A Mathematical Model of Substance Abuse–Driven Criminality: Exploring the Dynamics of Addiction and Crime

Year 2025, Volume: 8 Issue: 4, 201 - 217, 15.12.2025

Farai Nyabadza , Robert Kajambeu

https://doi.org/10.33187/jmsm.1731496

Abstract

Drug abuse and crime are deeply interconnected, forming a vicious cycle that exacerbates public health and criminal justice challenges. In South Africa’s Gauteng province, substance abuse remains a major socioeconomic burden with far-reaching consequences. This study develops a seven-compartment deterministic model using ordinary differential equations to analyze the dynamics between drug abuse and criminal activity. The model incorporates homogeneous population mixing and accounts for removal rates associated with drug-related crime and rehabilitation. Analytical results indicate two equilibrium states: a narcocriminality-free equilibrium and a persistent (endemic) equilibrium. This study establishes that the narcocriminality-free equilibrium is globally asymptotically stable when the drug abuse criminogenic growth number (DGN, $\mathcal{D}_0<1$), while the endemic equilibrium exists when $\mathcal{D}_0 > 1$. Sensitivity analysis identifies the initiation rate as the most influential parameter on $\mathcal{D}_0$, showing that $\mathcal{D}_0$ increases with the progression rates $\alpha$ (light drug users) and $\rho$ (heavy drug users). Conversely, $\mathcal{D}_0$ shows a decrease with higher incarceration rates ($\epsilon$, $\gamma_1$) and rehabilitation rates ($\gamma_2$). These findings have important policy implications related to early intervention strategies targeting the drug-crime cycle, and enhancing rehabilitation programs and incarceration efficacy to reduce drug-driven criminality.

Keywords

Drug abuse criminogenic growth number , Narcocriminality , Rehabilitation , Relapse rate , Sensitivity analysis , Simulations

References

• [1] T. O. Orwa, F. Nyabadza, Mathematical modelling and analysis of alcohol-methamphetamine co-abuse in the Western Cape Province of South Africa, Cogent Math. Stat., 6(1) (2019), Article ID 1641175. https://doi.org/10.1080/25742558.2019.1641175
• [2] J. Liebenberg, L. Du Toit-Prinsloo, V. Steenkamp, G. Saayman, Fatalities involving illicit drug use in Pretoria, South Africa, for the period 2003–2012, S. Afr. Med. J., 106(10) (2016), 1051–1055. https://doi.org/10.7196/SAMJ.2016.v106i10.11105
• [3] E. R. Bos, D. T. Jamison, F. Bainga, R. G. A. Feacham, M. Makgoba, K. J. Hofman et al., Disease and Mortality in Sub-Saharan Africa, World Bank, Washington, DC, 2006.
• [4] T. Jackson, A. Radunskaya, Applications of Dynamical Systems in Biology and Medicine, Vol. 158, Springer, New York, 2015.
• [5] K. K. Nantomah, B. Seidu, C. S. Bornaa, A mathematical model of drug-crime dynamics, Asian J. Math. Appl. 2022 (2022), Article ID 14.
• [6] Statista Research Department, Number of Drug-Related Crimes in South Africa 2013/2014–2023/2024, https://www.statista.com/statistics/1448515/number-of-drug-related-crimes-in-South-Africa/, Accessed: 12 December 2024.
• [7] S. Ramlagan, K. Peltzer, G. Matseke, Epidemiology of drug abuse treatment in South Africa, S. Afr. J. Psychiatry, 16(2) (2010), Article ID 172. https://doi.org/10.4102/sajpsychiatry.v16i2.172
• [8] 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. https://doi.org/10.1007/s11538-012-9790-5
• [9] A. S. Kalula, F. Nyabadza, A theoretical model for substance abuse in the presence of treatment, S. Afr. J. Sci., 108(3/4) (2012), 1–12. http://dx.doi.org/10.4102/sajs.v108i3/4.654
• [10] S. P. Gatyeni, Modelling in-and out-patient rehabilitation for substance abuse in dynamic environments, Ph.D. Thesis, University of Stellenbosch, 2015.
• [11] M. Ma, S. Liu, H. Xiang, J. Li, Dynamics of synthetic drugs transmission model with psychological addicts and general incidence rate, Physica A, 491 (2018), 641–649. https://doi.org/10.1016/j.physa.2017.08.128
• [12] B. Mataru, O. J. Abonyo, D. Malonza, Mathematical model for crimes in developing countries with some control strategies, J. Appl. Math., 2023(1) (2023), Article ID 8699882. https://doi.org/10.1155/2023/8699882
• [13] F. Nyabadza, L. A. Coetzee, A systems dynamic model for drug abuse and drug-related crime in the Western Cape Province of South Africa, Comput. Math. Methods Med., 2017(1) (2017), Article ID 4074197. https://doi.org/10.1155/2017/4074197
• [14] K. Manoj, A. Syed, Modelling and prevention of crime using age-structure and law enforcement, J. Math. Anal. Appl., 519(2) (2023), Article ID 126849. https://doi.org/10.1016/j.jmaa.2022.126849
• [15] A Kaur, M. Sadhwani, S. Abbas, Law enforcement: the key to a crime-free society, J. Math. Sociol., 46(4) (2022), 342–359. https://doi.org/10.1080/0022250X.2021.1941002
• [16] T. J. Prakash, B. Sarita, B. Kavita, A. Syed, Dynamical analysis and effects of law enforcement in a social interaction model, Physica A, 567 (2021), Article ID 125725. https://doi.org/10.1016/j.physa.2020.125725
• [17] Fatmawati, H. Tasman, A malaria model with controls on mass treatment and insecticide, Appl. Math. Sci., 7(68) (2013), 3379–3391. https://doi.org/10.12988/ams.2013.33180
• [18] Fatmawati, H. Tasman, An optimal treatment control of TB-HIV coinfection, Int. J. Math. Math. Sci., 2016 (2016), Article ID 8261208. https://doi.org/10.1155/2016/8261208
• [19] Fatmawati, H. Tasman, An optimal control strategy to reduce the spread of malaria resistance, Math. Biosci., 262 (2015), 73–79. https://doi.org/10.1016/j.mbs.2014.12.005
• [20] F. Nyabadza, S. D. Hove-Musekwa, From heroin epidemics to methamphetamine epidemics: Modelling substance abuse in a South African province, Math. Biosci., 225(2) (2010), 132–140. https://doi.org/10.1016/j.mbs.2010.03.002
• [21] M. Gossop, J. Marsden, D. Stewart, T. Kidd, The national treatment outcome research study (NTORS): 4–5 year follow-up results, Addiction, 98(3) (2003), 291–303. https://doi.org/10.1046/j.1360-0443.2003.00296.x
• [22] S. Belenko, M. Hiller, L. Hamilton, Treating substance use disorders in the criminal justice system, Curr. Psychiatry Rep., 15(11) (2013), Article ID 414. https://doi.org/10.1007/s11920-013-0414-z
• [23] T. L. Gasebonno, Modelling substance abuse in Botswana in presence of multiple amelioration stages and outpatient rehabilitation, optimal control and fractional-order dynamics, M.Sc. Thesis, Botswana International University of Science and Technology, 2020.
• [24] B. J. Bassingthwaighte, E. Butterworth, B. Jardine, G. M. Raymond, Compartmental modeling in the analysis of biological systems, In Computational Toxicology: Volume I, Springer, New York, 2012, pp. 391–438. https://doi.org/10.1007/978-1-62703-050-2 17
• [25] T. Decorte, Drug users’ perceptions of ’controlled’ and ’uncontrolled’ use, Int. J. Drug Policy, 12(4) (2001), 297–320. https://doi.org/10.1016/S0955-3959(01)00095-0
• [26] S. Abbas, J. P. Tripathi, A. A. Neha, Dynamical analysis of a model of social behavior: Criminal vs non-criminal population, Chaos Solitons Fractals, 98 (2017), 121–129. https://doi.org/10.1016/j.chaos.2017.03.027
• [27] A. Galassi, E. Mpofu, J. Athanasou, Therapeutic community treatment of an inmate population with substance use disorders: Post-release trends in rearrest, re-incarceration, and drug misuse relapse, Int. J. Environ. Res. Public Health, 12(6) (2015), 7059–7072. https://doi.org/10.3390/ijerph120607059
• [28] G. Birkhoff, G. C. Rota, Ordinary Differential Equations, Blaisdell Publishing Co., Waltham, 1969.
• [29] O. Diekmann, J. A. P. Heesterbeek, J. A. J. Metz, On the definition and the computation of the basic reproduction ratio 𝑅0 in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365–382. https://doi.org/10.1007/BF00178324
• [30] R. M. Anderson, R. M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, Oxford, 1991.
• [31] P. Van den Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180(1–2) (2002), 29–48. https://doi.org/10.1016/S0025-5564(02)00108-6
• [32] C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, CRC Press, Boca Raton, 1998.
• [33] D. I. Cartwright, M. J. Field, A refinement of the arithmetic mean-geometric mean inequality, Proc. Amer. Math. Soc., 71 (1978), 36–38. https://doi.org/10.1090/s0002-9939-1978-0476971-2
• [34] J. P. LaSalle, Stability theory for ordinary differential equations, J. Differ. Equations, 4(1) (1968), 57–65. https://dx.doi.org/10.1016/0022-0396(68)90048-X
• [35] N. I. Hamdan, A. Kilicman, Sensitivity analysis in a dengue fever transmission model: A fractional order system approach, J. Phys.: Conf. Ser., 1366(1) (2019), 012048. https://doi.org/10.1088/1742-6596/1366/1/012048
• [36] R. L. Iman, W. J. Conover, Small sample sensitivity analysis techniques for computer models, with an application to risk assessment, Commun. Stat., Theory Methods, 9(17) (1980), 1749–1842. https://doi.org/10.1080/03610928008827996
• [37] Crime Hub Data, Crime StatisticsWizard, https://crimehub.org/wizard#categories=c0fdb690-4174-4573-8f91-9e85c5f523dd&quarterly=false, Accessed October 2024.
• [38] ArcGIS Hub, Dataset: 24ced21ce3e04f4f80acd1393158d0c5, 2025. Available at: https://arc-gis-hub-home-arcgishub.hub.arcgis.com/datasets/24ced21ce3e04f4f80acd1393158d0c5/explore, Accessed October 2025.