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## 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  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 & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020356
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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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