Many infectious diseases exhibit seasonally varying incidence rates. In order to mathematically describe and predict the behavior of such diseases, the seasonal susceptible-infected-removed (Seasonal SIR) model, a system of ordinary differential equations, was formulated. It has been found in numerical simulations of the Seasonal SIR model that sub-harmonic bifurcations may occur, leading to the appearance of periodic modes with inter-outbreak periods on the scale of two time units or more in response to a varying forcing of period one. In the extremes of these long inter-outbreak period modes, it has been observed that periodic modes on the order of nine time units or more may be found. Additionally, these long inter-outbreak period modes exhibit a behaviour of increasing outbreak severity as the time between outbreaks is lengthened. (Continued Within)