Centrality-Based Traffic Restriction in Delayed Epidemic Networks

SIAM Journal on Applied Dynamical Systems, Volume 22, Issue 4, Page 3165-3207, December 2023.
Abstract. In an epidemic network, lags due to travel time between populations, latent period, and recovery period can significantly change the epidemic behavior and result in successive echoing waves of the spread between various population clusters. Moreover, external shocks to a given population can propagate to other populations within the network, potentially snowballing into waves of resurgent epidemics. The main objective of this study is to investigate the effect of time delay and small shocks/uncertainties on the linear susceptible-infectious-susceptible (SIS) dynamics of epidemic networks. In this regard, the asymptotic stability of this class of networks is first studied, and then its performance loss due to small shocks/uncertainties is evaluated based on the notion of the [math] norm. It is shown that network performance loss is correlated with the structure of the underlying graph, intrinsic time delays, epidemic characteristics, and external shocks. This performance measure is then used to develop an optimal traffic restriction algorithm for network performance enhancement, resulting in reduced infection in the metapopulation. A novel epidemic-based centrality index is also defined to evaluate the impact of every subpopulation on network performance, and its asymptotic behavior is investigated. It is shown that for specific choices of parameters, the output of the epidemic-based centrality index converges to the results obtained by local or eigenvector centralities. Moreover, given that epidemic-based centrality depends on the epidemic properties of the disease, it may yield distinct node rankings as the disease characteristics slowly change over time or as different types of infections spread. This node interlacing phenomenon is not observed in other centralities that rely solely on network structure. This unique characteristic of epidemic-based centrality enables it to adjust to various epidemic features. The derived centrality index is then adopted to improve the network robustness against external shocks on the epidemic network. The numerical results, along with the theoretical expectations, highlight the role of time delay as well as small shocks in investigating the most effective methods of epidemic containment.