American Institute of Mathematical Sciences

Advanced
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  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 & 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]

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

[8]

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

[9]

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

[10]

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

[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

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.  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]

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

[8]

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

[9]

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

[10]

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

[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

Figure 1.  Initial condition $ U_1(\theta) = \varphi(\theta) $, $ \theta \in [-10,0] $
Figure Options

Download full-size image

Download as PowerPoint slide

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)
Figure Options

Download full-size image

Download as PowerPoint slide

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
Figure Options

Download full-size image

Download as PowerPoint slide

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
Figure Options

Download full-size image

Download as PowerPoint slide

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)
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Bin Fang, Xue-Zhi Li, Maia Martcheva, Li-Ming Cai. Global stability for a heroin model with two distributed delays. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 715-733. doi: 10.3934/dcdsb.2014.19.715
[2]

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
[3]

Jingjing Wang, Zaiyun Peng, Zhi Lin, Daqiong Zhou. On the stability of solutions for the generalized vector quasi-equilibrium problems via free-disposal set. Journal of Industrial & Management Optimization, 2021, 17 (2) : 869-887. doi: 10.3934/jimo.2020002
[4]

Xiong Li. The stability of the equilibrium for a perturbed asymmetric oscillator. Communications on Pure & Applied Analysis, 2006, 5 (3) : 515-528. doi: 10.3934/cpaa.2006.5.515
[5]

Xiong Li. The stability of the equilibrium for a perturbed asymmetric oscillator. Communications on Pure & Applied Analysis, 2007, 6 (1) : 69-82. doi: 10.3934/cpaa.2007.6.69
[6]

Junya Nishiguchi. On parameter dependence of exponential stability of equilibrium solutions in differential equations with a single constant delay. Discrete & Continuous Dynamical Systems, 2016, 36 (10) : 5657-5679. doi: 10.3934/dcds.2016048
[7]

Abdennasser Chekroun, Mohammed Nor Frioui, Toshikazu Kuniya, Tarik Mohammed Touaoula. Mathematical analysis of an age structured heroin-cocaine epidemic model. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4449-4477. doi: 10.3934/dcdsb.2020107
[8]

Svetlana Bunimovich-Mendrazitsky, Yakov Goltser. Use of quasi-normal form to examine stability of tumor-free equilibrium in a mathematical model of bcg treatment of bladder cancer. Mathematical Biosciences & Engineering, 2011, 8 (2) : 529-547. doi: 10.3934/mbe.2011.8.529
[9]

Franco Maceri, Michele Marino, Giuseppe Vairo. Equilibrium and stability of tensegrity structures: A convex analysis approach. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 461-478. doi: 10.3934/dcdss.2013.6.461
[10]

L. Bakker. The Katok-Spatzier conjecture, generalized symmetries, and equilibrium-free flows. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1183-1200. doi: 10.3934/cpaa.2013.12.1183
[11]

Leonid Berezansky, Elena Braverman. Stability of linear differential equations with a distributed delay. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1361-1375. doi: 10.3934/cpaa.2011.10.1361
[12]

Edoardo Beretta, Dimitri Breda. Discrete or distributed delay? Effects on stability of population growth. Mathematical Biosciences & Engineering, 2016, 13 (1) : 19-41. doi: 10.3934/mbe.2016.13.19
[13]

Wei Wang, Kai Liu, Xiulian Wang. Sensitivity to small delays of mean square stability for stochastic neutral evolution equations. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2403-2418. doi: 10.3934/cpaa.2020105
[14]

Lili Liu, Xianning Liu, Jinliang Wang. Threshold dynamics of a delayed multi-group heroin epidemic model in heterogeneous populations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2615-2630. doi: 10.3934/dcdsb.2016064
[15]

Elena Braverman, Sergey Zhukovskiy. Absolute and delay-dependent stability of equations with a distributed delay. Discrete & Continuous Dynamical Systems, 2012, 32 (6) : 2041-2061. doi: 10.3934/dcds.2012.32.2041
[16]

Fuke Wu, Peter E. Kloeden. Mean-square random attractors of stochastic delay differential equations with random delay. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1715-1734. doi: 10.3934/dcdsb.2013.18.1715
[17]

Sibel Senan, Eylem Yucel, Zeynep Orman, Ruya Samli, Sabri Arik. A Novel Lyapunov functional with application to stability analysis of neutral systems with nonlinear disturbances. Discrete & Continuous Dynamical Systems - S, 2021, 14 (4) : 1415-1428. doi: 10.3934/dcdss.2020358
[18]

Emine Kaya, Eugenio Aulisa, Akif Ibragimov, Padmanabhan Seshaiyer. Stability analysis of inhomogeneous equilibrium for axially and transversely excited nonlinear beam. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1447-1462. doi: 10.3934/cpaa.2011.10.1447
[19]

Liangliang Ma. Stability of hydrostatic equilibrium to the 2D fractional Boussinesq equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021068
[20]

Xiao-Bing Li, Xian-Jun Long, Zhi Lin. Stability of solution mapping for parametric symmetric vector equilibrium problems. Journal of Industrial & Management Optimization, 2015, 11 (2) : 661-671. doi: 10.3934/jimo.2015.11.661

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (190)
  • HTML views (122)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences