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Global stability for a heroin model with two distributed delays
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 |
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. |
[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. |
[11] |
G. Mulone and B. Straughan, A note on heroin epidemics,, Mathematical Biosciences, 218 (2009), 138.
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.
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.
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.
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.
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: , (). 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).
|
[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. |
[24] |
N. MacDonald, Time Lags in Biological Models, Lecture Notes in Biomathematics,, Springer-Verlag, 27 (1978).
|
[25] |
J. K. Hale, Theory of Functional Differential Equations,, Second edition, (1977).
|
[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).
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[35] |
J. P. LaSalle, Stability of Dynamical Systems,, SIAM, (1976).
|
[36] |
A. M. Lyapunov, The General Problem of the Stability of Motion,, Taylor and Francis, (1992).
|
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. |
[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. |
[11] |
G. Mulone and B. Straughan, A note on heroin epidemics,, Mathematical Biosciences, 218 (2009), 138.
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.
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.
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.
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.
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: , (). 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).
|
[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. |
[24] |
N. MacDonald, Time Lags in Biological Models, Lecture Notes in Biomathematics,, Springer-Verlag, 27 (1978).
|
[25] |
J. K. Hale, Theory of Functional Differential Equations,, Second edition, (1977).
|
[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).
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[35] |
J. P. LaSalle, Stability of Dynamical Systems,, SIAM, (1976).
|
[36] |
A. M. Lyapunov, The General Problem of the Stability of Motion,, Taylor and Francis, (1992).
|
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