1. MMC
  2. All Volumes and Issues
  3. Volume 11, Number 2, 2024
  4. Analysis and optimal control problem for a fractional mathematical model of tuberculosis with smoking consideration

Analysis and optimal control problem for a fractional mathematical model of tuberculosis with smoking consideration

MMC.
2024;
Volume 11, Number 2
: pp. 492–504
Received: April 27, 2023
Revised: November 03, 2023
Accepted: November 08, 2023

El Baz O., Khaloufi I., Kouidere A., Laarabi H., Rachik M.  Analysis and optimal control problem for a fractional mathematical model of tuberculosis with smoking consideration.  Mathematical Modeling and Computing. Vol. 11, No. 2, pp. 492–504 (2024)

Authors:
  1. O. El Baz
  2. I. Khaloufi
  3. A. Kouidere
  4. H. Laarabi
  5. M. Rachik
1
Laboratory of Analysis Modeling and Simulation, Casablanca, Morocco
2
Laboratory of Analysis, Modeling and Simulation, Casablanca, Morocco
3
Laboratory of Analysis Modeling and Simulation, Casablanca, Morocco
4
Laboratory of Analysis, Modeling and Simulation, Casablanca, Morocco
5
Laboratory of Analysis, Modeling and Simulation, Casablanca, Morocco

This article studies a mathematical model of the fractional order of tuberculosis (TB).  It describes the dynamics of the spread of tuberculosis among smokers.  The purpose of this research is to protect vulnerable people against the virus.  According to the survey results, the required model has an equilibrium point: the disease-free equilibrium point $E_f$.  We also analyze the local stability of this equilibrium point of the model, using the basic reproduction number $\mathcal{R}_{0}$ calculated according to the new generation method.  In our model, we include three controls that represent: restricting individual contact, treatment, and sensitization.  This article aims at reducing the number of infected smokers and non-smokers using an optimal control strategy and a fractional derivation.  The maximum principle of Pontryagin is used to describe optimal controls with Caputo-derived fractional over time and the optimal system is resolved iteratively.  The numerical simulation is presented according to the method presented by Matlab.

Caputo fractional derivative
optimal control
tuberculosis
smoking
contagious virus
local stability
dynamic system
infectious diseases
stability
free equilibrium
Pontryagin maximum
  1. Maurya V., Vijayan V. K., Shah A.  Smoking and tuberculosis: an association overlooked.  The International Journal of Tuberculosis and Lung Disease.  6 (11), 942–951 (2002).
  2. Narasimhan P., Wood J., MacIntyre C. R., Mathai D.  Risk Factors for Tuberculosis.  Pulmonary Medicine.  2013, 828939 (2013).
  3. Raja A.  Immunology of tuberculosis.  The Indian Journal of Medical Research.  120 (4), 213–232 (2004).
  4. Jiménez-Fuentes M. Á., Rodrigo T., Altet M. N., Jiménez-Ruiz C. A., Casals M., Penas A., Mir I., Reina S. S., Riesco-Miranda J. A., Caylá J. A.  Factors associated with smoking among tuberculosis patients in Spain.  BMC Infectious Diseases.  16, 486 (2016).
  5. Kolappan C., Gopi P. G.  Tobacco smoking and pulmonary tuberculosis.  Thorax.  57 (11), 964–966 (2002).
  6. Basu S., Stuckler D., Bitton A., Glantz S. A.  Projected effects of tobacco smoking on worldwide tuberculosis control: mathematical modelling analysis.  BMJ. 343 (2011).
  7. Gorenflo R., Kilbas A. A., Mainardi F., Rogosin S. V.  Mittag–Leffler Functions, Related Topics and Applications. Springer (2020).
  8. Ahmed E., El-Sayed A. M. A., El-Saka H. A. A.  Equilibrium points, stability and numerical solutions of fractional-order predator–prey and rabies models.  Journal of Mathematical Analysis and Applications.  325 (1), 542–553 (2007).
  9. Li H.-L., Zhang L., Hu C., Jiang Y.-L., Teng Z.  Dynamical analysis of a fractional-order predator–prey model incorporating a prey refuge.  Journal of Applied Mathematics and Computing.  54, 435–449 (2017).
  10. Vargas-De-León C.  Volterra–type Lyapunov functions for fractional-order epidemic systems.  Communications in Nonlinear Science and Numerical Simulation.  24 (1–3), 75–85 (2015).
  11. Matignon D.  Stability results for fractional differential equations with applications to control processing.  Computational Engineering in Systems Applications.  2, 963–968 (1996).
  12. Kolappan C., Gopi P.  Tobacco smoking and pulmonary tuberculosis.  Thorax.  57 (11), 964–966 (2002).
  13. Bates M. N., Khalakdina A., Pai M., Chang L., Lessa F., Smith K.  Risk of Tuberculosis From Exposure to Tobacco Smoke. A Systematic Review and Meta-analysis.  Arch Intern Med.  167 (4), 335–42 (2007).
  14. Wen C.-P., Chan T.-C., Chan H.-T., Tsai M.-K., Cheng T.-Y., Tsai S.-P.  The reduction of tuberculosis risks by smoking cessation.  BMC Infectious Diseases.  10, 156 (2010).
  15. Zhang H., Xin H., Li X., Li H., Li M., Lu W., Bai L., Wang X., Liu J., Jin Q., et al.  A dose-response relationship of smoking with tuberculosis infection: A cross-sectional study among 21008 rural residents in China.  PloS One.  12 (4), e0175183 (2017).
  16. Bani-Yaghoub M., Gautam R., Shuai Z., Van Den Driessche P., Ivanek R.  Reproduction numbers for infections with free-living pathogens growing in the environment.  Journal of Biological Dynamics.  6 (2), 923–940 (2012).
  17. Kouidere A., Kada D., Balatif O., Rachik M., Naim M.  Optimal control approach of a mathematical modeling with multiple delays of the negative impact of delays in applying preventive precautions against the spread of the COVID-19 pandemic with a case study of Brazil and cost-effectiveness.  Chaos, Solitons & Fractals.  142, 110438 (2021).
  18. Pawar D., Patil W., Raut D.  Fractional-order mathematical model for analysing impact of quarantine on transmission of COVID-19 in India.  Mathematical Modeling and Computing.  8 (2), 253–266 (2021).
  19. Fadugba S., Ali F., Abubakar A.  Caputo fractional reduced differential transform method for SEIR epidemic model with fractional order.  Mathematical Modeling and Computing.  8 (3), 537–548 (2021).
  20. Fleming W. H., Rishel R. W.  Deterministic and Stochastic Optimal Control.  Vol. 1, Springer Science & Business Media (2012).
  21. Lukes D. L.  Differential Equations: Classical to Controlled (1982).
  22. Sweilam N. H., Al-Mekhlafi S. M., Assiri T., Atangana A.  Optimal control for cancer treatment mathematical model using Atangana–Baleanu–Caputo fractional derivative.  Advances in Difference Equations.  2020, 334 (2020).
  23. Khajji B., Boujallal L., Elhia M., Balatif O., Rachik M.  A fractional-order model for drinking alcohol behaviour leading to road accidents and violence.  Mathematical Modeling and Computing.  9 (3), 501–518 (2022).
  24. Pontryagin L. S.  Mathematical Theory of Optimal Processes. CRC Press (1987).
  25. Khaloufi I., Lafif M., Benfatah Y., Laarabi H., Bouyaghroumni J., Rachik M.  A continuous SIR mathematical model of the spread of infectious illnesses that takes human immunity into account.  Mathematical Modeling and Computing.  10 (1), 53–65 (2023).
  26. Elyoussoufi L., Kouidere A., Kada D., Balatif O., Daouia A., Rachik M.  On stability analysis study and strategies for optimal control of a mathematical model of hepatitis HCV with the latent state.  Mathematical Modeling and Computing.  10 (1), 101–118 (2023).
  27. Khaloufi I., Karim M., Rhila S. B., Laarabi H., Rachik M.  A mathematical model describing the correlation between smokers and tuberculosis patients.  Mathematics in Engineering, Science & Aerospace (MESA).  14 (2), 347–361 (2023).
  28. Lafif M., Khaloufi I., Benfatah Y., Bouyaghroumni J., Laarabi H., Rachik M.  A mathematical SIR model on the spread of infectious diseases considering human immunity.  Communications in Mathematical Biology and Neuroscience.  2022, 69 (2022).