A SIRS epidemic model with general seasonal variation in the contact rate is analysed. This SIRS model has a unique disease free equilibrium (DFE) which is globally asymptotically stable when the basic reproductive number is less than or equal to one in value. Four childhood infectious diseases are studied (measles, chickenpox, mumps, and rubella). The computer simulation results show that non-trivial periodic solutions are possible depending on the type of the seasonally varying contact transmission rate. Different periodic solutions with different periods are obtained for different transmission forms. The range of endemic periodic solutions obtained also vary from one disease transmission functional form to another. Apart from measles with a sinusoidal varying contact rate these results are completely original.
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Greenhalgh et al. (2003) studied this question.