Previous |  Up |  Next

Article

Full entry | Fulltext not available (moving wall 24 months)      Feedback
Keywords:
fractional order; optimal control; tuberculosis; sensitivity analysis
Summary:
A fractional model is developed to study the transmission dynamics of tuberculosis disease. The use of a fractional model provides a memory effect and long-term dynamics often observed in chronic infectious diseases such as tuberculosis, which is characterized by a prolonged incubation period and risks of reactivation. The basic reproduction number is computed and we derive the qualitative stability analysis of equilibria. A sensitivity analysis is conducted to assess the impact of the model parameters. Three control strategies are applied, namely treatment, vaccination, and infection rate management, to minimize the number of infected individuals. Numerical simulations are carried out to illustrate the theoretical results obtained.
References:
[1] Aparicio, J. P., Castillo-Chávez, C.: Mathematical modelling of tuberculosis epidemics. Math. Biosci. Eng. 6 (2009), 209-237. DOI 10.3934/mbe.2009.6.209 | MR 2532014 | Zbl 1167.92019
[2] Apelblat, A.: Differentiation of the Mittag-Leffler functions with respect to parameters in the Laplace transform approach. Mathematics 8 (2020), Article ID 657, 22 pages. DOI 10.3390/math8050657
[3] Balzotti, C., D'Ovidio, M., Lai, A. C., Loreti, P.: Effects of fractional derivatives in epidemic models. Available at https://arxiv.org/abs/2107.02877 (2021), 18 pages. DOI 10.48550/arXiv.2107.02877
[4] Bhatter, S., Kumawat, S., Purohit, S. D., Suthar, D. L.: Mathematical modeling of tuberculosis using Caputo fractional derivative: A comparative analysis with real data. Sci. Rep. 15 (2025), Article ID 12672, 19 pages. DOI 10.1038/s41598-025-97502-5
[5] Chasnov, J. R.: Introduction to Differential Equations. The Hong Kong University of Science and Technology, Hong Kong (2009), Available at {\def\let \relax \brokenlink{ https://www.ms.uky.}{edu/ ejwh226/Spring2018/Chasnov.pdf}}\kern0pt.
[6] Vlas, S. J. de, Verver, S., Cai, R., Vanhommerig, J. W., Hontelez, J., Coffeng, L., Noordegraaf-Schouten, M. V., Richardus, J. H.: Mathematical Modelling of Programmatic Screening Strategies for Latent Tuberculosis Infection in Countries with Low Tuberculosis Incidence: ECDC Technical Report. European Centre for Disease Prevention and Control, Stockholm (2018), Available at {\def{ }\let \relax \brokenlink{ https://www.ecdc.europa.eu/sites/default/}{files/documents/LTBI-math-models_Feb2018-edited.pdf}}\kern0pt
[7] Dicko, H., Traoré, A.: Analysis of tuberculosis model with the impact of hospital resources. J. Appl. Math. Comput. 71 (2025), 3305-3328. DOI 10.1007/s12190-024-02356-1 | MR 4910852 | Zbl 08080546
[8] Fleming, W. H., Rishel, R. W.: Deterministic and Stochastic Optimal Control. Applications of Mathematics 1. Springer, Berlin (1975). DOI 10.1007/978-1-4612-6380-7 | MR 0454768 | Zbl 0323.49001
[9] Hamdan, N., Kilicman, A.: A fractional order SIR epidemic model for dengue transmission. Chaos Solitons Fractals 114 (2018), 55-62. DOI 10.1016/j.chaos.2018.06.031 | MR 3856626 | Zbl 1415.92179
[10] Khan, A., Shah, K., Abdeljawad, T., Amacha, I.: Fractal fractional model for tuberculosis: Existence and numerical solutions. Sci. Rep. 14 (2024), Article ID 12211, 19 pages. DOI 10.1038/s41598-024-62386-4
[11] Kilbas, A. A., Rivero, M., Trujillo, J. J.: Existence and uniqueness theorems for differential equations of fractional order in weighted spaces of continuous functions. Fract. Calc. Appl. Anal. 6 (2003), 363-399. MR 2044306 | Zbl 1085.34002
[12] Kilbas, A. A., Trujillo, J. J.: Differential equations of fractional order: Methods, results and problems. I. Appl. Anal. 78 (2001), 153-192. DOI 10.1080/00036810108840931 | MR 1887959 | Zbl 1031.34002
[13] Li, Y., Liu, X., Yuan, Y., Li, J., Wang, L.: Global analysis of tuberculosis dynamical model and optimal control strategies based on case data in the United States. Appl. Math. Comput. 422 (2022), Article ID 126983, 21 pages. DOI 10.1016/j.amc.2022.126983 | MR 4376403 | Zbl 1510.92219
[14] Liu, L., Wang, Y.: A mathematical study of a TB model with treatment interruptions and two latent periods. Comput. Math. Methods Med. 2014 (2014), Article ID 932186, 15 pages. DOI 10.1155/2014/932186 | MR 3214480 | Zbl 1307.92346
[15] Marino, S., Hogue, I. B., Ray, C. J., Kirschner, D. E.: A methodology for performing global uncertainty and sensitivity analysis in systems biology. J. Theor. Biol. 254 (2008), 178-196. DOI 10.1016/j.jtbi.2008.04.011 | MR 2971156 | Zbl 1400.92013
[16] Maurya, J., Blyuss, K. B., Misra, A. K.: Modeling the impact of hospital beds and vaccination on the dynamics of an infectious disease. Math. Biosci. 368 (2024), Article ID 109133, 15 pages. DOI 10.1016/j.mbs.2023.109133 | MR 4682551 | Zbl 1539.92073
[17] Mondal, P. K., Kar, T. K.: Optimal treatment control and bifurcation analysis of a tuberculosis model with effect of multiple re-infections. Int. J. Dyn. Control 5 (2017), 367-380. DOI 10.1007/s40435-015-0176-z | MR 3656606
[18] Nandi, T. R., Saha, A. K., Roy, S.: Analysis of a fractional order epidemiological model for tuberculosis transmission with vaccination and reinfection. Sci. Rep. 14 (2024), Article ID 28290, 21 pages. DOI 10.1038/s41598-024-73392-x
[19] Olayiwola, M. O., Adedokun, K. A.: A novel tuberculosis model incorporating a Caputo fractional derivative and treatment effect via the homotopy perturbation method. Bull. Natl. Res. Cent. 47 (2023), Article ID 121, 16 pages. DOI 10.1186/s42269-023-01079-9
[20] K. Oshinubi, O. J. Peter, E. Addai, E. Mwizerwa, O. Babasola, I. V. Nwabufo, I. Sane, U. M. Adam, A. Adeniji, J. O. Agbaje: Mathematical modelling of tuberculosis outbreak in an East African country incorporating vaccination and treatment. Computation 11 (2023), Article ID 143, 21 pages. DOI 10.3390/computation11070143
[21] Pontryagin, L. S., Boltyanskii, V. G., Gamkrelidze, R. V., Mishchenko, E. F.: The Mathematical Theory of Optimal Processes. Interscience Publishers, New York (1962). MR 0166037 | Zbl 0102.32001
[22] Pooseh, S., Rodrigues, H. S., Torres, D. F. M.: Fractional derivatives in dengue epidemics. AIP Conf. Proc. 1389 (2011), 739-742. DOI 10.1063/1.3636838
[23] Routh, E. J.: A Treatise on the Stability of a Given State of Motion, Particularly Steady Motion: Being the Essay to Which the Adams Prize Was Adjudged in 1877 in the University of Cambridge. Macmillan, London (1877), Available at https://archive.org/details/atreatiseonstab00routgoog\kern0pt
[24] Tilahun, G. T., Wolle, G. A., Tofik, M.: Eco-epidemiological model and analysis of potato leaf roll virus using fractional differential equation. Arab J. Basic Appl. Sci. 28 (2021), 41-50. DOI 10.1080/25765299.2020.1865621
[25] Driessche, P. van den, Watmough, J.: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180 (2002), 29-48. DOI 10.1016/S0025-5564(02)00108-6 | MR 1950747 | Zbl 1015.92036
[26] Wojtak, W., Silva, C. J., Torres, D. F. M.: Uniform asymptotic stability of a fractional tuberculosis model. Math. Model. Nat. Phenom. 13 (2018), Article ID 9, 10 pages. DOI 10.1051/mmnp/2018015 | MR 3789859 | Zbl 1407.92133
[27] Organization, World Health: Guidelines on the Management of Latent Tuberculosis Infection. World Health Organization, Geneva (2015), Available at https://www.who.int/publications/i/item/9789241548908\kern0pt
[28] Organization, World Health: Global Tuberculosis Report. World Health Organization, Geneva (2023), Available at {\def{ }\let \relax \brokenlink{ https://www.who.int/teams/global-programme-}{on-tuberculosis-and-lung-health/data}} (Accessed November 7, 2023)\kern0pt.
[29] Yang, Y., Li, J., Ma, Z., Liu, L.: Global stability of two models with incomplete treatment for tuberculosis. Chaos Solitons Fractals 43 (2010), 79-85. DOI 10.1016/j.chaos.2010.09.002 | MR 2738396 | Zbl 1211.92038
[30] Yang, Y., Tang, S., Ren, X., Zhao, H., Guo, C.: Global stability and optimal control for a tuberculosis model with vaccination and treatment. Discrete Contin. Dyn. Syst., Ser. B 21 (2016), 1009-1022. DOI 10.3934/dcdsb.2016.21.1009 | MR 3503055 | Zbl 1333.34081
[31] Zhang, J., Li, Y., Zhang, X.: Mathematical modeling of tuberculosis data of China. J. Theor. Biol. 365 (2015), 159-163. DOI 10.1016/j.jtbi.2014.10.019 | MR 3284848 | Zbl 1314.92172
[32] Ziv, E., Daley, C. L., Blower, S.: Potential public health impact of new tuberculosis vaccines. Emerging Infectious Diseases 10 (2004), 1529-1535. DOI 10.3201/eid1009.030921
Partner of
EuDML logo