# American Institute of Mathematical Sciences

April  2021, 14(4): 1233-1244. doi: 10.3934/dcdss.2020356

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

 ◇. 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  April 2021 Early access  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.

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 and Continuous Dynamical Systems - S, 2021, 14 (4) : 1233-1244. 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. [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. [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. [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. [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. [6] I. I. Gikhman and A. V. Skorokhod, Stochastic differential equations, in The Theory of Stochastic Processes III, Springer, 2007,113–219. [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. [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. [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. [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. [11] J. D. Murray, Mathematical Biology. I, Interdisciplinary Applied Mathematics, 17, Springer-Verlag, New York, 2002. doi: 10.1007/b98868. [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. [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. [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. [15] L. Shaikhet, Lyapunov Functionals and Stability of Stochastic Functional Differential Equations, Springer, Cham, 2013. doi: 10.1007/978-3-319-00101-2. [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. [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.

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##### 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. [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. [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. [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. [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. [6] I. I. Gikhman and A. V. Skorokhod, Stochastic differential equations, in The Theory of Stochastic Processes III, Springer, 2007,113–219. [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. [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. [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. [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. [11] J. D. Murray, Mathematical Biology. I, Interdisciplinary Applied Mathematics, 17, Springer-Verlag, New York, 2002. doi: 10.1007/b98868. [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. [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. [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. [15] L. Shaikhet, Lyapunov Functionals and Stability of Stochastic Functional Differential Equations, Springer, Cham, 2013. doi: 10.1007/978-3-319-00101-2. [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. [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.
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
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, $E_1 = (N^*, C_o^*,C_r^*,C_b^*) = (0.3167,0.4189,0.1542,0.1101)$
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 $1.1 \%$ $1.0 \%$ $0.9 \%$ $1.1 \%$
 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 $1.1 \%$ $1.0 \%$ $0.9 \%$ $1.1 \%$
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 $\mu$ $1.587198 \,\, 10^{-3}$ $d$ $1.587198 \,\, 10^{-3}$ $\beta$ $5.013946 \,\, 10^{-3}$ $\varepsilon$ $5.855882 \,\, 10^{-6}$ $\gamma$ $1.003084 \,\, 10^{-3}$ $\sigma$ $1.137033 \,\, 10^{-3}$
 Model parameters Estimations $\mu$ $1.587198 \,\, 10^{-3}$ $d$ $1.587198 \,\, 10^{-3}$ $\beta$ $5.013946 \,\, 10^{-3}$ $\varepsilon$ $5.855882 \,\, 10^{-6}$ $\gamma$ $1.003084 \,\, 10^{-3}$ $\sigma$ $1.137033 \,\, 10^{-3}$
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