Pulse vaccination strategy in the SIR epidemic model

Theoretical results show that the measles ‘pulse’ vaccination strategy can be distinguished from the conventional strategies in leading to disease eradication at relatively low values of vaccination. Using the SIR epidemic model we showed that under a planned pulse vaccination regime the system converges to a stable solution with the number of infectious individuals equal to zero. We showed that pulse vaccination leads to epidemics eradication if certain conditions regarding the magnitude of vaccination proportion and on the period of the pulses are adhered to. Our theoretical results are confirmed by numerical simulations. The introduction of seasonal variation into the basic SIR model leads to periodic and chaotic dynamics of epidemics. We showed that under seasonal variation, in spite of the complex dynamics of the system, pulse vaccination still leads to epidemic eradication. We derived the conditions for epidemic eradication under various constraints and showed their dependence on the parameters of the epidemic. We compared effectiveness and cost of constant, pulse and mixed vaccination policies.

[1]  J. Yorke,et al.  Recurrent outbreaks of measles, chickenpox and mumps. I. Seasonal variation in contact rates. , 1973, American journal of epidemiology.

[2]  Petre Tautu,et al.  Mathematical Models in Medicine : Workshop, Mainz, March 1976 , 1976 .

[3]  William S. Blau.  THE EFFECT OF ENVIRONMENTAL DISTURBANCE ON A TROPICAL BUTTERFLY POPULATION , 1980 .

[4]  G. Iooss,et al.  Elementary stability and bifurcation theory , 1980 .

[5]  R. May,et al.  Directly transmitted infections diseases: control by vaccination. , 1982, Science.

[6]  D. Schenzle.  An age-structured model of pre- and post-vaccination measles transmission. , 1984, IMA journal of mathematics applied in medicine and biology.

[7]  Zvia Agur,et al.  Randomness, synchrony and population persistence , 1985 .

[8]  M. Kot,et al.  Nearly one dimensional dynamics in an epidemic. , 1985, Journal of theoretical biology.

[9]  J. Deneubourg,et al.  The effect of environmental disturbances on the dynamics of marine intertidal populations , 1985 .

[10]  I B Schwartz,et al.  Multiple stable recurrent outbreaks and predictability in seasonally forced nonlinear epidemic models , 1985, Journal of mathematical biology.

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

[12]  D. Zwillinger.  Handbook of differential equations , 1990 .

[13]  L. Olsen,et al.  Chaos versus noisy periodicity: alternative hypotheses for childhood epidemics. , 1990, Science.

[14]  J. Aron,et al.  Multiple attractors in the response to a vaccination program. , 1990, Theoretical population biology.

[15]  George Sugihara,et al.  Nonlinear forecasting as a way of distinguishing chaos from measurement error in time series , 1990, Nature.

[16]  R. May,et al.  Infectious Diseases of Humans: Dynamics and Control , 1991, Annals of Internal Medicine.

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

[18]  Bryan T. Grenfell,et al.  Chance and Chaos in Measles Dynamics , 1992 .

[19]  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.

[20]  Eric T. Funasaki,et al.  Invasion and Chaos in a Periodically Pulsed Mass-Action Chemostat , 1993 .

[21]  Ralf Engbert,et al.  Chance and chaos in population biology—Models of recurrent epidemics and food chain dynamics , 1994 .

[22]  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.

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

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