A mathematical model for the cannabis epidemic in a South African province with a non-linear incidence rate

Abstract

A deterministic mathematical model for the dynamics of cannabis use in a South Africa metropolis of Durban is proposed and analysed. To the analysis the model, the important threshold parameter ℛ0 (the basic reproduction number), is determined. It is proved that the model exhibits multiple cannabis persistent equilibria. For ℛ0 < 1, the model exhibits a backward bifurcation due to double exposure to cannabis sources and re-addiction in the population. More precisely, the locally asymptotically stable cannabis-free equilibrium co-exists with the locally asymptotically stable cannabis persistent equilibrium. Under this situation, the cannabis consumption will remain stay in the population even though ℛ0 < 1. In the absence of double exposure and re-addiction, it is shown that the cannabis-free equilibrium is globally asymptotically stable (GAS) when ℛ0 < 1, while the cannabis persistent equilibrium is GAS when ℛ0 > 1. The model is fitted into the available data and the values of the parameters involved in the model formulation are estimated. Sensitivity analysis of the model, using the parameters involved in the formulation of ℛ0 , is given. Numerical simulation to support the theoretical analysis of the model is provided.