In this paper, a nonlinear mathematical model of illicit drug use in a population is studied using dynamical system theory. The work is largely concerned with the analysis of asymptotic behaviour of solutions to a six-dimensional system of differential equations modeling the influence of illicit drug use in the population. The model is mathematically well-posed based on positivity and boundedness of solutions. A key threshold which measures the potential spread of the illicit drug use in the population is derived analytically. The model is shown to exhibit forward bifurcation property, implying the existence, uniqueness and local stability of an illicit drug-present equilibrium. Furthermore, the global asymptotic dynamics of the model around the illicit drug-free and drug-present equilibria are extensively investigated using appropriate Lyapunov functions. Numerical simulations are carried out to complement the obtained theoretical results, and to examine the effects of some parameters, such as influence rate, rehabilitation rates of drug users and relapse rate, on the dynamical spread of illicit drug use in the population. Measures to guide against the menace of the illicit drug use are suggested.

本文利用动力学系统理论研究了人口非法使用毒品的非线性数学模型。这项工作主要涉及分析微分方程的六维系统解的渐近行为，该系统模拟了非法药物使用对人群的影响。该模型在数学上基于正解和解的有界性。分析得出了衡量非法药物在人群中潜在传播程度的关键阈值。该模型显示出正向的分叉性质，这意味着非法药物存在平衡的存在，唯一性和局部稳定性。此外，使用适当的Lyapunov函数广泛研究了模型的围绕无毒和存在药物的非法平衡的全局渐近动力学。进行了数值模拟，以补充所获得的理论结果，并检验某些参数（如影响率，吸毒者的康复率和复发率）对人口中非法药物动态扩散的影响。建议采取一些措施来制止危害非法药物使用的威胁。