We study changes in the bifurcations of seasonally driven compartmental epidemic models, where the transmission rate is modulated temporally. In the presence of periodic modulation of the transmission rate, the dynamics varies from periodic to chaotic. The route to chaos is typically through period doubling bifurcation. There are coexisting attractors for some sets of parameters. However in the presence of quasiperiodic modulation, tori are created in place of periodic orbits and chaos appears via finite torus doublings. Strange nonchaotic attractors (SNAs) are created at the boundary of chaotic and torus dynamics. Multistability is found to be reduced as a function of quasiperiodic modulation strength. It is argued that occurrence of SNAs gives an opportunity of asymptotic predictability of epidemic growth even when the underlying dynamics is strange.