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

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  • 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.

    Mathematics Subject Classification: Primary: 34A08, 34D23; Secondary: 34Kxx, 34A08.

    Citation:

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  • Figure 1.  Number of TB infective cases in Khyber Pakhtunkhwa, Pakistan

    Figure 2.  Data fitting (red solid curve) of the number of cases using model (6), for TB Khyber Pakhtunkhwa, Pakistan, since 2002-2017

    Figure 3.  Simulations of the model (6), (a-e) and total TB infected individuals via A-B derivative (f), for σ = 1

    Figure 4.  Simulations of TB model (6), (a-e) and total TB infected individuals via A-B derivative (f), for σ = 0.95

    Figure 5.  Simulations of TB model (6), (a-e) and total TB infected individuals via A-B derivative (f), for $ \sigma = 0.90 $

    Figure 6.  Simulations of TB model (6), (a-e) and total TB infected individuals via A-B derivative for (f), σ = 0.85

    Figure 7.  Simulations of TB model (6), (a-e) and total TB infected individuals via A-B derivative (f), for $ \sigma = 0.80 $

    Table 1.  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
     | Show Table
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  • [1] World Health Organization Media Centre. Available, https://www.who.int/en/news-room/fact-sheets/detail/tuberculosis. Accessed 2018.
    [2] A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model, Therm. Sci., 20 (2016), 763-769. 
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