Primary and secondary substance use in the Western Cape Province of South Africa: A mathematical modelling approach

Abstract Substance abuse is an issue of considerable concern due to its devastating effects to families and communities in many countries. We formulate a mathematical model for multiple substance abuse using nonlinear ordinary differential equations. The model includes two classes of individuals in the substance using phase, namely, those using the primary substance of abuse only and those using the primary substance of abuse together with additional substance(s). An epidemic threshold value, , termed the initiation reproduction number, is proposed and defined herein in the substance-using context. It has been shown that the substance-free equilibrium is locally asymptotically stable when and the system has a unique positive substance-persistent equilibrium if and only if . The least-squares curve fit routine (lsqcurvefit) in Matlab with optimization has been used to fit the model to data on primary and secondary substance abusers in treatment centres of Cape Town, South Africa. The results indicate a continued decrease in the proportion of primary substance abusers in Cape Town whereas secondary substance abusers shall continue to increase but steadily. Numerical results suggest that there is need to increase treatment efforts targeting secondary substance abusers as this will have more impact in the fight against substance abuse.