A theoretical model for substance abuse in the presence of treatment

  • Asha Saidi Kalula University of Stellenbosch
  • Farai Nyabadza University of Stellenbosch
Keywords: core group model, backward bifurcation, relapse, treatment, stability, simulations


The production and use of addictive stimulants has been a major problem in South Africa. Although research has shown increased demand for drug abuse treatment, the actual size of the drug-abusing population remains unknown. Thus the prevalence of drug abuse requires estimation through available tools. Many questions remain unanswered with regard to interventions, new cases of substance abuse and relapse in recovering persons. A six-state compartmental model including a core and non-core group, with fast and slow progression to addiction, was formulated with the aim of qualitatively investigating the dynamics of substance abuse and predicting drug abuse trends. The analysis of the model was presented in terms of the substance abuse epidemic threshold R0. Numerical simulations were performed to fit the model to available data for methamphetamine use in the Western Cape and to determine the role played by some key parameters. The model was also fitted to data on methamphetamine users who enter rehabilitation using the least squares curve fitting method. It was shown that the model exhibits a backward bifurcation where a stable drug-free equilibrium coexists with a stable drug-persistent equilibrium for a certain defined range of values of R0. The stabilities of the model equilibria were ascertained and persistence conditions established. It was found that it is not sufficient to reduce R0 below unit to control the substance abuse epidemic. The reproduction number should be brought below a determined threshold, R0c. The results also suggested that the substance abuse epidemic can be reduced by intervention programmes targeted at light drug users and by increasing the uptake rate into treatment for those addicted. Projected trends showed a steady decline in the prevalence of methamphetamine abuse until 2015.


1. Bhatia NP, Szeg¨o GP, Stability theory of dynamical systems. Springer-Verlag, 1970.

2. Bhunu CP, Garira W, Mukandavire Z, Magombedze G, Modelling the effects of
pre-exposure and post-exposure vaccines in tuberculosis control, Journal of Theoretical
Biology, 254:633 (2008).

3. Burattini MN, Massad E, Coutinho FAB, A mathematical model of the impact of
crack-cocaine use on the prevalence of HIV/AIDS among drug users,
Mathematical and Computer Modelling, 28:21-29, (1998).

4. Castillo-Chavez C, Song B, Dynamical models of tuberculosis and their applications,
Mathematical Biosciences and Engineering, 1:361-404, (2004).

5. Cui J, Mu X, Wan H, Saturation recovery leads to multiple endemic equilibria and
backward bifurcation, Theoretical Biology, 254:275-283 (2008).

6. De Alarcon R, The spread of a heroin abuse in a community, Bulletin on Narcotics,
21:17-22, (1969).

7. Dushoff J, Incorporating immunological ideas in epidemiological models, Journal of
Theoretical Biology, 180:181 (1996).

8. Fenga Z, Castillo-Chavez C, Capurroe A, A model for tuberculosis with exogenous
Re-infection, Theoretical Population Biology, 57:235-247, (2000).

9. Garba S, Gumel A, Bakar M, Backward bifurcation in dengue transmission dynamics,
Mathematical Biosciences, 215:11-25, (2008).

10. K.P. Hadeler, C. Castillo-Chavez, Core group model for disease transmission,
Mathematical Bio- sciences, 128:41-55, (1995).

11. Hunt LG, Chambers CD, The heroin epidemics, New York: Spectrum Publications
Inc., 1976.

12. T.W. Lineberry, J.M. Bostwick, Methamphetamine abuse: a perfect storm of
complications, Mayo Clinic Proceedings, 81:77-84, (2006).

13.Mackintosh DR, Stewart GT, A mathematical model of a heroin epidemic: implications
for control policies, Journal of Epidemiology and Community Health, 33:299-304,
(1979) .

14.Mulone G, Straughan B, A note on heroin epidemics, Mathematical Biosciences,
218:138-141, (2009).

15.Nyabadza F, Hove-Musekwa SD, From heroin epidemics to methamphetamine
epidemics: Modelling substance abuse in a South African province, Mathematical
Biosciences, 225:132-140, (2010).

16.Parry CDH, Substance abuse intervention in South Africa, World Psychiatry, 4:34,

17.Rossi C, Operational models for the epidemics of problematic drug use: the mover –
stayer approach to heterogeneity, Socio-Economic Planning Sciences, 38:73-90, (2004).

18.Rossi C, The role of dynamic modelling in drug abuse epidemiology, Bulletin on
Narcotics, LIV:33- 44, (2002).

19.Sharomi O, Gumel AB, Curtailing smoking dynamics: A mathematical modelling
approach, Applied Mathematics and Computation, 195:475-499, (2008).

20. Sharomi O, Poddler CN, Gumel AB, et al., Role of incidence function in vaccine-
induced backward bifurcation in some HIV models, Mathematical Biosciences,
210:436-463, 2007.

21. van den Driessche P, Watmough J, Reproduction numbers and sub-threshold endemic
equilibria for compartmental models of disease transmission, Mathematical
Biosciences,180:29-48, (2002).

22. Wechsberg WM, Luseno WK, Karg RS et al., Alcohol, cannibis, and
methamphetamine use and other risk behaviours among black and coloured South
African Women: A small randomized trial in the Western Cape, International Journal
of Drug Policy, 19:130-139, (2008).

23. White E, Comiskey C, Heroin epidemics, treatment and ODE modelling, Mathematical
Biosciences, 208:312-324 (2007).