American Institute of Mathematical Sciences

Advanced
May  2014, 19(3): 715-733. doi: 10.3934/dcdsb.2014.19.715

Global stability for a heroin model with two distributed delays

Bin Fang 1, , Xue-Zhi Li 2, , Maia Martcheva 3, and  Li-Ming Cai 2,

1. 

Beijing Institute of Information and Control, Beijing 100037, China
2. 

Department of Mathematics, Xinyang Normal University, Xinyang 464000, China
3. 

Department of Mathematics, University of Florida, 358 Little Hall, PO Box 118105, Gainesville

Received  July 2013 Revised  October 2013 Published  February 2014

In this paper, we consider global stability for a heroin model with two distributed delays. The basic reproduction number of the heroin spread is obtained, which completely determines the stability of the equilibria. Using the direct Lyapunov method with Volterra type Lyapunov function, we show that the drug use-free equilibrium is globally asymptotically stable if the basic reproduction number is less than one, and the unique drug spread equilibrium is globally asymptotically stable if the basic reproduction number is greater than one.
Keywords: distributed delay, Heroin model, Lyapunov function, global stability., basic reproduction number.
Mathematics Subject Classification: Primary: 92D25; Secondary: 93D2.
Citation: 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
References:
[1]

, NIDA DrugFacts: Heroin, Report of National Institute on Drug Abuse., Available from: , ().   Google Scholar

[2]

K. A. Sporer, Acute heroin overdose, Ann. Intern. Med., 130 (1999), 584-590., Available from: , ().   Google Scholar

[3]

X. Li, Y. Zhou and B. Stanton, Illicit drug initiation among institutionalized drug users in China, Addiction, 97 (2002), 575-582., Available from: , ().   Google Scholar

[4]

J. Cohen, HIV/AIDS in China. Poised for takeoff? Science, 304 (2004), 1430-1432., Available from: , ().   Google Scholar

[5]

R. J. Garten, S. Lai, J. Zhang, W. Liu , J. Chen, D. Vlahov and X. F. Yu, Rapid transmission of hepatitis C virus among young injecting heroin users in Southern China, Int. J. Epidemiol., 33 (2004), 182-188., Available from: , ().   Google Scholar

[6]

C. Comiskey, National Prevalence of Problematic Opiate Use in Ireland,, EMCDDA Tech. Report, (1999).   Google Scholar

[7]

A. Kelly, M. Carvalho and C. Teljeur, Prevalence of Opiate Use in Ireland 2000-2001: A 3-Source Capture Recapture Study, A Report to the National Advisory Committee on Drugs, Sub-committee on Prevalence, Small Area Health Research Unit, Department of Public, 2003., Available from: , (): 2647.   Google Scholar

[8]

European Monitoring Centre for Drugs and Drug Addiction (EMCDDA), Annual Report, 2005., Available from: , ().   Google Scholar

[9]

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.  doi: 10.1136/jech.33.4.299.  Google Scholar

[10]

E. White and C. Comiskey, Heroin epidemics, treatment and ode modelling,, Mathematical Biosciences, 208 (2007), 312.  doi: 10.1016/j.mbs.2006.10.008.  Google Scholar

[11]

G. Mulone and B. Straughan, A note on heroin epidemics,, Mathematical Biosciences, 218 (2009), 138.  doi: 10.1016/j.mbs.2009.01.006.  Google Scholar

[12]

X. Y. Wang, J. Y. Yang and X. Z. Li, Dynamics of a heroin epidemic model with very population,, Applied Mathematics, 2 (2011), 732.  doi: 10.4236/am.2011.26097.  Google Scholar

[13]

G. P. Samanta, Dynamic behaviour for a nonautonomous heroin epidemic model with time delay,, J. Appl. Math. Comput., 35 (2011), 161.  doi: 10.1007/s12190-009-0349-z.  Google Scholar

[14]

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

[15]

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

[16]

H. Warburton, P. J. Turnbull and M. Hough, Occasional and Controlled Heroin Use: Not a Problem? A Report to Joseph Rowntree Foundation, UK, 2005., Available from: , ().   Google Scholar

[17]

L. D. Johnston, Reasons for Use, Abstention, and Quitting Illicit Drug Use by American Adolecents, A Report Commissioned by the Drugs-Violence Task Force of the National Sentencing Commission., Available from:, ().   Google Scholar

[18]

C. Comiskey, P. Kelly, Y. Leckey, L. McCulloch, B. O'Duill, R. D. Stapleton and E. White, The ROSIE Study: Drug Treatment Outcomes in Ireland, June, 2009., Available from: , ().   Google Scholar

[19]

L. Elveback et al., Stochastic two-agent epidemic simulation models for a 379 community of families,, Amer. J. Epidemiol., (1971), 267.   Google Scholar

[20]

N. T. J. Bailey, The Mathematical Theory of Infectious Diseases and Its Applications,, Second edition. Hafner Press (Macmillan Publishing Co., (1975).   Google Scholar

[21]

Z. Ma, Y. Zhou, W. Wang and Z. Jin, Mathematical Models and Dynamics of Infectious Diseases,, China Sciences Press, (2004).   Google Scholar

[22]

G. Samanta, Permanence and extinction for a nonautonomous SVIR epidemic model with distributed time delay,, World Journal of Modelling and Simulation, 8 (2012), 3.   Google Scholar

[23]

J. M. Cushing, Bifurcation of periodic solutions of integro-differential equations with applications to time delay models in population dynamics,, SIAM J. Appl. Math., 33 (1977), 640.  doi: 10.1137/0133045.  Google Scholar

[24]

N. MacDonald, Time Lags in Biological Models, Lecture Notes in Biomathematics,, Springer-Verlag, 27 (1978).   Google Scholar

[25]

J. K. Hale, Theory of Functional Differential Equations,, Second edition, (1977).   Google Scholar

[26]

R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control,, Oxford University, (1991).   Google Scholar

[27]

L. E. él'sgol'ts and S. B. Norkin, Introduction to the Theory and Application of Differential Equations with Deviating Arguments,, Academic Press, (1973).   Google Scholar

[28]

A. Korobeinikov, Stability of ecosystem: Global properties of a general predator-prey model,, Math. Med. Biol., 4 (2009), 309.  doi: 10.1093/imammb/dqp009.  Google Scholar

[29]

P. Magal, C. C. McCluskey and G. F. Webb, Lyapunov functional and global asymptotic stability for an infection-age model,, Applicable Analysis, 89 (2010), 1109.  doi: 10.1080/00036810903208122.  Google Scholar

[30]

C. C. McCluskey, Complete global stability for an SIR epidemic model with delay distributed or discrete,, Nonlinear Anal. RWA., 11 (2010), 55.  doi: 10.1016/j.nonrwa.2008.10.014.  Google Scholar

[31]

G. Huang, Y. Takeuchi and W. Ma, Lyapunov functionals for Delay differential equations model for viral infections,, SIAM J. Appl. Math., 70 (2010), 2693.  doi: 10.1137/090780821.  Google Scholar

[32]

G. Huang, Y. Takeuchi, W. Ma and D. Wei, Global Stability for delay SIR and SEIR epidemic models with nonlinear incidence rate,, Bull. Math. Biol., 72 (2010), 1192.  doi: 10.1007/s11538-009-9487-6.  Google Scholar

[33]

G. Huang, W. Ma and Y. Takeuchi, Global analysis for delay virus dynamics model with Beddington-DeAngelis functional response,, Appl. Math. Lett., 24 (2011), 1199.  doi: 10.1016/j.aml.2011.02.007.  Google Scholar

[34]

G. Huang and Y. Takeuchi, Global analysis on delay epidemiological dynamic models with nonlinear incidence,, J. Math. Biol., 63 (2011), 125.  doi: 10.1007/s00285-010-0368-2.  Google Scholar

[35]

J. P. LaSalle, Stability of Dynamical Systems,, SIAM, (1976).   Google Scholar

[36]

A. M. Lyapunov, The General Problem of the Stability of Motion,, Taylor and Francis, (1992).   Google Scholar

show all references

References:
[1]

, NIDA DrugFacts: Heroin, Report of National Institute on Drug Abuse., Available from: , ().   Google Scholar

[2]

K. A. Sporer, Acute heroin overdose, Ann. Intern. Med., 130 (1999), 584-590., Available from: , ().   Google Scholar

[3]

X. Li, Y. Zhou and B. Stanton, Illicit drug initiation among institutionalized drug users in China, Addiction, 97 (2002), 575-582., Available from: , ().   Google Scholar

[4]

J. Cohen, HIV/AIDS in China. Poised for takeoff? Science, 304 (2004), 1430-1432., Available from: , ().   Google Scholar

[5]

R. J. Garten, S. Lai, J. Zhang, W. Liu , J. Chen, D. Vlahov and X. F. Yu, Rapid transmission of hepatitis C virus among young injecting heroin users in Southern China, Int. J. Epidemiol., 33 (2004), 182-188., Available from: , ().   Google Scholar

[6]

C. Comiskey, National Prevalence of Problematic Opiate Use in Ireland,, EMCDDA Tech. Report, (1999).   Google Scholar

[7]

A. Kelly, M. Carvalho and C. Teljeur, Prevalence of Opiate Use in Ireland 2000-2001: A 3-Source Capture Recapture Study, A Report to the National Advisory Committee on Drugs, Sub-committee on Prevalence, Small Area Health Research Unit, Department of Public, 2003., Available from: , (): 2647.   Google Scholar

[8]

European Monitoring Centre for Drugs and Drug Addiction (EMCDDA), Annual Report, 2005., Available from: , ().   Google Scholar

[9]

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.  doi: 10.1136/jech.33.4.299.  Google Scholar

[10]

E. White and C. Comiskey, Heroin epidemics, treatment and ode modelling,, Mathematical Biosciences, 208 (2007), 312.  doi: 10.1016/j.mbs.2006.10.008.  Google Scholar

[11]

G. Mulone and B. Straughan, A note on heroin epidemics,, Mathematical Biosciences, 218 (2009), 138.  doi: 10.1016/j.mbs.2009.01.006.  Google Scholar

[12]

X. Y. Wang, J. Y. Yang and X. Z. Li, Dynamics of a heroin epidemic model with very population,, Applied Mathematics, 2 (2011), 732.  doi: 10.4236/am.2011.26097.  Google Scholar

[13]

G. P. Samanta, Dynamic behaviour for a nonautonomous heroin epidemic model with time delay,, J. Appl. Math. Comput., 35 (2011), 161.  doi: 10.1007/s12190-009-0349-z.  Google Scholar

[14]

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

[15]

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

[16]

H. Warburton, P. J. Turnbull and M. Hough, Occasional and Controlled Heroin Use: Not a Problem? A Report to Joseph Rowntree Foundation, UK, 2005., Available from: , ().   Google Scholar

[17]

L. D. Johnston, Reasons for Use, Abstention, and Quitting Illicit Drug Use by American Adolecents, A Report Commissioned by the Drugs-Violence Task Force of the National Sentencing Commission., Available from:, ().   Google Scholar

[18]

C. Comiskey, P. Kelly, Y. Leckey, L. McCulloch, B. O'Duill, R. D. Stapleton and E. White, The ROSIE Study: Drug Treatment Outcomes in Ireland, June, 2009., Available from: , ().   Google Scholar

[19]

L. Elveback et al., Stochastic two-agent epidemic simulation models for a 379 community of families,, Amer. J. Epidemiol., (1971), 267.   Google Scholar

[20]

N. T. J. Bailey, The Mathematical Theory of Infectious Diseases and Its Applications,, Second edition. Hafner Press (Macmillan Publishing Co., (1975).   Google Scholar

[21]

Z. Ma, Y. Zhou, W. Wang and Z. Jin, Mathematical Models and Dynamics of Infectious Diseases,, China Sciences Press, (2004).   Google Scholar

[22]

G. Samanta, Permanence and extinction for a nonautonomous SVIR epidemic model with distributed time delay,, World Journal of Modelling and Simulation, 8 (2012), 3.   Google Scholar

[23]

J. M. Cushing, Bifurcation of periodic solutions of integro-differential equations with applications to time delay models in population dynamics,, SIAM J. Appl. Math., 33 (1977), 640.  doi: 10.1137/0133045.  Google Scholar

[24]

N. MacDonald, Time Lags in Biological Models, Lecture Notes in Biomathematics,, Springer-Verlag, 27 (1978).   Google Scholar

[25]

J. K. Hale, Theory of Functional Differential Equations,, Second edition, (1977).   Google Scholar

[26]

R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control,, Oxford University, (1991).   Google Scholar

[27]

L. E. él'sgol'ts and S. B. Norkin, Introduction to the Theory and Application of Differential Equations with Deviating Arguments,, Academic Press, (1973).   Google Scholar

[28]

A. Korobeinikov, Stability of ecosystem: Global properties of a general predator-prey model,, Math. Med. Biol., 4 (2009), 309.  doi: 10.1093/imammb/dqp009.  Google Scholar

[29]

P. Magal, C. C. McCluskey and G. F. Webb, Lyapunov functional and global asymptotic stability for an infection-age model,, Applicable Analysis, 89 (2010), 1109.  doi: 10.1080/00036810903208122.  Google Scholar

[30]

C. C. McCluskey, Complete global stability for an SIR epidemic model with delay distributed or discrete,, Nonlinear Anal. RWA., 11 (2010), 55.  doi: 10.1016/j.nonrwa.2008.10.014.  Google Scholar

[31]

G. Huang, Y. Takeuchi and W. Ma, Lyapunov functionals for Delay differential equations model for viral infections,, SIAM J. Appl. Math., 70 (2010), 2693.  doi: 10.1137/090780821.  Google Scholar

[32]

G. Huang, Y. Takeuchi, W. Ma and D. Wei, Global Stability for delay SIR and SEIR epidemic models with nonlinear incidence rate,, Bull. Math. Biol., 72 (2010), 1192.  doi: 10.1007/s11538-009-9487-6.  Google Scholar

[33]

G. Huang, W. Ma and Y. Takeuchi, Global analysis for delay virus dynamics model with Beddington-DeAngelis functional response,, Appl. Math. Lett., 24 (2011), 1199.  doi: 10.1016/j.aml.2011.02.007.  Google Scholar

[34]

G. Huang and Y. Takeuchi, Global analysis on delay epidemiological dynamic models with nonlinear incidence,, J. Math. Biol., 63 (2011), 125.  doi: 10.1007/s00285-010-0368-2.  Google Scholar

[35]

J. P. LaSalle, Stability of Dynamical Systems,, SIAM, (1976).   Google Scholar

[36]

A. M. Lyapunov, The General Problem of the Stability of Motion,, Taylor and Francis, (1992).   Google Scholar

[1]

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
[2]

Hui Cao, Yicang Zhou. The basic reproduction number of discrete SIR and SEIS models with periodic parameters. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 37-56. doi: 10.3934/dcdsb.2013.18.37
[3]

Jinhu Xu, Yicang Zhou. Global stability of a multi-group model with vaccination age, distributed delay and random perturbation. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1083-1106. doi: 10.3934/mbe.2015.12.1083
[4]

Nicolas Bacaër, Xamxinur Abdurahman, Jianli Ye, Pierre Auger. On the basic reproduction number $R_0$ in sexual activity models for HIV/AIDS epidemics: Example from Yunnan, China. Mathematical Biosciences & Engineering, 2007, 4 (4) : 595-607. doi: 10.3934/mbe.2007.4.595
[5]

Yukihiko Nakata, Yoichi Enatsu, Yoshiaki Muroya. On the global stability of an SIRS epidemic model with distributed delays. Conference Publications, 2011, 2011 (Special) : 1119-1128. doi: 10.3934/proc.2011.2011.1119
[6]

Gerardo Chowell, R. Fuentes, A. Olea, X. Aguilera, H. Nesse, J. M. Hyman. The basic reproduction number $R_0$ and effectiveness of reactive interventions during dengue epidemics: The 2002 dengue outbreak in Easter Island, Chile. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1455-1474. doi: 10.3934/mbe.2013.10.1455
[7]

Deqiong Ding, Wendi Qin, Xiaohua Ding. Lyapunov functions and global stability for a discretized multigroup SIR epidemic model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 1971-1981. doi: 10.3934/dcdsb.2015.20.1971
[8]

Mostafa Adimy, Fabien Crauste. Modeling and asymptotic stability of a growth factor-dependent stem cell dynamics model with distributed delay. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 19-38. doi: 10.3934/dcdsb.2007.8.19
[9]

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
[10]

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
[11]

Amitava Mukhopadhyay, Andrea De Gaetano, Ovide Arino. Modeling the intra-venous glucose tolerance test: A global study for a single-distributed-delay model. Discrete & Continuous Dynamical Systems - B, 2004, 4 (2) : 407-417. doi: 10.3934/dcdsb.2004.4.407
[12]

C. Connell McCluskey. Global stability of an $SIR$ epidemic model with delay and general nonlinear incidence. Mathematical Biosciences & Engineering, 2010, 7 (4) : 837-850. doi: 10.3934/mbe.2010.7.837
[13]

C. Connell McCluskey. Global stability for an SEIR epidemiological model with varying infectivity and infinite delay. Mathematical Biosciences & Engineering, 2009, 6 (3) : 603-610. doi: 10.3934/mbe.2009.6.603
[14]

Yincui Yan, Wendi Wang. Global stability of a five-dimensional model with immune responses and delay. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 401-416. doi: 10.3934/dcdsb.2012.17.401
[15]

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

Tianhui Yang, Lei Zhang. Remarks on basic reproduction ratios for periodic abstract functional differential equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6771-6782. doi: 10.3934/dcdsb.2019166
[17]

Shengqiang Liu, Lin Wang. Global stability of an HIV-1 model with distributed intracellular delays and a combination therapy. Mathematical Biosciences & Engineering, 2010, 7 (3) : 675-685. doi: 10.3934/mbe.2010.7.675
[18]

Tomás Caraballo, Renato Colucci, Luca Guerrini. On a predator prey model with nonlinear harvesting and distributed delay. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2703-2727. doi: 10.3934/cpaa.2018128
[19]

Peter Giesl. Construction of a global Lyapunov function using radial basis functions with a single operator. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 101-124. doi: 10.3934/dcdsb.2007.7.101
[20]

Andrei Korobeinikov, Philip K. Maini. A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence. Mathematical Biosciences & Engineering, 2004, 1 (1) : 57-60. doi: 10.3934/mbe.2004.1.57

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (35)
  • HTML views (0)
  • Cited by (16)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences