Mathematical model analysis of a drug transmission with criminal law and its optimal control

The Indonesian government has made various efforts in drug prevention policies through criminal law. However, law enforcement officials still contemplate the Drugs Law is oriented towards imprisonment, so drug abuse is considered as a criminal act. In fact, the government has declared 2014 as the year to save the victims of drug abuse through rehabilitation. In this paper, a drug transmission model with the criminal law aspect is presented and analyzed. The optimal control strategy is then applied in the form of rehabilitation efforts. Based on the model analysis, we found two equilibriums, namely the drugs-free equilibrium and the drug addiction equilibrium. The stability of the equilibriums depends on the basic reproduction number. The spread of drug addicts persists in the population if the basic reproduction number greater than unity. Based on the simulation, it can be seen that criminal law give a significant impact to decrease the number of mild drug addicts and also the number of heavy levels of drug addicts. Furthermore, the existence of the optimal control variable is determined through the Pontryagin's Maximum Principle method. The comparison of simulation results with and without control shows that rehabilitation efforts can reduce drug addicts transmission.