The dynamics of a new SIR epidemic model concerning pulse vaccination strategy

Abstract A new SIR epidemic model with vertical and horizontal transmission is proposed, and the dynamics of this disease model under constant and pulse vaccination are analyzed. Firstly, global stability of the endemic equilibrium states of the model with constant vaccination is thereby established. Further, we show that there exists a stable ‘infection-free’ periodic solution when the period of impulsive effect is less than some critical value. The condition for the permanence of the system with pulse vaccination is also given, which implies the periodic bursts of epidemic occurs. Numerical simulation shows system with pulse vaccination has more complex dynamic behavior for positive periodic oscillation, ‘infection free’ quasi-periodic oscillation than system with constant vaccination. Finally, we compare validity of the strategy of pulse vaccination with no vaccination and constant vaccination, and conclude that pulse vaccination strategy is more effective than no vaccination and continuous vaccination.

[1]  P. Fine.  VECTORS AND VERTICAL TRANSMISSION: AN EPIDEMIOLOGIC PERSPECTIVE , 1975, Annals of the New York Academy of Sciences.

[2]  K. Cooke,et al.  Analysis of a model of a vertically transmitted disease , 1983, Journal of mathematical biology.

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

[4]  J. Hale,et al.  Ordinary Differential Equations , 2019, Fundamentals of Numerical Mathematics for Physicists and Engineers.

[5]  R. Anderson,et al.  Pulse mass measles vaccination across age cohorts. , 1993, Proceedings of the National Academy of Sciences of the United States of America.

[6]  Stavros Busenberg,et al.  MODELS OF VERTICALLY TRANSMITTED DISEASES WITH SEQUENTIAL-CONTINUOUS DYNAMICS , 1982 .

[7]  H. Bedford,et al.  Measles , 1889, BMJ.

[8]  J. Andrus,et al.  Eradication of poliomyelitis: progress in the Americas. , 1991, The Pediatric infectious disease journal.

[9]  Xin-zhu Meng,et al.  Global dynamics behaviors for new delay SEIR epidemic disease model with vertical transmission and pulse vaccination , 2007 .

[10]  Elaheh Pourabbas,et al.  A method to estimate the incidence of communicable diseases under seasonal fluctuations with application to cholera , 2001, Appl. Math. Comput..

[11]  Yasuhiro Takeuchi,et al.  Convergence results in SIR epidemic models with varying population sizes , 1997 .

[12]  T. John,et al.  Eradication of poliomyelitis , 1999, The Lancet.

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

[14]  Edward H. Twizell,et al.  An unconditionally convergent finite-difference scheme for the SIR model , 2003, Appl. Math. Comput..

[15]  V. Lakshmikantham,et al.  Nonlinear Phenomena in Mathematical Sciences , 1982 .

[16]  P E Fine,et al.  Towards a quantitative understanding of the epidemiology of Keystone virus in the eastern United States. , 1978, The American journal of tropical medicine and hygiene.

[17]  P K Maini,et al.  Pattern formation in a generalized chemotactic model , 1998, Bulletin of mathematical biology.

[18]  Fengyan Wang,et al.  Bifurcation and complexity of Monod type predator–prey system in a pulsed chemostat ☆ , 2006 .

[19]  M Rush,et al.  The epidemiology of measles in England and Wales: rationale for the 1994 national vaccination campaign. , 1994, Communicable disease report. CDR review.

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

[21]  Lansun Chen,et al.  Two profitless delays for the SEIRS epidemic disease model with nonlinear incidence and pulse vaccination , 2007, Appl. Math. Comput..

[22]  H. Hethcote.  Three Basic Epidemiological Models , 1989 .

[23]  A. Sabin,et al.  Measles, killer of millions in developing countries: strategy for rapid elimination and continuing control , 2004, European Journal of Epidemiology.