# American Institute of Mathematical Sciences

July  2021, 8(3): 277-297. doi: 10.3934/jdg.2021011

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

 1 Department of Mathematics, Faculty of Science, University Of Mauritius, Reduit, Mauritius 2 Laboratoire de Mathématiques de Versailles, UVSQ, CNRS, Université Paris-Saclay, 78035 Versailles, France

* Corresponding author: Muhammad Zaid Dauhoo

Received  February 2021 Revised  February 2021 Published  July 2021 Early access  March 2021

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.

Citation: Yusra Bibi Ruhomally, Muhammad Zaid Dauhoo, Laurent Dumas. A graph cellular automaton with relation-based neighbourhood describing the impact of peer influence on the consumption of marijuana among college-aged youths. Journal of Dynamics & Games, 2021, 8 (3) : 277-297. doi: 10.3934/jdg.2021011
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Types of neighbourhood in cellular automata
Schematic representation of the r-GCA model
The neighbourhood of a given individual within a population comprising of 900 individuals. Double arrows denote mutual influences (two-way relationship) and single arrows represent a one-way relationship with the individual. Four mutual influences are present in the neighbourhood of the individual
]">Figure 4.  Trends of the 4 categories of marijuana users for the period 1999-2017 in grades 7-12 according to [4]
] (solid lines)">Figure 5.  Superimposition of the evolution of the four categories of marijuana users (dotted lines) on the data collected from [4] (solid lines)
represents the initial configuration with 611 N, 110 E, 98 R and 81 A. The parameters used are $\alpha_{1} = 0.17, \alpha_{2} = 0.13, \alpha_{3} = 0.012, \gamma_{1} = 0.08, \gamma_{2} = 0.0101$ and $\gamma_{3} = 0.0124.$">Figure 6.  These figures show the snapshots of the r-GCA dynamics over 2555 time steps on a 30 $\times$ 30 grid. These figures illustrate the temporal evolution of the users. Figure 3 (a) represents the initial configuration with 611 N, 110 E, 98 R and 81 A. The parameters used are $\alpha_{1} = 0.17, \alpha_{2} = 0.13, \alpha_{3} = 0.012, \gamma_{1} = 0.08, \gamma_{2} = 0.0101$ and $\gamma_{3} = 0.0124.$
Mean evolution of the 4 categories of users over 2555 time steps
Superimposition of the temporal evolution of the 4 categories of users with campaigns of prevention subjected to the nonusers (dotted lines) on the evolution of the users without any campaign of prevention (solid lines)
Superimposition of the temporal evolution of the users with a campaign of prevention targeting the experimental and recreational users (dotted lines) on the evolution of the users without any campaign of prevention (solid lines)
Superimposition of the temporal evolution of the users with a campaign of prevention subjected to the non, experimental and recreational users (dotted lines) on the evolution of the users without any campaign of prevention (solid lines)
A comparison of the proportion of the 4 categories of users when $t = 1$ (Initial) and when $t = 2555$ (Final) for the 3 scenarios, with and without the enactment of campaigns of prevention
Neighbourhood specification
 Neighbourhood Neighbouring cells for $C_{i, j}$ Von Neumann $C_{i+1, j}$, $C_{i-1, j}$, $C_{i, j+1}$, $C_{i, j-1}$ Moore $C_{i+1, j}$, $C_{i-1, j}$, $C_{i, j+1}$, $C_{i, j-1}$, $C_{i+1, j+1}$, $C_{i-1, j-1}$, $C_{i-1, j+1}$, $C_{i+1, j-1}$ r-GCA $C_{i, j-1}$, $C_{i-2, j-1}$, $C_{i-2, j+1}$, $C_{i+1, j+2}$, $C_{i+2, j-1}$
 Neighbourhood Neighbouring cells for $C_{i, j}$ Von Neumann $C_{i+1, j}$, $C_{i-1, j}$, $C_{i, j+1}$, $C_{i, j-1}$ Moore $C_{i+1, j}$, $C_{i-1, j}$, $C_{i, j+1}$, $C_{i, j-1}$, $C_{i+1, j+1}$, $C_{i-1, j-1}$, $C_{i-1, j+1}$, $C_{i+1, j-1}$ r-GCA $C_{i, j-1}$, $C_{i-2, j-1}$, $C_{i-2, j+1}$, $C_{i+1, j+2}$, $C_{i+2, j-1}$
Definition of states and colour code of cells
 Type of user State Colour Nonuser - N 0 Green Experimental user - E 1 Blue Recreational user - R 2 Yellow Addict user - A 3 Red
 Type of user State Colour Nonuser - N 0 Green Experimental user - E 1 Blue Recreational user - R 2 Yellow Addict user - A 3 Red
Interpretation of the parameters involved in the r-GCA model
 Parameter Physical Meaning $\alpha_{1}(t)$ Influence rate of $E(t)$ on $N(t)$ $\alpha_{2}(t)$ Influence rate of $R(t)$ on $N(t)$ $\alpha_{3}(t)$ Influence rate of $R(t)$ on $E(t)$ $\alpha_{4}(t)$ Rate at which recreational users change to addicts $\gamma_{1}(t)$ Rate at which experimental users quit drugs $\gamma_{2(t)}$ Rate at which recreational users quit drugs $\gamma_{3}(t)$ Rate at which addicts quit drugs $\beta$ Proportion of nonusers moving into the population $\omega_{N}(t)\beta$ Proportion of nonusers moving out of the population $\omega_{E}(t)\beta$ Proportion of experimental users moving out of population $\omega_{R}(t)\beta$ Proportion of recreational users moving out of population $\omega_{A}(t)\beta$ Proportion of addicts moving out of the population
 Parameter Physical Meaning $\alpha_{1}(t)$ Influence rate of $E(t)$ on $N(t)$ $\alpha_{2}(t)$ Influence rate of $R(t)$ on $N(t)$ $\alpha_{3}(t)$ Influence rate of $R(t)$ on $E(t)$ $\alpha_{4}(t)$ Rate at which recreational users change to addicts $\gamma_{1}(t)$ Rate at which experimental users quit drugs $\gamma_{2(t)}$ Rate at which recreational users quit drugs $\gamma_{3}(t)$ Rate at which addicts quit drugs $\beta$ Proportion of nonusers moving into the population $\omega_{N}(t)\beta$ Proportion of nonusers moving out of the population $\omega_{E}(t)\beta$ Proportion of experimental users moving out of population $\omega_{R}(t)\beta$ Proportion of recreational users moving out of population $\omega_{A}(t)\beta$ Proportion of addicts moving out of the population
Parameter values obtained for the consumption of marijuana for the period 1999-2006, using genetic algorithm
 Parameter $\alpha_{1}$ $\alpha_{2}$ $\alpha_{3}$ $\alpha_{4}$ $\gamma_{1}$ $\gamma_{2}$ $\gamma_{3}$ Value $0.101$ $0.109$ $0.116$ $0.117$ $0.126$ $0.318$ $0.114$
 Parameter $\alpha_{1}$ $\alpha_{2}$ $\alpha_{3}$ $\alpha_{4}$ $\gamma_{1}$ $\gamma_{2}$ $\gamma_{3}$ Value $0.101$ $0.109$ $0.116$ $0.117$ $0.126$ $0.318$ $0.114$
Initial number of individuals that represent each state
 State N E R A Value $611$ $110$ $98$ $81$
 State N E R A Value $611$ $110$ $98$ $81$
Parameter values used for the scenario
 Parameter $\alpha_{1}$ $\alpha_{2}$ $\alpha_{3}$ $\alpha_{4}$ $\gamma_{1}$ $\gamma_{2}$ $\gamma_{3}$ Value $0.17$ $0.13$ $0.012$ $0.08$ $0.08$ $0.0101$ $0.0124$
 Parameter $\alpha_{1}$ $\alpha_{2}$ $\alpha_{3}$ $\alpha_{4}$ $\gamma_{1}$ $\gamma_{2}$ $\gamma_{3}$ Value $0.17$ $0.13$ $0.012$ $0.08$ $0.08$ $0.0101$ $0.0124$
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