The use of integrodifference equations in the study of the role of dispersal on populations with discrete generations has generated interesting mathematical problems and expanded our understanding of their spatio-temporal dynamics. Here, we use discrete-time epidemic models that can be reduced to a single map for the infectious class, It+1 =g(It), where g may or may not be monotone. We use new theoretical work, modeling, analysis and simulations to illustrate the role of g on disease dynamics in one and two spatial dimensions.