# American Institute of Mathematical Sciences

March  2020, 13(3): 937-956. doi: 10.3934/dcdss.2020055

## A fractional model for the dynamics of tuberculosis (TB) using Atangana-Baleanu derivative

 1 Department of Mathematics, University of Peshawar, Khyber Pakhtunkhwa, 25000, Pakistan 2 Department of Mathematics, City University of Science and Information Technology, Peshawar, Khyber Pakhtunkhwa, 25000, Pakistan 3 Department of Mathematics, University of Peshawar, Peshawar, Khyber Pakhtunkhwa, 25000, Pakistan 4 Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia

* Corresponding author: altafdir@gmail.com

Received  May 2018 Revised  July 2018 Published  March 2019

In the present paper, we explore the dynamics of fractional tuberculosis model with Atangana-Baleanu (A-B) derivative. The number of confirmed notified cases reported by national tuberculosis control program (NTP) Khyber Pakhtunkhwa, Pakistan, since 2002 to 2017 are used for our analysis and estimation of the model parameters. Initially, the essential properties of the model are presented. We prove the existence of the solution through fixed-point theory. Then, we show the uniqueness of the solution. Modified Adams-Bashforth technique is used to obtain the numerical solution of the fractional model. We obtain numerical results with different values of the fractional order parameters to show the importance of the newly proposed derivative, which provides useful information about the TB dynamics and its control.

Citation: Saif Ullah, Muhammad Altaf Khan, Muhammad Farooq, Ebraheem O. Alzahrani. A fractional model for the dynamics of tuberculosis (TB) using Atangana-Baleanu derivative. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 937-956. doi: 10.3934/dcdss.2020055
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##### References:
Number of TB infective cases in Khyber Pakhtunkhwa, Pakistan
Data fitting (red solid curve) of the number of cases using model (6), for TB Khyber Pakhtunkhwa, Pakistan, since 2002-2017
Simulations of the model (6), (a-e) and total TB infected individuals via A-B derivative (f), for σ = 1
Simulations of TB model (6), (a-e) and total TB infected individuals via A-B derivative (f), for σ = 0.95
Simulations of TB model (6), (a-e) and total TB infected individuals via A-B derivative (f), for $\sigma = 0.90$
Simulations of TB model (6), (a-e) and total TB infected individuals via A-B derivative for (f), σ = 0.85
Simulations of TB model (6), (a-e) and total TB infected individuals via A-B derivative (f), for $\sigma = 0.80$
Estimated parameters for the TB model (6)
 Parameter Description Baseline value Reference $\Lambda$ Recruitment rate 450,862.20088626 Estimated $\beta$ Transmission rate 0.5433 Fitted $\alpha$ Moving rate from $T$ to $R$ 0.3968 Fitted $\gamma$ Progression rate from $I$ to $T$ 0.2873 Fitted $\mu$ Natural death rate 1/67.7 [29] $\sigma_1$ Disease induced death rate of infected individuals 0.2202 Fitted $\sigma_2$ Disease induced death rate in $T$ 0.0550 Fitted $\delta$ Rate at which treated individuals leave the $T$ 1.1996 Fitted $\eta$ Rate of treatment failure 0.1500 Fitted $\epsilon$ Rate of progression from $L$ to $I$ 0.2007 Fitted
 Parameter Description Baseline value Reference $\Lambda$ Recruitment rate 450,862.20088626 Estimated $\beta$ Transmission rate 0.5433 Fitted $\alpha$ Moving rate from $T$ to $R$ 0.3968 Fitted $\gamma$ Progression rate from $I$ to $T$ 0.2873 Fitted $\mu$ Natural death rate 1/67.7 [29] $\sigma_1$ Disease induced death rate of infected individuals 0.2202 Fitted $\sigma_2$ Disease induced death rate in $T$ 0.0550 Fitted $\delta$ Rate at which treated individuals leave the $T$ 1.1996 Fitted $\eta$ Rate of treatment failure 0.1500 Fitted $\epsilon$ Rate of progression from $L$ to $I$ 0.2007 Fitted
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