The effect of impulsive vaccination on an SIR epidemic model

In this paper, an SIR epidemic model is constructed and analyzed. We get the result that if the parameters satisfy the condition @b>@a+@c+b, then the disease will be ultimately permanent. Under this condition, we consider how the impulsive vaccination affects the original system. The sufficient condition for the global asymptotical stability of the disease-eradication solution is obtained. We also get that if the impulsive vaccination rate is less than some value, the disease will be permanent, and the disease cannot be controlled. People can select appropriate vaccination rate according to our theoretical result to control diseases.

[1]  H. Hethcote.  A Thousand and One Epidemic Models , 1994 .

[2]  Denis Mollison,et al.  Simplifying simple epidemic models , 1984, Nature.

[3]  H. Hethcote,et al.  Population size dependent incidence in models for diseases without immunity , 1994, Journal of mathematical biology.

[4]  R R Kao,et al.  The dynamics of an infectious disease in a population with birth pulses. , 1998, Mathematical biosciences.

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

[6]  Herbert W. Hethcote,et al.  Dynamic models of infectious diseases as regulators of population sizes , 1992, Journal of mathematical biology.

[7]  Alberto d'Onofrio,et al.  Stability properties of pulse vaccination strategy in SEIR epidemic model. , 2002, Mathematical biosciences.

[8]  Roy M. Anderson,et al.  Population dynamics of fox rabies in Europe , 1981, Nature.

[9]  W. O. Kermack,et al.  A contribution to the mathematical theory of epidemics , 1927 .

[10]  Zhidong Teng,et al.  Global behavior and permanence of SIRS epidemic model with time delay , 2008 .

[11]  Yasuhiro Takeuchi,et al.  Global asymptotic stability of an SIR epidemic model with distributed time delay , 2001 .

[12]  P. Bacon,et al.  Population Dynamics of Rabies in Wildlife , 1985 .

[13]  Xinzhu Meng,et al.  GLOBAL DYNAMICAL BEHAVIORS FOR AN SIR EPIDEMIC MODEL WITH TIME DELAY AND PULSE VACCINATION , 2008 .

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

[15]  Norman T. J. Bailey,et al.  The Mathematical Theory of Infectious Diseases , 1975 .

[16]  Lansun Chen,et al.  A STAGE-STRUCTURED SI ECO-EPIDEMIOLOGICAL MODEL WITH TIME DELAY AND IMPULSIVE CONTROLLING* , 2008, J. Syst. Sci. Complex..

[17]  Shujing Gao,et al.  Pulse vaccination strategy in a delayed sir epidemic model with vertical transmission , 2006 .

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

[19]  Lansun Chen,et al.  The dynamics of a new SIR epidemic model concerning pulse vaccination strategy , 2008, Appl. Math. Comput..

[20]  H. Hethcote,et al.  Four SEI endemic models with periodicity and separatrices. , 1995, Mathematical biosciences.

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

[22]  Denis Mollison,et al.  Sensitivity analysis of simple endemic models , 1985 .

[23]  M. Li,et al.  Global dynamics of a SEIR model with varying total population size. , 1999, Mathematical Biosciences.