American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Machine-learning construction of a model for a macroscopic fluid variable using the delay-coordinate of a scalar observable
  • DCDS-S Home
  • This Issue
  • Next Article
    A novel model for the contamination of a system of three artificial lakes
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]

Mazyar Ghani Varzaneh, Sebastian Riedel. A dynamical theory for singular stochastic delay differential equations Ⅱ: nonlinear equations and invariant manifolds. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020304
[2]

Graeme D. Chalmers, Desmond J. Higham. Convergence and stability analysis for implicit simulations of stochastic differential equations with random jump magnitudes. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 47-64. doi: 10.3934/dcdsb.2008.9.47
[3]

Zhenyu Lu, Junhao Hu, Xuerong Mao. Stabilisation by delay feedback control for highly nonlinear hybrid stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4099-4116. doi: 10.3934/dcdsb.2019052
[4]

Tian Zhang, Huabin Chen, Chenggui Yuan, Tomás Caraballo. On the asymptotic behavior of highly nonlinear hybrid stochastic delay differential equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5355-5375. doi: 10.3934/dcdsb.2019062
[5]

Neville J. Ford, Stewart J. Norton. Predicting changes in dynamical behaviour in solutions to stochastic delay differential equations. Communications on Pure & Applied Analysis, 2006, 5 (2) : 367-382. doi: 10.3934/cpaa.2006.5.367
[6]

Evelyn Buckwar, Girolama Notarangelo. A note on the analysis of asymptotic mean-square stability properties for systems of linear stochastic delay differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1521-1531. doi: 10.3934/dcdsb.2013.18.1521
[7]

Tomás Caraballo, José Real, T. Taniguchi. The exponential stability of neutral stochastic delay partial differential equations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 295-313. doi: 10.3934/dcds.2007.18.295
[8]

Mogtaba Mohammed, Mamadou Sango. Homogenization of nonlinear hyperbolic stochastic partial differential equations with nonlinear damping and forcing. Networks & Heterogeneous Media, 2019, 14 (2) : 341-369. doi: 10.3934/nhm.2019014
[9]

Zhen Li, Jicheng Liu. Synchronization for stochastic differential equations with nonlinear multiplicative noise in the mean square sense. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5709-5736. doi: 10.3934/dcdsb.2019103
[10]

Viorel Barbu. Existence for nonlinear finite dimensional stochastic differential equations of subgradient type. Mathematical Control & Related Fields, 2018, 8 (3&4) : 501-508. doi: 10.3934/mcrf.2018020
[11]

Igor Chueshov, Michael Scheutzow. Invariance and monotonicity for stochastic delay differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1533-1554. doi: 10.3934/dcdsb.2013.18.1533
[12]

Tomás Caraballo, Leonid Shaikhet. Stability of delay evolution equations with stochastic perturbations. Communications on Pure & Applied Analysis, 2014, 13 (5) : 2095-2113. doi: 10.3934/cpaa.2014.13.2095
[13]

Yuri Bakhtin. Lyapunov exponents for stochastic differential equations with infinite memory and application to stochastic Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 697-709. doi: 10.3934/dcdsb.2006.6.697
[14]

Leonid Shaikhet. Stability of delay differential equations with fading stochastic perturbations of the type of white noise and poisson's jumps. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3651-3657. doi: 10.3934/dcdsb.2020077
[15]

Nguyen Dinh Cong, Nguyen Thi Thuy Quynh. Coincidence of Lyapunov exponents and central exponents of linear Ito stochastic differential equations with nondegenerate stochastic term. Conference Publications, 2011, 2011 (Special) : 332-342. doi: 10.3934/proc.2011.2011.332
[16]

Alexandra Rodkina, Henri Schurz. On positivity and boundedness of solutions of nonlinear stochastic difference equations. Conference Publications, 2009, 2009 (Special) : 640-649. doi: 10.3934/proc.2009.2009.640
[17]

Bahareh Akhtari, Esmail Babolian, Andreas Neuenkirch. An Euler scheme for stochastic delay differential equations on unbounded domains: Pathwise convergence. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 23-38. doi: 10.3934/dcdsb.2015.20.23
[18]

Weiyin Fei, Liangjian Hu, Xuerong Mao, Dengfeng Xia. Advances in the truncated Euler–Maruyama method for stochastic differential delay equations. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2081-2100. doi: 10.3934/cpaa.2020092
[19]

Zhen Wang, Xiong Li, Jinzhi Lei. Second moment boundedness of linear stochastic delay differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2963-2991. doi: 10.3934/dcdsb.2014.19.2963
[20]

David Lipshutz. Exit time asymptotics for small noise stochastic delay differential equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 3099-3138. doi: 10.3934/dcds.2018135

2019 Impact Factor: 1.233

Tools

Article outline

Figures and Tables

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences