Global analysis for a general epidemiological model with vaccination and varying population

An SIR model with vaccination and varying population is formulated. The global dynamics of this model and its corresponding proportionate system are investigated. The correlations between the two systems in terms of disease eradication and persistence are presented. Three critical vaccination rates ϕ1c, ϕ2c and ϕ3c are obtained. It is found that when ϕ>ϕ1c the disease can be eradicated by increasing the vaccination rate until it exceeds ϕ3c. When ϕ<ϕ1c, the disease can be controlled to an endemic level by taking the appropriate vaccination rate ϕ2c.