Global stability for a heroin model with two distributed delays

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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

Abstract
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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 and 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: http://www.drugabuse.gov/publications/drugfacts/heroin.
[2]	K. A. Sporer, Acute heroin overdose, Ann. Intern. Med., 130 (1999), 584-590. Available from: http://www.ncbi.nlm.nih.gov/pubmed/10189329.
[3]	X. Li, Y. Zhou and B. Stanton, Illicit drug initiation among institutionalized drug users in China, Addiction, 97 (2002), 575-582. Available from: http://www.ncbi.nlm.nih.gov/pubmed/12033658.
[4]	J. Cohen, HIV/AIDS in China. Poised for takeoff? Science, 304 (2004), 1430-1432. Available from: http://www.ncbi.nlm.nih.gov/pubmed/15178777.
[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: http://www.ncbi.nlm.nih.gov/pubmed/15075167.
[6]	C. Comiskey, National Prevalence of Problematic Opiate Use in Ireland, EMCDDA Tech. Report, 1999.
[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: http://www.drugsandalcohol.ie/5942/1/2647-2829.pdf.
[8]	European Monitoring Centre for Drugs and Drug Addiction (EMCDDA), Annual Report, 2005. Available from: http://www.drugsandalcohol.ie/11597/.html.
[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), 299-304. Available from:http://www.jstor.org/stable/25566135. doi: 10.1136/jech.33.4.299.
[10]	E. White and C. Comiskey, Heroin epidemics, treatment and ode modelling, Mathematical Biosciences, 208 (2007), 312-324. doi: 10.1016/j.mbs.2006.10.008.
[11]	G. Mulone and B. Straughan, A note on heroin epidemics, Mathematical Biosciences, 218 (2009), 138-141. doi: 10.1016/j.mbs.2009.01.006.
[12]	X. Y. Wang, J. Y. Yang and X. Z. Li, Dynamics of a heroin epidemic model with very population, Applied Mathematics, 2 (2011), 732-738. doi: 10.4236/am.2011.26097.
[13]	G. P. Samanta, Dynamic behaviour for a nonautonomous heroin epidemic model with time delay, J. Appl. Math. Comput., 35 (2011), 161-178. doi: 10.1007/s12190-009-0349-z.
[14]	J. Liu and T. 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.
[15]	G. Huang and A. Liu, A note on global stability for heroin epidemic model with distributed delay, Appl. Math. Lett., 26 (2013), 687-691. doi: 10.1016/j.aml.2013.01.010.
[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: http://www.doctordeluca.com/Library/AbstinenceHR/ControlledHeroinUse05.htm.
[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:http://www.monitoringthefuture.org/pubs/occpapers/occ44.pdf.
[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: http://www.drugs.ie/resourcesfiles/research/2009/ROSIE3-YearReport.pdf.
[19]	L. Elveback et al., Stochastic two-agent epidemic simulation models for a 379 community of families, Amer. J. Epidemiol., (1971), 267-280.
[20]	N. T. J. Bailey, The Mathematical Theory of Infectious Diseases and Its Applications, Second edition. Hafner Press (Macmillan Publishing Co., Inc.), New York, 1975.
[21]	Z. Ma, Y. Zhou, W. Wang and Z. Jin, Mathematical Models and Dynamics of Infectious Diseases, China Sciences Press, Beijing, 2004.
[22]	G. Samanta, Permanence and extinction for a nonautonomous SVIR epidemic model with distributed time delay, World Journal of Modelling and Simulation, 8 (2012), 3-18. Available from: http://www.worldacademicunion.com/journal/1746-7233WJMS/wjmsvol08no01paper01.pdf.
[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), 640-654. doi: 10.1137/0133045.
[24]	N. MacDonald, Time Lags in Biological Models, Lecture Notes in Biomathematics, Springer-Verlag, Berlin-New York, 27, 1978.
[25]	J. K. Hale, Theory of Functional Differential Equations, Second edition, Applied Mathematical Sciences, Vol.3., Springer-Verlag, New York-Heidelberg, 1977.
[26]	R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University, Oxford, 1991.
[27]	L. E. él'sgol'ts and S. B. Norkin, Introduction to the Theory and Application of Differential Equations with Deviating Arguments, Academic Press, New York-London, 1973.
[28]	A. Korobeinikov, Stability of ecosystem: Global properties of a general predator-prey model, Math. Med. Biol., 4 (2009), 309-321. doi: 10.1093/imammb/dqp009.
[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), 1109-1140. doi: 10.1080/00036810903208122.
[30]	C. C. McCluskey, Complete global stability for an SIR epidemic model with delay distributed or discrete, Nonlinear Anal. RWA., 11 (2010), 55-59. doi: 10.1016/j.nonrwa.2008.10.014.
[31]	G. Huang, Y. Takeuchi and W. Ma, Lyapunov functionals for Delay differential equations model for viral infections, SIAM J. Appl. Math., 70 (2010), 2693-2708. doi: 10.1137/090780821.
[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), 1192-1207. doi: 10.1007/s11538-009-9487-6.
[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), 1199-1203. doi: 10.1016/j.aml.2011.02.007.
[34]	G. Huang and Y. Takeuchi, Global analysis on delay epidemiological dynamic models with nonlinear incidence, J. Math. Biol., 63 (2011), 125-139. doi: 10.1007/s00285-010-0368-2.
[35]	J. P. LaSalle, Stability of Dynamical Systems, SIAM, Philadelphia, 1976.
[36]	A. M. Lyapunov, The General Problem of the Stability of Motion, Taylor and Francis, Ltd., London, 1992.

show all references

References:

[1]	, NIDA DrugFacts: Heroin, Report of National Institute on Drug Abuse. Available from: http://www.drugabuse.gov/publications/drugfacts/heroin.
[2]	K. A. Sporer, Acute heroin overdose, Ann. Intern. Med., 130 (1999), 584-590. Available from: http://www.ncbi.nlm.nih.gov/pubmed/10189329.
[3]	X. Li, Y. Zhou and B. Stanton, Illicit drug initiation among institutionalized drug users in China, Addiction, 97 (2002), 575-582. Available from: http://www.ncbi.nlm.nih.gov/pubmed/12033658.
[4]	J. Cohen, HIV/AIDS in China. Poised for takeoff? Science, 304 (2004), 1430-1432. Available from: http://www.ncbi.nlm.nih.gov/pubmed/15178777.
[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: http://www.ncbi.nlm.nih.gov/pubmed/15075167.
[6]	C. Comiskey, National Prevalence of Problematic Opiate Use in Ireland, EMCDDA Tech. Report, 1999.
[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: http://www.drugsandalcohol.ie/5942/1/2647-2829.pdf.
[8]	European Monitoring Centre for Drugs and Drug Addiction (EMCDDA), Annual Report, 2005. Available from: http://www.drugsandalcohol.ie/11597/.html.
[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), 299-304. Available from:http://www.jstor.org/stable/25566135. doi: 10.1136/jech.33.4.299.
[10]	E. White and C. Comiskey, Heroin epidemics, treatment and ode modelling, Mathematical Biosciences, 208 (2007), 312-324. doi: 10.1016/j.mbs.2006.10.008.
[11]	G. Mulone and B. Straughan, A note on heroin epidemics, Mathematical Biosciences, 218 (2009), 138-141. doi: 10.1016/j.mbs.2009.01.006.
[12]	X. Y. Wang, J. Y. Yang and X. Z. Li, Dynamics of a heroin epidemic model with very population, Applied Mathematics, 2 (2011), 732-738. doi: 10.4236/am.2011.26097.
[13]	G. P. Samanta, Dynamic behaviour for a nonautonomous heroin epidemic model with time delay, J. Appl. Math. Comput., 35 (2011), 161-178. doi: 10.1007/s12190-009-0349-z.
[14]	J. Liu and T. 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.
[15]	G. Huang and A. Liu, A note on global stability for heroin epidemic model with distributed delay, Appl. Math. Lett., 26 (2013), 687-691. doi: 10.1016/j.aml.2013.01.010.
[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: http://www.doctordeluca.com/Library/AbstinenceHR/ControlledHeroinUse05.htm.
[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:http://www.monitoringthefuture.org/pubs/occpapers/occ44.pdf.
[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: http://www.drugs.ie/resourcesfiles/research/2009/ROSIE3-YearReport.pdf.
[19]	L. Elveback et al., Stochastic two-agent epidemic simulation models for a 379 community of families, Amer. J. Epidemiol., (1971), 267-280.
[20]	N. T. J. Bailey, The Mathematical Theory of Infectious Diseases and Its Applications, Second edition. Hafner Press (Macmillan Publishing Co., Inc.), New York, 1975.
[21]	Z. Ma, Y. Zhou, W. Wang and Z. Jin, Mathematical Models and Dynamics of Infectious Diseases, China Sciences Press, Beijing, 2004.
[22]	G. Samanta, Permanence and extinction for a nonautonomous SVIR epidemic model with distributed time delay, World Journal of Modelling and Simulation, 8 (2012), 3-18. Available from: http://www.worldacademicunion.com/journal/1746-7233WJMS/wjmsvol08no01paper01.pdf.
[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), 640-654. doi: 10.1137/0133045.
[24]	N. MacDonald, Time Lags in Biological Models, Lecture Notes in Biomathematics, Springer-Verlag, Berlin-New York, 27, 1978.
[25]	J. K. Hale, Theory of Functional Differential Equations, Second edition, Applied Mathematical Sciences, Vol.3., Springer-Verlag, New York-Heidelberg, 1977.
[26]	R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University, Oxford, 1991.
[27]	L. E. él'sgol'ts and S. B. Norkin, Introduction to the Theory and Application of Differential Equations with Deviating Arguments, Academic Press, New York-London, 1973.
[28]	A. Korobeinikov, Stability of ecosystem: Global properties of a general predator-prey model, Math. Med. Biol., 4 (2009), 309-321. doi: 10.1093/imammb/dqp009.
[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), 1109-1140. doi: 10.1080/00036810903208122.
[30]	C. C. McCluskey, Complete global stability for an SIR epidemic model with delay distributed or discrete, Nonlinear Anal. RWA., 11 (2010), 55-59. doi: 10.1016/j.nonrwa.2008.10.014.
[31]	G. Huang, Y. Takeuchi and W. Ma, Lyapunov functionals for Delay differential equations model for viral infections, SIAM J. Appl. Math., 70 (2010), 2693-2708. doi: 10.1137/090780821.
[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), 1192-1207. doi: 10.1007/s11538-009-9487-6.
[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), 1199-1203. doi: 10.1016/j.aml.2011.02.007.
[34]	G. Huang and Y. Takeuchi, Global analysis on delay epidemiological dynamic models with nonlinear incidence, J. Math. Biol., 63 (2011), 125-139. doi: 10.1007/s00285-010-0368-2.
[35]	J. P. LaSalle, Stability of Dynamical Systems, SIAM, Philadelphia, 1976.
[36]	A. M. Lyapunov, The General Problem of the Stability of Motion, Taylor and Francis, Ltd., London, 1992.

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