Hopf bifurcation in epidemic models with a time delay in vaccination

Two SIR 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 Heterogeneity, pp. 79-101, Winnipeg: Wuerz Publishing) are considered but with a time delay in the vaccination term. This reflects the fact that real vaccines do not immediately confer permanent immunity. The population is divided into susceptible, infectious, 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 parameter, necessary and sufficient conditions for Hopf bifurcation to occur are derived. Numerical results indicate that that for diseases in human populations Hopf bifurcation is unlikely to occur at realistic parameter values if the death rate is a concave function of the population size.