American Institute of Mathematical Sciences

Advanced
doi: 10.3934/dcdss.2020356

A delayed nonlinear stochastic model for cocaine consumption: Stability analysis and simulation using real data

C. Burgos ◇, , J.-C. Cortés ◇,, , L. Shaikhet ♣, and  R.-J. Villanueva ◇,

◇. 

Instituto Universitario de Matemática Multidisciplinar, Universitat Politècnica de València, Camino de vera s/n, Valencia, 46022, Spain
♣. 

Department of Mathematics, Ariel University, Ariel 40700, Israel

* Corresponding author: jccortes@imm.upv.es

Received  August 2019 Revised  November 2019 Published  May 2020

In this paper we propose a stochastic mathematical model with distributed delay in order to describe the transmission dynamics of cocaine consumption in Spain. We investigate conditions to guarantee the stability in probability of the equilibrium points under stochastic perturbations via the white noise processes. The results are applied to the model cocaine consumption using data retrieved from the Spanish Drug National Plan, http://www.pnsd.mscbs.gob.es/. The obtained results may be useful for policy health authorities in order to improve the strategies against the drug consumption in the long-run.

Keywords: Nonlinear stochastic differential equations with delay, Lyapunov stability stochastic analysis, cocaine consumption dynamical model, Computational methods and simulations in nonlinear stochastic models.
Mathematics Subject Classification: Primary: 93E15, 93A30; Secondary: 65C30, 60H10, 34F05, 37Hxx, 81T80, 34F05, 37B25, 49J55.
Citation: C. Burgos, J.-C. Cortés, L. Shaikhet, R.-J. Villanueva. A delayed nonlinear stochastic model for cocaine consumption: Stability analysis and simulation using real data. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020356
References:
[1]

C. Burgos, J.-C. Cortés, L. Shaikhet and R.-J. Villanueva, A nonlinear dynamic age-structured model of e-commerce in Spain: Stability analysis of the equilibrium by delay and stochastic perturbations, Commun. Nonlinear Sci. Numer. Simul., 64 (2018), 149-158.  doi: 10.1016/j.cnsns.2018.04.022.  Google Scholar

[2]

A. Caselles, J. C. Micó and S. Amigò, Cocaine addiction and personality: A mathematical model, British J. Math. Statist. Psych., 63 (2010), 449-448.  doi: 10.1348/000711009X470768.  Google Scholar

[3]

N. A. Christakis and J. H. Folwer, Connected: The Surprising Power of Our Social Networks and How They Shape Our Lives, Little, Brown Spark, 2009. Google Scholar

[4]

Encuesta sobre alcohol y otras drogas en España, (EDADES 1995-2017). Survey from alcohol and other drugs in Spain, 2017. Available from: http://www.pnsd.mscbs.gob.es/profesionales/sistemasInformacion/sistemaInformacion/pdf/EDADES_2017_Informe.pdf. Google Scholar

[5]

E. Fridman and L. Shaikhet, Simple LMIs for stability of stochastic systems with delay term given by Stieltjes integral or with stabilizing delay, Systems Control Lett., 124 (2019), 83-91.  doi: 10.1016/j.sysconle.2018.12.007.  Google Scholar

[6]

I. I. Gikhman and A. V. Skorokhod, Stochastic differential equations, in The Theory of Stochastic Processes III, Springer, 2007,113–219. Google Scholar

[7]

F. Guerrero, F.-J. Santonja and R.-J. Villanueva, Analysing the Spanish smoke-free legislation of 2006: A new method to quantify its impact using a dynamic model, Inter. J. Drug Policy, 22 (2011), 247-251.  doi: 10.1016/j.drugpo.2011.05.003.  Google Scholar

[8]

F. Guerrero, F.-J. Santonja and R.-J. Villanueva, Solving a model for the evolution of smoking habit in Spain with homotopy analysis method, Nonlinear Anal. Real World Appl., 14 (2013), 549-558.  doi: 10.1016/j.nonrwa.2012.07.015.  Google Scholar

[9]

F. Guerrero and H. Vazquez-Leal, Application of multi-stage HAM-Padé to solve a model for the evolution of cocaine consumption in Spain, TWMS J. Pure Appl. Math., 5 (2014), 241-255.  doi: 10.1016/j.mcm.2010.02.032.  Google Scholar

[10]

C. Jacob and N. Khemka, Particle swarm optimization in Mathematica, as exploration kit for evolutionary optimization, Proceedings of the Sixth International Mathematica Symposium, 2004. Google Scholar

[11]

J. D. Murray, Mathematical Biology. I, Interdisciplinary Applied Mathematics, 17, Springer-Verlag, New York, 2002. doi: 10.1007/b98868.  Google Scholar

[12]

E. Sánchez, R.-J. Villanueva, F.-J. Santonja and M. Rubio, Predicting cocaine consumption in Spain: A mathematical modelling approach, Drugs: Education Prevention Policy, 18 (2011), 108-115.  doi: 10.3109/09687630903443299.  Google Scholar

[13]

F. J. Santonja, E. Sánchez, M. Rubio and J. L. Morera, Alcohol consumption in Spain and its economic cost: A mathematical modeling approach, Math. Comput. Modelling, 52 (2010), 999-1003.  doi: 10.1016/j.mcm.2010.02.029.  Google Scholar

[14]

F. J. Santonja, I. C. Lombana, M. Rubio, E. Sánchez and J. Villanueva, A network model for the short-term prediction of the evolution of cocaine consumption in Spain, Math. Comput. Modelling, 52 (2010), 1023-1029.  doi: 10.1016/j.mcm.2010.02.032.  Google Scholar

[15]

L. Shaikhet, Lyapunov Functionals and Stability of Stochastic Functional Differential Equations, Springer, Cham, 2013. doi: 10.1007/978-3-319-00101-2.  Google Scholar

[16]

Spanish INE: Indicadores Demográficos Básicos (BasicDemographic Indicators), 2017. Available from: http://www.ine.es/dyngs/INEbase/es/operacion.htm?c=Estadistica_C&cid=1254736177003&menu=resultados&idp=1254735573002. Google Scholar

[17]

G. Wanner and E. Hairer, Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems, Springer Series in Computational Mathematics, 14, Springer-Verlag, Berlin, 1991. doi: 10.1007/978-3-662-09947-6.  Google Scholar

show all references

References:
[1]

C. Burgos, J.-C. Cortés, L. Shaikhet and R.-J. Villanueva, A nonlinear dynamic age-structured model of e-commerce in Spain: Stability analysis of the equilibrium by delay and stochastic perturbations, Commun. Nonlinear Sci. Numer. Simul., 64 (2018), 149-158.  doi: 10.1016/j.cnsns.2018.04.022.  Google Scholar

[2]

A. Caselles, J. C. Micó and S. Amigò, Cocaine addiction and personality: A mathematical model, British J. Math. Statist. Psych., 63 (2010), 449-448.  doi: 10.1348/000711009X470768.  Google Scholar

[3]

N. A. Christakis and J. H. Folwer, Connected: The Surprising Power of Our Social Networks and How They Shape Our Lives, Little, Brown Spark, 2009. Google Scholar

[4]

Encuesta sobre alcohol y otras drogas en España, (EDADES 1995-2017). Survey from alcohol and other drugs in Spain, 2017. Available from: http://www.pnsd.mscbs.gob.es/profesionales/sistemasInformacion/sistemaInformacion/pdf/EDADES_2017_Informe.pdf. Google Scholar

[5]

E. Fridman and L. Shaikhet, Simple LMIs for stability of stochastic systems with delay term given by Stieltjes integral or with stabilizing delay, Systems Control Lett., 124 (2019), 83-91.  doi: 10.1016/j.sysconle.2018.12.007.  Google Scholar

[6]

I. I. Gikhman and A. V. Skorokhod, Stochastic differential equations, in The Theory of Stochastic Processes III, Springer, 2007,113–219. Google Scholar

[7]

F. Guerrero, F.-J. Santonja and R.-J. Villanueva, Analysing the Spanish smoke-free legislation of 2006: A new method to quantify its impact using a dynamic model, Inter. J. Drug Policy, 22 (2011), 247-251.  doi: 10.1016/j.drugpo.2011.05.003.  Google Scholar

[8]

F. Guerrero, F.-J. Santonja and R.-J. Villanueva, Solving a model for the evolution of smoking habit in Spain with homotopy analysis method, Nonlinear Anal. Real World Appl., 14 (2013), 549-558.  doi: 10.1016/j.nonrwa.2012.07.015.  Google Scholar

[9]

F. Guerrero and H. Vazquez-Leal, Application of multi-stage HAM-Padé to solve a model for the evolution of cocaine consumption in Spain, TWMS J. Pure Appl. Math., 5 (2014), 241-255.  doi: 10.1016/j.mcm.2010.02.032.  Google Scholar

[10]

C. Jacob and N. Khemka, Particle swarm optimization in Mathematica, as exploration kit for evolutionary optimization, Proceedings of the Sixth International Mathematica Symposium, 2004. Google Scholar

[11]

J. D. Murray, Mathematical Biology. I, Interdisciplinary Applied Mathematics, 17, Springer-Verlag, New York, 2002. doi: 10.1007/b98868.  Google Scholar

[12]

E. Sánchez, R.-J. Villanueva, F.-J. Santonja and M. Rubio, Predicting cocaine consumption in Spain: A mathematical modelling approach, Drugs: Education Prevention Policy, 18 (2011), 108-115.  doi: 10.3109/09687630903443299.  Google Scholar

[13]

F. J. Santonja, E. Sánchez, M. Rubio and J. L. Morera, Alcohol consumption in Spain and its economic cost: A mathematical modeling approach, Math. Comput. Modelling, 52 (2010), 999-1003.  doi: 10.1016/j.mcm.2010.02.029.  Google Scholar

[14]

F. J. Santonja, I. C. Lombana, M. Rubio, E. Sánchez and J. Villanueva, A network model for the short-term prediction of the evolution of cocaine consumption in Spain, Math. Comput. Modelling, 52 (2010), 1023-1029.  doi: 10.1016/j.mcm.2010.02.032.  Google Scholar

[15]

L. Shaikhet, Lyapunov Functionals and Stability of Stochastic Functional Differential Equations, Springer, Cham, 2013. doi: 10.1007/978-3-319-00101-2.  Google Scholar

[16]

Spanish INE: Indicadores Demográficos Básicos (BasicDemographic Indicators), 2017. Available from: http://www.ine.es/dyngs/INEbase/es/operacion.htm?c=Estadistica_C&cid=1254736177003&menu=resultados&idp=1254735573002. Google Scholar

[17]

G. Wanner and E. Hairer, Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems, Springer Series in Computational Mathematics, 14, Springer-Verlag, Berlin, 1991. doi: 10.1007/978-3-662-09947-6.  Google Scholar

Figure 1.  Compartmental diagram of the dynamic model for cocaine consumption depicted from equations (1). The boxes represent the four different subpopulations and the arrows the transitions between them
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Simulations of 500 trajectories of the approximated solution stochastic process modelling the dynamics of cocaine consumption in Spain according to stochastic system with delay (11). Those approximations have been constructed using the numerical scheme (16) taking $ \Delta t = 1 $ month and delay $ h = 12 $ months. Red line represents the average of the trajectories and the black one represents the equilibrium point, $ E_1 = (N^*, C_o^*,C_r^*,C_b^*) = (0.3167,0.4189,0.1542,0.1101) $
Figure Options

Download full-size image

Download as PowerPoint slide

Table 1.  Percentage of non-consumers, occasional consumers, regular consumers and habitual consumers of cocaine during the period $ 2001-2017 $ for Spanish population aged $ 15-64 $, [4]
Percentages Dec $ 2001 $ Dec $ 2003 $ Dec $ 2005 $ Dec $ 2007 $ Dec $ 2009 $
Non-consumers $ 91.4 \% $ $ 90.3\% $ $ 88.4\% $ $ 87.4\% $ $ 86.0\% $
Occasional consumers $ 4.8 \% $ $ 5.9 \% $ $ 7.0 \% $ $ 8.0 \% $ $ 10.2\% $
Regular consumers $ 2.5 \% $ $ 2.7 \% $ $ 3.0 \% $ $ 3.0 \% $ $ 2.6 \% $
Habitual consumers $ 1.3 \% $ $ 1.1 \% $ $ 1.6 \% $ $ 1.6 \% $ $ 1.2 \% $
Percentages Dec $2011 $ Dec $2013 $ Dec $2015 $ Dec $2017 $
Non-consumers $87.9 \%$ $86.7\%$ $88.3\%$ $86.9\%$
Occasional consumers $8.8 \%$ $10.2\%$ $8.9 \%$ $10.0\%$
Regular consumers $2.2 \%$ $2.1 \%$ $1.9 \%$ $2.0 \%$
Habitual consumers $1.1 \%$ $1.0 \%$ $0.9 \%$ $1.1 \%$
Table Options

Download as excel

Percentages Dec $ 2001 $ Dec $ 2003 $ Dec $ 2005 $ Dec $ 2007 $ Dec $ 2009 $
Non-consumers $ 91.4 \% $ $ 90.3\% $ $ 88.4\% $ $ 87.4\% $ $ 86.0\% $
Occasional consumers $ 4.8 \% $ $ 5.9 \% $ $ 7.0 \% $ $ 8.0 \% $ $ 10.2\% $
Regular consumers $ 2.5 \% $ $ 2.7 \% $ $ 3.0 \% $ $ 3.0 \% $ $ 2.6 \% $
Habitual consumers $ 1.3 \% $ $ 1.1 \% $ $ 1.6 \% $ $ 1.6 \% $ $ 1.2 \% $
Percentages Dec $2011 $ Dec $2013 $ Dec $2015 $ Dec $2017 $
Non-consumers $87.9 \%$ $86.7\%$ $88.3\%$ $86.9\%$
Occasional consumers $8.8 \%$ $10.2\%$ $8.9 \%$ $10.0\%$
Regular consumers $2.2 \%$ $2.1 \%$ $1.9 \%$ $2.0 \%$
Habitual consumers $1.1 \%$ $1.0 \%$ $0.9 \%$ $1.1 \%$
Table 2.  Values of the parameters that best fit model (2) with the data in Table 1 using PSO algorithm [10]. Recall that we assumed that $ \mu = d $
Model parameters Estimations
$ \mu $ $ 1.587198 \,\, 10^{-3} $
$ d $ $ 1.587198 \,\, 10^{-3} $
$ \beta $ $ 5.013946 \,\, 10^{-3} $
$ \varepsilon $ $ 5.855882 \,\, 10^{-6} $
$ \gamma $ $ 1.003084 \,\, 10^{-3} $
$ \sigma $ $ 1.137033 \,\, 10^{-3} $
Table Options

Download as excel

Model parameters Estimations
$ \mu $ $ 1.587198 \,\, 10^{-3} $
$ d $ $ 1.587198 \,\, 10^{-3} $
$ \beta $ $ 5.013946 \,\, 10^{-3} $
$ \varepsilon $ $ 5.855882 \,\, 10^{-6} $
$ \gamma $ $ 1.003084 \,\, 10^{-3} $
$ \sigma $ $ 1.137033 \,\, 10^{-3} $
[1]

Xin Zhao, Tao Feng, Liang Wang, Zhipeng Qiu. Threshold dynamics and sensitivity analysis of a stochastic semi-Markov switched SIRS epidemic model with nonlinear incidence and vaccination. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021010
[2]

Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020468
[3]

Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264
[4]

Tetsuya Ishiwata, Young Chol Yang. Numerical and mathematical analysis of blow-up problems for a stochastic differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 909-918. doi: 10.3934/dcdss.2020391
[5]

Pengyu Chen. Non-autonomous stochastic evolution equations with nonlinear noise and nonlocal conditions governed by noncompact evolution families. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020383
[6]

Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317
[7]

Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020047
[8]

Bixiang Wang. Mean-square random invariant manifolds for stochastic differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1449-1468. doi: 10.3934/dcds.2020324
[9]

Qingfeng Zhu, Yufeng Shi. Nonzero-sum differential game of backward doubly stochastic systems with delay and applications. Mathematical Control & Related Fields, 2021, 11 (1) : 73-94. doi: 10.3934/mcrf.2020028
[10]

Ténan Yeo. Stochastic and deterministic SIS patch model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021012
[11]

Zhenzhen Wang, Tianshou Zhou. Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020323
[12]

Guangjun Shen, Xueying Wu, Xiuwei Yin. Stabilization of stochastic differential equations driven by G-Lévy process with discrete-time feedback control. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 755-774. doi: 10.3934/dcdsb.2020133
[13]

Zhimin Li, Tailei Zhang, Xiuqing Li. Threshold dynamics of stochastic models with time delays: A case study for Yunnan, China. Electronic Research Archive, 2021, 29 (1) : 1661-1679. doi: 10.3934/era.2020085
[14]

Giuseppina Guatteri, Federica Masiero. Stochastic maximum principle for problems with delay with dependence on the past through general measures. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020048
[15]

Eduard Feireisl, Elisabetta Rocca, Giulio Schimperna, Arghir Zarnescu. Weak sequential stability for a nonlinear model of nematic electrolytes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 219-241. doi: 10.3934/dcdss.2020366
[16]

Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432
[17]

Yuan Tan, Qingyuan Cao, Lan Li, Tianshi Hu, Min Su. A chance-constrained stochastic model predictive control problem with disturbance feedback. Journal of Industrial & Management Optimization, 2021, 17 (1) : 67-79. doi: 10.3934/jimo.2019099
[18]

Mugen Huang, Moxun Tang, Jianshe Yu, Bo Zheng. A stage structured model of delay differential equations for Aedes mosquito population suppression. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3467-3484. doi: 10.3934/dcds.2020042
[19]

Xuhui Peng, Rangrang Zhang. Approximations of stochastic 3D tamed Navier-Stokes equations. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5337-5365. doi: 10.3934/cpaa.2020241
[20]

Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320

2019 Impact Factor: 1.233

Tools

Article outline

Figures and Tables

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences