Threshold and stability results for an sirs epidemic model with a general periodic vaccination strategy

An SIRS epidemic model with general periodic vaccination strategy is analyzed. This periodic vaccination strategy is discussed first for an SIRS model with seasonal variation in the contact rate of period T = 1 year. We start with the case where the vaccination strategy and the contact rate have the same period and then discuss the case where the period of the vaccination strategy is LT, where L is an integer. We investigate whether a periodic vaccination strategy may force the epidemic dynamics to have periodic behavior. We prove that our SIRS model has a unique periodic disease free solution (DFS) whose period is the same as that of the vaccination strategy, which is globally asymptotically stable when the basic reproductive number Ro is less than or equal to one in value. When R-0 > 1, we prove that there exists a non-trivial periodic solution of period the same as that of the vaccination strategy. Some persistence results are also discussed. Threshold conditions for these periodic vaccination strategies to ensure that R-0 <= 1 are derived.