Modelling and qualitative analysis of an illicit drugs model with saturated incidence rate and relapse

Volume 28, Issue 4, pp 373--392 http://dx.doi.org/10.22436/jmcs.028.04.06

Publication Date: July 26, 2022 Submission Date: September 26, 2021 Revision Date: February 11, 2022 Accteptance Date: May 20, 2022

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Authors

K. P. Tan - Faculty of Ocean Engineering Technology and Informatics, Universiti Malaysia Terengganu, 21030 Kuala Nerus, Terengganu, Malaysia. A. A. Mat Daud - Faculty of Ocean Engineering Technology and Informatics, Universiti Malaysia Terengganu, 21030 Kuala Nerus, Terengganu, Malaysia.

Abstract

Drug abuse is now regarded as a global issue that brings severe consequences on the health, social well-being, and economy. In this study, an illicit drugs model with saturated incidence rate and relapse of individuals who quit using drugs is proposed and analyzed qualitatively. This study aims to determine the behavior of a drug epidemic when the psychological or inhibitory effect and relapse are being considered as well as assist the policymakers in devising effective control measures. The basic reproduction number \(R_0\) is derived and used as a threshold parameter in the global stability analysis. It is found that the drug-free equilibrium is globally asymptotically stable when \(R_0\le 1\). This implies that we can eradicate a drug epidemic when the threshold \(R_0\) is less than or equal to one, irrespective of the initial population size of drug users. On the other hand, the drug persistent equilibrium is globally asymptotically stable when \(R_0>1\). This indicates that the phenomenon of drug use will remain in a community when the threshold \(R_0\) is greater than one, irrespective of the initial population size of drug users. Next, the sensitivity analysis is performed and the results show that the effective contact rate should be targeted to reduce its value. The numerical simulations are also carried out to illustrate the analytical results and investigate the relationship between the measure of psychological or inhibitory effect and the number of drug users.

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Keywords

• Illicit drugs model
• saturated incidence rate
• relapse
• global stability analysis
• sensitivity analysis

MSC

•  00A71
•  34C60
•  34A34

References

• [1] L. J. Allen, An introduction to mathematical biology, Prentice Hall, Upper Saddle River (2007)
  • View Article
  • Google Scholar


• [2] N. Chitnis, J. M. Hyman, J. M. Cushing, Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model, Bull. Math. Biol., 70 (2008), 1272--1296
  • View Article
  • MathSciNet
  • MATH


• [3] M. A. Crocq, Historical and Cultural Aspects of Man’s Relationship with Addictive Drugs, Dialogues Clin. Neurosci., 9 (2007), 355--361
  • View Article
  • Google Scholar


• [4] D. M. Hamby, A review of techniques for parameter sensitivity analysis of environmental models, Environ. Monit. Assess., 32 (1994), 135--154
  • View Article
  • Google Scholar


• [5] Z. Hu, W. Ma, S. Ruan, Analysis of SIR epidemic models with nonlinear incidence rate and treatment, Math. Biosci., 238 (2012), 12--20
  • View Article
  • Google Scholar


• [6] International Narcotics Control Board, Chapter 1: Economic Consequences of Drug Abuse, INCB Report, 2013 (2013), 6 pages
  • View Article


• [7] A. S. Kalula, F. Nyabadza, A Theoretical Model for Substance Abuse in the Presence of Treatment, S. Afr. J. Sci., 108 (2012), 96--107
  • View Article
  • Google Scholar


• [8] J. H. Liu, A first course in the qualitative theory of differential equations, Prentice Hall, Upper Saddle River (2003)
  • MathSciNet
  • Google Scholar


• [9] X. B Liu, L. J. Yang, Stability analysis of an SEIQV epidemic model with saturated incidence rate, Nonlinear Anal. Real World Appl., 13 (2012), 2671--2679
  • View Article
  • MathSciNet


• [10] F. Luo, M. Li, C. Florence, State-level economic costs of opioid use disorder and fatal opioid overdose–United States, Morb. Mortal. Wkly. Rep., 70 (2021), 541--552
  • View Article
  • Google Scholar


• [11] M. J. Ma, S. Y. Liu, J. Li, Does Media Coverage Influence the Spread of Drug Addiction?, Commun. Nonlinear Sci. Numer. Simul., 50 (2017), 169--179
  • View Article
  • MathSciNet


• [12] M. J. Ma, S. Y. Liu, H. Xiang, J. Li, Dynamics of synthetic drugs transmission model with psychological addicts and general incidence rate, Phys. A, 491 (2018), 641--649
  • View Article
  • MathSciNet


• [13] McDowell Group, The Economic Costs of Drug Misuse in Alaska, 2019 Update, (2020)
  • View Article


• [14] A. T. McLellan, D. C. Lewis, C. P. O’brien, H. D. Kleber, Drug Dependence, a Chronic Medical Illness: Implications for Treatment, Insurance, and Outcomes Evaluation, Jama, 284 (2000), 1689--1695
  • View Article
  • Google Scholar


• [15] Ministry of Home Affairs, Malaysia Country Report On Drug, Issues 2019, (2019)
  • View Article


• [16] G. Mulone, B. Straughan, A note on heroin epidemics, Math. Biosci., 218 (2009), 138--141
  • View Article
  • Google Scholar


• [17] J. Mushanyu, F. Nyabadza, A. G. R. Stewart, Modelling the trends of inpatient and outpatient rehabilitation for methamphetamine in the Western Cape province of South Africa, BMC Res. Notes, 8 (2015), 1--13
  • View Article
  • Google Scholar


• [18] S. Mushayabasa, G. Tapedzesa, Modeling Illicit Drug Use Dynamics and Its Optimal Control Analysis, Comput. Math. Methods Med., 2015 (2015), 11 pages
  • View Article
  • MathSciNet


• [19] H. J. B. Njagarah, F. Nyabadza, Modeling the Impact of Rehabilitation, Amelioration and Relapse on the Prevalence of Drug Epidemics, J. Biol. Systems, 21 (2013), 23 pages
  • View Article
  • MathSciNet


• [20] F. Nyabadza, S. D. Hove-Musekwa, From heroin epidemics to methamphetamine epidemics: Modelling substance abuse in a South African province, Math. Biosci., 225 (2010), 132--140
  • View Article
  • MathSciNet


• [21] 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
  • View Article
  • MathSciNet


• [22] S. Saha, G. P. Samanta, Synthetic drugs transmission: stability analysis and optimal control, Lett. Biomath., 6 (2019), 31 pages
  • View Article


• [23] Statista Research Department, Breakdown of drug offenders in Malaysia in 2020, by age bracket,, , 2021 ()
  • View Article


• [24] United Nations Office on Drugs and Crime, World Drug Report 2019, Booklet 1, , (2019)
  • View Article


• [25] United Nations Office on Drugs and Crime, World Drug Report 2021, Booklet 1, , (2021)
  • View Article


• [26] P. van den Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosc., 180 (2002), 29--48
  • View Article
  • Google Scholar


• [27] E. White, C. Comiskey, Heroin epidemics, treatment and ODE modelling, Math. Biosci., 208 (2007), 312--324
  • View Article
  • MathSciNet


• [28] S. A. Wirkus, R. J. Swift, A course in ordinary differential equations (2nd ed.), CRC Press, Boca Raton (2004)
  • View Article


• [29] X. Zhang, X. Liu, Backward Bifurcation of an Epidemic Model with Saturated Treatment Function, J. Math. Anal. Appl., 348 (2008), 433--443
  • View Article
  • MathSciNet