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

Yusra Bibi Ruhomally; Muhammad Zaid Dauhoo; Laurent Dumas

doi:10.3934/jdg.2021011

Article Contents

2021, Volume 8, Issue 3: 277-297. Doi: 10.3934/jdg.2021011

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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
^* Corresponding author: Muhammad Zaid Dauhoo

Received: January 31, 2021

Revised: January 31, 2021

Early access: March 2021

Published: July 2021

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Abstract

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.

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Mathematics Subject Classification: Primary: 37N99.

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Figure 1. Types of neighbourhood in cellular automata

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Figure 2. Schematic representation of the r-GCA model

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Figure 3. 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

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Figure 4. Trends of the 4 categories of marijuana users for the period 1999-2017 in grades 7-12 according to [4]

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Figure 5. Superimposition of the evolution of the four categories of marijuana users (dotted lines) on the data collected from [4] (solid lines)

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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. $

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Figure 7. Mean evolution of the 4 categories of users over 2555 time steps

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Figure 8. 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)

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Figure 9. 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)

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Figure 10. 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)

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Figure 11. 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

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Table 1. 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} $

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Table 2. 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

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Table 3. 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

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Table 4. 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 $

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Table 5. Initial number of individuals that represent each state

State	N	E	R	A
Value	$ 611 $	$ 110 $	$ 98 $	$ 81 $

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Table 6. 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 $

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