Analysis of a Diffusive Heroin Epidemic Model in a Heterogeneous Environment

This paper is concerned with a reaction-diffusion heroin model in a bound domain. The objective of this paper is to explore the threshold dynamics based on threshold parameter and basic reproduction number (BRN) ℜ0, and it is proved that if ℜ0<1, heroin spread will be extinct, while if ℜ0>1, heroin spread is uniformly persistent and there exists a positive heroin-spread steady state. We also obtain that the explicit formula of ℜ0 and global attractiveness of constant positive steady state (PSS) when all parameters are positive constants. Our simulation results reveal that compared to the homogeneous setting, the spatial heterogeneity has essential impacts on increasing the risk of heroin spread.

Numerical treatment of stochastic heroin epidemic model

We have presented the numerical analysis of a stochastic heroin epidemic model in this paper. The mean of stochastic heroin model is itself a deterministic solution. The effect of reproduction number has also been observed in the stochastic heroin epidemic model. We have developed some stochastic explicit and implicitly driven explicit methods for this model. But stochastic explicit methods have flopped for certain values of parameters. In support, some theorems and graphical illustrations are presented.

Global dynamics of a heroin epidemic model with age structure and nonlinear incidence

A heroin model with nonlinear incidence rate and age structure is investigated. The basic reproduction number is determined whether or not a heroin epidemic breaks out. By employing the Lyapunov functionals, the drug-free equilibrium is globally asymptotically stable if [Formula: see text]; while the drug spread equilibrium is also globally asymptotically stable if [Formula: see text]. Our results imply that improving detected rates and drawing up the efficient prevention play more important role than increasing the treatment for drug users.