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.