Hopf bifurcation in epidemic models with a time delay in vaccination

Citation
Qja. Khan et D. Greenhalgh, Hopf bifurcation in epidemic models with a time delay in vaccination, IMA J MATH, 16(2), 1999, pp. 113-142
Citations number
37
Categorie Soggetti
Multidisciplinary
Journal title
IMA JOURNAL OF MATHEMATICS APPLIED IN MEDICINE AND BIOLOGY
ISSN journal
02650746 → ACNP
Volume
16
Issue
2
Year of publication
1999
Pages
113 - 142
Database
ISI
SICI code
0265-0746(199906)16:2<113:HBIEMW>2.0.ZU;2-I
Abstract
Two STP models for the spread of infectious diseases which were originally suggested by Greenhalgh & Das (1995, Theor. Popul. Biol. 47, 129-179; 1995, Mathematical Population Dynamics: Analysis of Hetrerogeneity, pp. 79-101, Winnipeg: Wuerz Publishing) are considered but with a time delay in the vac cination term. This reflects the fact that real vaccines do not immediately confer permanent immunity. The population is divided into susceptible, inf ectious, and immune classes. The contact rate is constant in model I but it depends on the population size in model II. The death rate depends on the population size in both models. There is an additional mortality due to the disease, and susceptibles are vaccinated and may become permanently immune after a lapse of some time. Using the time delay as a bifurcation paramete r, necessary and sufficient conditions for Hopf bifurcation to occur are de rived. Numerical results indicate that that for diseases in human populatio ns Hopf bifurcation is unlikely to occur at realistic parameter values if t he death rate is a concave function of the population size.