1. MMC
  2. All Volumes and Issues
  3. Volume 10, Number 3, 2023
  4. Modeling and mathematical analysis of drug addiction with the study of the effect of psychological and biological treatment

Modeling and mathematical analysis of drug addiction with the study of the effect of psychological and biological treatment

MMC.
2023;
Volume 10, Number 3
: pp. 935–943
https://doi.org/10.23939/mmc2023.03.935
Received: November 17, 2022
Revised: June 17, 2023
Accepted: June 19, 2023

Mathematical Modeling and Computing, Vol. 10, No. 3, pp. 935–943 (2023)

Authors:
  1. E. M. Moumine
  2. O. Balatif
  3. M. Rachik
1
Laboratory of Analysis, Modeling and Simulation, Department of Mathematics and Computer Science, Faculty of Science Ben M'sik, University of Hassan II, Casablanca, Morocco
2
Laboratory of Fundamental Mathematics and Their Applications, Department of Mathematics, Faculty of Sciences El Jadida, Chouaib Doukkali University, El Jadida, Morocco
3
Laboratory of Analysis, modeling and simulation, Department of mathematics and computer sciences, Faculty of sciences Ben M'Sik, University Hassan II of Casablanca

In this article, we propose a discrete mathematical model which describes the propagation of the drug phenomenon in a human population.  The population is unscrewed in five compartments: "$S$" People likely to become drug addicts, "$M$" Moderate drug addicts, "$H$" Heavy drug addicts, "$T$" People receiving drug addiction treatment, "$R$" The recovered people who have completely abstained from drug addiction.  Our goal is to find a better strategy to reduce the number of heavy addicts and to maximize the number of people receiving full treatment.  The tools of optimal control theory were used in this study, in particular the Pontryagin maximum principle.

optimal control
mathematical model
discrete model
drug addiction
  1. 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).
  2. Sadki M., Harroudi S., Allali K.  Dynamical analysis of an HCV model with cell-to-cell transmission and cure rate in the presence of adaptive immunity.  Mathematical Modeling and Computing.  9 (3), 579–593 (2022).
  3. Bounkaicha C., Allali K., Tabit Y., Danane J.  Global dynamic of spatio-temporal fractional order SEIR model.  Mathematical Modeling and Computing.  10 (2), 299–310 (2023).
  4. Elkaf M., Allali K.  Fractional derivative model for tumor cells and immune system competition.  Mathematical Modeling and Computing.  10 (2), 288–298 (2023).
  5. Hu Z., Teng Z., Jiang H.  Stability analysis in a class of discrete SIRS epidemic model.  Nonlinear Analysis: Real World Applications.  13 (5), 2017–2033 (2012).
  6. Lenhart S., Workman J. T.  Optimal control applied to biological models.  Chapman and Hall/CRC, Boca Raton (2007).
  7. Mushayabasa S., Tapedzesa G.  Modeling illicit drug use dynamics and its optimal control analysis.  Computational and Mathematical Methods in Medicine.  2015, 383154 (2015).
  8. Rafal M. D., Stevens W. F.  Discrete dynamic optimization applied to on-line optimal control.  AlChE Journal.  14 (1), 85–91 (1968).
  9. Martin R. B.  Optimal control drug scheduling of cancer chemotherapy.  Automatica.  28 (6), 1113–1123 (1992).
  10. Mushayabasa S., Tapedzesa G.  Modeling Illicit Drug Use Dynamics and Its Optimal Control Analysis.  Computational and Mathematical Methods in Medicine.  2015, 383154 (2015).
  11. Behrens D. A., Caulkins J. P., Tragler G., Feichtinger G.  Optimal Control of Drug Epidemics: Prevent and Treat – But Not at the Same Time?  Management Science.  46 (3), 333–347 (2000).
  12. Khajji B., Moumine E. M, Ferjouchia H., Balatif O., Rachik M.  Optimal control and discrete-time modelling of alcohol model with physical and psychological complications.  Journal of Mathematical and Computational Science.  10 (5), 1969–1986 (2020).
  13. Khajji B., Kouidere A., Balatif O., Rachik M.  Mathematical modeling, analysis and optimal control of an alcohol drinking model with liver complication.  Communications in Mathematical Biology and Neuroscience.  2020, 32 (2020).
  14. Khajji B., Labzai A., Balatif O., Rachik M.  Mathematical modeling and analysis of an alcohol drinking model with the influence of alcohol treatment centers.  International Journal of Mathematics and Mathematical Sciences.  2020, 4903168 (2020).
  15. Moore H.  How to mathematically optimize drug regimens using optimal control.  Journal of Pharmacokinetics and Pharmacodynamics.  45, 127–137 (2018).
  16. Labzai A., Balatif O., Rachik M.  Optimal control strategy for a discrete time smoking model with specific saturated incidence rate.  Discrete Dynamics in Nature and Society.  2018, 5949303 (2018).
  17. Njagarah J. B. H., Nyabadza F.  Modelling the role of drug barons on the prevalence of drug epidemics.  Mathematical Biosciences and Engineering.  10 (3), 843–860 (2013).
  18. Labzai A., Kouidere A., Khajji B., Balatif O., Rachik M.  Mathematical Modeling and Optimal Control Strategy for a Discrete Time Drug Consumption Model.  Discrete Dynamics in Nature and Society.  2020, 5671493 (2020).
  19. Khajji B., Labzai A., Kouidere A., Balatif O., Rachik M.  A discrete mathematical modeling of the influence of alcohol treatment centers on the drinking dynamics using optimal control. Journal of Applied Mathematics.  2020, 9284698 (2020).
  20. Athans M., Falb P. L.  Optimal Control: An Introduction tothe Theory and Its Applications.  Dover Publications (2006).
  21. Liberzon D.  Calculus of Variations and Optimal Control Theory.  A Concise Introduction. Princeton University Press, Princeton, NJ, USA (2012).
  22. Chinchuluun A., Pardalos P. M., Enkhbat R., Tseveendori I. (Eds.)  Optimization and Optimal Control: Theory and Applications. Vol. 39. Springer (2010).
  23. Stengel R. F.  Optimal Control and Estimation. Dover, New York, USA (1994).
  24. Chen P., Islam S. M. N.  Optimal Control Models in Finance: A New Computation Approach. Springer (2005).
  25. Aniţa S., Arnăutu V., Capasso V.  An Introduction to Optimal Control Problems in Life Sciences and Economics: From Mathematical Models to Numerical Simulation with MATLAB. Birkhäuser (2011).
  26. La Torre D., Kunze H., Ruiz-Galan M., Malik T., Marsiglio S.  Optimal control: theory and application to science,engineering, and social sciences.  2015, 890527 (2015).
  27. Ding W., Hendon R., Cathey B., Lancaster E., Germick R.  Discrete time optimal control applied to pest control problems.  Involve, A Journal of Mathematics.  7 (4), 479–489 (2014).
  28. Guibout V., Bloch A. M.  A discrete maximum principle for solving optimal control problems.  2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601). 1806–1811 (2004).
  29. Hwang C. L., Fan L. T.  A discrete version of Pontryagin's maximum principle.  Operations Research.  15 (1), 139–146 (1967).
  30. Pontryagin L. S., Boltyanskii V. G., Gamkrelidze R. V., Mishchenko E. F.  The Mathematical Theory of Optimal Processes.  John Wiley & Sons, London (1962).