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, doi: 10.3934/dcdss.2020055
References:
[1]

World Health Organization Media Centre. Available, https://www.who.int/en/news-room/fact-sheets/detail/tuberculosis. Accessed 2018. Google Scholar

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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.   Google Scholar

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A. Atangana and I. Koca, Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order, App. Math. and Comp., 89 (2016), 447-454.  doi: 10.1016/j.chaos.2016.02.012.  Google Scholar

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A. Atangana, Non validity of index law in fractional calculus: A fractional differential operator with Markovian and non-Markovian properties, Physica A: Statistical Mechanics and its Applications, 505 (2018), 688-706.  doi: 10.1016/j.physa.2018.03.056.  Google Scholar

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A. Atangana and J. F. G Aguilar, Decolonisation of fractional calculus rules: Breaking commutativity and associativity to capture more natural phenomena, The Eur. Phy. Jour. Plus, 133 (2018), 166.  doi: 10.1140/epjp/i2018-12021-3.  Google Scholar

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O. J. J. Algahtani, Comparing the Atangana–Baleanu and Caputo–Fabrizio derivative with fractional order: Allen Cahn model, Fractals, 89 (2016), 552-559.  doi: 10.1016/j.chaos.2016.03.026.  Google Scholar

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B. S. T. Alkahtani, Chua's circuit model with Atangana-Baleanu derivative with fractional order, Fractals, 89 (2016), 547-551.   Google Scholar

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B. S. T. AlkahtaniA. Atangana and I. Koca, Novel analysis of the fractional Zika model using the Adams type predictor-corrector rule for non-singular and non-local fractional operators, Jour. Nonl. Sci. Appl., 10 (2017), 3191-3200.  doi: 10.22436/jnsa.010.06.32.  Google Scholar

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M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl, 2 (2015), 1-13.   Google Scholar

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C. C. Chavez and Z. Feng, To treat or not to treat: the case of tuberculosis, Jour. Math. Bio., 35 (1997), 629-656.  doi: 10.1007/s002850050069.  Google Scholar

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P. V. D. Driessche and J. Watmough, Reproduction number and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Bios., 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[12] Z. Feng and C. C. Chavez, Mathematical Models for the Disease Dynamics of Tubeculosis, Gordon and Breach Science Publishers, London, 1998.   Google Scholar
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M. A. KhanS. Ullah and M. F. Farooq, A new fractional model for tuberculosis with relapse via Atangana-Baleanu derivative, Chao. Solit. Frac., 116 (2018), 227-238.  doi: 10.1016/j.chaos.2018.09.039.  Google Scholar

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S. KimA. Aurelio and E. Jung, Mathematical model and intervention strategies for mitigating tuberculosis in the Philippines, J. Theoret. Biol., 443 (2018), 100-112.  doi: 10.1016/j.jtbi.2018.01.026.  Google Scholar

[15]

I. Koca, Modelling the spread of Ebola virus with Atangana-Baleanu fractional operators, The Eur. Phy. Jour. Plus, 133 (2018), 100.  doi: 10.1140/epjp/i2018-11949-4.  Google Scholar

[16]

J. Liu and T. Zhang, Global stability for a tuberculosis model, Math. Comp. Modelling, 54 (2011), 836-845.  doi: 10.1016/j.mcm.2011.03.033.  Google Scholar

[17]

L. LiuX. Q. Zhao and Y. Zhou, A tuberculosis model with seasonality, Chao. Soliton. Fract., 72 (2010), 931-952.  doi: 10.1007/s11538-009-9477-8.  Google Scholar

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S. C RevelleR. W. Lynn and F. Feldmann, Mathematical models for the economic allocation of tuberculosis control activities in developing nations, American Review of Respiratory Disease, 96 (1967), 893-909.   Google Scholar

[19]

S. G. Samko, A. A. Kilbas, I. O. Marichev and others, Fractional Integrals and Derivatives, Gordon and Breach Science Publishers, Yverdon, 1993.  Google Scholar

[20] J. Singh JD. KumarM. A. Qurashi and D. Baleanu, Fractional Differential Equations: an Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Academic press, 1999.   Google Scholar
[21]

M. Toufik and A. Atangana, New numerical approximation of fractional derivative with non-local and non-singular kernel: Application to chaotic models, Chaos Solitons Fract., 132 (2017), 1-16.   Google Scholar

[22]

M. J. TrauerT. J. McBryde and S. E. Denholmand, Construction of a mathematical model for tuberculosis transmission in highly endemic regions of the Asia-Pacific, J. of Theo. Bio., 385 (2014), 74-84.   Google Scholar

[23]

S. UllahM. A. Khan and M. F. Farooq, A new fractional model for the dynamics of Hepatitis B virus using Caputo–Fabrizio derivative, The Eur. Phy. Jour. Plus, 133 (2018), 237.  doi: 10.1140/epjp/i2018-12072-4.  Google Scholar

[24]

H. WaalerA. Geser and S. Andersen, The use of mathematical models in the study of the epidemiology of tuberculosis, American J. of Public Health and the Nations Health, 52 (1962), 1002-1013.   Google Scholar

[25]

S. R. Wallis, Mathematical models of tuberculosis reactivation and relapse, Front. in Microbiol., 7 (2016), 1-7.  doi: 10.3389/fmicb.2016.00669.  Google Scholar

[26]

Y. YangJ. LiZ. Ma and L. Liu, Global stability of two models with incomplete treatment for tuberculosis, Fractals, 43 (2010), 79-85.  doi: 10.1016/j.chaos.2010.09.002.  Google Scholar

[27]

J. ZhangY. Li and X. Zhang, Mathematical modeling of tuberculosis data of China, J. of Theor. Bio., 365 (2015), 159-163.  doi: 10.1016/j.jtbi.2014.10.019.  Google Scholar

[28]

National TB Control Program Pakistan (NTP)., http://www.ntp.gov.pk/webdatabase.php. Google Scholar

[29]

Pakistan Bureau of Statistics. Pakistans 6th census: Population of major cities 662 census, http://www.pbscensus.gov.pk/. Google Scholar

show all references

References:
[1]

World Health Organization Media Centre. Available, https://www.who.int/en/news-room/fact-sheets/detail/tuberculosis. Accessed 2018. Google Scholar

[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.   Google Scholar

[3]

A. Atangana and I. Koca, Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order, App. Math. and Comp., 89 (2016), 447-454.  doi: 10.1016/j.chaos.2016.02.012.  Google Scholar

[4]

A. Atangana, Non validity of index law in fractional calculus: A fractional differential operator with Markovian and non-Markovian properties, Physica A: Statistical Mechanics and its Applications, 505 (2018), 688-706.  doi: 10.1016/j.physa.2018.03.056.  Google Scholar

[5]

A. Atangana and J. F. G Aguilar, Decolonisation of fractional calculus rules: Breaking commutativity and associativity to capture more natural phenomena, The Eur. Phy. Jour. Plus, 133 (2018), 166.  doi: 10.1140/epjp/i2018-12021-3.  Google Scholar

[6]

O. J. J. Algahtani, Comparing the Atangana–Baleanu and Caputo–Fabrizio derivative with fractional order: Allen Cahn model, Fractals, 89 (2016), 552-559.  doi: 10.1016/j.chaos.2016.03.026.  Google Scholar

[7]

B. S. T. Alkahtani, Chua's circuit model with Atangana-Baleanu derivative with fractional order, Fractals, 89 (2016), 547-551.   Google Scholar

[8]

B. S. T. AlkahtaniA. Atangana and I. Koca, Novel analysis of the fractional Zika model using the Adams type predictor-corrector rule for non-singular and non-local fractional operators, Jour. Nonl. Sci. Appl., 10 (2017), 3191-3200.  doi: 10.22436/jnsa.010.06.32.  Google Scholar

[9]

M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl, 2 (2015), 1-13.   Google Scholar

[10]

C. C. Chavez and Z. Feng, To treat or not to treat: the case of tuberculosis, Jour. Math. Bio., 35 (1997), 629-656.  doi: 10.1007/s002850050069.  Google Scholar

[11]

P. V. D. Driessche and J. Watmough, Reproduction number and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Bios., 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[12] Z. Feng and C. C. Chavez, Mathematical Models for the Disease Dynamics of Tubeculosis, Gordon and Breach Science Publishers, London, 1998.   Google Scholar
[13]

M. A. KhanS. Ullah and M. F. Farooq, A new fractional model for tuberculosis with relapse via Atangana-Baleanu derivative, Chao. Solit. Frac., 116 (2018), 227-238.  doi: 10.1016/j.chaos.2018.09.039.  Google Scholar

[14]

S. KimA. Aurelio and E. Jung, Mathematical model and intervention strategies for mitigating tuberculosis in the Philippines, J. Theoret. Biol., 443 (2018), 100-112.  doi: 10.1016/j.jtbi.2018.01.026.  Google Scholar

[15]

I. Koca, Modelling the spread of Ebola virus with Atangana-Baleanu fractional operators, The Eur. Phy. Jour. Plus, 133 (2018), 100.  doi: 10.1140/epjp/i2018-11949-4.  Google Scholar

[16]

J. Liu and T. Zhang, Global stability for a tuberculosis model, Math. Comp. Modelling, 54 (2011), 836-845.  doi: 10.1016/j.mcm.2011.03.033.  Google Scholar

[17]

L. LiuX. Q. Zhao and Y. Zhou, A tuberculosis model with seasonality, Chao. Soliton. Fract., 72 (2010), 931-952.  doi: 10.1007/s11538-009-9477-8.  Google Scholar

[18]

S. C RevelleR. W. Lynn and F. Feldmann, Mathematical models for the economic allocation of tuberculosis control activities in developing nations, American Review of Respiratory Disease, 96 (1967), 893-909.   Google Scholar

[19]

S. G. Samko, A. A. Kilbas, I. O. Marichev and others, Fractional Integrals and Derivatives, Gordon and Breach Science Publishers, Yverdon, 1993.  Google Scholar

[20] J. Singh JD. KumarM. A. Qurashi and D. Baleanu, Fractional Differential Equations: an Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Academic press, 1999.   Google Scholar
[21]

M. Toufik and A. Atangana, New numerical approximation of fractional derivative with non-local and non-singular kernel: Application to chaotic models, Chaos Solitons Fract., 132 (2017), 1-16.   Google Scholar

[22]

M. J. TrauerT. J. McBryde and S. E. Denholmand, Construction of a mathematical model for tuberculosis transmission in highly endemic regions of the Asia-Pacific, J. of Theo. Bio., 385 (2014), 74-84.   Google Scholar

[23]

S. UllahM. A. Khan and M. F. Farooq, A new fractional model for the dynamics of Hepatitis B virus using Caputo–Fabrizio derivative, The Eur. Phy. Jour. Plus, 133 (2018), 237.  doi: 10.1140/epjp/i2018-12072-4.  Google Scholar

[24]

H. WaalerA. Geser and S. Andersen, The use of mathematical models in the study of the epidemiology of tuberculosis, American J. of Public Health and the Nations Health, 52 (1962), 1002-1013.   Google Scholar

[25]

S. R. Wallis, Mathematical models of tuberculosis reactivation and relapse, Front. in Microbiol., 7 (2016), 1-7.  doi: 10.3389/fmicb.2016.00669.  Google Scholar

[26]

Y. YangJ. LiZ. Ma and L. Liu, Global stability of two models with incomplete treatment for tuberculosis, Fractals, 43 (2010), 79-85.  doi: 10.1016/j.chaos.2010.09.002.  Google Scholar

[27]

J. ZhangY. Li and X. Zhang, Mathematical modeling of tuberculosis data of China, J. of Theor. Bio., 365 (2015), 159-163.  doi: 10.1016/j.jtbi.2014.10.019.  Google Scholar

[28]

National TB Control Program Pakistan (NTP)., http://www.ntp.gov.pk/webdatabase.php. Google Scholar

[29]

Pakistan Bureau of Statistics. Pakistans 6th census: Population of major cities 662 census, http://www.pbscensus.gov.pk/. Google Scholar

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
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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