Network inoculation: Heteroclinics and phase transitions in an epidemic model
Hui Yang, Tim Rogers and Thilo Gross
Chaos 26, 083116, 2016
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.