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

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  • 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.

    Mathematics Subject Classification: Primary: 26A33, 47H10.


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  • Figure 1.  The incidence data of TB from Khyber Pakhtunkhwa, Pakistan

    Figure 2.  The incidence data of TB from Khyber Pakhtunkhwa, Pakistan and the model fit for $ \tau = 1 $

    Figure 3.  Long term behavior of the CF model with realistic data when $ \tau = 1 $

    Figure 4.  Simulation of $ S $ with $ \tau $

    Figure 5.  Simulation of $ L $ with $ \tau $

    Figure 6.  Simulation of $I$ with $\tau$.

    Figure 7.  Simulation of $T$ with $\tau$.

    Figure 8.  Simulation of $ R $ with $ \tau $

    Figure 9.  Simulation of cumulative TB infected people with $ \tau $

    Figure 10.  The graphical result of the total infected people for several values of the parameter $ \gamma $ (treatment rate) and $ \tau $ (fractional parameter)

    Figure 11.  The graphical result of the total infective with TB individuals for various values of the parameter $ \eta $ (treatment failure rate) and $ \tau $ (fractional parameter)

    Table 1.  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 [44]
    $ \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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