Modelling substance abuse in Botswana in presence of multiple amelioration stages and outpatient rehabilitation, optimal control and fractional-order dynamics

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.