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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

  • * Corresponding author: Muhammad Zaid Dauhoo

    * Corresponding author: Muhammad Zaid Dauhoo 
Abstract / Introduction Full Text(HTML) Figure(11) / Table(6) Related Papers Cited by
  • 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.

    Mathematics Subject Classification: Primary: 37N99.

    Citation:

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  • Figure 1.  Types of neighbourhood in cellular automata

    Figure 2.  Schematic representation of the r-GCA model

    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

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

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

    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. $

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

    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)

    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)

    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)

    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

    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} $
     | Show Table
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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
     | Show Table
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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
     | Show Table
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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 $
     | Show Table
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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 $
     | Show Table
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