# American Institute of Mathematical Sciences

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

## Stability of stochastic heroin model with two distributed delays

 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 $01.01.2014.$

Citation: Miljana Jovanović, Vuk Vujović. Stability of stochastic heroin model with two distributed delays. Discrete & 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [4] R. Z. Hasminskiǐ, Stochastic Stability of Differential Equations, Mechanics and Analysis, 7. Sijthoff & Noordhoff, Alphen aan den Rijn—Germantown, Md., 1980. doi: 90286010079789028601000.  Google Scholar [5] H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653.  doi: 10.1137/S0036144500371907.  Google Scholar [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.  Google Scholar [7] [8] [9] [10] [11] V. B. Kolmanovskii and V. R. Nosov, Stability of Functional Differential Equations, Academic Press, 1986.   Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [15] X. R. Mao, Stochastic Differential Equations and Applications, Second edition. Horwood Publishing Limited, Chichester, 2008. doi: 10.1533/9780857099402.  Google Scholar [16] S. Mosel, Heroin relapse, Available from: http://drugabuse.com/library/heroin-relapse/. Google Scholar [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.  Google Scholar [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. Google Scholar [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. Google Scholar [20] Drug Facts., National Institute on Drug Abuse, Aviable from: http://teens.drugabuse.gov/sites/default/files/drugfacts_heroin_10_14.pdf. Google Scholar [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 Google Scholar [22] E. Patterson, Can you get addicted to Heroin after first use?, Aviable from: http://drugabuse.com/library/heroin-first-time-use/. Google Scholar [23] S. Rushing and J. McKinley, Non-natural death trends in the United States, On the Risk, 32 (2016), 54-60.   Google Scholar [24] L. Shaikhet, Lyapunov Functionals and Stability of Stochastic Functional Differential Equations, Springer, Cham, 2013. doi: 10.1007/978-3-319-00101-2.  Google Scholar [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. Google Scholar [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. Google Scholar [27] Statista, 2015, Aviable from: http://www.statista.com/. Google Scholar [28] M. Szalavitz, The Washington Post: Five myths about heroin, (2016). Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [4] R. Z. Hasminskiǐ, Stochastic Stability of Differential Equations, Mechanics and Analysis, 7. Sijthoff & Noordhoff, Alphen aan den Rijn—Germantown, Md., 1980. doi: 90286010079789028601000.  Google Scholar [5] H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653.  doi: 10.1137/S0036144500371907.  Google Scholar [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.  Google Scholar [7] [8] [9] [10] [11] V. B. Kolmanovskii and V. R. Nosov, Stability of Functional Differential Equations, Academic Press, 1986.   Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [15] X. R. Mao, Stochastic Differential Equations and Applications, Second edition. Horwood Publishing Limited, Chichester, 2008. doi: 10.1533/9780857099402.  Google Scholar [16] S. Mosel, Heroin relapse, Available from: http://drugabuse.com/library/heroin-relapse/. Google Scholar [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.  Google Scholar [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. Google Scholar [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. Google Scholar [20] Drug Facts., National Institute on Drug Abuse, Aviable from: http://teens.drugabuse.gov/sites/default/files/drugfacts_heroin_10_14.pdf. Google Scholar [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 Google Scholar [22] E. Patterson, Can you get addicted to Heroin after first use?, Aviable from: http://drugabuse.com/library/heroin-first-time-use/. Google Scholar [23] S. Rushing and J. McKinley, Non-natural death trends in the United States, On the Risk, 32 (2016), 54-60.   Google Scholar [24] L. Shaikhet, Lyapunov Functionals and Stability of Stochastic Functional Differential Equations, Springer, Cham, 2013. doi: 10.1007/978-3-319-00101-2.  Google Scholar [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. Google Scholar [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. Google Scholar [27] Statista, 2015, Aviable from: http://www.statista.com/. Google Scholar [28] M. Szalavitz, The Washington Post: Five myths about heroin, (2016). Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
Initial condition $U_1(\theta) = \varphi(\theta)$, $\theta \in [-10,0]$
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)
The graph of the deterministic model (1) and the stochastic trajectory of the number of susceptible individuals in USA from 1.1.2014
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
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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