A Risk-Structured Model for Understanding the Spread of Drug Abuse

  • J. MushanyuEmail author
  • F. Nyabadza
Original Paper


Drug abuse is an issue of considerable concern due to its correlation with negative effects such as delinquency, unemployment, divorce and health problems. Understanding the dynamics of drug abuse is important in developing effective prevention programs. We formulate a mathematical model for the spread of drug abuse using nonlinear ordinary differential equations. Susceptibility to drug use varies, due to differences in behavioral, social and environmental factors. Risk structure is included before initiation and after recovery to help differentiate those more likely (high risk) to abuse drugs and those less likely (low risk) to abuse drugs. The model allows back and forth transition between risk groups. It is shown that the model has multiple equilibria and using the center manifold theory, the model exhibits the phenomenon of backward bifurcation whose implications to rehabilitation are discussed. An epidemic threshold value, \({\mathcal {R}}_a\), termed the abuse reproduction number, is proposed and defined herein in the drug-using context. Sensitivity analysis of the abuse reproduction number and numerical simulations were performed. The results show that education about effective coping response and/or skills to deal with the risky situation may better equip individuals to stand against initiating or re-initiating into drug abuse.


Drug abuse Risk structure Reproduction number Treatment 



J. Mushanyu acknowledge, with thanks, the support of the Department of Mathematics, University of Zimbabwe. 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 (India) Private Ltd., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ZimbabweHarareZimbabwe
  2. 2.Department of Mathematical SciencesStellenbosch UniversityMatielandSouth Africa

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