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We formulate and discuss models for the spread of infectious diseases with variablepopulation sizes and vaccinations on the susceptible individuals. First, we assume that the susceptible individuals are vaccinatedcontinuously. We establish the threshold-like results for the existence and global stability of the disease-free and the endemicequilibriums for these systems. Especially, we prove the global stability of the endemic equilibriums by converting the systems intointegrodifferential equations. Second, we suppose that vaccinations occur once per time period. We obtain the existence and globalstability of the disease-free periodic solutions for such systems with impulsive effects. By a useful bifurcation theorem, we acquirethe existence of the endemic periodic solutions when the disease-related deaths do not occur. At last, we compare theresults with vaccinations and without vaccinations and illustrate our results by numerical simulations

Dynamical Models for Infectious Diseases with Varying Population Size and Vaccinations