Bifurcations in the Herd Immunity Threshold for Discrete-Time Models of Epidemic Spread*

We performed a complete 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. We observe this behavior in both network- and differential equation-based models, suggesting this behavior is a universal feature of all discrete-time SIR models. This suggests careful attention is needed in selecting the assumptions on how to model time and heterogeneity in the standard epidemic models that are used.