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Journal of Applied Nonlinear Dynamics 12(1) (2023) 133--146 | DOI:10.5890/JAND.2023.03.010

S. Olaniyi$^{1}$, J. O. Akanni$^{2,3}$, O. A. Adepoju$^{1}$

$^{1}$ Department of Pure and Applied Mathematics, Ladoke Akintola University of Technology, Ogbomoso, Nigeria

$^{2}$ Department of Mathematical and Computing Sciences, Koladaisi University Ibadan, Oyo State, Nigeria

$^{3}$ Department of Mathematics, Universitas Airlangga, Kampus C Mulyorejo Surabaya 60115, Indonesia

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Abstract

The problem of illicit drug use with its associated hazardous effects on social behavior and human population dynamics require an intervention. This paper presents the optimal intervention strategy for controlling the menace of illicit drug use at population levels. Two time-dependent intervention measures are incorporated into a nonlinear system of differential equations modelling illicit drug use population dynamics. The first measure aims to prevent the influence of drug users on non drug users while the second intervention measure is concerned with behavioral therapy. The control problem is analyzed using a popular Pontryagin's maximum principle to find the necessary condition for optimum solution. Assessment of single implementation of each the two measures and combination of both intervention measures are conducted to investigate the dynamic behaviour of illicit drug users in the population. Moreover, cost-effectiveness analysis based on incremental cost-effectiveness ratio is conducted to find the most cost effective intervention measure capable of averting good numbers of illicit drug users in the population at lowest cost. The results show that combination of both control variables reduce the spread of drug abuse mostly. However, single implementation of preventive control is the most cost-effective intervention.

Acknowledgments

\noindent Authors gratefully thank the handling editor and all the anonymous reviewers for their excellent comments and suggestions which helped in improving the quality of the manuscript.

References

1. [1]	United Nations Organizations on Drug Council (UNODC) (2005), World health organization expert committee on dependence producing drugs. Fourteenth report urban adolescents, Child Dev 61, 2032-2046.

2. [2]	World Health Organization~(WHO) (2019), Biregional Strategy for Harm Reduction 2005-2009:~HIV and Injecting Drug Use, Manila: WHO Regional Office for the Western Pacific, 2005.

3. [3]	Okoye, N.N. (2001), The adolescents and hard drugs: a psychological concern in R.U.N, in Okonkwo and R.O. Okoye (eds). The Nigerian adolescent in perspective, a publication of the Nigerian Society for Education.

4. [4]	Balogun, S.K. (2006), Chronic intake of separate and combined alcohol and nicotine on body maintenance among albinorats, Journal of Human Ecology, 19(1), 21-24.

5. [5]	Fawa, M.S. (2003), Drug abuse eradication programme in Schools: the relevance of team, approach alternative, in A. Garba (Ed) Youth and Drug Abuse in Nigeria, Strategies for Counselling, Management and Control, Matasa Press, Kano.

6. [6]	National Drug Law Enforcement Agency (NDLEA), Drug Data Collection and Research, Lagos: Drug Demand Reduction Unit, National Drug Law Enforcement Agency.

7. [7]	Okafor, I.P. (2020), Causes and consequences of drug abuse among youth in Kwara state, Nigeria, Canadian Journal of Family and Youth/Le Journal Canadien de Famille et de la Jeunesse,, 12(1), 147-162.

8. [8]	Rossi, C. (2012), The role of dynamic modelling in drug abuse epidemiology, Bulletin on Narcotics, 54(1-2), 33-44.

9. [9]	Alkhudhari, Z., Al-Sheikh, S., and Al-Tuwairqi, S. (2014), Stability analysis of a giving up smoking model, International Journal of Applied Mathematics Research, 3(2), 168-177.

10. [10]	Pang, L., Zhao, Z., Liu, S., and Zhang, X. (2015), A mathematical model approach for tobacco control in China, Applied Mathematics and Computation, 259, 497-509.

11. [11]	Nyabadza, F. and Coetzee, L. (2017), A systems dynamic model for drug abuse and drug-related crime in the western cape province of South Africa, Computational and Mathematical Methods in Medicine, 2017, 1-13.

12. [12]	Jung, J.H., Park, A., and Jung, I.H. (2018), Qualitative and sensitivity analysis of the effect of electronic cigarettes on smoking cessation, Computational and Mathematical Methods in Medicine, 2018, 1-11.

13. [13]	Akanni, J.O., Olaniyi, S., and Akinpelu, F.O. (2021), Global asymptotic dynamics of a nonlinear illicit drug use system, Journal of Applied Mathematics and Computing , 66(1), 39-60.

14. [14]	Khan, S.A., Shah, K., Zaman, G., and Jarad, F. (2019), Existence theory and numerical solutions to smoking model under Caputo-Fabrizio fractional derivative, Chaos: An Interdisciplinary Journal of Nonlinear Sciences, 29(1), 013128.

15. [15]	Saha, S. and Samanta, G.P. (2019), Synthetic drugs transmission: stability analysis and optimal control, Letters in Biomathematics, 6(2), 1-31.

16. [16]	Srivastav, A.K., Athithan, S., and Ghosh, M. (2020), Modelling and analysis of crime prediction and prevention, Social Network Analysis and Mining, 10, 26.

17. [17]	Akanni, J.O., Akinpelu, F.O., Olaniyi, S., Oladipo, A.T., and Ogunsola, A.W. (2020), Modelling financial crime population dynamics: optimal control and cost-effectiveness analysis, International Journal of Dynamics and Control, 8(2), 531-544.

18. [18]	Olaniyi, S. and Obabiyi, O.S. (2014), Qualitative analysis of malaria dynamics with nonlinear incidence function, Applied Mathematical Sciences, 8(78), 3889-3904.

19. [19]	Akanni, J.O. (2020), Asymptotic stability of illicit drug dynamics with banditry compartment, Applied Mathematics $\&$ Information Sciences, 14(5), 791-800.

20. [20]	Athithan, S., Ghosh M., and Li, X.Z. (2018), Mathematical modeling and optimal control of corruption dynamics, Asian-European Journal of Mathematics, 11(6), 1850090.

21. [21]	Mushayabasa, S. and Tapedzesa, S. (2005), Modeling illicit drug use dynamics and its optimal control analysis, Computational and Mathematical Methods in medicine, 2015, 1-11.

22. [22]	Olaniyi, S., Okosun, K.O., Adesanya, S.O., and Areo, E.A. (2018), Global stability and optimal control analysis of malaria dynamics in the presence of human travelers, The Open Infectious Diseases Journal, 10(1), 166-186.

23. [23]	Abimbade, S.F., Olaniyi, S., Ajala, O.A., and Ibrahim M.O. (2020), Optimal control analysis of a tuberculosis model with exogenous re-infection and incomplete treatment, Optimal Control Applications and Methods, 41(6), 2349-2368.

24. [24]	Ghosh, M., Olaniyi, S., and Obabiyi, O.S. (2020), Mathematical analysis of reinfection and relapse in malaria dynamics, Applied Mathematics and Computation, 373, 125044.

25. [25]	Goswami, N.K. and Shanmukha, B. (2020), Modeling and analysis of symptomatic and asymptomatic infections of Zika virus disease with non-monotonic incidence rate, Applied Mathematics $\&$ Information Sciences, 14(4), 655-671.

26. [26]	Okosun, K.O. (2020), On the dynamics malaria-dysentery co-infection model, Journal of Biological Systems, 28(2), 453-474.

27. [27]	Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., and Mishchenko, E.F. (1962), The Mathematical Theory of Optimal Processes, Wiley, New York.

28. [28]	Fleming, W.H. and Richel , R.W. (1975), Deterministic and Stochastic Optimal Control, Springer, New York.

29. [29]	Olaniyi, S., Okosun, K.O., Adesanya, S.O., and Lebelo, R.S. (2020), Modelling malaria dynamics with partial immunity and protected travellers: optimal control and cost-effectiveness analysis, Journal of Biological Dynamics, 14(1), 90-115.

30. [30]	Adepoju, O.A. and Olaniyi, S. (2021), Stability and optimal control of a disease model with vertical transmission and saturated incidence, Scientific African, 12, e00800

31. [31]	Ademosu, J.A., Olaniyi, S., and Adewale, O.S. (2021), Stability analysis and optimal measure for controlling eco-epidemiological dynamics of prey-predator model, Advances in Systems Science and Applications, 21(2), 83-103.

32. [32]	Lenhart, S. and Workman J.T. (2007), Optimal Control Applied to Biological Models, Chapman and Hall/CRC, London.

33. [33]	Abidemi, A. and Aziz, N.A.B. (2020), Optimal control strategies for dengue fever spread in Johor, Malaysia, Computer Methods and Programs in Biomedicine, 196, 105585.

34. [34]	Agusto, F.B. and ELmojtaba, I.M. (2017), Optimal control and cost-effective analysis of malaria/visceral leishmaniasis co-infection, Plos One, 12(2), e0171102.

35. [35]	Asamoah, J.K.K., Jin, Z., and Sun. G.Q. (2021), Non-seasonal and seasonal relapse model for Q fever disease with comprehensive cost-effectiveness analysis, Results in Physics, 22, 103889.

36. [36]	Okyere, E., Olaniyi, S., and Bonyah, E. (2020), Analysis of Zika virus dynamics with sexual transmission route using multiple optimal controls, Scientific African, 9, e00532.

37. [37]	Olaniyi, S., Obabiyi, O.S., Okosun, K.O., Oladipo, A.T., and Adewale, S.O. (2020), Mathematical modelling and optimal cost-effective control of COVID-19 transmission dynamics, The European Physical Journal Plus, 135(1), 938.

38. [38]	Tilahun, G.T., Makinde, O.D., and Malonza, D. (2017), Modelling and optimal control of typhoid fever spread disease with cost-effectiveness strategies, Computational and Mathematical Methods in Medicine, 2017, 1-16.

39. [39]	Abdo, M.S., Shah, K., Wahash, H.A., and Panchal, S.K. (2020), On a comprehensive model of the novel coronavirus (COVID-19) under Mittag-Leftler derivative, Chaos, Solitons and Fractals, 135, 109867.

40. [40]	Das, M., Samanta, G.P., and De la Sen, M. (2021), Stability analysis and optimal control of a fractional order synthetic drugs transmission model, Mathematics, 9(7), 703.