Dynamics of synthetic drug transmission models

Abstract

Abstract The deterministic and stochastic synthetic drug transmission models with relapse are formulated. For the deterministic model, the basic reproduction number R 0 is derived. We show that if R 0 < 1, the drug-free equilibrium is globally asymptotically stable and if R 0 > 1, there exists a unique drug-addition equilibrium which is globally asymptotically stable. For the stochastic model, we show there exists a unique global positive solution of the stochastic model for any positive initial value. Then by constructing some stochastic Lyapunov functions, we show that the solution of the stochastic model is going around each of the steady states of the corresponding deterministic model under certain parametric conditions. The sensitive analysis of the basic reproduction number R 0 indicates that it is helpful to reduce the relapse rate of people who have a history of drug abuse in the control of synthetic drug spreading. Numerical simulations are carried out and approve our results.