The NERA model incorporating cellular automata approach and the analysis of the resulting induced stochastic mean field

Abstract

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.