A stochastic process depicting the spreading dynamics of illicit drug consumption in a given population forms the crux of this work. A probabilistic cellular automaton (PCA) model is developed to examine the effects of the social interactions between nonusers and drug users. The model, called NERA, comprises four classes of individuals, namely, nonuser (N), experimental user (E), recreational user (R), and addict user (A). The stochastic process evolves in time by local transition rules. By means of dynamical simple mean field approximation, a nonlinear system of differential equations illustrating the dynamics of the PCA is obtained. The existence and uniqueness of a positive solution of the model is established, and the fixed points of the system are sought to perform the stability analysis. Furthermore, a stochastic mean field (SMF) approach to the NERA system is introduced. SMF extends the latter model to integrate the stochastic behaviour of drug consumers in a given environment. The SMF system is shown to exhibit a unique global solution which is stochastically ultimately bounded. Simulations of the cellular automaton and mean field analysis are used to study the evolution of the model. Verification and validation are carried out using data available on the consumption of cannabis in the state of Washington (Darnell and Bitney in \({I}-502\) evaluation and benefit-cost analysis: second required report, Washington state institute for public policy. Technical Report, 2017). These numerical experiments confirm that the NERA model can help in the analysis and quantification of the spatial dynamics of illicit drug usage in a given society and eventually provide insight to policy-makers on different steps to be taken to curb this social epidemic.

随机过程描述了给定人群中非法药物消费的传播动态，这是这项工作的关键。开发了概率细胞自动机（PCA）模型来检查非使用者和毒品使用者之间的社交互动的影响。称为NERA的模型包含四类个人，即非用户（N），实验用户（E），娱乐用户（R）和上瘾者（A）。随机过程会根据本地过渡规则随时间变化。通过动态简单平均场逼近，获得了说明PCA动力学的微分方程非线性系统。建立了模型正解的存在性和唯一性，并寻求系统的固定点进行稳定性分析。此外，介绍了一种用于NERA系统的随机平均场（SMF）方法。SMF扩展了后一种模型，以整合给定环境中吸毒者的随机行为。SMF系统显示出独特的全局解决方案，该解决方案随机地最终受到限制。利用元胞自动机的仿真和均值场分析来研究模型的演化。核查和确认使用的是华盛顿州大麻消费量的可用数据（达内尔和比尼 利用元胞自动机的仿真和均值场分析来研究模型的演化。核查和确认使用的是华盛顿州大麻消费量的可用数据（达内尔和比尼 利用元胞自动机的仿真和均值场分析来研究模型的演化。核查和确认使用的是华盛顿州大麻消费量的可用数据（达内尔和比尼\（{I} -502 \）评估和收益成本分析：第二份必要报告，华盛顿州公共政策研究所。技术报告，2017）。这些数值实验证实，NERA模型可以帮助分析和量化特定社会中非法药物使用的空间动态，并最终为决策者提供遏制此社会流行病的不同步骤的见解。