We study the NERA model that describes the dynamic evolution of illicit drug usage in a population. The model consists of nonusers (N) and three categories of drug users: the experimental (E) category, the recreational (R) category and the addict (A) category. Two epidemic threshold terms known as the reproduction numbers, R0 and µ are defined and derived. Sensitivity analysis of R0 on the parameters are performed in order to determine their relative importance to illicit drug prevalence. The local and global stability of the equilibrium states are also analysed. We also prove that a transcritical bifurcation occurs at R0 = 1. It is shown that an effective campaign of prevention can help to fight against the prevalence of illicit drug consumption. We demonstrate persistence when R0 > 1 and conditions for the extinction of drug consumption are also established. Numerical simulations are performed to verify our model. Our results show that the NERA model can assist policy makers in targeting prevention for maximum effectiveness and can be used to adopt evidence-based policies to better monitor and quantify drug use trends.

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 novel approach depicting the dynamics of marijuana usage to gauge the effects of peer influence in a school population, is the site of investigation. Consumption of drug is considered as a contagious social epidemic which is spread mainly by peer influences. A relation-based graph-CA (r-GCA) model consisting of 4 states namely, Nonusers (N), Experimental users (E), Recreational users (R) and Addicts (A), is formulated in order to represent the prevalence of the epidemic on a campus. The r-GCA model is set up by local transition rules which delineates the proliferation of marijuana use. Data available in [4] is opted to verify and validate the r-GCA. Simulations of the r-GCA system are presented and discussed. The numerical results agree quite accurately with the observed data. Using the model, the enactment of campaigns of prevention targeting N, E and R states respectively were conducted and analysed. The results indicate a significant decline in marijuana consumption on the campus when a campaign of prevention targeting the latter three states simultaneously, is enacted.

Recently, a mathematical model describing the illicit drug consumption in a population consisting of drug users and non-users has been proposed. The model describes the dynamics of non-users, experimental users, recreational users, and addict users within a population. The aim of this work is to propose a modified version of this model by analogy with the classical predator-prey models, in particular considering non-users as prey and users as predator. Hence, our model includes a stabilizing effect of the growth rate of the prey, and a destabilizing effect of the predator saturation. Functional responses of Verhulst and of Holling type II have been used for modeling these effects. To forecast the marijuana consumption in the states of Colorado and Washington, we used data from Hanley (2013) and a genetic algorithm to calibrate the parameters in our model. Assuming that the population of non-users increases in proportion with the demography, and following the seminal works of Sir Robert May (1976), we use the growth rate of non-users as the main bifurcation parameter. For the state of Colorado, the model first exhibits a limit cycle, which agrees quite accurately with the reported periodic data in Hanley (2013). By further increasing the growth rate of non-users, the population then enters into two chaotic regions, within which the evolution of the variables becomes unpredictable. For the state of Washington, the model also exhibits a periodic solution, which is again in good agreement with observed data. A chaotic region for Washington is likewise observed in the bifurcation diagram. Our research confirms that mathematical models can be a useful tool for better understanding illicit drug consumption, and for guiding policy-makers towards more effective policies to contain this epidemic.

The stochastic framework of the NERA model (N: Nonuser, E: Experimental user, R: Recreational user, A: Addict) depicting the dynamics of marijuana usage in the pre and post Initiative-502 (I-502) in Washington, is analysed. Randomness is introduced in (i) the degree of influence that E exerts on N in order to take into account the fluctuations in social interactions between nonusers and experimental users (S-NE) and (ii) the transition of R to A, accounting for the varying dopamine level in each individual of the R category (S-RA). The resulting stochastic model with the two nonlinear stochastic transitions, is termed as SNESRA. It is shown that SNESRA is stochastically ultimately bounded and has a unique global solution. The drug free equilibrium is proved to be $p^{th}$ moment exponentially stable under suitable conditions. Conditions for the extinction of drug consumption for SNESRA are established. SNESRA is validated using data available in ??, on the pervasiveness of marijuana use in Washington. Numerical simulations are performed to illustrate the theoretical results. The concept of targeted campaigns of prevention is explained and the numerical experiments conducted indicate a decline in marijuana consumption if targeted campaigns of prevention were enacted 1 year prior to I-502 in Washington.