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.

A graph cellular automaton with relation-based neighbourhood describing the impact of peer influence on the consumption of marijuana among college-aged youths

“…is considered to simulate the cellular automaton. The model is run for a certain number of time steps, with the grid being updated at every time step t. Each cell "interacts" with cells in its neighbourhood and is updated according to a set of predefined rules [22]. Within this model, the stochastic rules for the transitions in the r-GCA are set up according to the following assumption:…”

“…In contrast to the model of [22], in the present r-GCA system, with each site of the lattice being occupied by only one user that can be in one of the following four states: nonuser (0), experimental user (1), recreational user (2) and addict user (3), an indicator variable is created to represent the age of each cell which is coded as 12, 13, • • • , 18. The age of each cell is updated after every 365 time steps (1 year).…”

“…There exists few works dealing with the use of cellular automata to simulate the prevalence of illicit drug use within a given population. The work of [22] is a notable example whereby a drug model based on cellular automata, endowed with the traditional topology using the Von Neumann neighbourhood, was presented. As mentioned in [16], the usual topologies of CAs are chains and regular lattices.…”

“…The present work aims at modelling illicit drug usage within a campus, using the relation-based graph-cellular automata (r-GCA) approach. The model proposed in this study can be considered as a continuation and improvement of the system in [22], whereby this study enables the modelling in the relational space where the neighbourhood corresponds to the relationships established between the users. A relation-based graph-cellular automata (r-GCA) is set up by local transition rules that are capable of describing the social interactions and influences among both nonusers and drug users.…”

An Analysis of the Recreational Use of Marijuana Amongst the 21+ Population of the State of Washington in the Context of I-502 and its Aftermath

“…Also, through dynamic models, researchers can simulate policy experiments that are not possible in real life due to practical reasons since performing experimentation on a human population is often impossible, unethical and expensive [21]. The work of [22], [23], [18] and [24] are notable examples where the authors have analysed this social epidemic using the NERA model. The main objective of the present work is to use the NERA model in order to assess the impact of I-502 on recreational marijuana consumption, in the state of Washington.…”