Phase Transition of Epidemic Dynamics on Adaptive Networks

Mathematical models of disease propagation have been studied for a long time, and various models have been proposed. In the last decade, much attention has been paid to the epidemic dynamics on a network, and those researches have revealed that the network structure affects the epidemic dynamics on the network. Similarly, the spreading of an infectious disease on the network can have a strong effect on the dynamics of the network structure. For example, people tend to avoid contact with infectious individuals in real networks. Therefore, an epidemic model on an adaptive network has been introduced as a model that takes into account the interaction between the dynamics of networks and the dynamics on networks [1, 2]. The study of these models develops our understanding of the epidemic spreading on networks. In the previous study [2], the moment-closure approximation (MCA) and the degree-based mean-field (DBMF) theory have been used to analyze the epidemic dynamics and the network dynamics. These are common analytical approaches for dynamics on networks, however, these are often inaccurate due to the following assumptions. The DBMF theory is based on the assumption that there is no dynamical correlation. The MCA includes dynamical correlations at a pairwise level only. In order to improve accuracy, the approximate master equation (AME) approach, which is the extension of the DBMF theory, has been proposed [3, 4]. The variables of the AME have full information of those neighbors, namely, the dynamical correlation is taken into account. The AME approach provides a good prediction of the time evolution and good estimates of critical values for binary-state dynamics on a network. In this study, we extend the AME approach to threestate dynamics and analyze the adaptive SIRS (susceptibleinfected-recovered-susceptible) model of Shaw et al. [2].