Optimal Control of a Fractional-Order New Psychoactive Substance Model

Abstract

In this paper, we develop a fractional-order model of new psychoactive substance (NPS) transmission. We first prove the non-negativity and boundedness of the model. Then, prevention function and treatment function are introduced into the model to establish the fractional-order optimal control model. The existence of the optimal control pair is proved, and the solution of the model is obtained. Finally, the numerical simulation of the optimal control model is carried out. The results show that the fractional model is helpful for us to analyze the NPS dynamics model more deeply. Through the analysis of three control strategies, the optimal control strategy is the combination of prevention and treatment measures. In addition, it is worth noting that this paper obtains a new insight that prevention is more effective than treatment in the early stage of NPS transmission, and the control strategy at this time is to invest in maximum prevention measures and moderate treatment measures. However, when the spread of NPS shows an epidemic trend, the influence of preventive measures is greatly reduced, and the control strategy at this time is to focus on increasing the investigation of addicts and improving the treatment rate of psychological addicts.