Research Article

Year 2024,
Volume: 4 Issue: 3, 238 - 255, 30.09.2024
### Abstract

### References

- [1] World Health Organization, Global Tuberculosis Report 2021, (2021). https://www.who.int/teams/global-tuberculosis-programme/tb-reports/ global-tuberculosis-report-2021
- [2] World Health Organization, Global Tuberculosis Report 2022, (2022). https://www.who.int/teams/global-tuberculosis-programme/tb-reports/ global-tuberculosis-report-2022
- [3] World Health Organization, Latent Tuberculosis Infection: Updated and Consolidated Guidelines for Programmatic Management, (2023). https://www.who.int/tb/publications/201
- [4] World Health Organization, The END TB Strategy, (2015). https://www.who.int/ publications/i/item/WHO-HTM-TB-2015.19
- [5] Zumla, A., Raviglione, M., Hafner, R. and Von Reyn, C.F. Tuberculosis. The New England Journal of Medicine, 368(8), 745-755, (2013).
- [6] Dodd, P.J., Sismanidis, C. and Seddon, J.A. Global burden of drug-resistant tuberculosis in children: a mathematical modelling study. The Lancet Infectious Diseases, 16(10), 1193-1201, (2016).
- [7] Centers for Disease Control and Prevention, Tuberculosis (TB)-Data and Statistics, (2023). https://www.cdc.gov/tb/statistics/default.htm
- [8] Gupta, R.K., Lipman, M., Story, A., Hayward, A., De Vries, G., Van Hest, R. et al. Active case finding and treatment adherence in risk groups in the tuberculosis pre-elimination era. The International Journal of Tuberculosis and Lung Disease, 22(5), 479-487, (2018).
- [9] Goufo, E.F.D., Maritz, R. and Pene, M.K. A mathematical and ecological analysis of the effects of petroleum oil droplets breaking up and spreading in aquatic environments. International Journal of Environment and Pollution, 61(1), 64-71, (2017).
- [10] Atangana, A. and Doungmo Goufo, E.F. Computational analysis of the model describing HIV infection of CD4+ T cells. BioMed Research International, 2014, 618404, (2014).
- [11] Tchepmo Djomegni, P.M., Govinder, K.S. and Doungmo Goufo, E.F. Movement, competition and pattern formation in a two prey–one predator food chain model. Computational and Applied Mathematics, 37, 2445-2459, (2018).
- [12] Peter, O.J., Yusuf, A., Oshinubi, K., Oguntolu, F.A., Lawal, J.O., Abioye, A.I. et al. Fractional order of pneumococcal pneumonia infection model with Caputo-Fabrizio operator. Results in Physics, 29, 104581, (2021).
- [13] Atangana, A. and Qureshi, S. Mathematical modeling of an autonomous nonlinear dynamical system for malaria transmission using Caputo derivative. In Fractional Order Analysis: Theory, Methods and Applications (pp. 225-252). New York, United States: John Wiley & Sons, (2020).
- [14] Peter, O.J., Shaikh, A.S., Ibrahim, M.O., Nisar, K.S., Baleanu, D., Khan, I. et al. Analysis and dynamics of fractional order mathematical model of COVID-19 in Nigeria using AtanganaBaleanu operator. Computers, Materials, & Continua, 66(2), 1823-1848, (2021).
- [15] Peter, O.J., Qureshi, S., Yusuf, A., Al-Shomrani, M. and Idowu, A.A. A new mathematical model of COVID-19 using real data from Pakistan. Results in Physics, 24, 104098, (2021).
- [16] Khan, H., Gómez-Aguilar, J.F., Alkhazzan, A. and Khan, A. A fractional order HIV-TB coinfection model with nonsingular Mittag-Leffler law. Mathematical Methods in the Applied Sciences, 43(6), 3786-3806, (2020).
- [17] Akinpelu, F.O. and Ojo, M.M. Mathematical analysis of effect of isolation on the transmission of Ebola virus disease in a population. Asian Research Journal of Mathematics, 1(5), 1-12, (2016).
- [18] Ahmad, S., Ullah, A., Al-Mdallal, Q.M., Khan, H., Shah, K. and Khan, A. Fractional order mathematical modeling of COVID-19 transmission. Chaos, Solitons & Fractals, 139, 110256, (2020).
- [19] Arafa, A.A.M., Khalil, M. and Sayed, A. A non-integer variable order mathematical model of human immunodeficiency virus and malaria coinfection with time delay. Complexity, 2019, 4291017, (2019).
- [20] Ojo, M.M. and Goufo, E.F.D. Modeling, analyzing and simulating the dynamics of Lassa fever in Nigeria. Journal of the Egyptian Mathematical Society, 30, 1, (2022).
- [21] Demongeot, J., Griette, Q., Magal, P. and Webb, G. Modeling vaccine efficacy for COVID-19 outbreak in New York city. Biology, 11(3), 345, (2022).
- [22] Musa, S.S., Qureshi, S., Zhao, S., Yusuf, A., Mustapha, U.T. and He, D. Mathematical modeling of COVID-19 epidemic with effect of awareness programs. Infectious Disease Modelling, 6, 448-460, (2021).
- [23] Memon, Z., Qureshi, S. and Memon, B.R. Assessing the role of quarantine and isolation as control strategies for COVID-19 outbreak: a case study. Chaos, Solitons & Fractals, 144, 110655, (2021).
- [24] Yang, Y., Li, J., Ma, Z. and Liu, L. Global stability of two models with incomplete treatment for tuberculosis. Chaos, Solitons & Fractals, 43(1-12), 79-85, (2010).
- [25] Zhang, J., Li, Y. and Zhang, X. Mathematical modeling of tuberculosis data of China. Chaos, Solitons & Fractals, 365, 159-163, (2015).
- [26] Egonmwan, A.O. and Okuonghae, D. Analysis of a mathematical model for tuberculosis with diagnosis. Journal of Applied Mathematics and Computing, 59, 129-162, (2019). [CrossRef]
- [27] Ullah, I., Ahmad, S., Al-Mdallal, Q., Khan, Z.A., Khan, H. and Khan, A. Stability analysis of a dynamical model of tuberculosis with incomplete treatment. Advances in Difference Equations, 2020, 499, (2020).
- [28] Syahrini, I., Sriwahyuni, Halfiani, V., Yuni, S.M., Iskandar, T., Rasudin, et al. The epidemic of tuberculosis on vaccinated population. In Proceedings, Journal of Physics: Conference Series (Vol. 890, No. 1), p. 012017, (2017, September).
- [29] Okuonghae, D. A mathematical model of tuberculosis transmission with heterogeneity in disease susceptibility and progression under a treatment regime for infectious cases. Applied Mathematical Modelling, 37(10-11), 6786-6808, (2013).
- [30] Liu, J. and Zhang, T. Global stability for a tuberculosis model. Mathematical and Computer Modelling, 54(1-2), 836-845, (2011).
- [31] Andrawus, J., Eguda, F.Y., Usman, I.G., Maiwa, S.I., Dibal, I.M., Urum, T.G. et al. A mathematical model of a tuberculosis transmission dynamics incorporating first and second line treatment. Journal of Applied Sciences and Environmental Management, 24(5), 917-922, (2020).
- [32] Kasereka Kabunga, S., Doungmo Goufo, E.F. and Ho Tuong, V. Analysis and simulation of a mathematical model of tuberculosis transmission in Democratic Republic of the Congo. Advances in Difference Equations, 2020, 642, (2020).
- [33] Kim, S., De Los Reyes V, A.A. and Jung, E. Country-specific intervention strategies for top three TB burden countries using mathematical model. PloS One, 15(4), e0230964, (2020).
- [34] Nkamba, L.N., Manga, T.T., Agouanet, F. and Mann Manyombe, M.L. Mathematical model to assess vaccination and effective contact rate impact in the spread of tuberculosis. Journal of Biological Dynamics, 13(1), 26-42, (2019).
- [35] Gerberry, D.J. Practical aspects of backward bifurcation in a mathematical model for tuberculosis. Journal of Theoretical Biology, 388, 15-36, (2016).
- [36] Ludji, D.G., Sianturi, P. and Nugrahani, E. Dynamical system of the mathematical model for tuberculosis with vaccination. ComTech: Computer, Mathematics and Engineering Applications, 10(2), 59-66, (2019).
- [37] Mishra, B.K. and Srivastava, J. Mathematical model on pulmonary and multidrug-resistant tuberculosis patients with vaccination. Journal of the Egyptian Mathematical Society, 22(2), 311-316, (2014).
- [38] Olaniyi, S. Dynamics of Zika virus model with nonlinear incidence and optimal control strategies. Applied Mathematics & Information Sciences, 12(5), 969-982, (2018).
- [39] Peter, O.J., Oguntolu, F.A., Ojo, M.M., Olayinka Oyeniyi, A., Jan, R. and Khan, I. Fractional order mathematical model of monkeypox transmission dynamics. Physica Scripta, 97(8), 084005, (2022).
- [40] Abidemi, A., Zainuddin, Z.M. and Aziz, N.A.B. Impact of control interventions on COVID-19 population dynamics in Malaysia: a mathematical study. The European Physical Journal Plus, 136, 237, (2021).
- [41] Joshi, H. and Yavuz, M. Transition dynamics between a novel coinfection model of fractional order for COVID-19 and tuberculosis via a treatment mechanism. The European Physical Journal Plus, 138, 468, (2023).
- [42] Joshi, H., Jha, B.K. and Yavuz, M. Modelling and analysis of fractional-order vaccination model for control of COVID-19 outbreak using real data. Mathematical Biosciences and Engineering, 20(1), 213-240, (2023).
- [43] Joshi, H. Mechanistic insights of COVID-19 dynamics by considering the influence of neurodegeneration and memory trace. Physica Scripta, 99(3), 035254, (2024).
- [44] Allegretti, S., Bulai, I.M., Marino, R., Menandro, M.A. and Parisi, K. Vaccination effect conjoint to fraction of avoided contacts for a SARS-CoV-2 mathematical model. Mathematical Modelling and Numerical Simulation with Applications, 1(2), 56-66, (2021).
- [45] Bolaji, B., Onoja, T., Agbata, C., Omede, B.I. and Odionyenma, U.B. Dynamical analysis of HIV-TB co-infection transmission model in the presence of treatment for TB. Bulletin of Biomathematics, 2(1), 21-56, (2024).
- [46] Van den Driessche, P. and Watmough, J. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical Biosciences, 180(1- 2), 29-48, (2002).
- [47] Gumel, A.B. Causes of backward bifurcations in some epidemiological models. Journal of Mathematical Analysis and Applications, 395(1), 355-365, (2012).
- [48] Singer, B.H. and Kirschner, D.E. Influence of backward bifurcation on interpretation of R0 in a model of epidemic tuberculosis with reinfection. Mathematical Biosciences and Engineering, 1(1), 81-93, (2004).
- [49] Egbelowo, O.F., Munyakazi, J.B., Dlamini, P.G., Osaye, F.J. and Simelane, S.M. Modeling visceral leishmaniasis and tuberculosis co-infection dynamics. Frontiers in Applied Mathematics and Statistics, 9, 1153666, (2023).
- [50] La Salle, J.P. The Stability of Dynamical Systems. SIAM: United States of America, (1976).
- [51] Ojo, M.M., Peter, O.J., Goufo, E.F.D., Panigoro, H.S. and Oguntolu, F.A. Mathematical model for control of tuberculosis epidemiology. Journal of Applied Mathematics and Computing, 69, 69-87, (2023).

Tuberculosis (TB) remains a formidable global health challenge, demanding effective control strategies to alleviate its burden. In this study, we introduce a comprehensive mathematical model to unravel the intricate dynamics of TB transmission and assess the efficacy and cost-effectiveness of diverse intervention strategies. Our model meticulously categorizes the total population into seven distinct compartments, encompassing susceptibility, vaccination, diagnosed infectious, undiagnosed infectious, hospitalized, and recovered individuals. Factors such as susceptible individual recruitment, the impact of vaccination, immunity loss, and the nuanced dynamics of transmission between compartments are considered. Notably, we compute the basic reproduction number, providing a quantitative measure of TB transmission potential. Through this comprehensive model, our study aims to offer valuable insights into optimal control measures for TB prevention and control, contributing to the ongoing global efforts to combat this pressing health challenge.

- [1] World Health Organization, Global Tuberculosis Report 2021, (2021). https://www.who.int/teams/global-tuberculosis-programme/tb-reports/ global-tuberculosis-report-2021
- [2] World Health Organization, Global Tuberculosis Report 2022, (2022). https://www.who.int/teams/global-tuberculosis-programme/tb-reports/ global-tuberculosis-report-2022
- [3] World Health Organization, Latent Tuberculosis Infection: Updated and Consolidated Guidelines for Programmatic Management, (2023). https://www.who.int/tb/publications/201
- [4] World Health Organization, The END TB Strategy, (2015). https://www.who.int/ publications/i/item/WHO-HTM-TB-2015.19
- [5] Zumla, A., Raviglione, M., Hafner, R. and Von Reyn, C.F. Tuberculosis. The New England Journal of Medicine, 368(8), 745-755, (2013).
- [6] Dodd, P.J., Sismanidis, C. and Seddon, J.A. Global burden of drug-resistant tuberculosis in children: a mathematical modelling study. The Lancet Infectious Diseases, 16(10), 1193-1201, (2016).
- [7] Centers for Disease Control and Prevention, Tuberculosis (TB)-Data and Statistics, (2023). https://www.cdc.gov/tb/statistics/default.htm
- [8] Gupta, R.K., Lipman, M., Story, A., Hayward, A., De Vries, G., Van Hest, R. et al. Active case finding and treatment adherence in risk groups in the tuberculosis pre-elimination era. The International Journal of Tuberculosis and Lung Disease, 22(5), 479-487, (2018).
- [9] Goufo, E.F.D., Maritz, R. and Pene, M.K. A mathematical and ecological analysis of the effects of petroleum oil droplets breaking up and spreading in aquatic environments. International Journal of Environment and Pollution, 61(1), 64-71, (2017).
- [10] Atangana, A. and Doungmo Goufo, E.F. Computational analysis of the model describing HIV infection of CD4+ T cells. BioMed Research International, 2014, 618404, (2014).
- [11] Tchepmo Djomegni, P.M., Govinder, K.S. and Doungmo Goufo, E.F. Movement, competition and pattern formation in a two prey–one predator food chain model. Computational and Applied Mathematics, 37, 2445-2459, (2018).
- [12] Peter, O.J., Yusuf, A., Oshinubi, K., Oguntolu, F.A., Lawal, J.O., Abioye, A.I. et al. Fractional order of pneumococcal pneumonia infection model with Caputo-Fabrizio operator. Results in Physics, 29, 104581, (2021).
- [13] Atangana, A. and Qureshi, S. Mathematical modeling of an autonomous nonlinear dynamical system for malaria transmission using Caputo derivative. In Fractional Order Analysis: Theory, Methods and Applications (pp. 225-252). New York, United States: John Wiley & Sons, (2020).
- [14] Peter, O.J., Shaikh, A.S., Ibrahim, M.O., Nisar, K.S., Baleanu, D., Khan, I. et al. Analysis and dynamics of fractional order mathematical model of COVID-19 in Nigeria using AtanganaBaleanu operator. Computers, Materials, & Continua, 66(2), 1823-1848, (2021).
- [15] Peter, O.J., Qureshi, S., Yusuf, A., Al-Shomrani, M. and Idowu, A.A. A new mathematical model of COVID-19 using real data from Pakistan. Results in Physics, 24, 104098, (2021).
- [16] Khan, H., Gómez-Aguilar, J.F., Alkhazzan, A. and Khan, A. A fractional order HIV-TB coinfection model with nonsingular Mittag-Leffler law. Mathematical Methods in the Applied Sciences, 43(6), 3786-3806, (2020).
- [17] Akinpelu, F.O. and Ojo, M.M. Mathematical analysis of effect of isolation on the transmission of Ebola virus disease in a population. Asian Research Journal of Mathematics, 1(5), 1-12, (2016).
- [18] Ahmad, S., Ullah, A., Al-Mdallal, Q.M., Khan, H., Shah, K. and Khan, A. Fractional order mathematical modeling of COVID-19 transmission. Chaos, Solitons & Fractals, 139, 110256, (2020).
- [19] Arafa, A.A.M., Khalil, M. and Sayed, A. A non-integer variable order mathematical model of human immunodeficiency virus and malaria coinfection with time delay. Complexity, 2019, 4291017, (2019).
- [20] Ojo, M.M. and Goufo, E.F.D. Modeling, analyzing and simulating the dynamics of Lassa fever in Nigeria. Journal of the Egyptian Mathematical Society, 30, 1, (2022).
- [21] Demongeot, J., Griette, Q., Magal, P. and Webb, G. Modeling vaccine efficacy for COVID-19 outbreak in New York city. Biology, 11(3), 345, (2022).
- [22] Musa, S.S., Qureshi, S., Zhao, S., Yusuf, A., Mustapha, U.T. and He, D. Mathematical modeling of COVID-19 epidemic with effect of awareness programs. Infectious Disease Modelling, 6, 448-460, (2021).
- [23] Memon, Z., Qureshi, S. and Memon, B.R. Assessing the role of quarantine and isolation as control strategies for COVID-19 outbreak: a case study. Chaos, Solitons & Fractals, 144, 110655, (2021).
- [24] Yang, Y., Li, J., Ma, Z. and Liu, L. Global stability of two models with incomplete treatment for tuberculosis. Chaos, Solitons & Fractals, 43(1-12), 79-85, (2010).
- [25] Zhang, J., Li, Y. and Zhang, X. Mathematical modeling of tuberculosis data of China. Chaos, Solitons & Fractals, 365, 159-163, (2015).
- [26] Egonmwan, A.O. and Okuonghae, D. Analysis of a mathematical model for tuberculosis with diagnosis. Journal of Applied Mathematics and Computing, 59, 129-162, (2019). [CrossRef]
- [27] Ullah, I., Ahmad, S., Al-Mdallal, Q., Khan, Z.A., Khan, H. and Khan, A. Stability analysis of a dynamical model of tuberculosis with incomplete treatment. Advances in Difference Equations, 2020, 499, (2020).
- [28] Syahrini, I., Sriwahyuni, Halfiani, V., Yuni, S.M., Iskandar, T., Rasudin, et al. The epidemic of tuberculosis on vaccinated population. In Proceedings, Journal of Physics: Conference Series (Vol. 890, No. 1), p. 012017, (2017, September).
- [29] Okuonghae, D. A mathematical model of tuberculosis transmission with heterogeneity in disease susceptibility and progression under a treatment regime for infectious cases. Applied Mathematical Modelling, 37(10-11), 6786-6808, (2013).
- [30] Liu, J. and Zhang, T. Global stability for a tuberculosis model. Mathematical and Computer Modelling, 54(1-2), 836-845, (2011).
- [31] Andrawus, J., Eguda, F.Y., Usman, I.G., Maiwa, S.I., Dibal, I.M., Urum, T.G. et al. A mathematical model of a tuberculosis transmission dynamics incorporating first and second line treatment. Journal of Applied Sciences and Environmental Management, 24(5), 917-922, (2020).
- [32] Kasereka Kabunga, S., Doungmo Goufo, E.F. and Ho Tuong, V. Analysis and simulation of a mathematical model of tuberculosis transmission in Democratic Republic of the Congo. Advances in Difference Equations, 2020, 642, (2020).
- [33] Kim, S., De Los Reyes V, A.A. and Jung, E. Country-specific intervention strategies for top three TB burden countries using mathematical model. PloS One, 15(4), e0230964, (2020).
- [34] Nkamba, L.N., Manga, T.T., Agouanet, F. and Mann Manyombe, M.L. Mathematical model to assess vaccination and effective contact rate impact in the spread of tuberculosis. Journal of Biological Dynamics, 13(1), 26-42, (2019).
- [35] Gerberry, D.J. Practical aspects of backward bifurcation in a mathematical model for tuberculosis. Journal of Theoretical Biology, 388, 15-36, (2016).
- [36] Ludji, D.G., Sianturi, P. and Nugrahani, E. Dynamical system of the mathematical model for tuberculosis with vaccination. ComTech: Computer, Mathematics and Engineering Applications, 10(2), 59-66, (2019).
- [37] Mishra, B.K. and Srivastava, J. Mathematical model on pulmonary and multidrug-resistant tuberculosis patients with vaccination. Journal of the Egyptian Mathematical Society, 22(2), 311-316, (2014).
- [38] Olaniyi, S. Dynamics of Zika virus model with nonlinear incidence and optimal control strategies. Applied Mathematics & Information Sciences, 12(5), 969-982, (2018).
- [39] Peter, O.J., Oguntolu, F.A., Ojo, M.M., Olayinka Oyeniyi, A., Jan, R. and Khan, I. Fractional order mathematical model of monkeypox transmission dynamics. Physica Scripta, 97(8), 084005, (2022).
- [40] Abidemi, A., Zainuddin, Z.M. and Aziz, N.A.B. Impact of control interventions on COVID-19 population dynamics in Malaysia: a mathematical study. The European Physical Journal Plus, 136, 237, (2021).
- [41] Joshi, H. and Yavuz, M. Transition dynamics between a novel coinfection model of fractional order for COVID-19 and tuberculosis via a treatment mechanism. The European Physical Journal Plus, 138, 468, (2023).
- [42] Joshi, H., Jha, B.K. and Yavuz, M. Modelling and analysis of fractional-order vaccination model for control of COVID-19 outbreak using real data. Mathematical Biosciences and Engineering, 20(1), 213-240, (2023).
- [43] Joshi, H. Mechanistic insights of COVID-19 dynamics by considering the influence of neurodegeneration and memory trace. Physica Scripta, 99(3), 035254, (2024).
- [44] Allegretti, S., Bulai, I.M., Marino, R., Menandro, M.A. and Parisi, K. Vaccination effect conjoint to fraction of avoided contacts for a SARS-CoV-2 mathematical model. Mathematical Modelling and Numerical Simulation with Applications, 1(2), 56-66, (2021).
- [45] Bolaji, B., Onoja, T., Agbata, C., Omede, B.I. and Odionyenma, U.B. Dynamical analysis of HIV-TB co-infection transmission model in the presence of treatment for TB. Bulletin of Biomathematics, 2(1), 21-56, (2024).
- [46] Van den Driessche, P. and Watmough, J. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical Biosciences, 180(1- 2), 29-48, (2002).
- [47] Gumel, A.B. Causes of backward bifurcations in some epidemiological models. Journal of Mathematical Analysis and Applications, 395(1), 355-365, (2012).
- [48] Singer, B.H. and Kirschner, D.E. Influence of backward bifurcation on interpretation of R0 in a model of epidemic tuberculosis with reinfection. Mathematical Biosciences and Engineering, 1(1), 81-93, (2004).
- [49] Egbelowo, O.F., Munyakazi, J.B., Dlamini, P.G., Osaye, F.J. and Simelane, S.M. Modeling visceral leishmaniasis and tuberculosis co-infection dynamics. Frontiers in Applied Mathematics and Statistics, 9, 1153666, (2023).
- [50] La Salle, J.P. The Stability of Dynamical Systems. SIAM: United States of America, (1976).
- [51] Ojo, M.M., Peter, O.J., Goufo, E.F.D., Panigoro, H.S. and Oguntolu, F.A. Mathematical model for control of tuberculosis epidemiology. Journal of Applied Mathematics and Computing, 69, 69-87, (2023).

There are 51 citations in total.

Primary Language | English |
---|---|

Subjects | Biological Mathematics, Dynamical Systems in Applications |

Journal Section | Research Articles |

Authors | |

Publication Date | September 30, 2024 |

Submission Date | March 29, 2024 |

Acceptance Date | July 15, 2024 |

Published in Issue | Year 2024 Volume: 4 Issue: 3 |

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