Analysis of illegal drug transmission model using fractional delay differential equations

Komal Bansal; Trilok Mathur; Narinderjit Singh Sawaran Singh; Shivi Agarwal; Komal Bansal; Trilok Mathur; Narinderjit Singh Sawaran Singh; Shivi Agarwal

doi:10.3934/math.20221000

AIMS Mathematics

2022, Volume 7, Issue 10: 18173-18193. doi: 10.3934/math.20221000

Previous Article Next Article

Research article Special Issues

Analysis of illegal drug transmission model using fractional delay differential equations

1.
Department of Mathematics, Birla Institute of Technology and Science, Pilani, India
2.
Faculty of Data Science and Information Technology, INTI International University, Persiaran Perdana BBN, Putra Nilai, Negeri, 71800, Nilai, Negeri Sembilan, Malaysia

Received: 20 June 2022 Revised: 14 July 2022 Accepted: 17 July 2022 Published: 11 August 2022
MSC : 00A71, 26A33, 92D30, 93C43

The global burden of illegal drug-related death and disability continues to be a public health threat in developed and developing countries. Hence, a fractional-order mathematical modeling approach is presented in this study to examine the consequences of illegal drug usage in the community. Based on epidemiological principles, the transmission mechanism is the social interaction between susceptible and illegal drug users. A pandemic threshold value ($ \Lambda $) is provided for the illegal drug-using profession, which determines the stability of the model. The Lyapunov function is employed to determine the stability conditions of illegal drug addiction equilibrium point of society. Finally, the proposed model has been extended to include time lag in the delayed illegal drug transmission model. The characteristic equation of the endemic equilibrium establishes a set of appropriate conditions for ensuring local stability and the development of a Hopf bifurcation of the model. Finally, numerical simulations are performed to support the analytical results.
- fractional differential equation,
- illegal drug addiction,
- stability analysis,
- delayed model,
- Hopf bifurcation
Citation: Komal Bansal, Trilok Mathur, Narinderjit Singh Sawaran Singh, Shivi Agarwal. Analysis of illegal drug transmission model using fractional delay differential equations[J]. AIMS Mathematics, 2022, 7(10): 18173-18193. doi: 10.3934/math.20221000

Related Papers:

Abstract

The global burden of illegal drug-related death and disability continues to be a public health threat in developed and developing countries. Hence, a fractional-order mathematical modeling approach is presented in this study to examine the consequences of illegal drug usage in the community. Based on epidemiological principles, the transmission mechanism is the social interaction between susceptible and illegal drug users. A pandemic threshold value ($ \Lambda $) is provided for the illegal drug-using profession, which determines the stability of the model. The Lyapunov function is employed to determine the stability conditions of illegal drug addiction equilibrium point of society. Finally, the proposed model has been extended to include time lag in the delayed illegal drug transmission model. The characteristic equation of the endemic equilibrium establishes a set of appropriate conditions for ensuring local stability and the development of a Hopf bifurcation of the model. Finally, numerical simulations are performed to support the analytical results.

References

[1]	UNODC, World drug report, United Nations Office on Drugs and Crime, 2016.
[2]	WHO, HIV drug resistance surveillance guidance, World Health Organization, 2016.
[3]	A. Labzai, A. Kouidere, B. Khajji, O. Balatif, M. Rachik, Mathematical modeling and optimal control strategy for a discrete time drug consumption model, Discrete Dyn. Nat. Soc., 2020 (2020), 5671493. http://doi.org/10.1155/2020/5671493 doi: 10.1155/2020/5671493
[4]	F. Guerrero, F. J. Santonja, R. J. Villanueva, Analysing the Spanish smoke-free legislation of 2006: A new method to quantify its impact using a dynamic model, Int. J. Drug Policy, 22 (2011), 247–251. http://doi.org/10.1016/j.drugpo.2011.05.003 doi: 10.1016/j.drugpo.2011.05.003
[5]	Z. Y. Hu, Z. D. Teng, H. J. Jiang, Stability analysis in a class of discrete SIRS epidemic models, Nonlinear Anal.-Real, 13 (2012), 2017–2033. http://doi.org/10.1016/j.nonrwa.2011.12.024 doi: 10.1016/j.nonrwa.2011.12.024
[6]	J. B. H. Njagarah, F. Nyabadza, Modelling the role of drug barons on the prevalence of drug epidemics, Math. Biosci. Eng., 10 (2013), 843–860. http://doi.org/10.3934/mbe.2013.10.843 doi: 10.3934/mbe.2013.10.843
[7]	A. Labzai, O. Balatif, M. Rachik, Optimal control strategy for a discrete time smoking model with specific saturated incidence rate, Discrete Dyn. Nat. Soc., 2018 (2018), 5949303. http://doi.org/10.1155/2018/5949303 doi: 10.1155/2018/5949303
[8]	O. Latif, A. Labzai, M. Rachik, A discrete mathematical modeling and optimal control of the electoral behavior with regard to a political party, Discrete Dyn. Nat. Soc., 2018 (2018), 9649014. http://doi.org/10.1155/2018/9649014 doi: 10.1155/2018/9649014
[9]	M. J. Ma, S. Y. Liu, H. Xiang, J. Li, Dynamics of synthetic drugs transmission model with psychological addicts and general incidence rate, Physica A, 491 (2018), 641–649. http://doi.org/10.1016/j.physa.2017.08.128 doi: 10.1016/j.physa.2017.08.128
[10]	F. Nyabadza, J. B. H. Njagarah, R. J. Smith, Modelling the dynamics of crystal meth('tik') abuse in the presence of drug-supply chains in South Africa, Bull. Math. Biol., 75 (2013), 24–48. http://doi.org/10.1007/s11538-012-9790-5 doi: 10.1007/s11538-012-9790-5
[11]	P. Y. Liu, L. Zhang, Y. F. Xing, Modelling and stability of a synthetic drugs transmission model with relapse and treatment, J. Appl. Math. Comput., 60 (2019), 465–484. http://doi.org/10.1007/s12190-018-01223-0 doi: 10.1007/s12190-018-01223-0
[12]	S. Sangeeta, G. P. Samanta, Synthetic drugs transmission, Lett. Biomath., 6 (2019), 1–31. http://doi.org/10.30707/LiB6.2Saha doi: 10.30707/LiB6.2Saha
[13]	M. Das, G. P. Samanta, A prey-predator fractional order model with fear effect and group defense, Int. J. Dyn. Control, 9 (2021), 334–349. http://doi.org/10.1007/s40435-020-00626-x doi: 10.1007/s40435-020-00626-x
[14]	M. Das, G. P. Samanta, Stability analysis of a fractional ordered COVID-19 model, Comput. Math. Biophys., 9 (2021), 22–45. http://doi.org/10.1515/cmb-2020-0116 doi: 10.1515/cmb-2020-0116
[15]	M. Das, G. P. Samanta, Optimal control of fractional order COVID-19 epidemic spreading in Japan and India 2020, Biophys. Rev. Lett., 15 (2020), 207–236. http://doi.org/10.1142/S179304802050006X doi: 10.1142/S179304802050006X
[16]	K. S. Pritam, Sugandha, T. Mathur, S. Agarwal, Underlying dynamics of crime transmission with memory, Chaos Soliton. Fract., 146 (2021), 110838. http://doi.org/10.1016/j.chaos.2021.110838 doi: 10.1016/j.chaos.2021.110838
[17]	K. Bansal, S. Arora, K. S. Pritam, T. Mathur, S. Agarwal, Dynamics of crime transmission using fractional-order differential equations, Fractals, 30 (2022), 2250012. http://doi.org/10.1142/S0218348X22500128 doi: 10.1142/S0218348X22500128
[18]	G. S. Teodoro, J. A. T. Machado, E. C. de Oliveira, A review of definitions of fractional derivatives and other operators, J. Comput. Phys., 388 (2019), 195–208. https://doi.org/10.1016/j.jcp.2019.03.008 doi: 10.1016/j.jcp.2019.03.008
[19]	M. L. Du, Z. H. Wang, H. Y. Hu, Measuring memory with the order of fractional derivative, Sci. Rep., 3 (2013), 3431. https://doi.org/10.1038/srep03431 doi: 10.1038/srep03431
[20]	K. Diethelm, Efficient solution of multi-term fractional differential equations using $P{(EC)}^{m}E$ methods, Computing, 71 (2003), 305–319. http://doi.org/10.1007/s00607-003-0033-3 doi: 10.1007/s00607-003-0033-3
[21]	M. Das, A. Maitis, G. P. Samanta, Stability analysis of a prey-predator fractional order model incorporating prey refuge, Ecol. Genet. Genomics, 7-8 (2018), 33–46. http://doi.org/10.1016/j.egg.2018.05.001 doi: 10.1016/j.egg.2018.05.001
[22]	M. Das, G. P. Samanta, A delayed fractional order food chain model with fear effect and prey refuge, Math. Comput. Simul., 178 (2020), 218–245. http://doi.org/10.1016/j.matcom.2020.06.015 doi: 10.1016/j.matcom.2020.06.015
[23]	K. Diethelm, A. D. Freed, The FracPECE subroutine for the numerical solution of differential equations of fractional order, Forsch. und wissenschaftliches Rechnen, 1998, 57–71.
[24]	B. P. Moghaddam, J. A. T. Machado, Extended algorithms for approximating variable order fractional derivatives with applications, J. Sci. Comput., 71 (2017), 1351–1374. http://doi.org/10.1007/s10915-016-0343-1 doi: 10.1007/s10915-016-0343-1
[25]	B. P. Moghaddam, Z. S. Mostaghim, Modified finite difference method for solving fractional delay differential equations, Bol. Soc. Parana. Mat., 35 (2017), 49–58. http://dx.doi.org/10.5269/bspm.v35i2.25081 doi: 10.5269/bspm.v35i2.25081
[26]	J. A. T. Machado, B. P. Moghaddam, A robust algorithm for nonlinear variable-order fractional control systems with delay, Int. J. Nonlinear Sci. Numer. Simul., 19 (2018), 231–238. http://doi.org/10.1515/ijnsns-2016-0094 doi: 10.1515/ijnsns-2016-0094
[27]	S. Yaghoobi, B. P. Moghaddam, K. Ivaz, An efficient cubic spline approximation for variable-order fractional differential equations with time delay, Nonlinear Dyn., 87 (2017), 815–826. https://doi.org/10.1007/s11071-016-3079-4 doi: 10.1007/s11071-016-3079-4
[28]	F. K. Keshi, B. P. Moghaddam, A. Aghili, A numerical approach for solving a class of variable-order fractional functional integral equations, Comput. Appl. Math., 37 (2018), 4821–4834. http://doi.org/10.1007/s40314-018-0604-8 doi: 10.1007/s40314-018-0604-8
[29]	Q. X. Zhu, T. W. Huang, control of stochastic networked control systems with time-varying delays: The event-triggered sampling case, Int. J. Robust Nonlinear Control, 31 (2021), 9767–9781. http://doi.org/10.1002/rnc.5798 doi: 10.1002/rnc.5798
[30]	Y. Zhao, Q.X. Zhu, Stabilization of stochastic highly nonlinear delay systems with neutral-term, IEEE T. Automat. Contr., 2022. http://doi.org/10.1109/TAC.2022.3186827 doi: 10.1109/TAC.2022.3186827
[31]	X. T. Yang, H. Wang, Q. X. Zhu, Event-triggered predictive control of nonlinear stochastic systems with output delay, Automatica, 140 (2022), 110230. http://doi.org/10.1016/j.automatica.2022.110230 doi: 10.1016/j.automatica.2022.110230
[32]	N. D. Volkow, T. K. Li, Drug addiction: The neurobiology of behaviour gone awry, Nat. Rev. Neurosci., 5 (2004), 963–970. http://doi.org/10.1038/nrn1539 doi: 10.1038/nrn1539
[33]	M. A. Crocq, Historical and cultural aspects of man's relationship with addictive drugs, Dialogues Clin. Neuro., 9 (2007), 355–361. http://doi.org/10.31887/DCNS.2007.9.4/macrocq doi: 10.31887/DCNS.2007.9.4/macrocq
[34]	M. Costantini, I. Meco, A. Paradiso, Do inequality, unemployment and deterrence affect crime over the long run? Reg. Stud., 52 (2018), 558–571. http://doi.org/10.1080/00343404.2017.1341626 doi: 10.1080/00343404.2017.1341626
[35]	S. Kundu, S. Maitra, Dynamics of a delayed predator-prey system with stage structure and cooperation for preys, Chaos Soliton. Fract., 114 (2018), 453–460. http://doi.org/10.1016/j.chaos.2018.07.013 doi: 10.1016/j.chaos.2018.07.013
[36]	X. Y. Meng, J. G. Wang, Analysis of a delayed diffusive model with Beddington–DeAngelis functional response, Int. J. Biomath., 12 (2019), 1950047. http://doi.org/10.1142/S1793524519500475 doi: 10.1142/S1793524519500475
[37]	Z. Z. Zhang, Y. G. Wang, Hopf bifurcation of a heroin model with time delay and saturated treatment function, Adv. Differ. Equ., 2019 (2019), 64. http://doi.org/10.1186/s13662-019-2009-4 doi: 10.1186/s13662-019-2009-4
[38]	I. Podlubny, Fractional differential equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, San Diego: Academic Press, 1999.
[39]	K. S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, New York: Wiley, 1993.
[40]	E. C. de Oliveira, J. A. T. Machado, A review of definitions for fractional derivatives and integral, Math. Probl. Eng., 2014 (2014), 238459. http://doi.org/10.1155/2014/238459 doi: 10.1155/2014/238459
[41]	A. Atangana, J. F. Gómez-Aguilar, Fractional derivatives with no-index law property: Application to chaos and statistics, Chaos Soliton. Fract., 114 (2018), 516–535. http://doi.org/10.1016/j.chaos.2018.07.033 doi: 10.1016/j.chaos.2018.07.033
[42]	M. D. Ortigueira, J. A. T. Machado, A critical analysis of the Caputo–Fabrizio operator, Commun. Nonlinear Sci. Numer. Simul., 59 (2018), 608–611. http://doi.org/10.1016/j.cnsns.2017.12.001 doi: 10.1016/j.cnsns.2017.12.001
[43]	K. Diethelm, R. Garrappa, A. Giusti, M. Stynes, Why fractional derivatives with nonsingular kernels should not be used, Fract. Calc. Appl. Anal., 23 (2020), 610–634. http://doi.org/10.1515/fca-2020-0032 doi: 10.1515/fca-2020-0032
[44]	M. Kurulay, M. Bayram, Some properties of the Mittag-Leffler functions and their relation with the Wright functions, Adv. Differ. Equ., 2012 (2012), 181. https://doi.org/10.1186/1687-1847-2012-181 doi: 10.1186/1687-1847-2012-181
[45]	K. X. Li, J. G. Peng, Laplace transform and fractional differential equations, Appl. Math. Lett., 24 (2011), 2019–2023. http://doi.org/10.1016/j.aml.2011.05.035 doi: 10.1016/j.aml.2011.05.035
[46]	P. A. Naik, J. Zu, K. M. Owolabi, Global dynamics of a fractional order model for the transmission of HIV epidemic with optimal control, Chaos Soliton. Fract., 138 (2020), 109826. http://doi.org/10.1016/j.chaos.2020.109826 doi: 10.1016/j.chaos.2020.109826
[47]	F. Mainardi, On some properties of the Mittag-Leffler function ${E}_\alpha (-t^\alpha)$, completely monotone for $t > 0$ with $0< \alpha< 1$, DCDS-B, 19 (2014), 2267–2278. http://doi.org/10.3934/dcdsb.2014.19.2267 doi: 10.3934/dcdsb.2014.19.2267
[48]	P. van den Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29–48. http://doi.org/10.1016/S0025-5564(02)00108-6 doi: 10.1016/S0025-5564(02)00108-6
[49]	E. Ahmed, A. M. A. El-Sayed, H. A. A. El-Saka, On some Routh–Hurwitz conditions for fractional order differential equations and their applications in Lorenz, Rössler, Chua and Chen systems, Phys. Lett. A, 358 (2006), 1–4. http://doi.org/10.1016/j.physleta.2006.04.087 doi: 10.1016/j.physleta.2006.04.087
[50]	D. Y. Chen, R. F. Zhang, X. Z. Liu, X. Y. Ma, Fractional order Lyapunov stability theorem and its applications in synchronization of complex dynamical networks, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 4105–4121. http://doi.org/10.1016/j.cnsns.2014.05.005 doi: 10.1016/j.cnsns.2014.05.005
[51]	A. Boukhouima, K. Hattaf, El M. Lotfi, M. Mahrouf, D. F. M. Torres, N. Yousfi, Lyapunov functions for fractional-order systems in biology: Methods and applications, Chaos Soliton. Fract., 140 (2020), 110224. http://doi.org/10.1016/j.chaos.2020.110224 doi: 10.1016/j.chaos.2020.110224
[52]	N. Aguila-Camacho, M. A. Duarte-Mermoud, J. A. Gallegos, Lyapunov functions for fractional order systems, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 2951–2957. http://doi.org/10.1016/j.cnsns.2014.01.022 doi: 10.1016/j.cnsns.2014.01.022
[53]	T. Orwa, F. Nyabadza, J. A. Conejero, Mathematical modelling and analysis of alcohol-methamphetamine co-abuse in the Western Cape province of South Africa, Cogent Math. Stat., 6 (2019), 1641175. http://doi.org/10.1080/25742558.2019.1641175 doi: 10.1080/25742558.2019.1641175

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)