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July  2021, 14(7): 2101-2117. doi: 10.3934/dcdss.2021057

An analysis of tuberculosis model with exponential decay law operator

1. 

Department of Mathematics Education, Akenten Appiah Menka University of Skills Training and Entrepreneurial Development, Kumasi, Ghana

2. 

Department of Mathematics, Faculty of Science and Technology, Universitas Airlangga, Surabaya 60115, Indonesia

* Corresponding author: fatmawati@fst.unair.ac.id

Received  March 2020 Revised  May 2020 Published  May 2021

Fund Project: This research is supported by Universitas Airlangga with Riset Kolaborasi Mitra LN 2020

In this paper, we explore the dynamics of tuberculosis (TB) epidemic model that includes the recruitment rate in both susceptible and infected population. Stability and sensitivity analysis of the classical TB model is carried out. Caputo-Fabrizio (CF) operator is then used to explain the dynamics of the TB model. The concept of fixed point theory is employed to obtain the existence and uniqueness of the solution of the TB model in the light of CF operator. Numerical simulations based on Homotopy Analysis Transform Method (HATM) and padé approximations are performed to obtain qualitative information on the model. Numerical solutions depict that the order of the fractional derivative has great dynamics of the TB model.

Citation: Ebenezer Bonyah, Fatmawati. An analysis of tuberculosis model with exponential decay law operator. Discrete & Continuous Dynamical Systems - S, 2021, 14 (7) : 2101-2117. doi: 10.3934/dcdss.2021057
References:
[1]

A. O. Adebiyi, Mathematical Modeling of the Population Dynamics of Tuberculosis, An Unpublished Thesis, University of Cape Town, South Africa, 2016. Google Scholar

[2]

A. Athithan and M. Ghosh, Optimal control of tuberculosis with case detection and treatment, World Journal of Modelling and Simulation, 11 (2015), 111-122.   Google Scholar

[3]

I. A. Baba and B. Ghanbari, Existence and uniqueness of solution of a fractional order tuberculosis model, Eur. Phys. J. Plus., 134 (2019), 489. doi: 10.1140/epjp/i2019-13009-1.  Google Scholar

[4]

C. P. Bhunu, Mathematical analysis of a three-strain tuberculosis transmission model, Appl. Math. Model., 35 (2011), 4647-4660.  doi: 10.1016/j.apm.2011.03.037.  Google Scholar

[5]

E. Bonyah, A. Atangana and M. Chand, Analysis of 3D IS-LM macroeconomic system model within the scope of fractional calculus, Chao. Solit. Frac. X, 2 (2019), 100007. doi: 10.1016/j.csfx.2019.100007.  Google Scholar

[6]

S. Bushnaq, S. A. Khan, K. Shah and G. Zaman, Mathematical analysis of HIV/AIDS infection model with Caputo-Fabrizio fractional derivative, Cogent Math. Stat., 5 (2018), 1432521. doi: 10.1080/23311835.2018.1432521.  Google Scholar

[7]

M. Caputo and M. Fabrizio, A new definition of fractional derivative with- out singular kernel, Progr Fract Differ Appl., 1 (2015), 73-85.   Google Scholar

[8]

M. Caputo and M. Fabrizio, Applications of new time and spatial fractional derivatives with exponential kernels, Progr. Fract. Differ. Appl., 2 (2016), 1-11.  doi: 10.18576/pfda/020101.  Google Scholar

[9]

C. Castillo-Chavez, S. Blower, P. van den Driessche, D. Kirschner and A.-A. Yakubu, Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4613-0065-6.  Google Scholar

[10]

Fatmawati, U. D. Purwati, F. Riyudha and H. Tasman, Optimal control of a discrete age-structured model for tuberculosis transmission, Heliyon, 6 (2020), e03030. doi: 10.1016/j.heliyon.2019.e03030.  Google Scholar

[11]

M. A. KhanS. Ullah and M. 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

[12]

A. Khan, T. Abdeljawad, J. F. Gómez-Aguilar and H. Khan, Dynamical study of fractional order mutualism parasitism food web module, Chao. Solit. Frac., 134 (2020), 109685. doi: 10.1016/j.chaos.2020.109685.  Google Scholar

[13]

H. Khan, J. F. Gómez-Aguilar, A. Alkhazzan and A. Khan, A fractional order HIV-TB coinfection model with nonsingular Mittag-Leffler Law, Math Method Appl Sci., 43 (2020), 3786–3806. doi: 10.1002/mma.6155.  Google Scholar

[14]

A. KhanH. KhanJ. F. Gómez-Aguilar and T. Abdeljawad, Existence and Hyers-Ulam stability for a nonlinear singular fractional differential equations with Mittag-Leffler kernel, Chao. Solit. Frac., 127 (2019), 422-427.  doi: 10.1016/j.chaos.2019.07.026.  Google Scholar

[15]

A. KhanJ. F. Gómez-AguilarT. S. Khan and H. Khan, Stability analysis and numerical solutions of fractional order HIV/AIDS model, Chao. Solit. Frac., 122 (2019), 119-128.  doi: 10.1016/j.chaos.2019.03.022.  Google Scholar

[16]

S. KimA. A. de los ReyesV 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.  Google Scholar

[17]

D. Kumar, J. Singh, M. Al Qurashi and D. Baleanu, A new fractional SIRS-SI malaria disease model with application of vaccines, antimalarial drugs, and spraying, Adv. Differ. Equ., 2019 (2019), 278. doi: 10.1186/s13662-019-2199-9.  Google Scholar

[18]

J. Losada and J. J. Nieto, Properties of a new fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 2 (2015), 87-92.   Google Scholar

[19]

H. Yépez-Martínez and J. F. Gómez-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.  Google Scholar

[20]

R. Naresh and A. Tripath, Modelling and analysis of HIV–TB coinfection in a variable size population, Math. Model. Anal., 10 (2005), 275-286.  doi: 10.3846/13926292.2005.9637287.  Google Scholar

[21]

D. Okuonghae and S. E. Omosigho, Analysis of a mathematical model for tuberculosis: What could be done to increase case detection, J. Theor. Biol., 269 (2011), 31-45.  doi: 10.1016/j.jtbi.2010.09.044.  Google Scholar

[22]

S. Qureshi, E. Bonyah and A. A. Shaikh, Classical and contemporary fractional operators for modeling diarrhea transmission dynamics under real statistical data, Physica A., 535 (2019), 122496. doi: 10.1016/j.physa.2019.122496.  Google Scholar

[23]

J. Singh, D. Kumar, M. Al Qurashi and D. Baleanu, A new fractional model for giving up smoking dynamics, Adv. Differ. Equ., 2017 (2017), 88. doi: 10.1186/s13662-017-1139-9.  Google Scholar

[24]

N. H. Sweilam and S. M. Al-Mekhlafi, Numerical study for multi-strain tuberculosis (TB) model of variable-order fractional derivatives, J. Adv. Res., 7 (2016), 271-283.  doi: 10.1016/j.jare.2015.06.004.  Google Scholar

[25]

S. Ullah, M. A. Khan, M. Farooq, Z. Hammouch and D. Baleanu, A fractional model for the dynamics of Tuberculosis infection using Caputo-Fabrizio derivative, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 975–993. doi: 10.3934/dcdss.2020057.  Google Scholar

[26]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmition, Math. Biosci., 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[27]

R. S. Wallis, Mathematical models of tuberculosis reactivation and relapse, Frontiers in Microbiology, 17 (2016), 669. doi: 10.3389/fmicb.2016.00669.  Google Scholar

[28]

World Health Organization, Global tuberculosis report, 2017. Available from: https://www.who.int/tb/publications/global_report/gtbr2017_main_text.pdf. Accessed on Feb 24, 2018. Google Scholar

[29]

Y YangJ. WuJ. Li and X. Xu, Tuberculosis with relapse: A model, Math. Popul. Stud., 24 (2017), 3-20.  doi: 10.1080/08898480.2014.998550.  Google Scholar

[30]

A. Yusuf, S. Qureshi, M. Inc, A. I. Aliyu, D. Baleanu, and A. A. Shaikh, Two strain epidemic model involving fractional derivative with Mittag- Leffer kernel, Chaos, 28 (2018), 123121. doi: 10.1063/1.5074084.  Google Scholar

show all references

References:
[1]

A. O. Adebiyi, Mathematical Modeling of the Population Dynamics of Tuberculosis, An Unpublished Thesis, University of Cape Town, South Africa, 2016. Google Scholar

[2]

A. Athithan and M. Ghosh, Optimal control of tuberculosis with case detection and treatment, World Journal of Modelling and Simulation, 11 (2015), 111-122.   Google Scholar

[3]

I. A. Baba and B. Ghanbari, Existence and uniqueness of solution of a fractional order tuberculosis model, Eur. Phys. J. Plus., 134 (2019), 489. doi: 10.1140/epjp/i2019-13009-1.  Google Scholar

[4]

C. P. Bhunu, Mathematical analysis of a three-strain tuberculosis transmission model, Appl. Math. Model., 35 (2011), 4647-4660.  doi: 10.1016/j.apm.2011.03.037.  Google Scholar

[5]

E. Bonyah, A. Atangana and M. Chand, Analysis of 3D IS-LM macroeconomic system model within the scope of fractional calculus, Chao. Solit. Frac. X, 2 (2019), 100007. doi: 10.1016/j.csfx.2019.100007.  Google Scholar

[6]

S. Bushnaq, S. A. Khan, K. Shah and G. Zaman, Mathematical analysis of HIV/AIDS infection model with Caputo-Fabrizio fractional derivative, Cogent Math. Stat., 5 (2018), 1432521. doi: 10.1080/23311835.2018.1432521.  Google Scholar

[7]

M. Caputo and M. Fabrizio, A new definition of fractional derivative with- out singular kernel, Progr Fract Differ Appl., 1 (2015), 73-85.   Google Scholar

[8]

M. Caputo and M. Fabrizio, Applications of new time and spatial fractional derivatives with exponential kernels, Progr. Fract. Differ. Appl., 2 (2016), 1-11.  doi: 10.18576/pfda/020101.  Google Scholar

[9]

C. Castillo-Chavez, S. Blower, P. van den Driessche, D. Kirschner and A.-A. Yakubu, Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4613-0065-6.  Google Scholar

[10]

Fatmawati, U. D. Purwati, F. Riyudha and H. Tasman, Optimal control of a discrete age-structured model for tuberculosis transmission, Heliyon, 6 (2020), e03030. doi: 10.1016/j.heliyon.2019.e03030.  Google Scholar

[11]

M. A. KhanS. Ullah and M. 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

[12]

A. Khan, T. Abdeljawad, J. F. Gómez-Aguilar and H. Khan, Dynamical study of fractional order mutualism parasitism food web module, Chao. Solit. Frac., 134 (2020), 109685. doi: 10.1016/j.chaos.2020.109685.  Google Scholar

[13]

H. Khan, J. F. Gómez-Aguilar, A. Alkhazzan and A. Khan, A fractional order HIV-TB coinfection model with nonsingular Mittag-Leffler Law, Math Method Appl Sci., 43 (2020), 3786–3806. doi: 10.1002/mma.6155.  Google Scholar

[14]

A. KhanH. KhanJ. F. Gómez-Aguilar and T. Abdeljawad, Existence and Hyers-Ulam stability for a nonlinear singular fractional differential equations with Mittag-Leffler kernel, Chao. Solit. Frac., 127 (2019), 422-427.  doi: 10.1016/j.chaos.2019.07.026.  Google Scholar

[15]

A. KhanJ. F. Gómez-AguilarT. S. Khan and H. Khan, Stability analysis and numerical solutions of fractional order HIV/AIDS model, Chao. Solit. Frac., 122 (2019), 119-128.  doi: 10.1016/j.chaos.2019.03.022.  Google Scholar

[16]

S. KimA. A. de los ReyesV 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.  Google Scholar

[17]

D. Kumar, J. Singh, M. Al Qurashi and D. Baleanu, A new fractional SIRS-SI malaria disease model with application of vaccines, antimalarial drugs, and spraying, Adv. Differ. Equ., 2019 (2019), 278. doi: 10.1186/s13662-019-2199-9.  Google Scholar

[18]

J. Losada and J. J. Nieto, Properties of a new fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 2 (2015), 87-92.   Google Scholar

[19]

H. Yépez-Martínez and J. F. Gómez-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.  Google Scholar

[20]

R. Naresh and A. Tripath, Modelling and analysis of HIV–TB coinfection in a variable size population, Math. Model. Anal., 10 (2005), 275-286.  doi: 10.3846/13926292.2005.9637287.  Google Scholar

[21]

D. Okuonghae and S. E. Omosigho, Analysis of a mathematical model for tuberculosis: What could be done to increase case detection, J. Theor. Biol., 269 (2011), 31-45.  doi: 10.1016/j.jtbi.2010.09.044.  Google Scholar

[22]

S. Qureshi, E. Bonyah and A. A. Shaikh, Classical and contemporary fractional operators for modeling diarrhea transmission dynamics under real statistical data, Physica A., 535 (2019), 122496. doi: 10.1016/j.physa.2019.122496.  Google Scholar

[23]

J. Singh, D. Kumar, M. Al Qurashi and D. Baleanu, A new fractional model for giving up smoking dynamics, Adv. Differ. Equ., 2017 (2017), 88. doi: 10.1186/s13662-017-1139-9.  Google Scholar

[24]

N. H. Sweilam and S. M. Al-Mekhlafi, Numerical study for multi-strain tuberculosis (TB) model of variable-order fractional derivatives, J. Adv. Res., 7 (2016), 271-283.  doi: 10.1016/j.jare.2015.06.004.  Google Scholar

[25]

S. Ullah, M. A. Khan, M. Farooq, Z. Hammouch and D. Baleanu, A fractional model for the dynamics of Tuberculosis infection using Caputo-Fabrizio derivative, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 975–993. doi: 10.3934/dcdss.2020057.  Google Scholar

[26]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmition, Math. Biosci., 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[27]

R. S. Wallis, Mathematical models of tuberculosis reactivation and relapse, Frontiers in Microbiology, 17 (2016), 669. doi: 10.3389/fmicb.2016.00669.  Google Scholar

[28]

World Health Organization, Global tuberculosis report, 2017. Available from: https://www.who.int/tb/publications/global_report/gtbr2017_main_text.pdf. Accessed on Feb 24, 2018. Google Scholar

[29]

Y YangJ. WuJ. Li and X. Xu, Tuberculosis with relapse: A model, Math. Popul. Stud., 24 (2017), 3-20.  doi: 10.1080/08898480.2014.998550.  Google Scholar

[30]

A. Yusuf, S. Qureshi, M. Inc, A. I. Aliyu, D. Baleanu, and A. A. Shaikh, Two strain epidemic model involving fractional derivative with Mittag- Leffer kernel, Chaos, 28 (2018), 123121. doi: 10.1063/1.5074084.  Google Scholar

Figure 1.  Flow diagram of model for TB transmission
Figure 2.  Numerical simulation for TB model (16) with CF fractional using HATM at $ \sigma = 1, 0.95, 0.90, 0.85 $
Figure 3.  Numerical simulation for TB model (16) with CF fractional using padé approximations at $ \sigma = 1, 0.95, 0.90, 0.85 $
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