar  T. Abdeljawad and Q. M. Al-Mdallal, Discrete Mittag-Leffler kernel type fractional difference initial value problems and Gronwalls inequality, J. Comp. App. Math., 339 (2018), 218-230.  doi: 10.1016/j.cam.2017.10.021. Google Scholar  T. Abdeljawad and and F. Madjidi, Lyapunov type inequalities for fractional difference operators with discrete Mittag-Leffler kernels of order $2 show all references ##### References:   , World Health Organization Media Centre. Available: , Available from: http://apps.who.int/iris/bitstream/10665/136607/1/ccsbrief_pak_en.pdf.Accessed2016. Google Scholar  T. Abdeljawad and D. Baleanu, On fractional derivatives with exponential kernel and their discrete versions, J. Report. Math. Phy., 80 (2017), 11-27. doi: 10.1016/S0034-4877(17)30059-9.  Google Scholar  T. Abdeljawad, Fractional operators with exponential kernels and a Lyapunov type inequality, Phy. A: Stat. Mech. Appl., 313 (2017), 1-12. doi: 10.1186/s13662-017-1285-0.  Google Scholar  T. Abdeljawad, Q. M. Al-Mdallal and M. A. Hajji, Arbitrary order fractional difference operators with discrete exponential kernels and applications, Dis. Dyn. Nat. Soci., 2017 (2017), Art. ID 4149320, 8 pp. doi: 10.1155/2017/4149320.  Google Scholar  T. Abdeljawad and D. Baleanu, Monotonicity results for fractional difference operators with discrete exponential kernels, Adv. Differ. Equ., 78 (2017), 1-9. doi: 10.1186/s13662-017-1126-1.  Google Scholar  T. Abdeljawad and Q. M. Al-Mdallal, Discrete Mittag-Leffler kernel type fractional difference initial value problems and Gronwalls inequality, J. Comp. App. Math., 339 (2018), 218-230. doi: 10.1016/j.cam.2017.10.021. Google Scholar  T. Abdeljawad and and F. Madjidi, Lyapunov type inequalities for fractional difference operators with discrete Mittag-Leffler kernels of order$2 The incidence data of TB from Khyber Pakhtunkhwa, Pakistan and the model fit for $\tau = 1$ Long term behavior of the CF model with realistic data when $\tau = 1$ The graphical result of the total infected people for several values of the parameter $\gamma$ (treatment rate) and $\tau$ (fractional parameter) The graphical result of the total infective with TB individuals for various values of the parameter $\eta$ (treatment failure rate) and $\tau$ (fractional parameter)
Fitting of the model parameters and its estimations for The TB infected cases of Khyber Pakhtunkhwa, Pakistan
 Parameter Definition value Ref. $\Lambda$ Birth rate 450,862.20088626 Estimated $\beta$ Disease contact rate 0.5433 Fitted $\alpha$ Progression from $T$ class to $R$ 0.3968 Fitted $\gamma$ Transmission from $I$ class to $T$ 0.2873 Fitted $\mu$ Natural mortality rate 1/67.7  $\tau_1$ Disease related motility rate of infected individuals 0.2202 Fitted $\tau_2$ Disease related death rate in $T$ 0.0550 Fitted $\delta$ Leaving rate of the individuals from class $T$ 1.1996 Fitted $\eta$ Treatment failure rate 0.1500 Fitted $\epsilon$ Moving rate from $L$ class to $I$ 0.2007 Fitted
 Parameter Definition value Ref. $\Lambda$ Birth rate 450,862.20088626 Estimated $\beta$ Disease contact rate 0.5433 Fitted $\alpha$ Progression from $T$ class to $R$ 0.3968 Fitted $\gamma$ Transmission from $I$ class to $T$ 0.2873 Fitted $\mu$ Natural mortality rate 1/67.7  $\tau_1$ Disease related motility rate of infected individuals 0.2202 Fitted $\tau_2$ Disease related death rate in $T$ 0.0550 Fitted $\delta$ Leaving rate of the individuals from class $T$ 1.1996 Fitted $\eta$ Treatment failure rate 0.1500 Fitted $\epsilon$ Moving rate from $L$ class to $I$ 0.2007 Fitted
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