Acta Biotheoretica

, Volume 65, Issue 1, pp 37–61 | Cite as

On the Role of Imitation on Adolescence Methamphetamine Abuse Dynamics

  • J. Mushanyu
  • F. Nyabadza
  • G. Muchatibaya
  • A. G. R. Stewart
Regular Article


Adolescence methamphetamine use is an issue of considerable concern due to its correlation with later delinquency, divorce, unemployment and health problems. Understanding how adolescents initiate methamphetamine abuse is important in developing effective prevention programs. We formulate a mathematical model for the spread of methamphetamine abuse using nonlinear ordinary differential equations. It is assumed that susceptibles are recruited into methamphetamine use through imitation. An epidemic threshold value, \({\mathcal {R}}_a\), termed the abuse reproduction number, is proposed and defined herein in the drug-using context. The model is shown to exhibit the phenomenon of backward bifurcation. This means that methamphetamine problems may persist in the population even if \({\mathcal {R}}_a\) is less than one. Sensitivity analysis of \({\mathcal {R}}_a\) was performed to determine the relative importance of different parameters in methamphetamine abuse initiation. The model is then fitted to data on methamphetamine users less than 20 years old reporting methamphetamine as their primary substance of abuse in the treatment centres of Cape Town and parameter values that give the best fit are chosen. Results show that the proportion of methamphetamine users less than 20 years old reporting methamphetamine as their primary substance of abuse will continue to decrease in Cape Town of South Africa. The results suggest that intervention programs targeted at reducing adolescence methamphetamine abuse, are positively impacting methamphetamine abuse.


Methamphetamine Imitation Adolescence Reproduction number Least squares curve fitting 



The authors would like to thank the anonymous referees for their helpful comments and suggestions. J. Mushanyu, G. Muchatibaya and A.G.R. Stewart authors acknowledge, with thanks, the support of the Department of Mathematics, University of Zimbabwe. J. Mushanyu acknowledges financial support for his Doctor of Philosophy studies from the Deutscher Akademischer Austausch Dienst (DAAD) in-country scholarship. F. Nyabadza acknowledges with gratitude the support from National Research Foundation and Stellenbosch University for the production of this manuscript.


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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • J. Mushanyu
    • 1
  • F. Nyabadza
    • 2
  • G. Muchatibaya
    • 1
  • A. G. R. Stewart
    • 1
  1. 1.Department of MathematicsUniversity of ZimbabweMount Pleasant, HarareZimbabwe
  2. 2.Department of Mathematical SciencesStellenbosch UniversityMatielandSouth Africa

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