doi: 10.3934/dcdss.2020057

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
References:
[1]

, World Health Organization Media Centre. Available: , Available from: http://apps.who.int/iris/bitstream/10665/136607/1/ccsbrief_pak_en.pdf.Accessed2016.

[2]

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.

[3]

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.

[4]

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.

[5]

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.

[6]

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.

[7]

T. Abdeljawad and and F. Madjidi, Lyapunov type inequalities for fractional difference operators with discrete Mittag-Leffler kernels of order $2<a<5/2$, J. Spec. Top., 226 (2017), 3355-3368.

[8]

T. Abdeljawad and D. Baleanu, Monotonicity analysis of a nabla discrete fractional operator with discrete Mittag-Leffler kernel, Chao. Solit. Frac., 102 (2017), 106-110. doi: 10.1016/j.chaos.2017.04.006.

[9]

T. Abdeljawad and D. Baleanu, Integration by parts and its applications of a new nonlocal fractional derivative with Mittag-Leffler nonsingular kernel, J. Nonlinear Sci. Appl. 10 (2017), 1098–1107. doi: 10.22436/jnsa.010.03.20.

[10]

P. Agarwal, et al., Fractional differential equations for the generalized MittagLeffler function, Adv. Diff. Equa., 2018 (2018), 58.

[11]

J. F. G. Aguilar, Fundamental solutions to electrical circuits of non-integer order via fractional derivatives with and without singular kernels, Eur. Phys. J. Plus, 133 (2018), 1-20.

[12]

M. Q. Al-MdallalS. Ahmed and A. Omer, Fractional-order Legendre-collocation method for solving fractional initial value problems, Appl. Math. Comp., 321 (2018), 74-84. doi: 10.1016/j.amc.2017.10.012.

[13]

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. doi: 10.2298/TSCI160111018A.

[14]

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.

[15]

A. Atangana and K. M. Owolabi, New numerical approach for fractional differential equations, Math. Model. Nat. Phenom., 13 (218), Art. 3, 21 pp. doi: 10.2307/2152750.

[16]

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

[17]

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

[18]

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.

[19]

A. O. Egonmwan and D. Okuonghae, Analysis of a mathematical model for tuberculosis with diagnosis, J. Appl. Math. Comput., (2018), 1–34. doi: 10.1007/s12190-018-1172-1.

[20]

Z. Feng and C. C. Chavez, Mathematical Models for the Disease Dynamics of Tubeculosis, London: Gordon and Breach Science Publishers, 1998.

[21]

M. A Hajji and Q. Al-Mdallal, Numerical simulations of a delay model for immune system-tumor interaction, Sultan Qaboos Univ. Jour. Sci., 23 (2018), 19-31. doi: 10.24200/squjs.vol23iss1pp19-31.

[22]

Z. Hammouch and T. Mekkaoui, Chaos synchronization of a fractional nonautonomous system, Nonautonomous Dynamical Systems, 1 (2014), 61-71. doi: 10.2478/msds-2014-0001.

[23]

Z. Hammouch and T. Mekkaoui, Control of a new chaotic fractional-order system using Mittag–Leffler stability, Nonlinear Studies, 22 (2015), 565-577.

[24]

Z. Hammouch and T. Mekkaoui, Circuit design and simulation for the fractional-order chaotic behavior in a new dynamical system, Complex and Intelligent Systems, 4 (2018), 251-260. doi: 10.1007/s40747-018-0070-3.

[25]

F. JaradT. Abdeljawad and Z. Hammouch, On a class of ordinary differential equations in the frame of Atangana–Baleanu fractional derivative, Chaos, Solitons and Fractals, 117 (2018), 16-20. doi: 10.1016/j.chaos.2018.10.006.

[26]

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.

[27]

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

[28]

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.

[29]

J. Losada and J. J. Nieto, Properties of a New Fractional Derivative without Singular Kernel, Progr. Fract. Differ. Appl, 2 (2015), 87-92.

[30]

J. E. E. MartnezJ. F. G. AguilarC. C. RamnA. A. Melndez and P. P. Longoria, Synchronized bioluminescence behavior of a set of fireflies involving fractional operators of Liouville-Caputo type, Int. J. Biomath., 11 (2018), 1-24. doi: 10.1142/S1793524518500419.

[31]

J. E. E. Martnez, J. F. G. Aguilar, C. C. Ramn, A. A. Melndez and P. P. Longoria, A mathematical model of circadian rhythms synchronization using fractional differential equations system of coupled van der Pol oscillators, Int. J. Biomath., 11 (2018), 1850041, 25 pp. doi: 10.1142/S1793524518500146.

[32]

H. Y. Martnez and J. F. G. Aguilar, A new modified definition of Caputo Fabrizio fractional-order derivative and their applications to the Multi Step Homotopy Analysis Method (MHAM), J. Comp. Appl. Math., 346 (2019), 247-260. doi: 10.1016/j.cam.2018.07.023.

[33]

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.

[34]

K. M. Saad and J. F. G. Aguilar, Analysis of reaction-diffusion system via a new fractional derivative with non-singular kernel, Phy. A: Stat. Mech. Appl., 509 (2018), 703-716. doi: 10.1016/j.physa.2018.05.137.

[35]

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

[36] J. SinghJ. D. 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.
[37]

J. SinghD. KumarZ. Hammouch and A. Atangana, A fractional epidemiological model for computer viruses pertaining to a new fractional derivative, Applied Mathematics and Computation, 316 (2018), 504-515. doi: 10.1016/j.amc.2017.08.048.

[38]

S. UllahM. A. Khan and M. Farooq, A fractional model for the dynamics of TB virus, Chao. Solit. Fract., 116 (2018), 63-71. doi: 10.1016/j.chaos.2018.09.001.

[39]

S. Ullah, M. A. Khan and M. F. Farooq, Modeling and analysis of the fractional HBV model with Atangana-Baleanu derivative, The Eur. Phy. Jour. Plus, 133 (2018), 313. doi: 10.1140/epjp/i2018-12120-1.

[40]

H. WaalerA. Geser and S. Andersen, he 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.

[41]

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

[42]

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

[43]

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

[44]

Pakistan Bureau of Statistics. Pakistan's 6th census: Population of Major Cities 583 Census. 584, http://www.pbs.gov.pk/content/provisional-summary-results-6th-population-and-housing-census-2017-0, 2017.

show all references

References:
[1]

, World Health Organization Media Centre. Available: , Available from: http://apps.who.int/iris/bitstream/10665/136607/1/ccsbrief_pak_en.pdf.Accessed2016.

[2]

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.

[3]

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.

[4]

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.

[5]

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.

[6]

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.

[7]

T. Abdeljawad and and F. Madjidi, Lyapunov type inequalities for fractional difference operators with discrete Mittag-Leffler kernels of order $2<a<5/2$, J. Spec. Top., 226 (2017), 3355-3368.

[8]

T. Abdeljawad and D. Baleanu, Monotonicity analysis of a nabla discrete fractional operator with discrete Mittag-Leffler kernel, Chao. Solit. Frac., 102 (2017), 106-110. doi: 10.1016/j.chaos.2017.04.006.

[9]

T. Abdeljawad and D. Baleanu, Integration by parts and its applications of a new nonlocal fractional derivative with Mittag-Leffler nonsingular kernel, J. Nonlinear Sci. Appl. 10 (2017), 1098–1107. doi: 10.22436/jnsa.010.03.20.

[10]

P. Agarwal, et al., Fractional differential equations for the generalized MittagLeffler function, Adv. Diff. Equa., 2018 (2018), 58.

[11]

J. F. G. Aguilar, Fundamental solutions to electrical circuits of non-integer order via fractional derivatives with and without singular kernels, Eur. Phys. J. Plus, 133 (2018), 1-20.

[12]

M. Q. Al-MdallalS. Ahmed and A. Omer, Fractional-order Legendre-collocation method for solving fractional initial value problems, Appl. Math. Comp., 321 (2018), 74-84. doi: 10.1016/j.amc.2017.10.012.

[13]

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. doi: 10.2298/TSCI160111018A.

[14]

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.

[15]

A. Atangana and K. M. Owolabi, New numerical approach for fractional differential equations, Math. Model. Nat. Phenom., 13 (218), Art. 3, 21 pp. doi: 10.2307/2152750.

[16]

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

[17]

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

[18]

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.

[19]

A. O. Egonmwan and D. Okuonghae, Analysis of a mathematical model for tuberculosis with diagnosis, J. Appl. Math. Comput., (2018), 1–34. doi: 10.1007/s12190-018-1172-1.

[20]

Z. Feng and C. C. Chavez, Mathematical Models for the Disease Dynamics of Tubeculosis, London: Gordon and Breach Science Publishers, 1998.

[21]

M. A Hajji and Q. Al-Mdallal, Numerical simulations of a delay model for immune system-tumor interaction, Sultan Qaboos Univ. Jour. Sci., 23 (2018), 19-31. doi: 10.24200/squjs.vol23iss1pp19-31.

[22]

Z. Hammouch and T. Mekkaoui, Chaos synchronization of a fractional nonautonomous system, Nonautonomous Dynamical Systems, 1 (2014), 61-71. doi: 10.2478/msds-2014-0001.

[23]

Z. Hammouch and T. Mekkaoui, Control of a new chaotic fractional-order system using Mittag–Leffler stability, Nonlinear Studies, 22 (2015), 565-577.

[24]

Z. Hammouch and T. Mekkaoui, Circuit design and simulation for the fractional-order chaotic behavior in a new dynamical system, Complex and Intelligent Systems, 4 (2018), 251-260. doi: 10.1007/s40747-018-0070-3.

[25]

F. JaradT. Abdeljawad and Z. Hammouch, On a class of ordinary differential equations in the frame of Atangana–Baleanu fractional derivative, Chaos, Solitons and Fractals, 117 (2018), 16-20. doi: 10.1016/j.chaos.2018.10.006.

[26]

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.

[27]

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

[28]

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.

[29]

J. Losada and J. J. Nieto, Properties of a New Fractional Derivative without Singular Kernel, Progr. Fract. Differ. Appl, 2 (2015), 87-92.

[30]

J. E. E. MartnezJ. F. G. AguilarC. C. RamnA. A. Melndez and P. P. Longoria, Synchronized bioluminescence behavior of a set of fireflies involving fractional operators of Liouville-Caputo type, Int. J. Biomath., 11 (2018), 1-24. doi: 10.1142/S1793524518500419.

[31]

J. E. E. Martnez, J. F. G. Aguilar, C. C. Ramn, A. A. Melndez and P. P. Longoria, A mathematical model of circadian rhythms synchronization using fractional differential equations system of coupled van der Pol oscillators, Int. J. Biomath., 11 (2018), 1850041, 25 pp. doi: 10.1142/S1793524518500146.

[32]

H. Y. Martnez and J. F. G. Aguilar, A new modified definition of Caputo Fabrizio fractional-order derivative and their applications to the Multi Step Homotopy Analysis Method (MHAM), J. Comp. Appl. Math., 346 (2019), 247-260. doi: 10.1016/j.cam.2018.07.023.

[33]

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.

[34]

K. M. Saad and J. F. G. Aguilar, Analysis of reaction-diffusion system via a new fractional derivative with non-singular kernel, Phy. A: Stat. Mech. Appl., 509 (2018), 703-716. doi: 10.1016/j.physa.2018.05.137.

[35]

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

[36] J. SinghJ. D. 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.
[37]

J. SinghD. KumarZ. Hammouch and A. Atangana, A fractional epidemiological model for computer viruses pertaining to a new fractional derivative, Applied Mathematics and Computation, 316 (2018), 504-515. doi: 10.1016/j.amc.2017.08.048.

[38]

S. UllahM. A. Khan and M. Farooq, A fractional model for the dynamics of TB virus, Chao. Solit. Fract., 116 (2018), 63-71. doi: 10.1016/j.chaos.2018.09.001.

[39]

S. Ullah, M. A. Khan and M. F. Farooq, Modeling and analysis of the fractional HBV model with Atangana-Baleanu derivative, The Eur. Phy. Jour. Plus, 133 (2018), 313. doi: 10.1140/epjp/i2018-12120-1.

[40]

H. WaalerA. Geser and S. Andersen, he 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.

[41]

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

[42]

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

[43]

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

[44]

Pakistan Bureau of Statistics. Pakistan's 6th census: Population of Major Cities 583 Census. 584, http://www.pbs.gov.pk/content/provisional-summary-results-6th-population-and-housing-census-2017-0, 2017.

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
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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