
-
Previous Article
A mean field game model for the evolution of cities
- JDG Home
- This Issue
-
Next Article
On cooperative fuzzy bubbly games
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 |
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 [
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. |









Neighbourhood | Neighbouring cells for |
Von Neumann | |
Moore | |
r-GCA |
Neighbourhood | Neighbouring cells for |
Von Neumann | |
Moore | |
r-GCA |
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 |
Parameter | Physical Meaning |
Influence rate of |
|
Influence rate of |
|
Influence rate of |
|
Rate at which recreational users change to addicts | |
Rate at which experimental users quit drugs | |
Rate at which recreational users quit drugs | |
Rate at which addicts quit drugs | |
Proportion of nonusers moving into the population | |
Proportion of nonusers moving out of the population | |
Proportion of experimental users moving out of population | |
Proportion of recreational users moving out of population | |
Proportion of addicts moving out of the population |
Parameter | Physical Meaning |
Influence rate of |
|
Influence rate of |
|
Influence rate of |
|
Rate at which recreational users change to addicts | |
Rate at which experimental users quit drugs | |
Rate at which recreational users quit drugs | |
Rate at which addicts quit drugs | |
Proportion of nonusers moving into the population | |
Proportion of nonusers moving out of the population | |
Proportion of experimental users moving out of population | |
Proportion of recreational users moving out of population | |
Proportion of addicts moving out of the population |
Parameter | |||||||
Value |
Parameter | |||||||
Value |
State | N | E | R | A |
Value |
State | N | E | R | A |
Value |
Parameter | |||||||
Value |
Parameter | |||||||
Value |
[1] |
Petr Kůrka. On the measure attractor of a cellular automaton. Conference Publications, 2005, 2005 (Special) : 524-535. doi: 10.3934/proc.2005.2005.524 |
[2] |
Gelasio Salaza, Edgardo Ugalde, Jesús Urías. Master--slave synchronization of affine cellular automaton pairs. Discrete and Continuous Dynamical Systems, 2005, 13 (2) : 491-502. doi: 10.3934/dcds.2005.13.491 |
[3] |
Akane Kawaharada. Singular function emerging from one-dimensional elementary cellular automaton Rule 150. Discrete and Continuous Dynamical Systems - B, 2022, 27 (4) : 2115-2128. doi: 10.3934/dcdsb.2021125 |
[4] |
Kazuhiko Kuraya, Hiroyuki Masuyama, Shoji Kasahara. Load distribution performance of super-node based peer-to-peer communication networks: A nonstationary Markov chain approach. Numerical Algebra, Control and Optimization, 2011, 1 (4) : 593-610. doi: 10.3934/naco.2011.1.593 |
[5] |
Sho Nanao, Hiroyuki Masuyama, Shoji Kasahara, Yutaka Takahashi. Queueing analysis of data block synchronization mechanism in peer-to-peer based video streaming system. Journal of Industrial and Management Optimization, 2011, 7 (3) : 699-716. doi: 10.3934/jimo.2011.7.699 |
[6] |
Quoc T. Luu, Paul DuChateau. The relative biologic effectiveness versus linear energy transfer curve as an output-input relation for linear cellular systems. Mathematical Biosciences & Engineering, 2009, 6 (3) : 591-602. doi: 10.3934/mbe.2009.6.591 |
[7] |
Chun-Xiang Guo, Guo Qiang, Jin Mao-Zhu, Zhihan Lv. Dynamic systems based on preference graph and distance. Discrete and Continuous Dynamical Systems - S, 2015, 8 (6) : 1139-1154. doi: 10.3934/dcdss.2015.8.1139 |
[8] |
Jianli Xiang, Guozheng Yan. The uniqueness of the inverse elastic wave scattering problem based on the mixed reciprocity relation. Inverse Problems and Imaging, 2021, 15 (3) : 539-554. doi: 10.3934/ipi.2021004 |
[9] |
Xinxin Tan, Shujuan Li, Sisi Liu, Zhiwei Zhao, Lisa Huang, Jiatai Gang. Dynamic simulation of a SEIQR-V epidemic model based on cellular automata. Numerical Algebra, Control and Optimization, 2015, 5 (4) : 327-337. doi: 10.3934/naco.2015.5.327 |
[10] |
Yair Daon, Georg Stadler. Mitigating the influence of the boundary on PDE-based covariance operators. Inverse Problems and Imaging, 2018, 12 (5) : 1083-1102. doi: 10.3934/ipi.2018045 |
[11] |
Blaine Keetch, Yves van Gennip. A Max-Cut approximation using a graph based MBO scheme. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 6091-6139. doi: 10.3934/dcdsb.2019132 |
[12] |
Suh-Yuh Yang, Cheng-Hsiung Hsu. Existence of monotonic traveling waves in modified RTD-based cellular neural networks. Conference Publications, 2005, 2005 (Special) : 930-939. doi: 10.3934/proc.2005.2005.930 |
[13] |
Chunfeng Liu, Yuanyuan Liu, Jufeng Wang. A revised imperialist competition algorithm for cellular manufacturing optimization based on product line design. Journal of Industrial and Management Optimization, 2021 doi: 10.3934/jimo.2021175 |
[14] |
Hem Joshi, Suzanne Lenhart, Kendra Albright, Kevin Gipson. Modeling the effect of information campaigns on the HIV epidemic in Uganda. Mathematical Biosciences & Engineering, 2008, 5 (4) : 757-770. doi: 10.3934/mbe.2008.5.757 |
[15] |
Lars Eirik Danielsen. Graph-based classification of self-dual additive codes over finite fields. Advances in Mathematics of Communications, 2009, 3 (4) : 329-348. doi: 10.3934/amc.2009.3.329 |
[16] |
Pengyu Yan, Shi Qiang Liu, Cheng-Hu Yang, Mahmoud Masoud. A comparative study on three graph-based constructive algorithms for multi-stage scheduling with blocking. Journal of Industrial and Management Optimization, 2019, 15 (1) : 221-233. doi: 10.3934/jimo.2018040 |
[17] |
Shuichiro Senda, Hiroyuki Masuyama, Shoji Kasahara. A stochastic fluid model for on-demand peer-to-peer streaming services. Numerical Algebra, Control and Optimization, 2011, 1 (4) : 611-626. doi: 10.3934/naco.2011.1.611 |
[18] |
Mingfang Ding, Yanqun Liu, John Anthony Gear. An improved targeted climbing algorithm for linear programs. Numerical Algebra, Control and Optimization, 2011, 1 (3) : 399-405. doi: 10.3934/naco.2011.1.399 |
[19] |
Chloe A. Fletcher, Jason S. Howell. Dynamic modeling of nontargeted and targeted advertising strategies in an oligopoly. Journal of Dynamics and Games, 2017, 4 (2) : 97-124. doi: 10.3934/jdg.2017007 |
[20] |
A. K. Misra, Anupama Sharma, Jia Li. A mathematical model for control of vector borne diseases through media campaigns. Discrete and Continuous Dynamical Systems - B, 2013, 18 (7) : 1909-1927. doi: 10.3934/dcdsb.2013.18.1909 |
Impact Factor:
Tools
Metrics
Other articles
by authors
[Back to Top]