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We performed a thorough sensitivity analysis of the herd immunity threshold for discrete-time SIR compartmental models with a static network structure. We find unexpectedly that these models violate classical intuition which holds that the herd immunity threshold should monotonically increase with the transmission parameter. We find the existence of bifurcations in the herd immunity threshold in the high transmission probability regime. The extent of these bifurcations is modulated by the graph heterogeneity, the recovery parameter, and the network size. In the limit of large, well-mixed networks, the behavior approaches that of difference equation models, suggesting this behavior is a universal feature of all discrete-time SIR models. These results suggest careful attention is needed in both selecting the assumptions on how to model time and heterogeneity in epidemiological models and the subsequent conclusions that can be drawn.