Mapping out emerging network structures in dynamic network models coupled with epidemics

We consider the susceptible infected susceptible (SIS) epidemic on a dynamic network model with addition and deletion of links depending on node status. We analyse the resulting pairwise model using classical bifurcation theory to map out the spectrum of all possible epidemic behaviours. However, the major focus of the chapter is on the evolution and possible equilibria of the resulting networks. Whereas most studies are driven by determining system-level outcomes, e.g., whether the epidemic dies out or becomes endemic, with little regard for the emerging network structure, here, we want to buck this trend by augmenting the system-level results with mapping out of the structure and properties of the resulting networks. We find that depending on parameter values the network can become disconnected and show bistable-like behaviour whereas the endemic steady state sees the emergence of networks with qualitatively different degree distributions. In particular, we observe de-phased oscillations of both prevalence and network degree during which there is role reversal between the level and nature of the connectivity of susceptible and infected nodes. We conclude with an attempt at describing what a potential bifurcation theory for networks would look like. István Z. Kiss Department of Mathematics, School of Mathematical and Physical Sciences, University of Sussex, Falmer, Brighton BN1 9QH, UK, e-mail: [email protected] Luc Berthouze Centre for Computational Neuroscience and Robotics, University of Sussex, Falmer, Brighton, BN1 9QH, UK e-mail: [email protected] Joel C. Miller School of Mathematical Sciences and School of Biological Sciences, Monash University, Clayton, VIC Australia, Monash Academy for Cross and Interdisciplinary Mathematics, and Institute for Disease Modeling, Bellevue, WA USA e-mail: [email protected] Péter L. Simon Institute of Mathematics, Eötvös Loránd University Budapest, and Numerical Analysis, and Large Networks Research Group, Hungarian Academy of Sciences, Hungarye-mail: [email protected]