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July  2020, 25(7): 2407-2432. doi: 10.3934/dcdsb.2020016

Stability of stochastic heroin model with two distributed delays

Miljana Jovanović , and  Vuk Vujović

Faculty of Sciences and Mathematics, University of Niš, Višegradska 33, 18000 Niš, Serbia

* Corresponding author: Miljana Jovanović

Received  May 2019 Revised  July 2019 Published  July 2020 Early access  April 2020

Fund Project: The first author is supported by Grant No 174007 of MNTRS

In this paper a stability of stochastic heroin model with two distributed delays is studied. Precisely, the deterministic model for dynamics of heroin users is extended by random perturbation that briefly describe how a environmental fluctuations lead an individual to become a heroin user. By using a suitable Lyapunov function stability conditions for heroin use free equilibrium are obtained. Furthermore, asymptotic behavior around the heroin spread equilibrium of the deterministic model is investigated by using appropriate Lyapunov functional. Theoretical studies, based on real data, are applied on modeling of number of heroin users in the USA from $ 01.01.2014. $

Keywords: Distributed delay, heroin free equilibrium, heroin spread equilibrium, Lyapunov functional, mean square stability.
Mathematics Subject Classification: Primary: 60H10; Secondary: 92D30, 93E15.
Citation: Miljana Jovanović, Vuk Vujović. Stability of stochastic heroin model with two distributed delays. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2407-2432. doi: 10.3934/dcdsb.2020016
References:
[1]

D. Baǐnov and P. Simeonov, Integral Inequalities and Applications, Mathematics and its Applications, 57. Kluwer Academic Publishers Group, Dordrecht, 1992. doi: 10.1007/978-94-015-8034-2.

[2]

E. Beretta, V. Kolmanovskii and L. Shaikhet, Stability of epidemic model with time delays influenced by stochastic perturbations, Math. Comput. Simul., 45 (1998), 269-277.  doi: 10.1016/S0378-4754(97)00106-7.

[3]

B. Fang, X. Z. Li, M. Martcheva and L. M. Cao, Global stability for a heroin model with two distributed delay, Discrete Continuous Dynam. Systems-B, 19 (2014), 715-733.  doi: 10.3934/dcdsb.2014.19.715.

[4]

R. Z. Hasminskiǐ, Stochastic Stability of Differential Equations, Mechanics and Analysis, 7. Sijthoff & Noordhoff, Alphen aan den Rijn—Germantown, Md., 1980. doi: 90286010079789028601000.

[5]

H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653.  doi: 10.1137/S0036144500371907.

[6]

G. Huang and A. P. Liu, A note on global stability for a heroin epidemic model with distributed time delay, Appl. Math. Lett., 26 (2013), 687-691.  doi: 10.1016/j.aml.2013.01.010.

[7]

http://www.addictions.com/heroin/what-is-the-heroin-relapse-rate/.

[8]

http://www.betterhealth.vic.gov.au.

[9]

http://drugabuse.com/library/opiate-relapse/.

[10]

http://www.drugabuse.gov/drugs-abuse/opioids/opioid-overdose-crisis#five.

[11] V. B. Kolmanovskii and V. R. Nosov, Stability of Functional Differential Equations, Academic Press, 1986. 
[12]

R. N. Lipari and A. Hughes, The NSDUH Report: Trends in Heroin Use in the United States: 2002 to 2013, The CBHSQ Report, Substance Abuse and Mental Health Services Administration, 2015.

[13]

J. L. Liu and T. L. Zhang, Global behaviour of a heroin epidemic model with distributed delays, Appl. Math. Lett., 24 (2011), 1685-1692.  doi: 10.1016/j.aml.2011.04.019.

[14]

D. R. Mackintosh and G. T. Stewart, A mathematical model of a heroin epidemic: Implications for control policies, Journal of Epidemiology and Community Health, 33 (1979), 299-304.  doi: 10.1136/jech.33.4.299.

[15]

X. R. Mao, Stochastic Differential Equations and Applications, Second edition. Horwood Publishing Limited, Chichester, 2008. doi: 10.1533/9780857099402.

[16]

S. Mosel, Heroin relapse, Available from: http://drugabuse.com/library/heroin-relapse/.

[17]

G. Mulon and B. Straugham, A note on heroin epidemics, Math. Biosci., 218 (2009), 138-141.  doi: 10.1016/j.mbs.2009.01.006.

[18]

Epidemiologic Trends in Drug Abuse-Proceedings of the Community Epidemiology Work Group, National Institute on Drug Abuse, Bethesda, MD: National Institute on Drug Abuse, January 2012.

[19]

Drugs, Brains and Behaviour: The Science of Addiction, National Institute on Drug Abuse, Aviable from: http://www.drugabuse.gov/publications/drugs-brains-behavior-science-addiction.

[20]

Drug Facts., National Institute on Drug Abuse, Aviable from: http://teens.drugabuse.gov/sites/default/files/drugfacts_heroin_10_14.pdf.

[21]

Heroin: What are the treatments for heroin use disorder?, National Institute on Drug Abuse, Aviable from: http://www.drugabuse.gov/publications/research-reports/heroin/overview

[22]

E. Patterson, Can you get addicted to Heroin after first use?, Aviable from: http://drugabuse.com/library/heroin-first-time-use/.

[23]

S. Rushing and J. McKinley, Non-natural death trends in the United States, On the Risk, 32 (2016), 54-60. 

[24]

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

[25]

Substance Abuse and Mental Health Services Administration, Results from the 2012 National Survey on Drug Use and Health: Summary of National Findings, NSDUH Series H-46, HHS Publication No. (SMA), (2013), 13–4795.

[26]

Substance Abuse and Mental Health Services Administration (US); Office of the Surgeon General (US). Facing Addiction in America: The Surgeon General's Report on Alcohol, Drugs and Health [Internet], Washington (DC): US Department of Health and Human Services; Chapter 4, Early Intermention, Treatment and Management of Substance Use Disorders, 2016.

[27]

Statista, 2015, Aviable from: http://www.statista.com/.

[28]

M. Szalavitz, The Washington Post: Five myths about heroin, (2016).

[29]

E. Tornatore, S. M. Buccellato and P. Vetro, Stability of a stochastic SIR system, Physica A, 354 (2005), 111-126.  doi: 10.4064/am34-4-2.

[30]

D. Waldorf and P. Biernacki, Natural recovery from heroin addiction: A review of the incidence literature, Journal of Drug Issues, 9 (2) (1979), 281-289.  doi: 10.1177/002204267900900212.

[31]

E. White and C. Comiskey, Heroin epidemics, treatment and ODE modelling, Math. Biosci., 208 (2007), 312-324.  doi: 10.1016/j.mbs.2006.10.008.

[32]

L. T. Wu, G. E. Woody, C. Yang, P. Mannelli and D. G. Blazer, Differences in onset and abuse/dependence episodes between prescription opioids and heroin: Results from the National Epidemiologic Survey on Alcohol and Related Conditions, Substance Abuse and Rehabilitation, 2 (2011), 77-88.  doi: 10.2147/SAR.S18969.

show all references

References:
[1]

D. Baǐnov and P. Simeonov, Integral Inequalities and Applications, Mathematics and its Applications, 57. Kluwer Academic Publishers Group, Dordrecht, 1992. doi: 10.1007/978-94-015-8034-2.

[2]

E. Beretta, V. Kolmanovskii and L. Shaikhet, Stability of epidemic model with time delays influenced by stochastic perturbations, Math. Comput. Simul., 45 (1998), 269-277.  doi: 10.1016/S0378-4754(97)00106-7.

[3]

B. Fang, X. Z. Li, M. Martcheva and L. M. Cao, Global stability for a heroin model with two distributed delay, Discrete Continuous Dynam. Systems-B, 19 (2014), 715-733.  doi: 10.3934/dcdsb.2014.19.715.

[4]

R. Z. Hasminskiǐ, Stochastic Stability of Differential Equations, Mechanics and Analysis, 7. Sijthoff & Noordhoff, Alphen aan den Rijn—Germantown, Md., 1980. doi: 90286010079789028601000.

[5]

H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653.  doi: 10.1137/S0036144500371907.

[6]

G. Huang and A. P. Liu, A note on global stability for a heroin epidemic model with distributed time delay, Appl. Math. Lett., 26 (2013), 687-691.  doi: 10.1016/j.aml.2013.01.010.

[7]

http://www.addictions.com/heroin/what-is-the-heroin-relapse-rate/.

[8]

http://www.betterhealth.vic.gov.au.

[9]

http://drugabuse.com/library/opiate-relapse/.

[10]

http://www.drugabuse.gov/drugs-abuse/opioids/opioid-overdose-crisis#five.

[11] V. B. Kolmanovskii and V. R. Nosov, Stability of Functional Differential Equations, Academic Press, 1986. 
[12]

R. N. Lipari and A. Hughes, The NSDUH Report: Trends in Heroin Use in the United States: 2002 to 2013, The CBHSQ Report, Substance Abuse and Mental Health Services Administration, 2015.

[13]

J. L. Liu and T. L. Zhang, Global behaviour of a heroin epidemic model with distributed delays, Appl. Math. Lett., 24 (2011), 1685-1692.  doi: 10.1016/j.aml.2011.04.019.

[14]

D. R. Mackintosh and G. T. Stewart, A mathematical model of a heroin epidemic: Implications for control policies, Journal of Epidemiology and Community Health, 33 (1979), 299-304.  doi: 10.1136/jech.33.4.299.

[15]

X. R. Mao, Stochastic Differential Equations and Applications, Second edition. Horwood Publishing Limited, Chichester, 2008. doi: 10.1533/9780857099402.

[16]

S. Mosel, Heroin relapse, Available from: http://drugabuse.com/library/heroin-relapse/.

[17]

G. Mulon and B. Straugham, A note on heroin epidemics, Math. Biosci., 218 (2009), 138-141.  doi: 10.1016/j.mbs.2009.01.006.

[18]

Epidemiologic Trends in Drug Abuse-Proceedings of the Community Epidemiology Work Group, National Institute on Drug Abuse, Bethesda, MD: National Institute on Drug Abuse, January 2012.

[19]

Drugs, Brains and Behaviour: The Science of Addiction, National Institute on Drug Abuse, Aviable from: http://www.drugabuse.gov/publications/drugs-brains-behavior-science-addiction.

[20]

Drug Facts., National Institute on Drug Abuse, Aviable from: http://teens.drugabuse.gov/sites/default/files/drugfacts_heroin_10_14.pdf.

[21]

Heroin: What are the treatments for heroin use disorder?, National Institute on Drug Abuse, Aviable from: http://www.drugabuse.gov/publications/research-reports/heroin/overview

[22]

E. Patterson, Can you get addicted to Heroin after first use?, Aviable from: http://drugabuse.com/library/heroin-first-time-use/.

[23]

S. Rushing and J. McKinley, Non-natural death trends in the United States, On the Risk, 32 (2016), 54-60. 

[24]

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

[25]

Substance Abuse and Mental Health Services Administration, Results from the 2012 National Survey on Drug Use and Health: Summary of National Findings, NSDUH Series H-46, HHS Publication No. (SMA), (2013), 13–4795.

[26]

Substance Abuse and Mental Health Services Administration (US); Office of the Surgeon General (US). Facing Addiction in America: The Surgeon General's Report on Alcohol, Drugs and Health [Internet], Washington (DC): US Department of Health and Human Services; Chapter 4, Early Intermention, Treatment and Management of Substance Use Disorders, 2016.

[27]

Statista, 2015, Aviable from: http://www.statista.com/.

[28]

M. Szalavitz, The Washington Post: Five myths about heroin, (2016).

[29]

E. Tornatore, S. M. Buccellato and P. Vetro, Stability of a stochastic SIR system, Physica A, 354 (2005), 111-126.  doi: 10.4064/am34-4-2.

[30]

D. Waldorf and P. Biernacki, Natural recovery from heroin addiction: A review of the incidence literature, Journal of Drug Issues, 9 (2) (1979), 281-289.  doi: 10.1177/002204267900900212.

[31]

E. White and C. Comiskey, Heroin epidemics, treatment and ODE modelling, Math. Biosci., 208 (2007), 312-324.  doi: 10.1016/j.mbs.2006.10.008.

[32]

L. T. Wu, G. E. Woody, C. Yang, P. Mannelli and D. G. Blazer, Differences in onset and abuse/dependence episodes between prescription opioids and heroin: Results from the National Epidemiologic Survey on Alcohol and Related Conditions, Substance Abuse and Rehabilitation, 2 (2011), 77-88.  doi: 10.2147/SAR.S18969.

Figure 1.  Initial condition $ U_1(\theta) = \varphi(\theta) $, $ \theta \in [-10,0] $
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Figure 2.  CDF of truncated Weibull distribution over interval $ [0,10] $ with parameters $ (0.25,1.3) $ (left) and CDF of double truncated Cauchy distribution over interval $ [0,0.25] $ with parameters $ (-0.011,0.005) $ (right)
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Figure 3.  The graph of the deterministic model (1) and the stochastic trajectory of the number of susceptible individuals in USA from 1.1.2014
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Figure 4.  The graph of the deterministic model (1) and the stochastic trajectory of the number of heroin users not in treatment in USA from 1.1.2014
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Figure 5.  Stochastic trajectories of the number of susceptible individuals and heroin users not in treatment in USA from 1.1.2014. (left); stochastic trajectory of the number of heroin users not in treatment and real data (right)
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