Dynamical aspects of epidemic models with reinfection due to waning immunity are not fully understood. In this talk we would like to introduce a mathematical model formulated by a system of delay differential equations in order to give a possible explanation of periodic outbreak of mycoplasma pneumoniae observed in Japan. Stability analysis and numerical studies for periodic solutions will be presented together with biological interpretations. Subsequently we study a general SIS type epidemic model formulated by a scalar integral equation, motivated by a recent open problem regarding global stability of an equilibrium.