American Institute of Mathematical Sciences

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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

Yusra Bibi Ruhomally 1, , Muhammad Zaid Dauhoo 1,, and  Laurent Dumas 2,

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.

Keywords: Marijuana, peer influence, relation-based, graph cellular automaton, targeted campaigns of prevention.
Mathematics Subject Classification: Primary: 37N99.
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 and Games, 2021, 8 (3) : 277-297. doi: 10.3934/jdg.2021011
References:
[1]

N.-R. Badurally Adam, M. Z. Dauhoo and O. Kavian, An analysis of the dynamical evolution of experimental, recreative and abusive marijuana consumption in the states of Colorado and Washington beyond the implementation of I–502, J. Math. Sociol., 39 (2015), 257-279.  doi: 10.1080/0022250X.2015.1077240.

[2]

A. Bakhtiari, Social Influences Among Drug Users and Mean Field Approximation of Cellular Automata, Ph.D thesis, Simon Fraser University, 2009.

[3]

D. A. Behrens and G. Tragler, The dynamic process of dynamic modelling: The cocaine epidemic in the United States, Bulletin on Narcotics, 53 (2001), 65-78. 

[4]

A. Boak, H. A. Hamilton, E. M. Adlaf and R. E. Mann, Drug use among Ontario students, 1977–2017: Detailed findings from the Ontario Student and Drug Use Health Survey (OSDUHS), Centre for Addiction and Mental Health.

[5]

V. Dabbaghian, V. Spicer, S. K. Singh, P. Borwein and P. Brantingham, The social impact in a high-risk community: A cellular automata model, J. Comput. Sci., 2 (2011), 238-246.  doi: 10.1016/j.jocs.2011.05.008.

[6]

S. Y. Del Valle, J. M. Hyman, H. W. Hethcote and S. G. Eubank, Mixing patterns between age groups in social networks, Social Networks, 29 (2007), 539-554.  doi: 10.1016/j.socnet.2007.04.005.

[7]

M. Z. Dauhoo, B. S. N. Korimboccus and S. B. Issack, On the dynamics of illicit drug consumption in a given population, IMA J. Appl. Math., 78 (2013), 432-448.  doi: 10.1093/imamat/hxr058.

[8]

S. T. Ennett and K. E. Bauman, Adolescent Social Networks: Friendship Cliques, Social Isolates, and Drug Use Risk, University of North Carolina at Chapel Hill, 2000.

[9]

P. Ghosh, A. Mukhopadhyay, A. Chanda, P. Mondal and A. Akhand, Application of Cellular automata and Markov-chain model in geospatial environmental modeling - A review, Remote Sensing Appl.: Soc. Environ., 5 (2017), 64-77.  doi: 10.1016/j.rsase.2017.01.005.

[10]

R. Gikonyo and K. Njagi, The influence of demographic factors on peer pressure among secondary school adolescents in Nyahururu Laikipia county, Res. Hummanities Soc. Sci., 6 (2016), 2224-5766. 

[11]

A. Gragnani, S. Rinaldi and G. Feichtinger, Dynamics of drug consumption: A theoretical model, Socio-Economic Planning Sci., 31 (1997), 127-137.  doi: 10.1016/S0038-0121(96)00020-1.

[12]

D. Grass, J. P. Caulkins, G. Feichtinger, G. Tragler and D. A. Behrens, Optimal Control of Nonlinear Processes, Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-77647-5.

[13]

L. Johnston, R. Miech, P. O'Malley, J. Bachman, J. Schulenberg and M. Patrick, Monitoring the future national survey results on drug use, 1975-2019: Overview, key findings on adolescent drug use.,

[14]

J. J. Kari, Basic concepts of cellular automata, in Handbook of Natural Computing, Springer, Berlin, Heidelberg, (2012), 3–24. doi: 10.1007/978-3-540-92910-9_1.

[15]

P.-Y. Louis and F. R. Nardi, Probabilistic Cellular Automata. Theory, Applications and Future Perspectives, Emergence, Complexity and Computation, 27, Springer, Cham, 2018. doi: 10.1007/978-3-319-65558-1.

[16]

K. Małecki, Graph cellular automata with relation-based neighbourhoods of cells for complex systems modelling: A case of traffic simulation, Symmetry, 9 (2017), 322. doi: 10.3390/sym9120322.

[17]

M. J. F. Martłnez, E. G. Merino, E. G. Sánchez, J. E. G. Sánchez, A. M. del Rey and G. R. Sánchez, A graph cellular automata model to study the spreading of an infectious disease, in Advances in Artificial Intelligence, Lecture Notes in Computer Science, 7629, Springer, Berlin, Heidelberg, (2012), 458–468. doi: 10.1007/978-3-642-37807-2_39.

[18]

E. R. Oetting and F. Beauvais, Peer cluster theory: Drugs and the adolescent, J. Counsel. Develop., 65 (1986), 17-22.  doi: 10.1002/j.1556-6676.1986.tb01219.x.

[19]

R. L. Pacula and R. Smart, Medical marijuana and marijuana legalization, Annual Review of Clinical Psychology, 13 (2017), 397-419.  doi: 10.1146/annurev-clinpsy-032816-045128.

[20]

P. Rinaldi, D. Dalponte, M. Vénere and A. Clausse et al., Graph-based cellular automata for simulation of surface flows in large plains, Asian J. Appl. Sci., 5 (2012), 224-231.  doi: 10.3923/ajaps.2012.224.231.

[21]

Y. B. Ruhomally, N. Banon Jahmeerbaccus and M. Z. Dauhoo, The deterministic evolution of illicit drug consumption within a given population, in CIMPA School on Mathematical Models in Biology and Medicine, ESAIM Proc. Surveys, 62, EDP Sci., Les Ulis, (2018), 139–157. doi: 10.1051/proc/201862139.

[22]

Y. B. Ruhomally and M. Z. Dauhoo, The NERA model incorporating cellular automata approach and the analysis of the resulting induced stochastic mean field, Comput. Appl. Math., 39 (2020), 327-356.  doi: 10.1007/s40314-020-01378-2.

[23]

Y. B. Ruhomally, M. Z. Dauhoo and L. Dumas, An analysis of the recreational use of marijuana amongst the 21+ population of the state of Washington in the context of I-502 and its aftermath, Neural, Parallel and Scientific Computations, 28 (2020), 273-304. 

[24]

J. Schulenberg, L. Johnston, P. O'Malley, J. Bachman, R. Miech and M. Patrick, Monitoring the Future National Survey Results on Drug Use, 1975-2018: Volume II, college students and adults ages 19-60, 2019.

[25]

B. Song, M. Castillo-Garsow, K. R. Rios-Soto, M. Mejran, L. Henso and C. Castillo-Chavez, Raves, clubs and ecstasy: The impact of peer pressure, Math. Biosci. Eng., 3 (2006), 249-266.  doi: 10.3934/mbe.2006.3.249.

[26]

L. Steinberg and K. C. Monahan, Age differences in resistance to peer influence, Develop. Psych., 43 (2007), 1531-1543.  doi: 10.1037/0012-1649.43.6.1531.

[27]

UN Office on Drugs and Crime, World Drug Report 2020, 2020. Available from: https://wdr.unodc.org/wdr2020/index.html.

show all references

References:
[1]

N.-R. Badurally Adam, M. Z. Dauhoo and O. Kavian, An analysis of the dynamical evolution of experimental, recreative and abusive marijuana consumption in the states of Colorado and Washington beyond the implementation of I–502, J. Math. Sociol., 39 (2015), 257-279.  doi: 10.1080/0022250X.2015.1077240.

[2]

A. Bakhtiari, Social Influences Among Drug Users and Mean Field Approximation of Cellular Automata, Ph.D thesis, Simon Fraser University, 2009.

[3]

D. A. Behrens and G. Tragler, The dynamic process of dynamic modelling: The cocaine epidemic in the United States, Bulletin on Narcotics, 53 (2001), 65-78. 

[4]

A. Boak, H. A. Hamilton, E. M. Adlaf and R. E. Mann, Drug use among Ontario students, 1977–2017: Detailed findings from the Ontario Student and Drug Use Health Survey (OSDUHS), Centre for Addiction and Mental Health.

[5]

V. Dabbaghian, V. Spicer, S. K. Singh, P. Borwein and P. Brantingham, The social impact in a high-risk community: A cellular automata model, J. Comput. Sci., 2 (2011), 238-246.  doi: 10.1016/j.jocs.2011.05.008.

[6]

S. Y. Del Valle, J. M. Hyman, H. W. Hethcote and S. G. Eubank, Mixing patterns between age groups in social networks, Social Networks, 29 (2007), 539-554.  doi: 10.1016/j.socnet.2007.04.005.

[7]

M. Z. Dauhoo, B. S. N. Korimboccus and S. B. Issack, On the dynamics of illicit drug consumption in a given population, IMA J. Appl. Math., 78 (2013), 432-448.  doi: 10.1093/imamat/hxr058.

[8]

S. T. Ennett and K. E. Bauman, Adolescent Social Networks: Friendship Cliques, Social Isolates, and Drug Use Risk, University of North Carolina at Chapel Hill, 2000.

[9]

P. Ghosh, A. Mukhopadhyay, A. Chanda, P. Mondal and A. Akhand, Application of Cellular automata and Markov-chain model in geospatial environmental modeling - A review, Remote Sensing Appl.: Soc. Environ., 5 (2017), 64-77.  doi: 10.1016/j.rsase.2017.01.005.

[10]

R. Gikonyo and K. Njagi, The influence of demographic factors on peer pressure among secondary school adolescents in Nyahururu Laikipia county, Res. Hummanities Soc. Sci., 6 (2016), 2224-5766. 

[11]

A. Gragnani, S. Rinaldi and G. Feichtinger, Dynamics of drug consumption: A theoretical model, Socio-Economic Planning Sci., 31 (1997), 127-137.  doi: 10.1016/S0038-0121(96)00020-1.

[12]

D. Grass, J. P. Caulkins, G. Feichtinger, G. Tragler and D. A. Behrens, Optimal Control of Nonlinear Processes, Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-77647-5.

[13]

L. Johnston, R. Miech, P. O'Malley, J. Bachman, J. Schulenberg and M. Patrick, Monitoring the future national survey results on drug use, 1975-2019: Overview, key findings on adolescent drug use.,

[14]

J. J. Kari, Basic concepts of cellular automata, in Handbook of Natural Computing, Springer, Berlin, Heidelberg, (2012), 3–24. doi: 10.1007/978-3-540-92910-9_1.

[15]

P.-Y. Louis and F. R. Nardi, Probabilistic Cellular Automata. Theory, Applications and Future Perspectives, Emergence, Complexity and Computation, 27, Springer, Cham, 2018. doi: 10.1007/978-3-319-65558-1.

[16]

K. Małecki, Graph cellular automata with relation-based neighbourhoods of cells for complex systems modelling: A case of traffic simulation, Symmetry, 9 (2017), 322. doi: 10.3390/sym9120322.

[17]

M. J. F. Martłnez, E. G. Merino, E. G. Sánchez, J. E. G. Sánchez, A. M. del Rey and G. R. Sánchez, A graph cellular automata model to study the spreading of an infectious disease, in Advances in Artificial Intelligence, Lecture Notes in Computer Science, 7629, Springer, Berlin, Heidelberg, (2012), 458–468. doi: 10.1007/978-3-642-37807-2_39.

[18]

E. R. Oetting and F. Beauvais, Peer cluster theory: Drugs and the adolescent, J. Counsel. Develop., 65 (1986), 17-22.  doi: 10.1002/j.1556-6676.1986.tb01219.x.

[19]

R. L. Pacula and R. Smart, Medical marijuana and marijuana legalization, Annual Review of Clinical Psychology, 13 (2017), 397-419.  doi: 10.1146/annurev-clinpsy-032816-045128.

[20]

P. Rinaldi, D. Dalponte, M. Vénere and A. Clausse et al., Graph-based cellular automata for simulation of surface flows in large plains, Asian J. Appl. Sci., 5 (2012), 224-231.  doi: 10.3923/ajaps.2012.224.231.

[21]

Y. B. Ruhomally, N. Banon Jahmeerbaccus and M. Z. Dauhoo, The deterministic evolution of illicit drug consumption within a given population, in CIMPA School on Mathematical Models in Biology and Medicine, ESAIM Proc. Surveys, 62, EDP Sci., Les Ulis, (2018), 139–157. doi: 10.1051/proc/201862139.

[22]

Y. B. Ruhomally and M. Z. Dauhoo, The NERA model incorporating cellular automata approach and the analysis of the resulting induced stochastic mean field, Comput. Appl. Math., 39 (2020), 327-356.  doi: 10.1007/s40314-020-01378-2.

[23]

Y. B. Ruhomally, M. Z. Dauhoo and L. Dumas, An analysis of the recreational use of marijuana amongst the 21+ population of the state of Washington in the context of I-502 and its aftermath, Neural, Parallel and Scientific Computations, 28 (2020), 273-304. 

[24]

J. Schulenberg, L. Johnston, P. O'Malley, J. Bachman, R. Miech and M. Patrick, Monitoring the Future National Survey Results on Drug Use, 1975-2018: Volume II, college students and adults ages 19-60, 2019.

[25]

B. Song, M. Castillo-Garsow, K. R. Rios-Soto, M. Mejran, L. Henso and C. Castillo-Chavez, Raves, clubs and ecstasy: The impact of peer pressure, Math. Biosci. Eng., 3 (2006), 249-266.  doi: 10.3934/mbe.2006.3.249.

[26]

L. Steinberg and K. C. Monahan, Age differences in resistance to peer influence, Develop. Psych., 43 (2007), 1531-1543.  doi: 10.1037/0012-1649.43.6.1531.

[27]

UN Office on Drugs and Crime, World Drug Report 2020, 2020. Available from: https://wdr.unodc.org/wdr2020/index.html.

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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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} $
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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Type of user State Colour
Nonuser - N 0 Green
Experimental user - E 1 Blue
Recreational user - R 2 Yellow
Addict user - A 3 Red
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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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
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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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 $
Table 5.  Initial number of individuals that represent each state
State N E R A
Value $ 611 $ $ 110 $ $ 98 $ $ 81 $
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State N E R A
Value $ 611 $ $ 110 $ $ 98 $ $ 81 $
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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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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