Global analysis of SIS epidemic models with variable total population size

Epidemic models of SIS type without vaccination and with vaccination are analyzed to determine the thresholds, equilibria, and stabilities. The vaccinations have two different formulations. One of SIS models with vaccination is a delay differential equation, another is an ordinary differential equation. For the delay differential equation, the asymptotic behavior is globally asymptotically stable convergence to a disease-free equilibrium, below the threshold, and to an endemic equilibrium, above a threshold. For the ordinary differential equations, the asymptotic behavior is globally asymptotically stable convergence to a disease-free equilibrium below the threshold, the sufficient conditions of the globally asymptotically stable convergence to an endemic equilibrium are obtained above the threshold, and the endemic equilibrium is locally asymptotically stable above the threshold.