Abstract:
Substance abuse in Botswana is a growing problem with a variety of drugs being abused
including cocaine, heroin, methamphetamine, marijuana and alcohol among others. At the
front line of attempting to address the drug abuse problem is the Botswana Substance Abuse
and Support Network (BoSASNet) which operates outpatient rehabilitation and support ser vices that constitute its clinical programme. Due to the criminalisation and stigma associated
with drug use and dealing in drugs, the number of individuals who seek rehabilitation from
BoSASNet is believed to be much lower than the actual number of drug users who need the
service. We developed mathematical models to; analyse the dynamics of substance abuse
in Botswana, optimal control of substance in Botswana and fractional order model of sub stance abuse in Botswana to account for dependence of future values of drug users on the
present and previous states. In this study, we derived three basic reproduction numbers R0,
ROP
0 and RFDE
0 where R0 is for the model with multiple amelioration stages and out-patient
rehabilitation, ROP
0 is for optimal control model and RFDE
0 is for the fractional-order model.
The developed model has a globally asymptotically stable drug free equilibrium when the
drug use threshold is less than unity. Sensitivity analysis of the parameters to the model
output was carried out using the Latin hypercube Sampling scheme. Our results show that,
the parameters with the highest positive partial rank correlation coefficients (PRCCs) are pa rameters related to contact between potential drug users and individuals using drugs. The
parameters related to amelioration processes, quitting and rehabilitation were observed to
have the highest negative PRCCs. Therefore, if the processes described by parameters with
negative PRCCs are enhanced, it increases the likelihood of containing the drug epidemic.
We recommend embarking on inpatient rehabilitation as this prevents contact between drug
users in treatment and susceptible individuals reducing initiations, and accelerating quitting
due to reduced accesses to substances. Our results in the optimal control model show that;
if we implement the controls on the susceptible population, light drug users and accelerate
the quitting rate from rehabilitation, then drug use can be contained. Our assessment in frac tional order model shows that, if the contact rate is reduced at a very high fractional order
value, then the progression rate of drug users can be reduced/contained. A high fractional order of the value 1 predicts high peaks of the drug users in the short term but low values in
the long term dynamics. The reverse is observed for reduced fractional-orders.
