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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  July 2021 Early access  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 and 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 [1] A. O. Adebiyi, Mathematical Modeling of the Population Dynamics of Tuberculosis, An Unpublished Thesis, University of Cape Town, South Africa, 2016. [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. [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. [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. [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. [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. [7] M. Caputo and M. Fabrizio, A new definition of fractional derivative with- out singular kernel, Progr Fract Differ Appl., 1 (2015), 73-85. [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. [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. [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. [11] M. A. Khan, S. 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. [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. [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. [14] A. Khan, H. Khan, J. 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. [15] A. Khan, J. F. Gómez-Aguilar, T. 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. [16] S. Kim, A. 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. [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. [18] J. Losada and J. J. Nieto, Properties of a new fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 2 (2015), 87-92. [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. [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. [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. [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. [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. [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. [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. [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. [27] R. S. Wallis, Mathematical models of tuberculosis reactivation and relapse, Frontiers in Microbiology, 17 (2016), 669. doi: 10.3389/fmicb.2016.00669. [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. [29] Y Yang, J. Wu, J. Li and X. Xu, Tuberculosis with relapse: A model, Math. Popul. Stud., 24 (2017), 3-20.  doi: 10.1080/08898480.2014.998550. [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.

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. [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. [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. [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. [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. [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. [7] M. Caputo and M. Fabrizio, A new definition of fractional derivative with- out singular kernel, Progr Fract Differ Appl., 1 (2015), 73-85. [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. [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. [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. [11] M. A. Khan, S. 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. [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. [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. [14] A. Khan, H. Khan, J. 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. [15] A. Khan, J. F. Gómez-Aguilar, T. 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. [16] S. Kim, A. 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. [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. [18] J. Losada and J. J. Nieto, Properties of a new fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 2 (2015), 87-92. [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. [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. [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. [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. [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. [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. [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. [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. [27] R. S. Wallis, Mathematical models of tuberculosis reactivation and relapse, Frontiers in Microbiology, 17 (2016), 669. doi: 10.3389/fmicb.2016.00669. [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. [29] Y Yang, J. Wu, J. Li and X. Xu, Tuberculosis with relapse: A model, Math. Popul. Stud., 24 (2017), 3-20.  doi: 10.1080/08898480.2014.998550. [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.
Flow diagram of model for TB transmission
Numerical simulation for TB model (16) with CF fractional using HATM at $\sigma = 1, 0.95, 0.90, 0.85$
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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