A note on global stability for a heroin epidemic model with distributed delay

Abstract By using the direct Lyapunov method and constructing appropriate Lyapunov functional, the global stability for the heroin epidemic model with distributed delay is investigated. It is shown that the disease endemic equilibrium of the system is globally asymptotically stable whenever it exists. This improves the related results presented in [J. Liu, T. Zhang, Global behavior of a heroin epidemic model with distributed delay, Appl. Math. Lett. 24 (2011) 1685–1692].

[1]  Philip K Maini,et al.  A lyapunov function and global properties for sir and seir epidemiological models with nonlinear incidence. , 2004, Mathematical biosciences and engineering : MBE.

[2]  Andrei Korobeinikov,et al.  Global Properties of Infectious Disease Models with Nonlinear Incidence , 2007, Bulletin of mathematical biology.

[3]  Yasuhiro Takeuchi,et al.  Global analysis on delay epidemiological dynamic models with nonlinear incidence , 2011, Journal of mathematical biology.

[4]  C. Connell McCluskey,et al.  Complete global stability for an SIR epidemic model with delay — Distributed or discrete , 2010 .

[5]  Andrei Korobeinikov,et al.  Global asymptotic properties of staged models with multiple progression pathways for infectious diseases. , 2011, Mathematical biosciences and engineering : MBE.

[6]  Jack K. Hale,et al.  Introduction to Functional Differential Equations , 1993, Applied Mathematical Sciences.

[7]  C. McCluskey,et al.  Global stability for an SEIR epidemiological model with varying infectivity and infinite delay. , 2009, Mathematical biosciences and engineering : MBE.

[8]  Tailei Zhang,et al.  Global behaviour of a heroin epidemic model with distributed delays , 2011, Appl. Math. Lett..

[9]  Dynamics of a Heroin Epidemic Model with Very Population , 2011 .

[10]  Andrei Korobeinikov,et al.  Global asymptotic properties of virus dynamics models with dose-dependent parasite reproduction and virulence and non-linear incidence rate. , 2009, Mathematical medicine and biology : a journal of the IMA.

[11]  Gang Huang,et al.  Global analysis for delay virus dynamics model with Beddington-DeAngelis functional response , 2011, Appl. Math. Lett..

[12]  Gang Huang,et al.  Lyapunov Functionals for Delay Differential Equations Model of Viral Infections , 2010, SIAM J. Appl. Math..

[13]  Andrei Korobeinikov,et al.  Lyapunov Functions and Global Stability for SIR and SIRS Epidemiological Models with Non-Linear Transmission , 2006, Bulletin of mathematical biology.

[14]  E. White,et al.  Heroin epidemics, treatment and ODE modelling. , 2007, Mathematical biosciences.

[15]  Yasuhiro Takeuchi,et al.  Global Stability for Delay SIR and SEIR Epidemic Models with Nonlinear Incidence Rate , 2010, Bulletin of mathematical biology.

[16]  Brian Straughan,et al.  A note on heroin epidemics. , 2009, Mathematical biosciences.

[17]  Anping Liu,et al.  Qualitative analysis of the SICR epidemic model with impulsive vaccinations , 2013 .

[18]  Y. Takeuchi,et al.  Stability conditions for a class of delay differential equations in single species population dynamics , 2012 .