Global attractivity and permanence of a SVEIR epidemic model with pulse vaccination and time delay

In this study, we propose a new SVEIR epidemic disease model with time delay, and analyze the dynamic behavior of the model under pulse vaccination. Pulse vaccination is an effective strategy for the elimination of infectious disease. Using the discrete dynamical system determined by the stroboscopic map, we obtain an 'infection-free' periodic solution. We also show that the 'infection-free' periodic solution is globally attractive when some parameters of the model under appropriate conditions. The permanence of the model is investigated analytically. Our results indicate that a large vaccination rate or a short pulse of vaccination or a long latent period is a sufficient condition for the extinction of the disease.

[1]  Maia Martcheva,et al.  Vaccination strategies and backward bifurcation in an age-since-infection structured model. , 2002, Mathematical biosciences.

[2]  D J Nokes,et al.  The control of childhood viral infections by pulse vaccination. , 1995, IMA journal of mathematics applied in medicine and biology.

[3]  B. Shulgin,et al.  Pulse vaccination strategy in the SIR epidemic model , 1998, Bulletin of mathematical biology.

[4]  Xue-Zhi Li,et al.  An epidemic model of a vector-borne disease with direct transmission and time delay , 2008 .

[5]  M. E. Alexander,et al.  A Vaccination Model for Transmission Dynamics of Influenza , 2004, SIAM J. Appl. Dyn. Syst..

[6]  D J Nokes,et al.  Vaccination in pulses: a strategy for global eradication of measles and polio? , 1997, Trends in microbiology.

[7]  Lansun Chen,et al.  Modeling and analysis of a predator-prey model with disease in the prey. , 2001, Mathematical biosciences.

[8]  S. N. Stepanov,et al.  Markov models with retrials: The calculation of stationary performance measures based on the concept of truncation , 1999 .

[9]  B. Hersh,et al.  Review of regional measles surveillance data in the Americas, 1996–99 , 2000, The Lancet.

[10]  M. Langlais,et al.  A remark on a generic SEIRS model and application to cat retroviruses and fox rabies , 2000 .

[11]  Alberto d’Onofrio,et al.  Pulse vaccination strategy in the sir epidemic model: Global asymptotic stable eradication in presence of vaccine failures , 2002 .

[12]  K. L. Cooke,et al.  Analysis of an SEIRS epidemic model with two delays , 1996, Journal of mathematical biology.

[13]  Graeme C. Wake,et al.  Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models , 2002, Appl. Math. Lett..

[14]  P Cull,et al.  Global stability of population models. , 1981, Bulletin of mathematical biology.

[15]  P. van Damme,et al.  Ten-year antibody persistence induced by hepatitis A and B vaccine (Twinrix) in adults. , 2007, Travel medicine and infectious disease.

[16]  A. Zanetti,et al.  Long-term immunogenicity of hepatitis B vaccination in a cohort of Italian healthy adolescents. , 2007, Vaccine.

[17]  Alberto d'Onofrio,et al.  Mixed pulse vaccination strategy in epidemic model with realistically distributed infectious and latent times , 2004, Appl. Math. Comput..

[18]  Julien Arino,et al.  Global Results for an Epidemic Model with Vaccination that Exhibits Backward Bifurcation , 2003, SIAM J. Appl. Math..

[19]  Alberto d'Onofrio,et al.  Vaccination policies and nonlinear force of infection: generalization of an observation by Alexander and Moghadas (2004) , 2005, Appl. Math. Comput..

[20]  D. Bainov,et al.  Impulsive Differential Equations: Periodic Solutions and Applications , 1993 .

[21]  Zvia Agur,et al.  Theoretical examination of the pulse vaccination policy in the SIR epidemic model , 2000 .

[22]  Liancheng Wang,et al.  Global Dynamics of an SEIR Epidemic Model with Vertical Transmission , 2001, SIAM J. Appl. Math..

[23]  Lansun Chen,et al.  Impulsive vaccination of sir epidemic models with nonlinear incidence rates , 2004 .

[24]  V. Lakshmikantham,et al.  Theory of Impulsive Differential Equations , 1989, Series in Modern Applied Mathematics.