Stochastic modelling of marijuana use in Washington: pre- and post-Initiative-502 (I-502)

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 pth moment exponentially stable under suitable conditions. Conditions for the extinction of drug consumption for SNESRA are established. SNESRA is validated using data available in Ruhomally et al. (2020), 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.