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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.

The experimental actualisation of organoids modelling organs from brains to pancreases has revealed that much of the diverse morphologies of organs are emergent properties of simple intercellular "rules" and not the result of top-down orchestration. In contrast to other organs, the initial plexus of the vascular system is formed by aggregation of cells in the process known as vasculogenesis. Here we study this self-assembling process of blood vessels in three dimensions through a set of simple rules that align intercellular apical-basal and planar cell polarity. We demonstrate that a fully connected network of tubes emerges above a critical initial density of cells. Through planar cell polarity our model demonstrates convergent extension, and this polarity furthermore allows for both morphology-maintaining growth and growth-induced buckling. We compare this buckling to the special vasculature of Islets of Langerhans in the pancreas and suggest that the mechanism behind the vascular density-maintaining growth of these islets could be the result of growth-induced buckling.