Qualitative analyses of SIS epidemic model with vaccination and varying total population size

An SIS epidemic model with vaccination, temporary immunity, and varying total population size is studied. Three threshold parameters R0, R1, and R2 are identified. The disease-free equilibrium is globally stable if R0 ≤ 1 and unstable if R0 > 1, the endemic equilibrium is globally stable if R0 > 1. The disease cannot break out if R1 < 1, the disease may break out when the fractions of the susceptible and the infectious satisfy some condition if R1 > 1 and R0 ≤ 1. The population becomes extinct ultimately and the disease always exists in the population if R0 > 1 and R2 ≤ 1. There is a really endemic disease if R0 > 1 and R2 > 1. The global stability of the disease-free equilibrium and the existence and global stability of the endemic equilibrium are proved by means of LaSalleʹs invariance principle, the method of estimating values and Stokesʹ theorem, respectively. The results with vaccination and without vaccination are compared, the measures and effects of vaccination are discussed.