In epidemiological modelling, dynamics on networks, and in particular adaptive and heterogeneous networks have recently received much interest. Here we present a detailed analysis of a previously proposed model that combines heterogeneity in the individuals with adaptive rewiring of the network structure in response to a disease. We show that in this model qualitative changes in the dynamics occur in two phase transitions. In a macroscopic description one of these corresponds to a local bifurcation whereas the other one corresponds to a non-local heteroclinic bifurcation. This model thus provides a rare example of a system where a phase transition is caused by a non-local bifurcation, while both micro- and macro-level dynamics are accessible to mathematical analysis. The bifurcation points mark the onset of a behaviour that we call network inoculation. In the respective parameter region exposure of the system to a pathogen will lead to an outbreak that collapses, but leaves the network in a configuration where the disease cannot reinvade, despite every agent returning to the susceptible class. We argue that this behaviour and the associated phase transitions can be expected to occur in a wide class of models of sufficient complexity.
Figure 1: Simplified sketch of the phase portait in the epidemic model. Shown is a flow field (thin blue arrows) the attracting endemic state (black circle), a saddle point (grey circle) and a manifold of disease-free steady states (strong grey/black line), which can be stable (black) or unstable (grey). Depending on the initial value of the x-axis we can distinguish between stable disease-free, outbreak and collapse, and endemic behavior, indicated by labels on the axis. The behaviour changes at two threshold values (T1, T2) which are marked by a local change in the stability of the manifold and the heteroclinic connection. We note that this sketch has been simplified from the situation in the epidemic model. If the x-axis were the a–a link density [aa] the type II behavior would occur for intermidate values whereas the type III behavior would occur at high values, which is harder to visualize in a 2d-plot, but qualitatively similar.